HP 12C Platinum Solutions Handbook Pt_solutions Handbook_English_E C00367123
User Manual: HP hp 12c pt_solutions handbook_English_E
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HP 12C Platinum
Solutions Handbook
HP would like to thank the following for their contribution:
Tony Hutchins, Wellington, NZ
Luiz Vieira, UNIPAC, Brazil
Gene Wright, Lipscomb University, USA
NOTICE
Hewlett-Packard Company makes no express or implied warranty with regard to the keystroke
procedures and program material offered or their merchantability or their fitness for any
particular purpose. The keystroke procedures and program material are made available
solely on an "as is" basis, and the entire risk as to their quality and performance is with the
user. Should the keystroke procedures or program material prove defective, the user (and not
Hewlett-Packard Company nor any other party) shall bear the entire cost of all necessary
correction and all incidental and consequential damages. Hewlett-Packard Company shall not
be liable for any incidental or consequential damages in connection with or arising out of the
furnishing, use, or performance of the keystroke procedures or program material.
© Hewlett-Packard Development Company, L.P.
rev. 03.04
Introduction
About This Handbook
This HP 12C Platinum Solutions Handbook has been designed to supplement the HP 12C
Platinum Owner's Handbook by providing a variety of applications in the financial area.
Programs and/or step-by-step keystroke procedures with corresponding examples in each
specific topic are explained. We hope that this book will serve as a reference guide to
many of your problems and will show you how to redesign our examples to fit your
specific needs.
This book expands the original HP-12C Solutions handbook with additional solutions in
algebraic mode. It contains the same RPN program keystrokes and RPN step-by-step
procedure keystrokes, in columns headed “12c platinum / 12C RPN Keystrokes”. The
alternative algebraic keystrokes are tablulated under “12c platinum ALG Keystrokes”. In
program listings the “Display” columns show the keycodes as seen on the HP 12C
Platinum.
Appendix A also contains algebraic listings for all the RPN programs given in Part III of
the HP 12C Platinum Owner’s Handbook.
Presentation of Algebraic and RPN
The conventions used to differentiate between RPN and ALG mode are:
1. Program Listings
Complete and separate listings are given for all programs. They appear side by side
in two columns with RPN on the left and Algebraic on the right.
2. Step-by-Step Keystroke Procedures
As for programs separate columns are used, with the RPN keystrokes on the left
and the Algebraic keystrokes on the right.
3. Program Instructions
Program instruction steps are generally the same for both modes. Any differences
are shown by clearly framing alternative steps and annotating the step or steps in
the first frame as RPN, and those in the second as ALG.
4. Text
Occasionally there are small differences which need to be indicated in the text
2
Introduction
3
itself and the ALG alternative is then indicated parenthetically.
5.
Usage of \(³)
To activate the ³ key it is sufficient just to press \, with the HP 12C
Platinum in ALG mode. In step-by-step and program instructions where the only
difference between the modes is that \ is used in RPN mode and ³ is used
in ALG mode, \(³) has been used to indicate both alternatives.
Using the RPN Programs on the HP-12C
Apart from GTO instructions, the keystrokes given in this book are exactly the same for
the HP 12C Platinum and the HP-12C. There are two notational differences to bear in
mind when typing the RPN programs into the HP-12C:
1. One keycode, for F, is different.
2. Line numbers tabulated as 000 to 099 refer to lines displayed as 00 to 99 on the
HP-12C. The relevant two digit line numbers should be used when typing GTO
instructions on the HP-12C.
Notes:
1. All display columns in the examples this book show 2 decimals. This is set by
pressing f2.
All programs that do rounding, amortization or depreciation will give slightly
different answers if other than 2 decimals are showing.
2. Three of the original programs have been updated:
a. The last program in the Real Estate section (ATNCPR) now takes the capital
gains tax rate as a separate input, and an extra example has been added
showing a different type of tax basis.
b. In the Personal Finance section the IRA program now handles explicit
inflation input and withdrawal tax rate input and the Stock Portfolio program
handles stock prices with decimal fractions rather than fractions expressed in
terms of eighths.
3. Market data (i.e.: interest rates; real estate values, growth rates and rents; taxes;
expenses; etc.) used in the examples in this book do not necessarily represent
typical current actual data, or reflect recent market trends.
Contents
Introduction ........................................................................................................................ 2
About This Handbook..................................................................................................... 2
Presentation of Algebraic and RPN ................................................................................ 2
Using the RPN Programs on the HP-12C ....................................................................... 3
Contents.............................................................................................................................. 4
Real Estate .......................................................................................................................... 7
Refinancing..................................................................................................................... 7
Wrap-Around Mortgage ................................................................................................. 8
Income Property Cash Flow Analysis........................................................................... 12
Before-Tax Cash Flows ............................................................................................ 12
Before-Tax Reversions (Resale Proceeds)................................................................ 13
After-Tax Cash Flows............................................................................................... 14
After-Tax Net Cash Proceeds of Resale.................................................................... 19
Lending............................................................................................................................. 24
Loan With a Constant Amount Paid Towards Principal ............................................... 24
Add-On Interest Rate Converted to APR...................................................................... 25
APR Converted to Add-On Interest Rate...................................................................... 26
Add-On Rate Loan with Credit Life ............................................................................. 27
Interest Rebate - Rule of 78's........................................................................................ 30
Graduated Payment Mortgages..................................................................................... 32
Variable Rate Mortgages .............................................................................................. 36
Skipped Payments......................................................................................................... 38
Savings ............................................................................................................................. 40
Initial Deposit with Periodic Deposits .......................................................................... 40
Number of Periods to Deplete a Savings Account or to Reach a Specified Balance .... 41
Periodic Deposits and Withdrawals .............................................................................. 42
Savings Account Compounded Daily ........................................................................... 44
Compounding Periods Different From Payment Periods.............................................. 46
Investment Analysis.......................................................................................................... 49
Lease vs. Purchase ........................................................................................................ 49
Break-Even Analysis .................................................................................................... 53
Operating Leverage ...................................................................................................... 59
Profit and Loss Analysis ............................................................................................... 61
Securities and Options ...................................................................................................... 65
After-Tax Yield ............................................................................................................ 65
Discounted Notes.......................................................................................................... 67
Black-Scholes Formula for Valuing European Options................................................ 69
4
Contents
5
Forecasting........................................................................................................................74
Simple Moving Average ...............................................................................................74
Seasonal Variation Factors Based on Centered Moving Averages ...............................78
Gompertz Curve Trend Analysis...................................................................................83
Forecasting with Exponential Smoothing .....................................................................87
Pricing Calculations ..........................................................................................................92
Markup and Margin Calculations..................................................................................92
Calculations of List and Net prices With Discounts......................................................95
Statistics............................................................................................................................98
Curve Fitting .................................................................................................................98
Exponential Curve Fit ...............................................................................................98
Logarithmic Curve Fit.............................................................................................102
Power Curve Fit ......................................................................................................104
Standard Error of the Mean.........................................................................................105
Mean, Standard Deviation, Standard Error for Grouped Data ....................................106
Chi-Square Statistics ...................................................................................................109
Normal Distribution ....................................................................................................112
Covariance ..................................................................................................................114
Permutations ...............................................................................................................116
Combinations ..............................................................................................................117
Random Number Generator ........................................................................................119
Personal Finance .............................................................................................................121
Homeowners Monthly Payment Estimator .................................................................121
Tax-Free Individual Retirement (IRA) or Keogh Plan................................................124
Stock Portfolio Evaluation and Analysis.....................................................................127
Canadian Mortgages .......................................................................................................131
Periodic Payment Amount ..........................................................................................131
Number of Periodic Payments to Fully Amortize a Mortgage ....................................132
Effective Interest Rate (Yield) ....................................................................................132
Balance Remaining at End of Specified Period...........................................................132
Miscellaneous .................................................................................................................134
Learning Curve for Manufacturing Costs ...................................................................134
Queuing and Waiting Theory......................................................................................138
Appendix A : Programs from Part III of the Owner's Handbook ....................................145
About this Appendix ...................................................................................................145
Algebraic Mode Programs ..........................................................................................146
Section 12: The Rent or Buy Decision ....................................................................146
Section 13: Straight-Line Depreciation...................................................................147
Section 13: Declining-Balance Depreciation ..........................................................148
Section 13: Sum-of-the-Years-Digits Depreciation.................................................149
Section 13: Full- and Partial- Year Depreciation with Crossover ...........................150
Section 14: Lease with Advance Payments - Solving For Payment ........................151
Section 14: Lease with Advance Payments - Solving For Yield .............................151
Section 14: Advance Payments With Residual - Solving for Payment ...................152
6
Contents
Section 15: Nominal Rate Converted to Effective Rate ......................................... 152
Section 16: 30/360 Day Basis Bonds ..................................................................... 153
Section 16: Annual Coupon Bonds ......................................................................... 154
Appendix B: Formulas Used........................................................................................... 155
Real Estate .................................................................................................................. 155
Wrap-Around Mortgage ......................................................................................... 155
After-Tax Cash Flows............................................................................................. 155
After-Tax Net Cash Proceeds of Resale.................................................................. 155
Lending....................................................................................................................... 156
Loans With a Constant Amount Paid Towards Principal........................................ 156
Add-On Interest Rate to APR ................................................................................. 156
Add-On to APR with Credit Life............................................................................ 156
Rule of 78's Rebate ................................................................................................. 156
Graduated Payment Mortgage ................................................................................ 157
Skipped Payments................................................................................................... 157
Savings ....................................................................................................................... 158
Compounding Periods Different From Payment Periods ........................................ 158
Investment Analysis.................................................................................................... 158
Lease vs. Purchase .................................................................................................. 158
Break-Even Analysis and Operating Leverage ....................................................... 158
Profit and Loss Analysis ......................................................................................... 158
Securities and Options ................................................................................................ 159
Discounted Notes.................................................................................................... 159
Black-Scholes Formula for Valuing European Options.......................................... 159
Forecasting ................................................................................................................. 160
Simple Moving Average ......................................................................................... 160
Seasonal Variation Factors Based on a Centered Moving Average ........................ 160
Gompertz Curve Trend Analysis ............................................................................ 160
Forecasting With Exponential Smoothing .............................................................. 161
Pricing Calculations.................................................................................................... 161
Markup and Margin Calculations ........................................................................... 161
Calculations of List and Net Prices with Discounts ................................................ 162
Statistics...................................................................................................................... 162
Exponential Curve Fit............................................................................................. 162
Logarithmic Curve Fit ............................................................................................ 163
Power Curve Fit...................................................................................................... 163
Standard Error of the Mean..................................................................................... 163
Mean, Standard Deviation, Standard Error for Grouped Data ................................ 163
Personal Finance......................................................................................................... 164
Tax-Free Retirement Account (IRA) or Keogh Plan .............................................. 164
Stock Portfolio Evaluation and Analysis ................................................................ 164
Portfolio beta coefficient: ....................................................................................... 164
Canadian Mortgages ................................................................................................... 164
Miscellaneous ............................................................................................................. 165
Learning Curve for Manufacturing Cost................................................................. 165
Queuing and Waiting Theory.................................................................................. 165
Subject Index .................................................................................................................. 166
Real Estate
Refinancing
It can be mutually advantageous to both borrower and lender to refinance an existing
mortgage which has an interest rate substantially below the current market rate, with a
loan at a below-market rate. The borrower has the immediate use of tax-free cash, while
the lender has substantially increased debt service on a relatively small cash outlay.
To find the benefits to both borrower and lender:
1. Calculate the monthly payment on the existing mortgage.
2. Calculate the monthly payment on the new mortgage.
3. Calculate the net monthly payment received by the lender (and paid by the borrower)
by adding the figure found in Step 1 to the figure found in Step 2.
4. Calculate the Net Present Value (NPV) to the lender of the net cash advanced.
5. Calculate the yield to the lender as an IRR.
6. Calculate the NPV to the borrower of the net cash received.
Example: An investment property has an existing mortgage which originated 8 years ago
with an original term of 25 years, fully amortized in level monthly payments at 6.5%
interest. The current balance is $133,190.
Although the going current market interest rate is 11.5%, the lender has agreed to
refinance the property with a $200,000, 17 year, level-monthly-payment loan at 9.5%
interest.
What are the NPV and effective yield to the lender on the net amount of cash actually
advanced?
What is the NPV to the borrower on this amount if he can earn a 15.25% equity yield rate
on the net proceeds of the loan?
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARG
17gA
6.5gC
133190$
P?0
12c platinum
ALG Keystrokes
gÂ
fCLEARG
17gA
6.5gC
133190$
P?0
7
Display
-1,080.33
Comments
Monthly payment on
existing mortgage
received by lender.
8 Real Estate
12c platinum / 12C
12c platinum
RPN Keystrokes
ALG Keystrokes
9.5gC
9.5gC
200000Þ$
200000Þ$
P
P
1,979.56
:0+P
+:0P
899.23
:$
133190+?0
:$
+133190³?0
-66,810.00
11.5gC$
11.5gC$$
-80,425.02
:0-
-:0³
-13,615.02
:0$¼
12§
15.25gC$
:0$¼
§12³
15.25gC$$
14.83
-65,376.72
:0-
-:0³
1,433.28
Display
Comments
Monthly payment on
new mortgage.
Net monthly payment (to
lender).
Net amount of cash
advanced (by lender).
Present value of net
monthly payment.
NPV to lender of net
cash advanced.
% nominal yield (IRR).
Present value of net
monthly payment at
15.25%.
NPV to borrower.
Wrap-Around Mortgage
A wrap-around mortgage is essentially the same as a refinancing mortgage, except that the
new mortgage is granted by a different lender, who assumes the payments on the existing
mortgage, which remains in full force. The new (second) mortgage is thus "wrapped
around" the existing mortgage. The "wrap-around" lender advances the net difference
between the new (second) mortgage and the existing mortgage in cash to the borrower,
and receives as net cash flow, the difference between debt service on the new (second)
mortgage and debt service on the existing mortgage.
When the terms of the original mortgage and the wrap-around are the same, the
procedures in calculating NPV and IRR to the lender and NPV to the borrower are exactly
the same as those presented in the preceding section on refinancing.
Example 1: A mortgage loan on an income property has a remaining balance of
$200,132.06. When the load originated 8 years ago, it had a 20 year term with full
amortization in level monthly payments at 6.75% interest.
A lender has agreed to "wrap" a $300,000 second mortgage at 10%, with full amortization
in level monthly payments over 12 years. What is the effective yield (IRR) to the lender
on the net cash advanced?
Real Estate
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARG
20\
8-gA
12c platinum
Display
ALG Keystrokes
gÂ
fCLEARG
20144.00
8³gA
6.75gC
200132.06$
P?0
6.75gC
200132.06$
P?0
0.56
200,132.06
-2,031.55
10gC
10gC
0.83
300000Þ$
300000Þ$
-300,000.00
P
P
3,585.23
:0+P
+:0P
1,553.69
:$
200132.06+$
¼12§
:$
+200132.06$ -99,867.94
15.85
¼§12³
9
Comments
Total number of months
remaining in original loan
(into n).
Monthly interest rate (into i).
Loan amount (into PV).
Monthly payment on existing
mortgage (calculated).
Monthly interest on wraparound.
Amount of wrap-around (into
PV).
Monthly payment on wraparound (calculated).
Net monthly payment
received (into PMT).
Net cash advanced (into PV).
Nominal yield (IRR) to lender
(calculated).
Sometimes the wrap around mortgage will have a longer payback period than the original
mortgage, or a balloon payment may exist.
PV1
BAL
PMT2 for n2 Years
(+)
...
PMT1 for n1 Years
(–)
PV2
...
10 Real Estate
Where:
n1 = number of years remaining in original mortgage
PMT1 = yearly payment of original mortgage
PV1 = remaining balance of original mortgage
n2 = number of years in wrap-around mortgage
PMT2 = yearly payment of wrap-around mortgage
PV2 = total amount of wrap-around mortgage
BAL = balloon payment
Example 2: A customer has an existing mortgage with a balance of $125,010, a
remaining term of 200 months, and a $1051.61 monthly payment. He wishes to obtain a
$200,000, 9 ½% wrap-around with 240 monthly payments of $1681.71 and a balloon
payment at the end of the 240th month of $129,963.35. If you, as a lender, accept the
proposal, what is your rate of return?
$125010
240 mos.
$1681.71
$ 129963.35
$1681.71
...
$1681.71
...
$ -1051.61
$ -1051.61
200 mos.
$ -200000
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARG
200000Þ\
125010+gJ
1051.61Þ\
1681.71+
12c platinum
ALG Keystrokes
gÂ
fCLEARG
200000Þ
+125010gJ
1051.61Þ
+1681.71gK
Display
Comments
-74,990.00
Net investment.
630.10
Net cash flow
received by lender.
Real Estate
12c platinum / 12C
RPN Keystrokes
gK99ga
~gK
~ga
~gK
2ga
12c platinum
ALG Keystrokes
99ga
~gK
~ga
~gK
2ga
gFgK
1681.71gK
1,681.71
39ga
39ga
39.00
~129963.35+
gK
fL12§
~+129963.35
gK
fL§12³
131,645.06
11.84
Display
11
Comments
The above cash flow
occurs 200 times.
Next cash flow
received by lender.
Cash flow occurs 39
times.
Final cash flow.
Rate of return to
lender.
If you, as a lender, know the yield on the entire transaction, and you wish to obtain the
payment amount on the wrap-around mortgage to achieve this yield, use the following
procedure. Once the monthly payment is known, the borrower's periodic interest rate may
also be determined.
1. Press the g and press fCLEARG.
2.
Key in the remaining periods of the original mortgage and press n.
3.
4.
Key in the desired annual yield and press gC.
Key in the monthly payment to be made by the lender on the original mortgage and
press ÞP.
5.
Press $.
6.
RPN: Key in the net amount of cash advanced and press +Þ$.
6.
ALG: Press +, key in the net amount of cash advanced and press ³Þ$.
7.
Key in the total term of the wrap-around mortgage and press n.
8.
If a balloon payment exists, key it in and press M.
9.
Press P to obtain the payment amount necessary to achieve the desired yield.
10. Key in the amount of the wrap-around mortgage and press Þ$¼ to obtain the
borrower's periodic interest rate.
Example 3: Your firm has determined that the yield on a wrap-around mortgage should
be 12% annually. In the previous example, what monthly payment must be received to
achieve this yield on a $200,000 wrap-around? What interest rate is the borrower paying?
12 Real Estate
12c platinum / 12C
12c platinum
RPN Keystrokes
ALG Keystrokes
gÂ
gÂ
fCLEARG
fCLEARG
200n12gC
200n12gC
Display
Comments
Number of periods and
monthly interest rate.
Monthly payment.
1051.61ÞP
$74990+
Þ$
1051.61ÞP
$+74990³
Þ$
-165,776.92
Present value of
payments plus cash
advanced.
240n
129963.35MP
240n
129963.35MP
1,693.97
Monthly payment
received by lender.
200000Þ$¼
12§
200000Þ$¼
§12³
9.58
Annual interest rate paid
by borrower.
Income Property Cash Flow Analysis
Before-Tax Cash Flows
The before-tax cash flows applicable to real estate analysis and problems are:
•
Potential Gross Income
•
Effective Gross Income
•
Net Operating Income (also called Net Income Before Recapture)
•
Cash Throw-off to Equity (also called Gross Spendable Cash)
The derivation of these cash flows follows a set sequence:
1. Calculate Potential Gross Income by multiplying the rent per unit times the number
of units, times the number of rental payment periods per year. This gives the rental
income the property would generate if it were fully occupied.
2. Deduct Allowance for Vacancy and Rental Loss. This is usually expressed as a
percentage. The result is Rent Collections (which is also Effective Gross Income if
there is no "Other Income").
3. Add "Other Income" such as receipts from concessions (laundry equipment, etc.),
produced from sources other than the rental office space. This is Effective Gross
Income.
4. Deduct Operating Expenses. These are expenditures the landlord-investor must
make, by contract or custom, to preserve the property and keep in capable of
producing the gross income. The result is the Net Operating Income.
5. Deduct Annual Debt Service on the mortgage. This produces Cash Throw-Off to
Equity.
Real Estate
13
Thus:
Effective Gross Income =Potential Gross Income - Vacancy Loss + Other Income.
Net Operating Income =Effective Gross Income - Operating Expenses.
Cash Throw-Off =Net Operating Income - Annual Dept Service.
Example: A 60 unit apartment building has rentals of $250 per unit per month. With a
5% vacancy rate, the annual operating cost is $76,855.
The property has just been financed with a $700,000 mortgage, fully amortized in a level
monthly payments at 11.5% over 20 years.
a. What is the Effective Gross Income?
b. What is the Net Operating Income?
c. What is the Cash Throw-Off to Equity?
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARG
60\
250§12§
5b7685520gA
11.5gC
700000$
P12§
+
12c platinum
Display
ALG Keystrokes
gÂ
fCLEARG
60§
180,000.00
250§12171,000.00
5b94,145.00
76855³
20gAd
11.5gCd
700000$
-89,580.09
P§12+
4,564.91
~³
Comments
Potential Gross Income.
Effective Gross Income.
Net Operating Income.
Annual Debt Service.
Cash Throw-Off.
Before-Tax Reversions (Resale Proceeds)
The reversion receivable at the end of the income projection period is usually based on
forecast or anticipated resale of the property at that time. The before tax reversion amount
applicable to real estate analysis and problems are:
•
Sale Price.
•
Cash Proceeds of Resale.
•
Outstanding Mortgage Balance.
•
Net Cash Proceeds of Resale to Equity.
The derivation of these reversions is as follows:
1. Forecast or estimate Sales Price. Deduct sales and Transaction Costs. The result is
the Proceeds of Resale.
14 Real Estate
2.
Calculate the Outstanding Balance of the Mortgage at the end of the Income
Projection Period and subtract it from Proceeds of Resale. The result is Net Cash
Proceeds of Resale.
Thus:
Cash Proceeds of Resale = Sales Price - Transaction Costs.
Net Cash Proceeds of Resale = Cash Proceeds of Resale - Outstanding Mortgage Balance.
Example: The apartment property in the preceding example is expected to be resold in 10
years. The anticipated resale price is $800,000. The transaction costs are expected to be
7% of the resale price. The mortgage is the same as that indicated in the preceding
example.
•
What will the Mortgage Balance be in 10 years?
•
What are the Cash Proceeds of Resale and Net Cash Proceeds of Resale?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
gÂ
gÂ
fCLEARG
fCLEARG
240.00
20gA
20gA
0.96
11.5gC
11.5gC
700000$
700000$
-7,465.01
P
P
120.00
10gA
10gA
-530,956.57
M
MM
800000\
800000744,000.00
7b7b+
213,043.43
+
~³
Comments
Mortgage term.
Mortgage rate.
Property value.
Monthly payment.
Projection period.
Mortgage balance in 10 years.
Estimated resale.
Cash Proceeds of Resale.
Net Cash Proceeds of Resale.
After-Tax Cash Flows
The After-Tax Cash Flow (ATCF) is found for the each year by deducting the Income
Tax Liability for that year from the Cash Throw Off.
Where Taxable Income = Net Operating Income - interest - depreciation,
Tax Liability = Taxable Income x Marginal Tax Rate,
and
After Tax Cash Flow = Cash Throw Off - Tax Liability.
The After-Tax Cash Flow for the initial and successive years may be calculated by the
following HP 12C Platinum program. This program calculates the Net Operating Income
using the Potential Gross Income, operational cost and vacancy rate. The Net Operating
Real Estate
15
income is readjusted each year from the growth rates in Potential Gross Income and
operational costs.
The user is able to change the method of finding the depreciation from declining balance
to straight line. To make the change, key in fV at line 032 (ALG: 026) of the
program in place of f#.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
0
n
?1
:7
Æ
2
z
?7
1
?+1
1
2
f!
?0
:5
n
:¼
:6
¼
d
?6
d
:$
:4
$
d
?4
d
gm
g(036
:1
f#
?-0
0
g(017
DISPLAY
000,
001,
0
002,
11
003,
44
1
004,
45
7
005,
26
006,
2
007,
10
008,
44
7
009,
1
010,44 40
1
011,
1
012,
2
013,
42 11
014,
44
0
015,
45
5
016,
11
017,
45 12
018,
45
6
019,
12
020,
33
021,
44
6
022,
33
023,
45 13
024,
45
4
025,
13
026,
33
027,
44
4
028,
33
029,
43 35
030,43,33,036
031,
45
1
032,
42 25
033,44 30
0
034,
0
035,43,33,017
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
0
n
1
2
f!
?0
:5
n
:¼
:6
¼
d
?6
d
:$
:4
$
d
?4
d
gm
g(030
1
?+1
:1
f#
?-0
0
g(009
:2
:.0
b
:3
DISPLAY
000,
001,
0
002,
11
003,
1
004,
2
005,
42 11
006,
44
0
007,
45
5
008,
11
009,
45 12
010,
45
6
011,
12
012,
33
013,
44
6
014,
33
015,
45 13
016,
45
4
017,
13
018,
33
019,
44
4
020,
33
021,
43 35
022,43,33,030
023,
1
024,44 40
1
025,
45
1
026,
42 25
027,44 30
0
028,
0
029,43,33,009
030,
45
2
031,
30
032,45 48
0
033,
25
034,
30
035,
45
3
16 Real Estate
12c platinum / 12C
RPN KEYSTROKES
n
:2
:8
b
?+2
d
:.0
b
:3
:9
b
?+3
d
1
:7
?§0
§
:P
1
2
§
+
:0
:1
gu
~
t
g(009
fs
n: Used
FV: 0
R3: Oper. cost
R7: Tax Rate
DISPLAY
036,
11
037,
45
2
038,
45
8
039,
25
040,44 40
2
041,
33
042,45 48
0
043,
25
044,
30
045,
45
3
046,
45
9
047,
25
048,44 40
3
049,
33
050,
30
051,
1
052,
45
7
053,44 20
0
054,
30
055,
20
056,
45 14
057,
1
058,
2
059,
20
060,
40
061,
45
0
062,
30
063,
45
1
064,
43 31
065,
34
066,
31
067,43,33,009
12c platinum
ALG KEYSTROKES
³
?+0
+
:P
gA
³
:7
b
?§0
d
:0
³
:1
gu
~
t
:2
+
:8
b
³
?2
:3
+
:9
b
³
?3
g(001
fs
REGISTERS
i: Annual %
PV: Used
R0: Used
R1: Counter
R4: Dep. value
R5: Dep. Life
R8: % gr. (PGI)
R9: % gr. (op)
DISPLAY
036,
36
037,44 40
0
038,
40
039,
45 14
040,
43 11
041,
36
042,
45
7
043,
25
044,44 20
0
045,
33
046,
30
047,
45
0
048,
36
049,
45
1
050,
43 31
051,
34
052,
31
053,
45
2
054,
40
055,
45
8
056,
25
057,
36
058,
44
2
059,
45
3
060,
40
061,
45
9
062,
25
063,
36
064,
44
3
065,43,33,001
PMT: Monthly
R2: PGI
R6: Factor (DB)
R.0: Vacancy rt.
Real Estate
17
Program Instructions:
1.
Press g and press fCLEARH.
2.
Key in loan values:
●
Key in annual interest rate and press gC
●
Key in principal to be paid and press $
●
Key in monthly payment and press ÞP
(If any of the values are not known, they should be solved for.)
3.
Key in Potential Gross Income (PGI) and press ?2.
4.
Key in Operational cost and press ?3.
5.
Key in depreciable value and press ?4.
6.
Key in depreciable life and press ?5.
7.
Key in factor (for declining balance only) and press ?6.
8.
9.
Key in the Marginal Tax Rate (as a percentage) and press ?7.
Key in the growth rate in Potential Gross Income (0 for no growth) and press
?8.
10.
Key in the growth rate in operational cost (0 if no growth) and press ?9.
11.
Key in the vacancy rate (0 for no vacancy rate) and press ?.0.
12.
RPN: Key in the desired depreciation function at line 032 in the program.
12.
ALG: Key in the desired depreciation function at line 026 in the program.
13.
Press t to compute ATCF. The display will pause showing the year and then
will stop with the ATCF for that year. The Y-register contains the year.
14.
Continue pressing t to compute successive After-Tax Cash Flows.
Example 1: A triplex was recently purchased for $100,000 with a 30 year loan at 12.25%
and a 20% down payment. Not including a 5% annual vacancy rate, the potential gross
income is $9,900 with an annual growth rate of 6%. Operating expenses are $3,291.75
with a 2.5% growth rate. The depreciable value is $75,000 with a projected useful life of
$20 years. Assuming a 125% declining balance depreciation, what are the After-Tax Cash
Flows for the first 10 years if the investors Marginal Tax Rate is 35%?
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARH
100000\
20b-$
12.25gC
30gA
P
9900?2
12c platinum
ALG Keystrokes
gÂ
fCLEARH
10000020b$
12.25gC
30gA
PP
9900?2
Display
80,000.00
1.02
360
-838.32
9,900.00
Comments
Mortgage amount.
Monthly interest rate.
Mortgage term.
Monthly payment.
Potential Gross Income.
18 Real Estate
12c platinum / 12C
RPN Keystrokes
3291.75?3
75000?4
20?5
125?6
35?7
6?8
12c platinum
ALG Keystrokes
3291.75?3
75000?4
20?5
125?6
35?7
6?8
2.5?9
5?.0
t
2.5?9
5?.0
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
Display
3,291.75
75,000.00
20.00
125.00
35.00
6.00
2.50
5.00
1.00
-1,020.88
2.00
-822.59
3.00
-598.85
4.00
-348.94
5.00
-72.16
6.00
232.35
7.00
565.48
8.00
928.23
9.00
1,321.62
10.00
1,746.81
Comments
1st year operating cost.
Depreciable value.
Useful life.
Declining balance factor.
Marginal Tax Rate.
Potential Gross Income
growth rate.
Operating cost growth.
Vacancy rate.
Year 1
ATCF1
Year 2
ATCF2
Year 3
ATCF3
Year 4
ATCF4
Year 5
ATCF5
Year 6
ATCF6
Year 7
ATCF7
Year 8
ATCF8
Year 9
ATCF9
Year 10
ATCF10
Example 2: An office building was purchased for $1,400,000. The value of depreciable
improvements is $1,200,000 with a 35 year economic life. Straight line depreciation will
be used. The property is financed with a $1,050,000 loan. The terms of the loan are 9.5%
interest and $9,173.81 monthly payments for 25 years. The office building generates a
Potential Gross Income of $175,200 which grows at a 3.5% annual rate. The operating
cost is $40,296.00 with a 1.6% annual growth rate. Assuming a Marginal Tax Rate of
50% and a vacancy rate of 7%, what are the After-Tax Cash Flows for the first 5 years?
Real Estate
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARH
1050000$
9173.81ÞP
9.5gC
25gA
175200?2
12c platinum
ALG Keystrokes
gÂ
fCLEARH
1050000$
9173.81ÞP
9.5gC
25gA
175200?2
40296?3
40296?3
1200000?4
35?5
50?7
3.5?8
1200000?4
35?5
50?7
3.5?8
1.6?9
1.6?9
7?.0
g(031
fsfV
7?.0
g(025
fsfV
fst
fst
t
t
t
t
t
t
t
t
Display
19
Comments
Potential Gross
Income.
1st year operating
40,296.00
cost.
1,200,000.00 Depreciable value.
Depreciable life.
35.00
Marginal tax rate.
50.00
Potential Gross
3.50
Income growth rate.
Operating cost
1.60
growth rate.
Vacancy rate.
7.00
Go to dep. step.
7.00
032, 42 23 RPN:Change to SL
026, 42 23 ALG:depreciation
Year 1
1.00
ATCF1
18,021.07
Year 2
2.00
ATCF2
20,014.26
Year 3
3.00
ATCF3
22,048.90
Year 4
4.00
ATCF4
24,123.14
Year 5
5.00
ATCF5
26,234.69
175,200.00
After-Tax Net Cash Proceeds of Resale
The After-Tax Net Cash Proceeds of Resale (ATNCPR) is the after-tax reversion to
equity; generally, the estimated resale price of the property less commissions, outstanding
debt and any tax claim.
The After-Tax Net Cash Proceeds can be found using the HP 12C Platinum program
which follows.
20 Real Estate
This program uses declining balance depreciation to find the amount of depreciation from
purchase to sale. This amount is used to determine the excess depreciation (which is equal
to the amount of actual depreciation minus the amount of the straight line depreciation).
The Marginal Tax Rate (MTR) that the user inputs is applied to this excess depreciation.
The Capital Gains Tax Rate (CGTR) that the user inputs is applied to the capital gain
from purchase to sale less the expenses of sale (i.e. the NCPR or Net cash Proceeds of
Resale), plus the straight line depreciation.
The user may change to a different depreciation method by keying in the desired function
at line 026 (ALG: 029) in place of f#.
In addition the user may nullify the straight line depreciation by keying in a 0 at line 035
(ALG: 039) in place of fV. This means that all of the actual depreciation from
purchase to sale is then treated as "excess" or unrecaptured depreciation. This is
illustrated below in Example 2.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
gÂ
?2
gA
d
b
?0
~
:7
b
?1
:P
fB
P
M
?+0
fCLEARG
:3
$
:4
n
:5
¼
:2
f#
d
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,44
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
43
44
43
44
45
44
45
42
40
42
45
45
45
45
42
8
2
11
33
25
30
0
34
30
7
25
1
14
14
14
15
0
34
3
13
4
11
5
12
2
25
33
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
gÂ
?2
gA
d
~
~
b
?0
~
§
:7
³
?1
:P
fB
P
M
?+0
fCLEARG
:3
$
:4
n
:5
¼
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,44
021,
022,
023,
024,
025,
026,
027,
43
44
43
44
45
44
45
42
40
42
45
45
45
8
2
11
33
34
30
34
25
30
0
34
20
7
36
1
14
14
14
15
0
34
3
13
4
11
5
12
Real Estate
12c platinum / 12C
RPN KEYSTROKES
:3
~
:6
b
?+1
:2
fV
:2
§
:7
:6
b
?+1
:0
:1
g(000
fs
n: Used
FV: Used
R3: Dep. value
R7: CGTR
DISPLAY
028,
45
3
029,
34
030,
30
031,
45
6
032,
25
033,44 40
1
034,
45
2
035,
42 23
036,
45
2
037,
20
038,
45
7
039,
45
6
040,
30
041,
25
042,44 40
1
043,
45
0
044,
45
1
045,
30
046,43,33,000
12c platinum
ALG KEYSTROKES
:2
f#
~
:3
~
§
:6
³
?+1
:2
fV
§
:2
³
:7
:6
§
~
+
:1
§
1
b
?1
:0
~
³
g(000
fs
REGISTERS
i: Used
PV: Used
R0: NCPR
R1: Tax paid
R4: Dep. life
R5: Factor
R8-R.3: Unused
Program Instructions:
1.
Key in the program and press fCLEARH.
21
DISPLAY
028,
45
2
029,
42 25
030,
34
031,
45
3
032,
30
033,
34
034,
20
035,
45
6
036,
36
037,44 40
1
038,
45
2
039,
42 23
040,
20
041,
45
2
042,
36
043,
45
7
044,
30
045,
45
6
046,
20
047,
34
048,
40
049,
45
1
050,
20
051,
1
052,
25
053,
30
054,
44
1
055,
45
0
056,
34
057,
36
058,43,33,000
PMT: Used
R2: Desired yr.
R6: MTR
22 Real Estate
2.
Key in the loan values:
•
Key in annual interest rate and press gC.
•
Key in mortgage amount and press $.
•
Key in monthly payment and press ÞP.
(If any of the values are unknown, they should be solved for and if one has
to be solved for then the correct payment mode needs to be set)
3.
Key in depreciable value and press ?3.
4.
Key in depreciable life in years and press ?4.
Key in accelerated depreciation factor for the declining balance method and press
?5.
5.
6.
Key in your Marginal Tax Rate as a percentage and press ?6.
7.
Key in the Capital Gains Tax Rate as a percentage and press ?7.
8.
Key in the purchase price and press \(³).
9.
Key in the sale price and press \(³).
10.
Key in the % commission charged on the sale and press \(³).
RPN: If a dollar value is desired instead of a commission rate, key in gÂ,
which does not affect the register values, at line 005 of the program.
ALG: If a dollar value is desired instead of a commission rate, key in gÂ,
which does not affect the register values, at line 008 of the program.
11.
Key in the number of years after purchase and press t. The ATNCPR is
displayed.
12.
To see the NCPR press :0 and to see the tax due press :1.
Example 1: An apartment complex, purchased for $900,000 ten years ago, is sold for
$1,750,000. The closing cost is 8% of the sale price, the income tax rate is 48% and the
capital gains tax rate is 19.2%.
A $700,000 loan for 20 years at 9.5% annual interest was used to purchase the complex.
When it was purchased the depreciable value was $750,000 with a useful life of 25 years.
Using 125% declining balance depreciation, what are the After-Tax Net Cash Proceeds in
year 10?
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARH
700000$
9.5gC
20gA
12c platinum
ALG Keystrokes
gÂ
fCLEARH
700000$
9.5gC
20gA
0.00
700,000.00
0.79
240.00
P
750000?3
PP
750000?3
-6,524.92
750,000.00
Display
Comments
Mortgage.
Monthly interest.
Number of
payments.
Monthly payment.
Depreciable value.
Real Estate
12c platinum / 12C
RPN Keystrokes
25?4
125?5
48?6
19.2?7
12c platinum
ALG Keystrokes
25?4
125?5
48?6
19.2?7
25.00
125.00
48.00
19.20
900000\
1750000\
8\
10t
:0
:1
900000³
1750000³
8³
10t
:0
:1
900,000.00
1,750,000.00
8.00
911,372.04
1,105,746.74
194,374.70
Display
23
Comments
Depreciable life.
Factor.
Marginal Tax Rate.
Capital Gains Tax
Rate.
Purchase price.
Sale price.
Commission rate.
ATNCPR.
NCPR.
Tax due on resale.
Example 2: Now, re-do the previous example assuming all depreciation is treated as
excess or unrecaptured depreciation, with MTR=25% and CGTR=15%.
First the fV in the program must be replaced with 0. This may be done as follows:
RPN: Press g(034 fs 0 fs.
ALG: Press g(038 fs 0 fs.
The data stored in registers R3-R5 need not be re-entered.
12c platinum / 12C
RPN Keystrokes
fCLEARG
700000$
9.5gC
20gA
12c platinum
ALG Keystrokes
fCLEARG
700000$
9.5gC
20gA
0.00
700,000.00
0.79
240.00
P
25?6
15?7
PP
25?6
15?7
-6,524.92
25.00
15.00
900000\
1750000\
8\
10t
:0
:1
900000³
1750000³
8³
10t
:0
:1
900,000.00
1,750,000.00
8.00
924,009.92
1,105,746.74
181,736.83
Display
Comments
Mortgage.
Monthly interest.
Number of
payments.
Monthly payment.
Marginal Tax Rate.
Capital Gains Tax
Rate.
Purchase price.
Sale price.
Commission rate.
ATNCPR.
NCPR.
Tax due on resale.
Lending
Loan With a Constant Amount Paid Towards
Principal
This type of loan is structured such that the principal is repaid in equal installments with
the interest paid in addition. Therefore each periodic payment has a constant amount
applied toward the principle and a varying amount of interest.
Loan Reduction Schedule
If the constant periodic payment to principal, annual interest rate, and loan amount are
known, the total payment, interest portion of each payment, and remaining balance after
each successive payment may be calculated as follows:
1.
RPN Mode:
Key in the constant periodic payment to principal and press ?0.
2.
3.
Key in periodic interest rate and press \\\.
Key in the loan amount. If you wish to skip to another time period, press \.
Then key in the number of payments to be skipped, and press :0§-.
4.
Press ~b to obtain the interest portion of the payment.
5.
Press :0+ to obtain the total payment.
6.
7.
Press O:0- to obtain the remaining balance of the loan.
Return to step 4 for each successive payment.
1.
ALG Mode:
Key in the constant periodic payment to principal and press ?0.
2.
Key in the loan amount and press ?1.
3.
Key in periodic interest rate and press ?2.
If you wish to skip to another time period, key in the number of payments to be
skipped, and press §:0³?-1.
4.
Press :1§:2b+ to obtain the interest portion of the payment.
5.
Press :0?-1³ to obtain the total payment.
6.
Press :1 to obtain the remaining balance of the loan.
Return to step 4 for each successive payment.
7.
24
Lending
25
Example 1: A $60,000 land loan at 10% interest calls for equal semi-annual principal
payments over a 6-year maturity. What is the loan reduction schedule for the first year?
(Constant payment to principal is $5000 semi-annually). What is the fourth year's
schedule (skip 4 payments)?
12c platinum / 12C
12c platinum
RPN Keystrokes
ALG Keystrokes
5000?0
5000?0
10\2z\
60000?1
10z2³?2
\\
60000~b
:1§:2b+
:0+
:0?-1³
O:0:1
§:2b+
~b
:0+
:0?-1³
O:0:1
5.00
3,000.00
8,000.00
55,000.00
2,750.00
7,750.00
50,000.00
Semi-annual interest rate.
First payment's interest.
Total first payment.
Remaining balance.
Second payment's interest.
Total second payment.
Remaining balance after
the first year.
4:0§~b
:0+
O:0~b
:0+
O:0-
1,500.00
6,500.00
25,000.00
1,250.00
6,250.00
20,000.00
Seventh payment's interest.
Total seventh payment.
Remaining balance.
Eighth payment's interest.
Total eighth payment.
Remaining balance after
fourth year.
4§:0³?-1
:1§:2b+
:0?-1³
:1
§:2b+
:0?-1³
:1
Display
Comments
Add-On Interest Rate Converted to APR
An add-on interest rate determines what portion of the principal will be added on for
repayment of a loan. This sum is then divided by the number of months in a loan to
determine the monthly payment. For example, a 10% add-on rate for 36 months on $3000
means add one-tenth of $3000 for 3 years (300 x 3) - usually called the "finance charge" for a total of $3900. The monthly payment is $3900/36.
This keystroke procedure converts an add-on interest rate to a annual percentage rate
when the add-on rate and number of months are known.
26
Lending
1.
RPN Mode:
Press g and press fCLEARG .
2.
Key in the number of months in the loan and press n\:gA .
3.
Key in the add-on rate and press §.
4.
Key in the amount of the loan and press $*~b+.
5.
Press ~zÞP.
6.
Press ¼12§ to obtain the APR.
1.
ALG Mode:
Press g and press fCLEARG .
2.
Key in the number of months in the loan and press n³:gA§ .
3.
Key in the add-on rate and press ³ .
4.
Key in the amount of the loan and press $*+~bz .
5.
Press ~³ÞP .
6.
Press ¼§12³ to obtain the APR.
Example 1: Calculate the APR and monthly payment of a 12% $1000 add-on loan which
has a life of 18 months.
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARG
18n\
:gA12§
1000$~b+
~zÞP
¼12§
12c platinum
ALG Keystrokes
gÂ
fCLEARG
18n³
:gA§12³
1000$+~bz
~³ÞP
¼§12³
Display
Comments
1,180.00 Amount of loan.
Monthly payment.
-65.56
Annual Percentage Rate.
21.64
APR Converted to Add-On Interest Rate
Given the number of months and annual percentage rate, this procedure calculates the
corresponding add-on interest rate.
1. Press g and press fCLEARG .
*
Positive for cash received; negative for cash paid out.
Lending
27
2. Enter the following information:
a. Key in number of months of loan and press n .
b. Key in APR and press gC .
c. Key in 100 and press $P .
3. RPN: Press :$:nz+Þ12§ to obtain the add-on rate.
3. ALG: Press :$z:n+~§12³Þ to obtain the add-on rate.
Example 1: What is the equivalent add-on rate for an 18 month loan with an APR of
14%?
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARG
18n14gC
100$P:$
:nz+Þ
12§
12c platinum
ALG Keystrokes
gÂ
fCLEARG
18n14gC
100$P:$
z:n+~
§12³Þ
Display
7.63
Comments
Add-On Interest Rate.
Add-On Rate Loan with Credit Life
This HP 12C Platinum program calculates the monthly payment amount, credit life
amount (an optional insurance which cancels any remaining indebtedness at the death of
the borrower), total finance charge, and annual percentage rate (APR) for an add-on
interest rate (AIR) loan. The monthly payment is rounded (in normal manner) to the
nearest cent. If other rounding techniques are used, slightly different results may occur.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
gÂ
1
:0
1
2
0
0
z
?4
:2
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
43
45
44
45
8
1
0
1
2
0
0
10
4
2
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
gÂ
:0
n
:gA
b
?4
§
:2
³
:4
DISPLAY
000,
001,
43
002,
45
003,
004,45,43
005,
006,
44
007,
008,
45
009,
010,
45
8
0
11
11
25
4
20
2
36
4
28
Lending
12c platinum / 12C
RPN KEYSTROKES
§
gF
:1
§
:4
§
:4
:1
§
1
+
~
z
:3
§
:0
z
fB
Þ
P
t
:P
:0
§
Þ
$
:$
:2
b
:0
§
1
2
z
?5
Æ
2
§
gT
gm
g(061
:5
DISPLAY
011,
20
012,
30
013,
43 40
014,
45
1
015,
20
016,
45
4
017,
20
018,
30
019,
45
4
020,
45
1
021,
20
022,
1
023,
40
024,
34
025,
10
026,
45
3
027,
20
028,
45
0
029,
10
030,
42 14
031,
16
032,
14
033,
31
034,
45 14
035,
45
0
036,
20
037,
16
038,
13
039,
45 13
040,
45
2
041,
25
042,
45
0
043,
20
044,
1
045,
2
046,
10
047,
44
5
048,
26
049,
2
050,
20
051,
43 24
052,
43 35
053,43,33,061
054,
45
5
12c platinum
ALG KEYSTROKES
§
:1
+
1
³
y
~
§
:0
z
:3
³
y
fB
Þ
P
t
Þ
§
:0
§
:2
§
:4
³
?5
§
Æ
2
³
gT
gm
g(052
1
b
+
:5
³
fB
?5
:5
t
+
DISPLAY
011,
20
012,
45
1
013,
40
014,
1
015,
36
016,
22
017,
30
018,
34
019,
20
020,
45
0
021,
10
022,
45
3
023,
36
024,
22
025,
42 14
026,
16
027,
14
028,
31
029,
16
030,
20
031,
45
0
032,
20
033,
45
2
034,
20
035,
45
4
036,
36
037,
44
5
038,
20
039,
26
040,
2
041,
36
042,
43 24
043,
43 35
044,43,33,052
045,
1
046,
25
047,
40
048,
45
5
049,
36
050,
42 14
051,
44
5
052,
45
5
053,
31
054,
40
Lending
12c platinum / 12C
RPN KEYSTROKES
.
0
1
+
fB
?5
:5
t
:$
~
:3
Þ
t
:5
:3
+
$
:0
n
¼
:gC
g(000
fs
n: N
FV: 0
R3: Loan
DISPLAY
055,
48
056,
0
057,
1
058,
40
059,
42 14
060,
44
5
061,
45
5
062,
31
063,
45 13
064,
34
065,
30
066,
45
3
067,
30
068,
16
069,
31
070,
45
5
071,
45
3
072,
40
073,
13
074,
45
0
075,
11
076,
12
077,45,43 12
078,43,33,000
i: i
R0: N
R4: N/1200
12c platinum
ALG KEYSTROKES
:3
$
M
Þ
t
0
M
¼
:gC
g(000
fs
REGISTERS
PV: Used
R1: AIR
R5: Used
29
DISPLAY
055,
45
3
056,
13
057,
15
058,
16
059,
31
060,
0
061,
15
062,
12
063,45,43 12
064,43,33,000
PMT: PMT
R2: CL (%)
R6-R9: Unused
Program Instructions:
1. Key in the program.
2. Press fCLEARG.
3.
Key in the number of monthly payments in the loan and press ?0.
4.
Key in the annual add-on interest rate as a percentage and press ?1.
5.
Key in the credit life as a percentage and press ?2.
6.
Key in the loan amount and press ?3.
7.
Press t to find the monthly payment amount.
8.
Press t to obtain the amount of credit life.
30
9.
Lending
Press t to calculate the total finance charge.
10. Press t to calculate the annual percentage rate.
11. For a new loan return to step 3.
Example 1: You wish to quote a loan on a $3100 balance, payable over 36 months at an
add-on rate of 6.75%. Credit life (CL) is 1%. What are the monthly payment amount,
credit life amount, total finance charge, and APR?
12c platinum / 12C
RPN Keystrokes
fCLEARG
36?0
6.75?1
1?2
3100?3
t
t
t
t
12c platinum
Display
ALG Keystrokes
fCLEARG
36.00
36?0
6.75
6.75?1
1.00
1?2
3100.00
3100?3
-107.42
t
116.02
t
-651.10
t
12.39
t
Comments
Months.
Add-on interest rate.
Credit life (%).
Loan.
Monthly payment.
Credit life.
Total finance charge.
APR.
Interest Rebate - Rule of 78's
This procedure finds the unearned interest rebate, as well as the remaining principal
balance due for a prepaid consumer loan using the Rule of 78's. The known values are the
current installment number, the total number of installments for which the loan was
written, and the total finance charge (amount of interest). The information is entered as
follows:
RPN Mode:
1. Key in number of months in the loan and press ?1.
2. Key in payment number when prepayment occurs and press -?2 1+.
3. Key in total finance charge and press §:1\§:1+z:2§ to
obtain the unearned interest (rebate).
4. Key in periodic payment amount and press :2§~- to obtain the amount of
principal outstanding.
ALG Mode:
1. Key in number of months in the loan and press ?1-.
2. Key in payment number when prepayment occurs and press³?2+1§.
3. Key in total finance charge and press ³:1g’+:1z~~§:2
³ to obtain the unearned interest (rebate).
4. Key in periodic payment amount and press §:2-~³ to obtain the amount
of principal outstanding.
Lending
31
Example 1: A 30 month $1000 loan having a finance charge of $180, is being repaid at
$39.33 per month. What is the rebate and balance due after the 25th regular payment?
12c platinum / 12C
RPN Keystrokes
30?1
25-?2
1+180§
:1\
§:1+
z:2§
39.33:2§
~-
12c platinum
Display
ALG Keystrokes
30?1
-25³?2
+1§180³
:1g’+:1
z~~§
5.81
:2³
39.33§:2
190.84
-~³
Comments
Rebate.
Outstanding principal.
The following HP 12C Platinum program can be used to evaluate the previous example.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
?0
d
?2
d
?1
:2
?2
1
+
:0
§
:1
\
§
:1
+
z
:2
§
t
:2
§
~
-
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
44
44
44
45
44
45
45
45
45
45
0
33
2
33
1
2
30
2
1
40
0
20
1
36
20
1
40
10
2
20
31
2
20
34
30
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
?0
d
~
?1
?2
~
?-2
:1
g’
+
:1
³
:2
g’
+
:2
z
~
§
:0
³
t
§
:2
-
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,44
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
44
44
44
30
45
43
45
45
43
45
45
45
0
33
34
1
2
34
2
1
20
40
1
36
2
20
40
2
10
34
20
0
36
31
20
2
30
32
Lending
12c platinum / 12C
12c platinum
DISPLAY
DISPLAY
RPN KEYSTROKES
ALG KEYSTROKES
026,43,33,000 ~
026,
34
g(000
027,
36
fs
³
028,43,33,000
g(000
fs
N: Unused
FV: Unused
R3-R.6: Unused
REGISTERS
i: Unused
PV: Unused
R0: Fin. charge
R1: # months
PMT: Unused
R2: Payment #
Program Instructions:
1. Key in the program.
2. Key in the number of months in the loan and press \(³).
3.
Key in the payment number when prepayment occurs and press \(³).
4.
Key in the total finance charge and press t to obtain the unearned interest
(rebate).
5.
Key in the periodic payment amount and press t to find the amount of principal
outstanding.
For a new case return to step 2.
6.
12c platinum / 12C
12c platinum
RPN Keystrokes ALG Keystrokes
30\
30³
25\
25³
180t
180t
39.33t
39.33t
Display
5.81
190.84
Comments
Rebate.
Outstanding principal.
Graduated Payment Mortgages
The Graduated Payment Mortgage is designed to meet the needs of young home buyers
who currently cannot afford high mortgage payments, but who have the potential of
increasing earning in the years to come.
Under the Graduated Payment Mortgage plan, the payments increase by a fixed
percentage at the end of each year for a specified number of years. Thereafter, the
payment amount remains constant for remaining life of the mortgage.
The result is that the borrower pays a reduced payment (a payment which is less than a
traditional mortgage payment) in the early years, and in the later years makes larger
payments than he would with a traditional loan. Over the entire term of the mortgage, the
borrower would pay more than he would with conventional financing.
Lending
33
Given the term of the mortgage (in years), the annual percentage rate, the loan amount,
the percentage that the payments increase, and the number of years that the payments
increase, the following HP 12C Platinum program determines the monthly payments and
remaining balance for each year until the level payment is reached.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
gÂ
?2
~
1
b
1
+
?0
:n
:2
gA
:¼
gC
:$
?3
1
Þ
P
$
Þ
M
1
gA
:P
:0
z
P
$
Þ
M
1
?+1
:1
:2
-
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
028,
029,
030,
031,
032,
033,44
034,
035,
036,
43
44
44
45
45
43
45
43
45
44
43
45
45
40
45
45
8
2
34
1
25
1
40
0
11
2
30
11
12
12
13
3
1
16
14
13
16
15
1
11
14
0
10
14
13
16
15
1
1
1
2
30
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
gÂ
?2
~
b
+
1
³
y
?0
:n
:2
³
gA
:¼
gC
:$
?3
?4
1
P
$
Þ
M
1
gA
:P
§
:0
P
$
Þ
M
1
?+1
:1
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
028,
029,
030,
031,
032,
033,
034,
035,44
036,
43
44
44
45
45
43
45
43
45
44
44
43
45
45
40
45
8
2
34
25
40
1
36
22
0
11
30
2
36
11
12
12
13
3
4
1
14
13
16
15
1
11
14
20
0
14
13
16
15
1
1
1
34
Lending
12c platinum / 12C
RPN KEYSTROKES
gm
g(040
g(025
:3
:$
z
?4
:3
$
1
?3
:3
t
:4
1
:0
:1
q
z
§
Þ
fB
P
t
M
M
fB
t
Þ
$
1
?+3
?-1
:1
gm
g(074
g(048
:4
Þ
t
g(076
fs
DISPLAY
037,
43 35
038,43,33,040
039,43,33,025
040,
45
3
041,
45 13
042,
10
043,
44
4
044,
45
3
045,
13
046,
1
047,
44
3
048,
45
3
049,
31
050,
45
4
051,
1
052,
45
0
053,
45
1
054,
21
055,
10
056,
20
057,
16
058,
42 14
059,
14
060,
31
061,
15
062,
15
063,
42 14
064,
31
065,
16
066,
13
067,
1
068,44 40
3
069,44 30
1
070,
45
1
071,
43 35
072,43,33,074
073,43,33,048
074,
45
4
075,
16
076,
31
077,43,33,076
12c platinum
ALG KEYSTROKES
:2
³
gm
g(043
g(027
:$
?z4
:3
$
1
?3
:3
t
:0
q
:1
§
:4
³
fB
t
P
M
fB
t
Þ
$
1
?+3
?-1
:1
gm
g(072
g(049
:4
t
g(073
fs
DISPLAY
037,
30
038,
45
2
039,
36
040,
43 35
041,43,33,043
042,43,33,027
043,
45 13
044,44 10
4
045,
45
3
046,
13
047,
1
048,
44
3
049,
45
3
050,
31
051,
45
0
052,
21
053,
45
1
054,
20
055,
45
4
056,
36
057,
42 14
058,
31
059,
14
060,
15
061,
42 14
062,
31
063,
16
064,
13
065,
1
066,44 40
3
067,44 30
1
068,
45
1
069,
43 35
070,43,33,072
071,43,33,049
072,
45
4
073,
31
074,43,33,073
Lending
n: Used
FV: Used
R3: Used
i: i/12
R0: Used
R4: Level Pmt.
REGISTERS
PV: Used
R1: Used
R5-R9: Unused
35
PMT: Used
R2: Used
Program Instructions:
1. Key in the program.
2. Press fCLEARH.
3. Key in the term of the loan and press n.
4. Key in the annual interest rate and press ¼.
5. Key in the total loan amount and press $.
6. Key in the rate of graduation (as a percent) and press \(³).
7. Key in the number of years for which the loan graduates and press t. The
following information will be displayed for each year until a level payment is
reached.
a. The current year.
Then press t to continue.
b. The monthly payment for the current year.
Then press t to continue.
c. The remaining balance to be paid on the loan at the end of the current year.
Then press t to return to step a. unless the level payment is reached.
If the level payment has been reached, the program will stop, displaying
the monthly payment over the remaining term of the loan.
8. For a new case press g(000 and return to step 2.
Example: A young couple recently purchased a new house with a Graduated Payment
Mortgage. The loan is for $50,000 over a period of 30 years at an annual interest rate of
12.5%. The monthly payments will be graduating at an annual rate of 5% for the first 5
years and then will be level for the remaining 25 years. What are the monthly payment
amount for the first 6 years?
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
30n
30n
12.5¼
12.5¼
50000$
50000$
5\
5³
5t
5t
t
t
t
t
0.00
30.00
12.50
50,000.00
5.00
1.00
-448.88
-50,914.67
t
t
t
2.00
-471.33
-51,665.07
t
t
t
Display
Comments
Term
Annual interest rate
Loan amount
Rate of graduation
Year 1
1st year monthly payment.
Remaining balance after 1st
year.
Year 2
2nd year monthly payment.
Remaining balance after 2nd
year.
36
Lending
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
3.00
t
t
-494.89
t
t
-52,215.34
t
t
t
t
t
t
t
t
4.00
-519.64
-52,523.86
t
t
t
t
t
t
5.00
-545.62
-52,542.97
t
t
-572.90
Comments
Year 3
3rd year monthly payment.
Remaining balance after 3rd
year.
Year 4
4th year monthly payment.
Remaining balance after 4th
year.
Year 5
5th year monthly payment.
Remaining balance after 5th
year.
Monthly payment for
remainder of term.
Variable Rate Mortgages
As its name suggests, a variable rate mortgage is a mortgage loan which provides for
adjustment of its interest rate as market interest rates change. As a result, the current
interest rate on a variable rate mortgage may differ from its origination rate (i.e., the rate
when the loan was made). This is the difference between a variable rate mortgage and the
standard fixed payment mortgage, where the interest rate and the monthly payment are
constant throughout the term.
Under the agreement of the variable rate mortgage, the mortgage is examined periodically
to determine any rate adjustments. The rate adjustment may be implemented in two ways:
1. Adjusting the monthly payment.
2. Modifying the term of the mortgage.
The period and limits to interest rate increases vary from state to state.
Each periodic adjustment may be calculated by using the HP 12C Platinum with the
following keystroke procedure. The original terms of the mortgage are assumed to be
known.
1. Press g and press fCLEARG.
2. Key in the remaining balance of the loan and press $.
The remaining balance is the difference between the loan amount and the total
principal from the payments which have been made.
To calculate the remaining balance, do the following:
a. Key in the previous remaining balance. If this is the first mortgage adjustment,
this value is the original amount of the loan. Press $.
b. Key in the annual interest rate before the adjustment (as a percentage) and press
gC.
Lending
37
c. Key in the number of years since the last adjustment. If this is the first mortgage
adjustment, then key in the number of years since the origination of the mortgage.
Press gA.
d. Key in the monthly payment over this period and press ÞP.
e. Press M to find the remaining balance, then press fCLEARGÞ$.
3. Key in the adjusted annual interest rate (as a percentage) and press gC.
To calculate the new monthly payment:
a. Key in the remaining life of the mortgage (years) and press gA.
b. RPN: Press P to find the new monthly payment.
b. ALG: Press PP to find the new monthly payment.
To calculate the revised remaining term of the mortgage:
a. Key in the present monthly payment and press P.
b. RPN: Press n12z to find the remaining term of the mortgage in years.
b. ALG: Press nz12³ to find the remaining term of the mortgage in years.
Example: A homeowner purchased his house 3 years ago with a $50,000 variable rate
mortgage. With a 30 year term, his current monthly payment is $495.15. When the
interest rate is adjusted from 11.5% to 11.75%, what will the monthly payment be? If the
monthly payment remained unchanged, find the revised remaining term on the mortgage.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
gÂ
gÂ
fCLEARG
fCLEARG
50000$
50000$
11.5gC
11.5gC
3gA
3gA
495.15ÞP
495.15ÞP
M
M
fCLEARG
fCLEARG
Þ$
Þ$
11.75gC
11.75gC
30\330-3³
gA
gA
P
PP
495.15ÞP
495.15ÞP
n12z
nz12³
Display
Comments
50,000.00
0.96
36.00
-495.15
-49,316.74
Original amount of loan.
Original monthly interest rate.
Period.
Previous monthly payment.
49,316.74
0.98
27.00
324.00
-504.35
-495.15
31.67
Remaining balance.
Adjusted monthly interest.
Remaining life of mortgage.
New monthly payment.
Previous monthly payment.
New remaining term (years).
38
Lending
Skipped Payments
Sometimes a loan (or lease) may be negotiated in which a specific set of monthly
payments are going to be skipped each year. Seasonally is usually the reason for such an
agreement. For example, because of heavy rainfall, a bulldozer cannot be operated in
Oregon during December, January, and February, and the lessee wishes to make payments
only when his machinery is being used. He will make nine payments per year, but the
interest will continue to accumulate over the months in which a payment is not made.
To find the monthly payment amount necessary to amortize the loan in the specified
amount of time, information is entered as follows:
1. Press g and press fCLEARG.
2.
Key in the number of the last payment period before payments close the first time
and press n.
3.
Key in the annual interest rate as a percentage and press gC1PM.
4.
RPN Mode:
Press Þ$12:n-n0PM?0:n.
5.
Key in the number of payments which are skipped and press n1P0$M?+0.
6.
Press 0P12n100$M:$+ÞfCLEARG¼
7.
Key in the total number of years in the loan and press n.
8.
Key in the loan amount and press $P:0z to obtain the monthly payment
amount when the payment is made at the end of the month.
9.
Press ÞM0P1n.
10. Key in the annual interest rate as a percent and press gC$ to find the
monthly payment amount when the payment is made at the beginning of the month.
4.
5.
ALG Mode:
Press Þ$12-:nn0PM?0:n-.
Key in the number of payments which are skipped and press
n1P0$M?+0.
6.
Press 0P12n100$M+:$³ÞfCLEARG¼
7.
Key in the total number of years in the loan and press n.
8.
Key in the loan amount and press $Pz:0³ to obtain the monthly
payment amount when the payment is made at the end of the month.
9.
Press ÞM0P1n.
10. Key in the annual interest rate as a percent and press gC$$ to find the
monthly payment amount when the payment is made at the beginning of the month.
Lending
39
Example: A bulldozer worth $100,000 is being purchased in September. The first
payment is due one month later, and payments will continue over a period of 5 years. Due
to the weather, the machinery will not be used during the winter months, and the
purchaser does not wish to make payments during January, February, and March (months
4 thru 6). If the current interest rate is 14%, what is the monthly payment necessary to
amortize the loan?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
gÂ
gÂ
fCLEARG
fCLEARG
3.00
3n
3n
14gC
1PMÞ$
12:n-n
0PM?0
:n3-n
1P0$M
?+0
0P12n
100$M
:$+Þ
fCLEARG¼
5n100000$
P:0z
1.17
14gC
1PMÞ$
12-:nn
-3.37
0PM?0
:n-3n
1P0$M
-6.18
?+0
0P12n
100$M
+:$³Þ
fCLEARG¼ 14.93
5n100000$
3,119.89
Pz:0³
Comments
Number of payments made
before a group of payments is
skipped.
Monthly interest rate.
FV of 3 monthly unit PMTs.
FV of 6 monthly unit PMTs.
Effective annual interest rate
Monthly payment in arrears.
Savings
Initial Deposit with Periodic Deposits
Given an initial deposit into a savings account, and a series of periodic deposits coincident
with the compounding period, the future value (or accumulated amount) may be
calculated as follows:
1.
Press g and press fCLEARG.
2.
Key in the initial investment and press Þ$.
3.
Key in the number of additional periodic deposits and press n.
4.
Key in the periodic interest rate and press ¼.
5.
Key in the periodic deposit and press ÞP.
6.
Press M to determine the value of the account at the end of the time period.
Example: You have just opened a savings account with a $200 deposit. If you deposit
$50 a month, and the account earns 5 ¼ % compounded monthly, how much will you
have in 3 years?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
gÂ
gÂ
fCLEARG
fCLEARG
200Þ$
200Þ$
3gA
3gA
5.25gC
5.25gC
2,178.94
50ÞPM
50ÞPM
Comments
Value of the account.
Note: If the periodic deposits do not coincide with the compounding periods, the account
must be evaluated in another manner. First, find the future value of the initial deposits and
store it. Then use the procedure for compounding periods different from payment periods
to calculate the future value of the periodic deposits. Recall the future value of the initial
deposit and add to obtain the value of the account.
40
Savings
41
Number of Periods to Deplete a Savings
Account or to Reach a Specified Balance
Given the current value of a savings account, the periodic interest rate, the amount of the
periodic withdrawal, and a specified balance, this procedure determines the number of
periods to reach that balance (the balance is zero if the account is depleted).
1.
Press g and press fCLEARG.
2.
Key in the value of the savings account and press Þ$.
3.
Key in the periodic interest rate and press ¼.
4.
Key in the amount of the periodic withdrawal and press P.
5.
Key in the amount remaining in the account and press M.
This step may be omitted if the account is depleted (FV=0).
6.
Press n to determine the number of periods to reach the desired balance.
Example: Your savings account presently contains $18,000 and earns 5 ½% compounded
monthly. You wish to withdraw $300 a month until the account is depleted. How long
will this take? If you wish to reduce the account to $5000, how many withdrawals can you
make?
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARG
18000Þ$
5.5gC
300Pn
5000Mn
12c platinum
ALG Keystrokes
gÂ
fCLEARG
18000Þ$
5.5gC
300Pn
5000Mn
Display
71.00
53.00
Comments
Months to deplete account.
Months to reduce the account to
$5000.
42
Savings
Periodic Deposits and Withdrawals
This section is presented as a guideline for evaluating a savings plan when deposits and
withdrawals occur at irregular intervals. One problem is given, and a step by step method
for setting up and solving the problem is presented:
Example: You are presently depositing $50 and the end of each month into a local
savings and loan, earning 5 ½% compounded monthly. Your current balance is $1023.25.
How much will you have accumulated in 5 months?
The cash flow diagram looks like this:
FV = ?
1
2
- 50
3
- 50
4
- 50
5
- 50
- 50
PV = - 1023.25
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
gÂ
gÂ
fCLEARG
fCLEARG
50ÞP
50ÞP
5.5gC
5.5gC
1023.25Þ$
1023.25Þ$
1,299.22
5nM
5nM
Comments
Amount in account.
Now suppose that at the beginning of the 6th month you withdrew $80.
What is the new balance?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
1,219.22
80-80³
Comments
New balance.
Savings
43
You increase your monthly deposit to $65. How much will you have in 3 months?
The cash flow diagram looks like this:
FV = ?
1
2
- 65
3
- 65
- 65
PV = - 1219.22
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
Þ$
Þ$
65ÞP
65ÞP
1,431.95
3nM
3nM
Comments
Account balance.
Suppose that for 2 months you decide not to make a periodic deposit. What is the balance
in the account?
FV = ?
1
2
PV = - 1431.95
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
Þ$2n
Þ$2n
1,445.11
0PM
0PM
Comments
Account balance.
44
Savings
This type of procedure may be continued for any length of time, and may be modified to
meet the user's particular needs.
Savings Account Compounded Daily
This HP 12C Platinum program determines the value of a savings account when interest is
compounded daily, based on a 365 day year. The user is able to calculate the total amount
remaining in the account after a series of transactions on specified dates.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
Þ
$
d
3
6
5
z
¼
d
?0
:$
Þ
t
?2
d
?1
:0
:1
gÒ
n
M
fB
M
\
:$
+
?+3
:M
:2
+
Þ
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,44
028,
029,
030,
031,
44
45
44
44
45
45
43
42
45
40
45
45
16
13
33
3
6
5
10
12
33
0
13
16
31
2
33
1
0
1
26
11
15
14
15
36
13
40
3
15
2
40
16
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
Þ
$
d
z
3
6
5
¼
d
?0
:$
Þ
t
?2
d
?1
:0
:1
gÒ
n
M
fB
M
+
:$
³
?+3
:M
Þ
:2
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,44
028,
029,
030,
031,
44
45
44
44
45
45
43
42
45
40
45
45
16
13
33
10
3
6
5
12
33
0
13
16
31
2
33
1
0
1
26
11
15
14
15
40
13
36
3
15
16
30
2
Savings
12c platinum / 12C
RPN KEYSTROKES
$
:1
?0
:$
Þ
g(013
fs
N: ∆days
FV: Used
R3: Interest
DISPLAY
032,
13
033,
45
1
034,
44
0
035,
45 13
036,
16
037,43,33,013
i: i/365
R0: Initial date
R4-R.4: Unused
12c platinum
ALG KEYSTROKES
$
:1
?0
:$
Þ
g(013
fs
REGISTERS
PV: Used
R1: Next date
45
DISPLAY
032,
13
033,
45
1
034,
44
0
035,
45 13
036,
16
037,43,33,013
PMT: 0
R2: $ amount
Program Instructions:
1. Key in the program.
2. Press fCLEARH and press gÕ.
3.
Key in the date (MM.DDYYYY) of the first transaction and press \(³).
4.
Key in the annual nominal interest rate as a percentage and press \(³).
5.
Key in the amount of the initial deposit and press t.
6.
Key in the date of the next transaction and press \(³).
Key in the amount of the transaction (positive for money deposited, negative for cash
withdrawn) and press t to determine the amount in the account.
Repeat steps 6 and 7 for subsequent transactions.
7.
8.
9.
To see the total interest to date, press :3.
10. For a new case press f(000 and go to step 2.
Example: Compute the amount remaining in this 5.25% account after the
following transactions:
1.
2.
3.
4.
5.
6
January 19, 2003 deposit $125.00
February 24, 2003 deposit $60.00
March 16, 2003 deposit $70.00
April 6, 2003 withdraw $50.00
June 1, 2003 deposit $175.00
July 6, 2003 withdraw $100.00
46
Savings
12c platinum / 12C
RPN Keystrokes
fCLEARH
gÕ
1.192003\
5.25\
125t
2.242003\
60t
12c platinum
Display
ALG Keystrokes
fCLEARH
gÕ
1.192003³
5.25³
125.00
125t
2.242003³
185.65
60t
3.162003\
70t
3.162003³
70t
256.18
Balance in account,
March 16, 2003.
4.062003\
50Þt
4.062003³
50Þt
206.95
Balance in account,
April 6, 2003.
6.012003\
175t
6.012003³
175t
383.62
Balance in account,
June 1, 2003.
7.062003\
100Þt
7.062003³
100Þt
285.56
:3
:3
5.56
Balance in account,
July 6, 2003.
Total interest.
Comments
Initial Deposit.
Balance in account,
February 24, 2003.
Compounding Periods Different From
Payment Periods
In financial calculations involving a series of payments equally spaced in time with
periodic compounding, both periods of time are normally equal and coincident. This
assumption is preprogrammed into the HP 12C Platinum.
In savings plans however, money may become available for deposit or investment at a
frequency different from the compounding frequencies offered. The HP 12C Platinum can
easily be used in these calculations. However, because of the assumptions mentioned the
periodic interest rate must be adjusted to correspond to an equivalent rate for the payment
period.
Payments deposited for a partial compounding period will accrue simple interest for the
remainder of the compounding period. This is often the case, but may not be true for all
institutions.
These procedures present solutions for future value, payment amount, and number of
payments. In addition, it should be noted that only annuity due (payments at the beginning
of payment period) calculations are shown since this is the most common in savings plan
calculations.
Savings
47
To calculate the equivalent payment period interest rate, information is entered as follows:
1. Press g× and press fCLEARG.
RPN Mode:
2. Key in the annual interest rate (as a percent) and press \.
3. Key in the number of compounding periods per year and press nz¼.
ALG Mode:
2. Key in the number of compounding periods per year and press n.
3. Key in the annual interest rate (as a percent) and press z:n¼.
4. Key in 1 and press $M.
5. Key in the number of payments (deposits) per year and press
n¼fCLEARG¼.
The interest rate which corresponds to the payment period is now in register "i" and you
are ready to proceed.
Example 1: Solving for future value.
Starting today you make monthly deposits of $25 into an account paying 5% compounded
daily (365-day basis). At the end of 7 years, how much will you receive from the account?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
g×
g×
fCLEARG
fCLEARG
5\
365n
365nz¼
5z:n¼
1$M
1$M
12n¼
12n¼
fCLEARG¼ fCLEARG¼ 0.42
7gA
25ÞP
M
7gA
25ÞP
M
2,519.61
Comments
Equivalent periodic interest
rate.
Future value.
48
Savings
Example 2: Solving for payment amount.
For 8 years you wish to make weekly deposits in a savings account paying 5.5%
compounded quarterly. What amount must you deposit each week to accumulate $6000.
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
g×
g×
fCLEARG
fCLEARG
5.5\
4n
4nz¼
5.5z:n¼
1$M
1$M
52n¼
52n¼
fCLEARG¼ fCLEARG¼ 0.11
8\52§n
6000MP
8§52n
6000MP
-11.49
Comments
Equivalent periodic interest
rate.
Periodic payment.
Example 3: Solving for number of payment periods.
You can make weekly deposits of $10 in to an account paying 5.25% compounded daily
(365-day basis). How long will it take you to accumulate $1000?
12c platinum / 12C
12c platinum
RPN Keystrokes ALG Keystrokes
g×
g×
fCLEARG
fCLEARG
5.25\
365n
365nz¼
5.25z:n¼
1$M
1$M
52n¼
52n¼
fCLEARG¼ fCLEARG¼
0.10
Equivalent periodic interest
rate.
10ÞP
1000Mn
96.00
Weeks.
10ÞP
1000Mn
Display
Comments
Investment Analysis
Lease vs. Purchase
An investment decision frequently encountered is the decision to lease or purchase capital
equipment or buildings. Although a thorough evaluation of a complex acquisition usually
requires the services of a qualified accountant, it is possible to simplify a number of the
assumptions to produce a first approximation.
The following HP 12C Platinum program assumes that the purchase is financed with a
loan and that the loan is made for the term of the lease. The tax advantages of interest
paid, depreciation, and the investment credit which accrues from ownership are compared
to the tax advantage of treating the lease payment as an expense. The resulting cash flows
are discounted to the present at the firm's after-tax cost of capital.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
1
?+0
:3
§
?8
1
f!
?1
:$
?9
:P
?.0
:n
?.1
:¼
?.2
:5
$
:6
n
DISPLAY
000,
001,
002,
003,44
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,44
015,
016,44
017,
018,44
019,
020,
021,
022,
40
45
44
42
44
45
44
45
48
45
48
45
48
45
45
30
1
0
3
30
20
8
1
11
1
13
9
14
0
11
1
12
2
5
13
6
11
49
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
~
³
?8
1
?+0
f!
?1
:$
?9
:¼
?.2
:5
$
:6
n
:7
¼
:0
fÝ
?+1
:9
DISPLAY
000,
001,
002,
003,
004,
005,
006,44
007,
008,
009,
010,
011,
012,44
013,
014,
015,
016,
017,
018,
019,
020,
021,44
022,
44
40
42
44
45
44
45
48
45
45
45
45
42
40
45
30
34
36
8
1
0
11
1
13
9
12
2
5
13
6
11
7
12
0
24
1
9
50
Investment Analysis
12c platinum / 12C
RPN KEYSTROKES
:7
¼
:0
fÝ
?+1
:9
$
:.0
P
:.1
n
:.2
¼
:1
:3
§
:P
:8
:4
:0
q
z
?+2
g(000
fs
023,
45
7
024,
12
025,
45
0
026,
42 24
027,44 40
1
028,
45
9
029,
13
030,45 48
0
031,
14
032,45 48
1
033,
11
034,45 48
2
035,
12
036,
45
1
037,
45
3
038,
20
039,
45 14
040,
30
041,
45
8
042,
30
043,
45
4
044,
45
0
045,
21
046,
10
047,44 40
2
048,43,33,000
n: Used
FV: 0
R3: Tax
R7: Factor (DB)
R.1: Used
i: Used
R0: Used
R4: Discount
R8: Used
R.2: Used
DISPLAY
12c platinum
ALG KEYSTROKES
$
:.2
¼
:1
:8
§
:3
+
:8
:P
³
:4
q
:0
z
~
~
³
?+2
g(000
fs
REGISTERS
PV: Used
R1: Used
R5: Dep. Value
R9: Used
R.3: Unused
Program Instructions:
1. Key in the program.
RPN: - Select the depreciation function and key in at line 26.
ALG: - Select the depreciation function and key in at line 20.
2. Press g and press fCLEARH.
3. Input the following information for the purchase of the loan:
- Key in the number of years for amortization and press n.
DISPLAY
023,
13
024,45 48
2
025,
12
026,
45
1
027,
30
028,
45
8
029,
20
030,
45
3
031,
40
032,
45
8
033,
30
034,
45 14
035,
36
036,
45
4
037,
21
038,
45
0
039,
10
040,
34
041,
34
042,
36
043,44 40
2
044,43,33,000
PMT: Used
R2: Purch. Adv.
R6: Dep. life
R.0: Used
Investment Analysis
4.
5.
5.
6.
7.
8.
9.
9.
10.
11.
12.
13.
13.
14.
51
- Key in the annual interest rate and press ¼.
- Key in the loan amount (purchase price) and press Þ$.
- Press P to find the annual payment.
Key in the marginal effective tax rate* and press ?3.
RPN: Key in the discount rate or cost of capital* and press \1+?4.
ALG: Key in the discount rate or cost of capital* and press +1³?4.
Key in the depreciable value and press ?5.
Key in the depreciable life and press ?6.
For declining balance depreciation, key in the depreciation factor (as a percentage)
and press ?7.
RPN: Key in the total first lease payment (including any advance payments) and
press \1:3-§?2.
ALG: Key in the total first lease payment (including any advance payments) and
press ³1-:3§~³?2.
Key in the first year's maintenance expense that would be anticipated if the asset was
owned and press \(³) . If the lease contract does not include maintenance, then
it is not a factor in the lease vs. purchase decision and 0 expense should be used.
Key in the next lease payment and press t. During any year in which a lease
payment does not occur (e.g. the last several payments of an advance payment
contract) use 0 for the payment.
Repeat steps 10 and 11 for all maintenance expenses and lease payments over the
term of the analysis.
Optional - If the investment tax credit is taken, key in the amount of the credit after
finishing steps 10 and 11 for the year in which the credit is taken and press
g(043 (ALG:036) t . Continue steps 10 and 11 for the remainder of the
term.
RPN: After all the lease payments and expenses have been entered (steps 10 and 11),
key in the lease buy back option and press \1:3-§g(043t. If no
buy back option exists, use the estimated salvage value of the purchased equipment at
the end of the term.
ALG: After all the lease payments and expenses have been entered (steps 10 and 11),
key in the lease buy back option and press ³1:3§~³g(036t. If no buy back option exists, use the estimated
salvage value of the purchased equipment at the end of the term.
To find the net advantage of owning press :2. A negative value represents a net
lease advantage.
Example: Home Style Bagel Company is evaluating the acquisition of a mixer which can
be leased for $1700 a year with the first and last payments in advance and a $750 buy
back option at the end of 10 years (maintenance is included).
The same equipment could be purchased for $10,000 with a 12% loan amortized over 10
years. Ownership maintenance is estimated to be 2% of the purchase price per year for the
first for years. A major overhaul is predicted for the 5th year at a cost of $1500.
Subsequent yearly maintenance of 3% is estimated for the remainder of the 10 year term.
The company would use sum of the years digits depreciation on a 10 year life with $1500
*
Key in as a decimal (e.g., 5% as .05).
52
Investment Analysis
salvage value. An accountant informs management to take the 10% capital investment tax
credit at the end of the second year and to figure the cash flows at a 48% tax rate. The
after tax cost of capital (discounting rate) is 5 percent.
Because lease payments are made in advance and standard loan payments are made in
arrears the following cash flow schedule is appropriate for a lease with the last payment in
advance.
Year
0
1
2
3
4
5
6
7
8
9
10
Maintenance
200
200
200
200
1500
300
300
300
300
300
Lease Payment
1700+1700
1700
1700
1700
1700
1700
1700
1700
1700
0
0
12c platinum / 12C
RPN Keystrokes
gÂ
fCLEARH
10n12¼
10000Þ$
12c platinum
ALG Keystrokes
gÂ
fCLEARH
10n12¼
10000Þ$
P
.48?3
.05\1+?4
10000\
1500-?5
10?6
1700\+
1:3-§?2
200\
1700t
P
.48?3
.05+1³?4
100001500³?5
10?6
1700+³
1-:3§~
³?2
200³
1700t
200\
1700t
1000g(043
t
200\
1700t
200\
1700t
200³
1700t
1000g(036
t
200³
1700t
200³
1700t
Tax Credit
Buy Back
1000
Display
750
Comments
0.00
1,769.84
0.48
1.05
Always use negative loan
amount.
Purchase payment.
Marginal tax rate.
Discounting factor.
8,500.00
10.00
3,400.00
1,768.00
Depreciable value.
Depreciable life.
1st lease payment.
After-tax expense.
312.36
Present value of 1st year's
net purchase.
200.43
1,000.00
907.03
2nd year's advantage.
Tax credit.
Present value of tax credit.
95.05
3rd year.
-4.38
4th year.
-10,000.00
Investment Analysis
12c platinum / 12C
RPN Keystrokes
1500\
1700t
300\
1700t
300\
1700t
300\
1700t
300\0t
300\0t
750\
1:3-§
12c platinum
ALG Keystrokes
1500³
1700t
300³
1700t
300³
1700t
300³
1700t
300³0t
300³0t
750³
1-:3§~³
g(043t
:2
g(036t
:2
239.43
-150.49
Display
53
Comments
-628.09
5th year.
-226.44
6th year.
-309.48
7th year.
-388.81
-1,034.72
-1,080.88
750.00
390.00
8th year.
9th year.
10th year.
Buy back.
After tax buy back
expense.
Present value.
Net lease advantage.
Break-Even Analysis
Break-even analysis is basically a technique for analyzing the relationships among fixed
costs, variable costs, and income. Until the break-even point is reached at the intersection
of the total sales revenue and total cost lines, the producer operates at a loss. After the
break-even point each unit produced and sold makes a profit. Break-even analysis may be
represented as follows.
Profit
Variable
Costs
Break-even Point
Fixed Costs
54
Investment Analysis
The variables are: fixed costs (F), Sales price per unit (P), variable cost per unit (V),
number of units sold (U), and gross profit (GP). One can readily evaluate GP, U or P
given the four other variables. To calculate the break-even volume, simply let the gross
profit equal zero and calculate the number of units sold (U).
To calculate the break-even volume:
RPN Mode:
1. Key in the fixed costs and press \.
2. Key in the unit price and press \.
3. Key in the variable cost per unit and press -.
4. Press z to calculate the break-even volume.
ALG Mode:
1. Key in the fixed costs and press ³.
2. Key in the unit price and press -.
3. Key in the variable cost per unit and press z.
4. Press ~~³ to calculate the break-even volume.
To calculate the gross profit at a given volume:
RPN Mode:
1. Key in the unit price and press \.
2. Key in the variable cost per unit and press -.
3. Key in the number of units sold and press §.
4. Key in the fixed cost and press - to calculate the gross profit.
ALG Mode:
1. Key in the unit price and press -.
2. Key in the variable cost per unit and press §.
3. Key in the number of units sold and press -.
4. Key in the fixed cost and press ³ to calculate the gross profit.
To calculate the sales volume needed to achieve a specified gross profit:
RPN Mode:
1. Key in the desired gross profit and press \.
2. Key in the fixed cost and press +.
3. Key in sales price per unit and press \.
4. Key in the variable cost per unit and press -.
5. Press z to calculate the sales volume.
Investment Analysis
55
ALG Mode:
1. Key in the desired gross profit and press +.
2. Key in the fixed cost and press ³.
3. Key in sales price per unit and press -.
4. Key in the variable cost per unit and press z.
5. Press ~~³ to calculate the sales volume.
To calculate the required sales price to achieve a given gross profit at a specified sales
volume:
RPN Mode:
1. Key in the fixed costs and press \.
2. Key in the gross desired and press +.
3. Key in the specified sales volume in units and press z.
4. Key in the variable cost per unit and press + to calculate the required sales price per
unit.
ALG Mode:
1. Key in the fixed costs and press +.
2. Key in the gross desired and press z.
3. Key in the specified sales volume in units and press +.
4. Key in the variable cost per unit and press ³ to calculate the required sales price per
unit.
Example 1: The E.Z. Sells company markets textbooks on salesmanship. The fixed cost
involved in setting up to print the books are $12,000. The variable cost per copy,
including printing and marketing the books are $6.75 per copy. The sales price per copy is
$13.00. How many copies must be sold to break even?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
12,000.00
12000\
12000³
13.00
13\
136.75-z
6.75z~~³ 1,920.00
Comments
Fixed cost.
Sales price.
Break-even volume.
56
Investment Analysis
Find the gross profit if 2500 units are sold.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
13\
136.756.75§
2500§
25001200012000³
Display
13.00
6.25
15,625.00
3,625.00
Comments
Sales price.
Profit per unit.
Gross profit.
If a gross profit of $4500 is desired at a sales volume of 2500 units, what should the sales
price be?
12c platinum / 12C
RPN Keystrokes
12000\
4500+
2500z
6.75+
12c platinum
ALG Keystrokes
12000+
4500z
2500+
6.75³
Display
12,000.00
16,500.00
6.60
13.35
Comments
Fixed cost.
Sales price per unit to achieve
desired gross profit.
For repeated calculation the following HP 12C Platinum program can be used.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:3
:2
g(000
:4
§
:1
g(000
:5
:1
+
~
z
g(000
:1
:5
+
DISPLAY
000,
001,
45
3
002,
45
2
003,
30
004,43,33,000
005,
45
4
006,
20
007,
45
1
008,
30
009,43,33,000
010,
45
5
011,
45
1
012,
40
013,
34
014,
10
015,43,33,000
016,
45
1
017,
45
5
018,
40
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:3
:2
³
g(000
§
:4
:1
³
g(000
:5
+
:1
z
~
³
g(000
DISPLAY
000,
001,
45
3
002,
30
003,
45
2
004,
36
005,43,33,000
006,
20
007,
45
4
008,
30
009,
45
1
010,
36
011,43,33,000
012,
45
5
013,
40
014,
45
1
015,
10
016,
34
017,
36
018,43,33,000
Investment Analysis
12c platinum / 12C
RPN KEYSTROKES
:4
z
:2
+
g(000
fs
n: Unused
FV: Unused
R3: P
DISPLAY
019,
45
4
020,
10
021,
45
2
022,
40
023,43,33,000
I: Unused
R0: Unused
R4: U
12c platinum
ALG KEYSTROKES
:1
+
:5
z
:4
+
:2
³
g(000
fs
REGISTERS
PV: Unused
R1: F
R5: GP
57
DISPLAY
019,
45
1
020,
40
021,
45
5
022,
10
023,
45
4
024,
40
025,
45
2
026,
36
027,43,33,000
PMT: Unused
R2: V
R6-R.6: Unused
Program Instructions:
1. Key in the program and store the known variables as follows:
a. Key in the fixed costs, F and press ?1.
b. Key in the variable costs per unit, V and press ?2.
c. Key in the unit price, P (if known) and press ?3.
d. Key in the sales volume, U, in units (if known) and press ?4.
e. Key in the gross profit, GP, (if known) and press ?5.
2. To calculate the sales volume to achieve a desired gross profit:
a. Store values as shown in 1a, 1b, and 1c.
b. Key in the desired gross profit (zero for break even) and press ?5.
c. RPN: Press tg(010t to calculate the required volume.
c. ALG: Press tg(012t to calculate the required volume.
3. To calculate the gross profit at a given sales volume.
a. Store values as shown in 1a, 1b, 1c, and 1d.
b. RPN: Press tg(005t to calculate gross profit.
b. ALG: Press tg(006t to calculate gross profit.
4. To calculate the sales price per unit to achieve a desired gross profit at a specified
sales volume:
a. Store values as shown in 1a, 1b, 1d, and 1e.
b. RPN: Press g(016t to calculate the required sales price.
b. ALG: Press g(019t to calculate the required sales price.
58
Investment Analysis
Example 2: A manufacturer of automotive accessories produces rear view mirrors. A new
line of mirrors will require fixed costs of $35,000 to produce. Each mirror has a variable
cost of $8.25. The price of mirrors is tentatively set at $12.50 each. What volume is
needed to break even?
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
35000?1
35000?1
8.25?2
8.25?2
12.5?3
12.5?3
0?5
0?5
tg(010t tg(012t
Display
35,000.00
8.25
12.50
0.00
8,235.29
Comments
Fixed cost.
Variable cost.
Sales price.
Break-even volume is between
8,235 and 8,236 units.
What would be the gross profit if the price is raised to $14.00 and the sales volume is
10,000 units?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
14.00
14?3
14?3
10,000.00
10000?4
10000?4
tg(005t tg(006t 22,500.00
Comments
Sales price.
F and V are already stored.
Volume.
Gross Profit.
Investment Analysis
59
Operating Leverage
The degree of operating leverage (OL) at a point is defined as the ratio of the percentage
change in net operating income to the percentage change in units sold. The greatest degree
of operating leverage is found near the break-even point where a small change in sales
may produce a very large increase in profits. Likewise, firms with a small degree of
operating leverage are operating farther form the break-even point, and they are relatively
insensitive to changes in sales volume.
The necessary inputs to calculate the degree of operating leverage and fixed costs (F),
sales price per unit (P), variable cost per unit (V) and number of units (U).
The operating leverage may be readily calculated as follows:
RPN Mode:
1. Key in the sales price per unit and press \.
2. Key in the variable cost per unit and press -.
3. Key in the number of units and press §\\.
4. Key in the fixed cost and press -z to obtain the operating leverage.
ALG Mode:
1. Key in the sales price per unit and press -.
2. Key in the variable cost per unit and press §.
3. Key in the number of units and press ³.
4. Key in the fixed cost and press àbÞy to obtain the operating leverage.
Example 1: For the data given in example 1 of the Break-Even Analysis section,
calculate the operating leverage at 2000 units and at 5000 units when the sales price is $13
a copy.
12c platinum / 12C
RPN Keystrokes
13\
6.752000§\
\12000-z
13\
6.755000§\
\12000-z
12c platinum
ALG Keystrokes
136.75§
2000³12000
àbÞy
136.75§
5000³12000
àbÞy
Display
Comments
13.00
6.25
Price per copy.
Profit per copy.
25.00
13.00
6.25
Close to break-even point.
Price per copy.
Profit per copy.
1.62
Operating further from the
breakeven point and less
sensitive to changes in sales
volume.
60
Investment Analysis
For repeated calculations the following HP 12C Platinum program can be used:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:3
:2
§
\
\
:1
z
g(000
fs
n: Unused
FV: Unused
R3: P
DISPLAY
000,
001,
45
3
002,
45
2
003,
30
004,
20
005,
36
006,
36
007,
45
1
008,
30
009,
10
010,43,33,000
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:3
:2
§
~
³
:1
à
b
Þ
y
g(000
fs
REGISTERS
i: Unused
PV: Unused
R0: Unused
R1: F
R4-R.8: Unused
DISPLAY
000,
001,
45
3
002,
30
003,
45
2
004,
20
005,
34
006,
36
007,
45
1
008,
24
009,
25
010,
16
011,
22
012,43,33,000
PMT: Unused
R2: V
Program Instructions:
1. Key in the program.
2. Key in and store input variables F, V and P as described in the Break-Even Analysis
program.
3. Key in the sales volume and press t to calculate the operating leverage.
4. To calculate a new operating leverage at a different sales volume, key in the new sales
volume and press t.
Example 2: For the figures given in example 2 of the Break-Even Analysis section,
calculate the operating leverage at a sales volume of 9,000 and 20,000 units if the sales
price is $12.50 per unit.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
35000?1
35000?1
8.25?2
8.25?2
12.5?3
12.5?3
9000t
9000t
35,000.00
8.25
12.50
11.77
20000t
1.70
20000t
Display
Comments
Fixed costs.
Variable cost.
Sales price.
Operating leverage near breakeven.
Operating leverage further
from break-even.
Investment Analysis
61
Profit and Loss Analysis
The HP 12C Platinum may be programmed to perform simplified profit and loss analysis
using the standard profit income formula and can be used as a dynamic simulator to
quickly explore ranges of variables affecting the profitability of a marketing operation.
The program operates with net income return and operating expenses as percentages. Both
percentage figures are based on net sales price.
It may also be used to simulate a company wide income statement by replacing list price
with gross sales and manufacturing cost with cost of goods sold.
Any of the five variables: a) list price, b) discount (as a percentage of list price), c)
manufacturing cost, d) operating expense (as a percentage), e) net profit after tax (as a
percentage) may be calculated if the other four are known.
Since the tax rage varies from company to company, provision is made for inputting your
applicable tax rate. The example problem uses a tax rate of 48%.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:5
:6
z
:4
+
Þ
:0
+
:0
z
g(000
:3
:1
:2
:0
z
Þ
1
+
§
t
z
Þ
1
DISPLAY
000,
001,
45
5
002,
45
6
003,
10
004,
45
4
005,
40
006,
16
007,
45
0
008,
40
009,
45
0
010,
10
011,43,33,000
012,
45
3
013,
45
1
014,
45
2
015,
45
0
016,
10
017,
16
018,
1
019,
40
020,
20
021,
31
022,
10
023,
16
024,
1
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:5
z
:6
+
:4
1
~
b
³
g(000
:3
:1
:2
~
~
b
³
t
g(027
~
z
~
DISPLAY
000,
001,
45
5
002,
10
003,
45
6
004,
40
005,
45
4
006,
30
007,
1
008,
34
009,
25
010,
36
011,43,33,000
012,
45
3
013,
45
1
014,
45
2
015,
34
016,
30
017,
34
018,
25
019,
36
020,
31
021,43,33,027
022,
34
023,
10
024,
34
62
Investment Analysis
12c platinum / 12C
RPN KEYSTROKES
+
:0
§
g(000
z
Þ
:1
+
:1
z
:0
§
g(000
:5
:6
z
g(000
:4
:6
§
g(000
fs
n: Unused
FV: Unused
R3: mfg. cost
R7-R.3: Unused
DISPLAY
025,
40
026,
45
0
027,
20
028,43,33,000
029,
10
030,
16
031,
45
1
032,
40
033,
45
1
034,
10
035,
45
0
036,
20
037,43,33,000
038,
45
5
039,
45
6
040,
10
041,
30
042,43,33,000
043,
45
5
044,
30
045,
45
6
046,
20
047,43,33,000
i: Unused
R0: 100
R4: % op. exp.
12c platinum
ALG KEYSTROKES
³
:1
~
à
Þ
g(000
:5
z
:6
~
~
³
g(000
:4
§
:6
³
g(000
fs
REGISTERS
PV: Unused
R1: list price
R5: % net profit
DISPLAY
025,
36
026,
45
1
027,
34
028,
24
029,
16
030,43,33,000
031,
45
5
032,
10
033,
45
6
034,
30
035,
34
036,
34
037,
36
038,43,33,000
039,
30
040,
45
4
041,
20
042,
45
6
043,
36
044,43,33,000
PMT: Unused
R2: % discount
R6: 1-% tax
Program Instructions:
1. Key in the program and press fCLEARH, then key in 100 and press ?0.
2. RPN: Key in 1 and press \, then key in your appropriate tax rate as a decimal and
press -?6.
2. ALG: Key in 1 and press -, then key in your appropriate tax rate as a decimal and
press ³?6.
3. a. Key in the list price in dollars (if known) and press ?1.
b. Key in the discount in percent (if known) and press ?2.
c.
Key in the manufacturing cost in dollars (if known) and press ?3.
d.
Key in the operating expense in percent (if known) and press ?4.
e.
Key in the net profit after tax in percent (if known) and press ?5.
Investment Analysis
4. To calculate list price:
a. Do steps 2 and 3b, c, d, e above.
b.
RPN: Press :3tz1g(014tzg(000.
b.
ALG: Press :3t~z~³1g(014t
~z~ ³g(000.
5. To calculate discount:
a. Do steps 2 and 3a, c, d, e above.
b.
RPN: Press :3tg(029t.
b.
ALG: Press :3tg(022t.
6. To calculate manufacturing cost:
a. Do steps 2 and 3a, b, d, e, above.
b.
RPN: Press g(013tg(001t§.
b.
ALG: Press g(013tg(001t§~³.
7. To calculate operating expense:
a. Do steps 2 and 3a, b, c, e, above.
b.
RPN: Press g(012ttg(038t.
b.
ALG: Press g(012ttg(031t.
8. To calculate net profit after tax:
a. Do steps 2 and 3a, b, c, d, above.
b.
RPN: Press g(012ttg(043t.
b.
ALG: Press g(012ttg(039t.
Example: What is the net return on an item that is sold for $11.98, discounted through
distribution an average of 35% and has a manufacturing cost of $2.50? The standard
company operating expense is 32% of net shipping (sales) price and tax rate is 48%.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
100?0
100?0
1\.48-?6 1-.48³?6
11.98?1
11.98?1
35?2
35?2
2.50?3
2.50?3
32?4
32?4
g(012tt g(012tt
g(043t
g(039t
Display
100.00
0.52
11.98
35.00
2.50
32.00
67.90
18.67
Comments
48% tax rate.
List price ($).
Discount (%).
Manufacturing cost ($).
Operating expenses (%).
Net profit (%).
63
64
Investment Analysis
If manufacturing expenses increase to $3.25, what is the effect on net profit?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
3.25
3.25?3
3.25?3
g(012tt g(012tt 58.26
13.66
g(043t
g(039t
Comments
Manufacturing cost.
Net profit reduced to 13.66%
If the manufacturing cost is maintained at $3.25, how high could the overhead (operating
expense) be before the product begins to lose money?
12c platinum / 12C
12c platinum
Display
RPN Keystrokes ALG Keystrokes
0.00
0?5
0?5
g(012tt g(012tt 58.26
58.26
g(038t
g(031t
Comments
Maximum operating expense
(%).
At 32% operating expense and $3.25 manufacturing cost, what should the list price be to
generate 20% net profit?
12c platinum / 12C
12c platinum
Display
Comments
RPN Keystrokes
ALG Keystrokes
20.00
20?5
20?5
11.00
:3tz
:3t~z~³
List price ($).
1g(014tz 1g(014t~z~³ 16.93
What reduction in manufacturing cost would achieve the same result without necessitating
an increase in list price above $11.98?
12c platinum / 12C
12c platinum
Display
Comments
RPN Keystrokes
ALG Keystrokes
7.79
g(013t
g(013t
Manufacturing cost ($).
g(001t§ g(001t§~³ 2.30
Securities and Options
After-Tax Yield
The following HP 12C Platinum program calculates the after-tax yield to maturity of a
bond held for more than one year. The calculation assumes an actual/actual day basis. For
after-tax computations, the interest or coupon payments are considered income, while the
difference between the bond's face value and its purchase price is considered capital gains.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
fCLEARG
?7
d
?6
:2
:1
:4
b
:2
~
Æ
2
z
?0
:3
:5
b
:0
z
P
:1
:0
z
$
:6
:7
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
028,
029,
42 34
44 7
33
44 6
45 2
45 1
30
45 4
25
45 2
34
30
26
2
10
44 0
45 3
45 5
25
30
45 0
10
14
45 1
45 0
10
13
45 6
45 7
65
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
fCLEARG
?7
d
?6
:2
:1
:4
b
+
:1
³
b
?0
:3
:5
b
z
:0
P
:1
z
:0
$
:6
:7
fS
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
028,
029,
42
44
44
45
45
45
45
44
45
45
45
45
45
45
45
42
34
7
33
6
2
30
1
30
4
25
40
1
36
25
0
3
30
5
25
10
0
14
1
10
0
13
6
7
22
66
Securities and Options
12c platinum / 12C
12c platinum
DISPLAY
DISPLAY
RPN KEYSTROKES
ALG KEYSTROKES
030,
42 22 g(000
030,43,33,000
fS
031,43,33,000 fs
g(000
fs
n: Unused
FV: 0
R3: Coupon rate
R7: Used
REGISTERS
i: Yield
PV: Used
R0: Used
R1: Purchase price
R4: Capital rate R5: Income rate
R8-R.5: Unused
PMT: Used
R2: Sales price
R6: Used
Program Instructions:
1. Key in the program.
2. Key in the purchase price and press ?1.
3.
Key in the sales price and press ?2.
4.
Key in the annual coupon rate (as a percentage) and press ?3.
5.
Key in capital gains tax rate (as a percentage) and press ?4.
6.
Key in the income tax rate (as a percentage) and press ?5.
7.
Press gÕ.
8.
Key in the purchase date (MM.DDYYYY) and press \(³).
9.
Key in the assumed sell date (MM.DDYYYY) and press t to find the after-tax
yield (as a percentage).
10. For the same bond but different date return to step 8.
11. For a new case return to step 2.
Example: You can buy a 7% bond on October 1, 2003 for $70 and expect to sell it in 5
years for $90. What is your net (after-tax) yield over the 5-year period if interim coupon
payments are considered as income, and your tax bracket is 50%?
(One-half of the long term capital gain is taxable at 50%, so the tax on capital gains alone
is 25%)
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
70?1
70?1
90?2
90?2
7?3
7?3
25?4
25?4
50?5
50?5
gÕ
gÕ
10.012003\
10.012003³
10.012008t
10.012008t
Display
Comments
70.00
90.00
7.00
25.00
50.00
Purchase price.
Selling price.
Annual coupon rate.
Capital gains tax rate.
Income tax rate.
10.01
8.53
Purchase Date.
% after tax yield.
Securities and Options
67
Discounted Notes
A note is a written agreement to pay a sum of money plus interest at a certain rate. Notes
do not have periodic coupons, since all interest is paid at maturity.
A discounted note is a note that is purchased below its face value. The following HP 12C
Platinum program finds the price and/or yield* of a discounted note.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:1
:2
gÒ
:3
z
:5
b
1
~
:4
§
?5
t
:1
:2
gÒ
:3
~
z
:4
:5
z
1
§
Æ
2
§
g(000
fs
*
DISPLAY
000,
001,
45 1
002,
45 2
003,
43 26
004,
45 3
005,
10
006,
45 5
007,
25
008,
1
009,
34
010,
30
011,
45 4
012,
20
013,
44 5
014,
31
015,
45 1
016,
45 2
017,
43 26
018,
45 3
019,
34
020,
10
021,
45 4
022,
45 5
023,
10
024,
1
025,
30
026,
20
027,
26
028,
2
029,
20
030,43,33,000
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:1
:2
gÒ
z
:3
§
:5
³
:4
~
b
³
?5
t
:1
:2
gÒ
:5
:4
à
~
d
z
~
§
:3
³
g(000
fs
The yield is a reflection of the return on an investment.
DISPLAY
000,
001,
45
1
002,
45
2
003,
43 26
004,
10
005,
45
3
006,
20
007,
45
5
008,
36
009,
45
4
010,
30
011,
34
012,
25
013,
36
014,
44
5
015,
31
016,
45
1
017,
45
2
018,
43 26
019,
45
5
020,
45
4
021,
24
022,
34
023,
33
024,
10
025,
34
026,
20
027,
45
3
028,
36
029,43,33,000
68
Securities and Options
n: Unused
FV: Unused
R3: 360 or 360
REGISTERS
i: Unused
PV: Unused
R0: Unused
R1: Settl. date
R4: redemp. Value
R5: dis./price
PMT: Unused
R2: Mat. Date
R6-R.5: Unused
Program Instructions:
1. Key in the program.
2. Press gÕ.
3. Key in the settlement date (MM.DDYYYY) and press ?1.
4. Key in the maturity date (MM.DDYYYY) and press ?2.
5. Key in the number of days in a year (360 or 365) and press ?3.
6. Key in the redemption value per $100 and press ?4.
7. To calculate the purchase price:
a.
Key in the discount rate and press ?5.
b.
Press t to calculate the purchase price.
c. Press t to calculate the yield.
d. For a new case, go to step 3.
8. To calculate the yield when the price is known:
a.
Key in the price and press ?5.
b.
RPN: Press g(015t to calculate the yield.
b.
ALG: Press g(016t to calculate the yield.
c.
For a new case, go to step 3.
Example 1: Calculate the price and yield on this bill: settlement date October 8, 2002;
maturity date March 21, 2003; discount rate 7.80%. Compute on a 360 day basis.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
gÕ
gÕ
10.082002?1
10.082002?1
3.212003?2
3.212003?2
360?3
360?3
100?4
100?4
7.8?5
7.8?5
t
t
t
t
Display
10.08
3.21
360.00
100.00
7.80
96.45
8.09
Comments
Settlement date.
Maturity date.
360 day basis.
Redemption value per $100.
Discount rate.
Price.
Yield.
Securities and Options
69
Example 2: Determine the yield of this security; settlement date June 25, 2002; maturity
date September 10, 2002; price $99.45; redemption value $101.33. Assume 360 day basis.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
6.252002?1
6.252002?1
9.102002?2
9.102002?2
360?3
360?3
101.33?4
101.33?4
99.45?5
99.45?5
g(015t
g(016t
Display
6.25
9.10
360.00
101.33
99.45
8.84
Comments
Settlement date.
Maturity date.
360 day basis.
Redemption value per $100.
Price.
Yield.
Black-Scholes Formula for Valuing European
Options
This program implements the Black-Scholes formula which has been used extensively in
option markets worldwide since its publication in the early 1970’s. The five inputs are
simply keyed into the five financial variables and then t displays the call option value,
and ~ shows the put option value. The option values produced are accurate to at least
the nearest cent for asset and strike prices under $100.
Reference: Hutchins, 2003, Black-Scholes takes over the HP12C, HPCC (www.hpcc.org)
DataFile,V22,N3 pp13-21.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:n
:¼
b
Þ
g>
:M
§
?4
~
gr
:P
b
?3
:$
:4
z
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
45
45
43
45
44
43
45
44
45
45
11
12
25
16
22
15
20
4
34
21
14
25
3
13
4
10
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:n
§
:¼
b
³
Þ
g>
§
:M
³
?4
:n
gr
§
:P
b
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
45
45
43
45
44
45
43
45
11
20
12
25
36
16
22
20
15
36
4
11
21
20
14
25
70
Securities and Options
12c platinum / 12C
RPN KEYSTROKES
g°
~
z
gF
2
?5
z
+
?6
:3
?3
\
§
gr
gF
2
z
Þ
g>
~
3
.
0
0
6
z
1
+
y
§
gF
gF
1
8
7
§
2
4
§
8
7
+
DISPLAY
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
028,
029,
030,
031,
032,
033,
034,
035,
036,
037,
038,
039,
040,
041,
042,
043,
044,
045,
046,
047,
048,
049,
050,
051,
052,
053,
054,
055,
056,
057,
058,
059,
060,
43
43
44
44
45
44
43
43
43
43
43
23
34
10
40
2
5
10
40
6
3
30
3
36
20
21
40
2
10
16
22
34
3
48
0
0
6
10
1
40
22
20
40
40
1
8
7
20
2
4
30
20
8
7
40
12c platinum
ALG KEYSTROKES
³
?3
:$
z
:4
³
g°
z
~
³
:3
z
2
?5
+
~
?6
:3
³
?3
g’
z
2
³
Þ
g>
:3
g’
gr
z
3
.
0
0
6
+
1
³
y
?2
§
~
³
DISPLAY
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
028,
029,
030,
031,
032,
033,
034,
035,
036,
037,
038,
039,
040,
041,
042,
043,
044,
045,
046,
047,
048,
049,
050,
051,
052,
053,
054,
055,
056,
057,
058,
059,
060,
44
45
45
43
45
44
44
45
44
43
43
45
43
43
44
36
3
13
10
4
36
23
10
34
36
3
10
2
5
40
34
30
6
3
36
3
20
10
2
36
16
22
3
20
21
10
3
48
0
0
6
40
1
36
22
2
20
34
36
Securities and Options
12c platinum / 12C
RPN KEYSTROKES
§
.
2
b
:3
~
?3
O
~
go
g(077
1
?-3
Þ
?§3
~
:5
gm
g(089
:6
:3
:4
§
?6
O
?5
~
g(028
~
:3
:$
?-4
§
:6
?+4
:4
~
?5
fs
DISPLAY
061,
20
062,
48
063,
2
064,
25
065,
45
3
066,
34
067,
44
3
068,
35
069,
34
070,
43 34
071,43,33,077
072,
1
073,44 30
3
074,
16
075,44 20
3
076,
34
077,
45
5
078,
43 35
079,43,33,089
080,
45
6
081,
45
3
082,
45
4
083,
20
084,
44
6
085,
35
086,
44
5
087,
34
088,43,33,028
089,
34
090,
45
3
091,
45 13
092,44 30
4
093,
20
094,
45
6
095,
30
096,44 40
4
097,
45
4
098,
34
099,
44
5
12c platinum
ALG KEYSTROKES
1
8
7
§
:2
2
4
§
:2
+
8
7
§
~
§
.
2
b
³
:3
~
?3
O
~
go
g(093
1
?-3
Þ
?§3
~
:5
gm
g(106
:6
:3
§
:4
³
71
DISPLAY
061,
1
062,
8
063,
7
064,
20
065,
45
2
066,
30
067,
2
068,
4
069,
20
070,
45
2
071,
40
072,
8
073,
7
074,
20
075,
34
076,
20
077,
48
078,
2
079,
25
080,
36
081,
45
3
082,
34
083,
44
3
084,
35
085,
34
086,
43 34
087,43,33,093
088,
1
089,44 30
3
090,
16
091,44 20
3
092,
34
093,
45
5
094,
43 35
095,43,33,106
096,
45
6
097,
45
3
098,
20
099,
45
4
100,
36
72
Securities and Options
12c platinum / 12C
RPN KEYSTROKES
DISPLAY
12c platinum
ALG KEYSTROKES
?6
O
?5
~
g(037
~
:$
?-4
§
:3
:6
³
?+4
:4
~
?5
g(000
fs
REGISTERS
n: Term to expiry i: Interest rate (%)
PV: Stock price
FV: Strike price R0: Unused
R1: Unused
R3: N(d1)
R4: Put value
R5: Call value
R7-R.9: Unused
DISPLAY
101,
44
6
102,
35
103,
44
5
104,
34
105,43,33,037
106,
34
107,
45 13
108,44 30
4
109,
20
110,
45
3
111,
30
112,
45
6
113,
36
114,44 40
4
115,
45
4
116,
34
117,
44
5
118,43,33,000
PMT: Volatility (%)
R2: Unused
R6: Q·N(d2)
Note: The n, i and PMT values must all be based on the same time unit (for example: n is
measured in years or months and i and PMT are rates per year or per month). i is a
continuous percentage rate. PMT is the standard deviation of the continuous percentage
stock return (as observed over the time unit). For sensible output, all inputs should be
positive. The PMT=0 case can be simulated by using a PMT arbitrarily close to 0.
Program Instructions
1. Key in the program.
2. Enter the five inputs into the five financial registers. These values are preserved by the
program.
a. Key in the unexpired term of the option and press n.
b. Key in the risk-free interest rate as a percentage and press ¼.
c. Key in the current (or spot) stock price and press $.
d. Key in the volatility assumption as a percentage and press P.
e. Key in the strike price and press M.
3. Press R/S. The Call value is displayed. Press ~ to see the Put value.
Securities and Options
73
Example 1: An option has 6 months to run and a strike price of $45. Find Call and Put
values assuming a spot price of $52, return volatility of 20.54% per month and a risk-free
interest rate of ½% per month. Show how to re-scale n, i and PMT to use a yearly time
unit, and how to re-scale them back again to the original monthly basis.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
6n
6n
.5¼
.5¼
52$
52$
20.54P
20.54P
45M
45M
t
t
~
~
:gAn
:gAn
:gC¼
:gC¼
:P
:P§
12gr§P
12grP
t
t
:ngA
:ngA
:¼gC
:¼gC
:P
:Pz
12grzP
12grP
Display
Comments
6.00
0.50
52.00
20.54
45.00
14.22
5.89
0.50
6.00
Time to expiry (months).
Interest rate (% per month).
Stock price.
Volatility (% per month).
Strike price.
Call value.
Put value.
Years to expiry.
Yearly interest rate %.
71.15
14.22
6.00
0.50
Yearly volatility %.
Call value (unchanged).
Months to expiry.
Monthly interest rate %.
20.54
Monthly volatility %.
The next example is Example 12.7 from Options, Futures, and Other Derivatives (5th
Edition) by John C. Hull (Prentice Hall, 2002).
Example 2: The stock price six months from the expiration of an option is $42, the
exercise price of the option is $40, the risk-free interest rate is 10% per annum, and the
volatility is 20% per annum. Find Call and Put values.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
.5n
.5n
10¼
10¼
42$
42$
20P
20P
40M
40M
t
t
~
~
Display
0.50
10.00
42.00
20.00
40.00
4.76
0.81
Comments
Time to expiry (years).
Interest rate (% per year).
Stock price.
Volatility (% per year).
Strike price.
Call value.
Put value.
Forecasting
Simple Moving Average
Moving averages are often useful in recording of forecasting sales figures, expenses or
manufacturing volume. There are many different types of moving average calculations.
An often used, straightforward method of calculation is presented here.
In a moving average a specified number of data points are averaged. When there is a new
piece of input data, the oldest piece of data is discarded to make room for the latest input.
This replacement scheme makes the moving average a valuable tool in following trends.
The fewer the number of data points, the more trend sensitive the average becomes. With
a large number of data points, the average behaves more like a regular average,
responding slowly to new input data.
A simple moving average may be calculated with your HP 12C Platinum as follows.
1. Press fCLEARH.
2. Key in the first m data points (where m is the number of data points in the average)
and press _ after each entry.
3. Press gÖ to obtain the first average.
4. Key in the oldest (first value) entered in step 2 and press g^.
5. Key in the newest data point (m + 1) and press _.
6. Press gÖ to obtain the next value of the moving average.
7. Repeat steps 4 through 5 for the remaining data.
Example 1: An electronics sales firm wished to calculate a 3-month moving average for
the dollar volume of components sold each month. Sales for the first six months of this
year were:
January
February
March
$211,570
112,550
190,060
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
211570_
211570_
112550_
112550_
190060_
190060_
gÖ
gÖ
74
April
May
June
Display
131,760
300,500
271,120
Comments
0.00
1.00
2.00
3.00
171,393.33 3-month average for March.
Forecasting
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
211570g^
211570g^
131760_
131760_
gÖ
gÖ
112550g^
112550g^
300500_
300500_
gÖ
gÖ
190060g^
190060g^
271120_
271120_
gÖ
gÖ
Display
75
Comments
2.00
3.00
144,790.00 3-month average for April.
2.00
3.00
207,440.00 3-month average for May.
2.00
3.00
234,460.00 3-month average for June.
For repeated calculations the following HP 12C Platinum program can be used for up to a
12 element moving average:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:1
:2
?1
+
:3
?2
+3
:4
?3
+4
:5
?4
+5
:6
?5
+6
:7
?6
+7
:8
?7
+8
:9
?
+9
:.0
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,45
45
45
44
45
44
45
44
45
44
45
44
45
44
45
44
45
44
48
1
2
1
40
3
2
40
4
3
40
5
4
40
6
5
40
7
6
40
8
7
40
9
8
40
0
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:1
+
:2
?1
+
:3
?23
+
:4
?34
+
:5
?45
+
:6
?56
+
:7
?67
+
:8
?78
+
:9
?89
+
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
45
45
44
45
44
45
44
45
44
45
44
45
44
45
44
45
44
1
40
2
1
40
3
2
40
4
3
40
5
4
40
6
5
40
7
6
40
8
7
40
9
8
40
76
Forecasting
12c platinum / 12C
RPN KEYSTROKES
?9
+10
:.1
?.0
+11
:.2
?.1
+12
:0
z
t
?m*
g(001
fs
n: Unused
FV: Unused
R3: X3
R7: X7
R.1: X11
DISPLAY
027,
44
9
028,
40
029,45 48
1
030,44 48
0
031,
40
032,45 48
2
033,44 48
1
034,
40
035,
45
0
036,
10
037,
31
038,
44 -039,43,33,001
i: Unused
R0: m
R4: X4
R8: X8
R.2: X12
12c platinum
ALG KEYSTROKES
:.0
?910
+
:.1
?.011
+
:.2
?.112
z
:0
³
t
?m*
g(001
fs
REGISTERS
PV: Unused
R1: X1
R5: X5
R9: X9
R.3-R.4: Unused
DISPLAY
027,45 48
0
028,
44
9
029,
40
030,45 48
1
031,44 48
0
032,
40
033,45 48
2
034,44 48
1
035,
10
036,
45
0
037,
36
038,
31
039,
44 -040,43,33,001
PMT: Unused
R2: X2
R6: X6
R.0: X.0
This program can be used for a moving average of 2 to 12 elements. It may be shortened
considerably for moving averages with less than 12 elements. To do this, key in the
program, as shown, from line 01 until you reach a + (ALG: a ? command)
superscripted with the number of elements you desire. Key in this line, then skip the rest
of the program down to line 35. Then key in lines 035 through 039 (ALG:040), being sure
to specify the register number at line 038 (ALG:039), ? m, corresponding to the
number of elements you are using. (For instance, for a 5 element moving average, key in
lines 01 through 13 then go to line 35 in the listing and key in the balance of the program.
Obviously the program listing line 38 (ALG:039), ? m becomes the displayed line 017,
?5 in RPN and 018, ?5 in ALG).
*
At step 038 (ALG:039), m=number of elements in the moving average, i.e. for a 5 element moving
average line 038 (ALG:039) would be ?5 and for a 12 element average line 38 (ALG:039) would
be ?.2
Forecasting
77
Program Instructions:
1. Key in the program.
2. Press fCLEARH. Key in the number of elements, m, and press ?0.
3. Key in the first data point and press ?1.
4. Key in the second data point and press ?2.
5. Continue as above, keying in and storing each data point in its appropriate register
until m data points have been stored.
6. Press g(000t to calculate the first moving average.
7. Key in the next data point and press t to calculate the next moving average.
8. Repeat step 7 for each new data point.
Example 2: Calculate the 3-element moving average for the data given in example 1.
Your modified program listing will look like this:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:1
:2
?1
+
:3
?2
+
:0
z
t
?3
g(001
fs
12c platinum / 12C
RPN Keystrokes
fCLEARH
3?0
211570?1
112550?2
190060?3
g(000t
131760t
DISPLAY
000,
001,
45
1
002,
45
2
003,
44
1
004,
40
005,
45
3
006,
44
2
007,
40
008,
45
0
009,
10
010,
31
011,
44
3
012,43,33,001
12c platinum
ALG Keystrokes
fCLEARH
3?0
211570?1
112550?2
190060?3
g(000t
131760t
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:1
+
:2
?1
+
:3
?2
z
:0
³
t
?3
g(001
fs
Display
DISPLAY
000,
001,
45
1
002,
40
003,
45
2
004,
44
1
005,
40
006,
45
3
007,
44
2
008,
10
009,
45
0
010,
36
011,
31
012,
44
3
013,43,33,001
Comments
0.00
3.00
211,570.00
112,550.00
190,060.00
171,393.33 3-month average for March.
144,790.00 3-month average for April.
78
Forecasting
12c platinum / 12C
12c platinum
Display
Comments
RPN Keystrokes ALG Keystrokes
207,440.00 3-month average for May.
300500t
300500t
234,460.00 3-month average for June.
271120t
271120t
Seasonal Variation Factors Based on
Centered Moving Averages
Seasonal variation factors are useful concepts in many types of forecasting. There are
several methods of developing seasonal moving averages, on the of more common ways
being to calculate them as a ratio of the periodic value to a centered moving average for
the same period.
For instance, to determine the sales for the 3rd quarter of a given year a centered moving
average for that quarter would be calculated from sales figures from the 1st, 2nd, 3rd and
4th quarters of the year and the 1st quarter of the following year. The seasonal variation
factor for that 3rd quarter would then be the ratio of the actual sales in the 3rd quarter to
the centered moving average for that quarter.
While quarterly seasonal variations are commonly used, the HP 12C Platinum can also be
programmed to calculate monthly seasonal variations using a centered 12 month moving
average. Programs for both of these calculations are represented here:
An HP 12C Platinum program to calculate the quarterly seasonal variations based on a
centered 4-point moving average is:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:1
2
z
:2
?1
+
:3
?2
+
:4
?3
+
:5
?4
2
z
+
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
45
45
44
45
44
45
44
45
44
1
2
10
2
1
40
3
2
40
4
3
40
5
4
2
10
40
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:1
z
2
+
:2
?1
+
:3
?2
+
:4
?3
³
:5
?4
z
2
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
45
45
44
45
44
45
44
45
44
1
10
2
40
2
1
40
3
2
40
4
3
36
5
4
10
2
Forecasting
12c platinum / 12C
RPN KEYSTROKES
4
z
t
:2
Z
t
?5
g(001
fs
n: Unused
FV: Unused
R3: X3
DISPLAY
018,
4
019,
10
020,
31
021,
45
2
022,
23
023,
31
024,
44
5
025,43,33,001
i: Unused
R0: n
R4: X4
12c platinum
ALG KEYSTROKES
+
~
z
4
³
t
:2
Z
t
?5
g(001
fs
REGISTERS
PV: Unused
R1: X1
R5: X5
79
DISPLAY
018,
40
019,
34
020,
10
021,
4
022,
36
023,
31
024,
45
2
025,
23
026,
31
027,
44
5
028,43,33,001
PMT: Unused
R2: X2
R6-R.6: Unused
Program Instructions:
1. Key in the program.
2. Press fCLEARH.
3. Key in the quarterly sales figures starting with the first quarter:
a. Key in 1st quarter sales and press ?1.
b. Key in 2nd quarter sales and press ?2.
c. Key in 3rd quarter sales and press ?3.
d. Key in 4th quarter sales and press ?4.
e. Key in the 1st quarter sales for the next year and press ?5.
4. Press g(000t to calculate the centered moving average for the 3rd quarter of
the first year.
5. Press t to calculate the seasonal variation for this quarter.
6. Key in the next quarter's sales and press t to calculate the moving average for the
next quarter.
7. Press t to calculate the seasonal variation.
8. Repeat steps 6 and 7 for the balance of the data.
80
Forecasting
Example: Econo-Wise Home Appliance Company had quarterly sales for the years 2000
thru 2002 as follows:
Quarterly
2000
2001
2002
Sales (IN $K)
1st 2nd
397 376
455 390
513 434
3rd
460
530
562
4th
501
560
593
Find the centered 4-quarter moving average and seasonal variation factor for each quarter.
12c platinum / 12C
RPN Keystrokes
fCLEARH
397?1
376?2
460?3
501?4
455?5
g(000t
12c platinum
ALG Keystrokes
fCLEARH
397?1
376?2
460?3
501?4
455?5
g(000t
0.00
397.00
376.00
460.00
501.00
455.00
440.75
t
390t
t
530t
t
560t
t
513t
t
434t
t
562t
t
593t
t
t
390t
t
530t
t
560t
t
513t
t
434t
t
562t
t
593t
t
104.37
449.75
111.40
460.25
98.86
476.38
81.87
491.00
107.94
503.75
111.17
513.25
99.95
521.38
83.24
Display
Comments
Centered 4-element average for
3rd quarter, 2000.
Seasonal variation factor.
4th quarter, 2000.
1st quarter, 2001.
2nd quarter, 2001.
3rd quarter, 2001.
4th quarter, 2001.
1st quarter, 2002.
2nd quarter, 2002.
Now, what is the average of each quarter's seasonal variation for the two years?
12c platinum / 12C
RPN Keystrokes
fCLEAR²
98.86_
99.95_
gÖ
12c platinum
ALG Keystrokes
fCLEAR²
98.86_
99.95_
gÖ
Display
0.00
1.00
2.00
99.41
Comments
1st quarter average seasonal
variation, %.
Forecasting
12c platinum / 12C
RPN Keystrokes
fCLEAR²
81.87_
83.24_
gÖ
12c platinum
ALG Keystrokes
fCLEAR²
81.87_
83.24_
gÖ
0.00
1.00
2.00
82.56
fCLEAR²
104.37_
107.94_
gÖ
fCLEAR²
104.37_
107.94_
gÖ
0.00
1.00
2.00
106.16
fCLEAR²
111.4_
111.17_
gÖ
fCLEAR²
111.4_
111.17_
gÖ
0.00
1.00
2.00
111.29
Display
81
Comments
2nd quarter average seasonal
variation, %.
3rd quarter average seasonal
variation, %.
4th quarter average seasonal
variation, %.
An HP 12C Platinum program to calculate a centered 12 month moving average and
seasonal variation factor is as follows:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:1
2
z
:2
?1
+
:3
?2
+
:4
?3
+
:5
?4
+
:6
?5
+
:7
?6
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
45
45
44
45
44
45
44
45
44
45
44
45
44
1
2
10
2
1
40
3
2
40
4
3
40
5
4
40
6
5
40
7
6
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:1
z
2
+
:2
?1
+
:3
?2
+
:4
?3
+
:5
?4
+
:6
?5
+
:7
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
45
45
44
45
44
45
44
45
44
45
44
45
1
10
2
40
2
1
40
3
2
40
4
3
40
5
4
40
6
5
40
7
82
Forecasting
12c platinum / 12C
RPN KEYSTROKES
+
:8
?7
+
:9
?8
+
:.0
?9
+
:.1
?.0
+
:.2
?.1
+
:.3
?.2
2
z
+
:0
z
t
:6
Z
t
?.3
g(001
fs
n: Unused
FV: Unused
R3: X3
R7: X7
R.1: X11
DISPLAY
021,
40
022,
45
8
023,
44
7
024,
40
025,
45
9
026,
44
8
027,
40
028,45 48
0
029,
44
9
030,
40
031,45 48
1
032,44 48
0
033,
40
034,45 48
2
035,44 48
1
036,
40
037,45 48
3
038,44 48
2
039,
2
040,
10
041,
40
042,
45
0
043,
10
044,
31
045,
45
6
046,
23
047,
31
048,44 48
3
048,43,33,001
i: Unused
R0: n
R4: X4
R8: X8
R.2: X12
12c platinum
ALG KEYSTROKES
?6
+
:8
?7
+
:9
?8
+
:.0
?9
+
:.1
?.0
+
:.2
?.1
³
:.3
?.2
z
2
+
~
z
:0
³
t
:6
Z
t
?.3
g(001
fs
REGISTERS
PV: Unused
R1: X1
R5: X5
R9: X9
R.3: X13
DISPLAY
021,
44
6
022,
40
023,
45
8
024,
44
7
025,
40
026,
45
9
027,
44
8
028,
40
029,45 48
0
030,
44
9
031,
40
032,45 48
1
033,44 48
0
034,
40
035,45 48
2
036,44 48
1
037,
36
038,45 48
3
039,44 48
2
040,
10
041,
2
042,
40
043,
34
044,
10
045,
45
0
046,
36
047,
31
048,
45
6
049,
23
050,
31
051,44 48
3
052,43,33,001
PMT: Unused
R2: X2
R6: X6
R.0: X10
Forecasting
83
Program Instructions:
1. Key in the program.
2. Press fCLEARH.
3. Key in 12 and press ?0.
4. Key in the values for the first 13 months, storing them one at a time in registers 1
through .3; i.e.
Key in the 1st month and press ?1.
Key in the 2nd month and press ?2, etc.,
Key in the 10th month and press ?.0, etc.,
Key in the 13th month and press ?.3.
5. Press g(000t to calculate the centered moving average for the 7th month.
6. Press t to calculate the seasonal variation for that month.
7. Key in the value for the next month (14th) and press t to calculate the moving
average for the next month (8th).
8. Repeat steps 6 and 7 for the balance of the data.
These programs may be customized by the user for different types of centered moving
averages. Inspection of the programs will show how they can be modified.
Gompertz Curve Trend Analysis
A useful curve for evaluating sales trends, etc., is the Gompertz curve. This is a "growth"
curve having a general "S" shape and may be used to describe series of data where the
early rate of growth is small, then accelerates for a period of time and then slows again as
the time grows long. The sales curve for many products follow this trend during the
introductory, growth and maturity phases.
The data points to be fit to a Gompertz curve should be equally spaced along the x (or
time) axis and all the data points must be positive. The points are divided serially into 3
groups for data entry.
The following HP 12C Platinum program processes the data, fits it to a Gompertz curve
and calculates estimated values for future data points. The 3 constants which characterize
the curve are available to the user if desired.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
g°
?+3
d
DISPLAY
000,
001,
43
002,44 40
003,
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
23 g°
3 ?+3
33 d
DISPLAY
000,
001,
43
002,44 40
003,
23
3
33
84
Forecasting
12c platinum / 12C
RPN KEYSTROKES
g°
?+2
d
g°
?+1
1
?+4
:4
g(000
:3
:2
:2
:1
z
:4
y
q
?6
:1
:3
§
:2
\
§
:1
:3
+
:2
2
§
z
:4
z
g>
?7
:6
1
:6
DISPLAY
004,
43 23
005,44 40
2
006,
33
007,
43 23
008,44 40
1
009,
1
010,44 40
4
011,
45
4
012,43,33,000
013,
45
3
014,
45
2
015,
30
016,
45
2
017,
45
1
018,
30
019,
10
020,
45
4
021,
22
022,
21
023,
44
6
024,
45
1
025,
45
3
026,
20
027,
45
2
028,
36
029,
20
030,
30
031,
45
1
032,
45
3
033,
40
034,
45
2
035,
2
036,
20
037,
30
038,
10
039,
45
4
040,
10
041,
43 22
042,
44
7
043,
45
6
044,
1
045,
30
046,
45
6
12c platinum
ALG KEYSTROKES
g°
?+2
d
g°
?+1
1
?+4
:4
g(000
:2
:1
³
?8
:3
:2
z
?9
~
³
q
:4
y
³
?6
:9
:8
³
:1
§
:3
:2
g’
z
~
z
:4
³
g>
?7
DISPLAY
004,
43 23
005,44 40
2
006,
33
007,
43 23
008,44 40
1
009,
1
010,44 40
4
011,
45
4
012,43,33,000
013,
45
2
014,
30
015,
45
1
016,
36
017,
44
8
018,
45
3
019,
30
020,
45
2
021,
10
022,
44
9
023,
34
024,
36
025,
21
026,
45
4
027,
22
028,
36
029,
44
6
030,
45
9
031,
30
032,
45
8
033,
36
034,
45
1
035,
20
036,
45
3
037,
30
038,
45
2
039,
43 20
040,
10
041,
34
042,
10
043,
45
4
044,
36
045,
43 22
046,
44
7
Forecasting
12c platinum / 12C
RPN KEYSTROKES
:4
q
1
\
§
z
:6
z
:2
:1
§
g>
?5
t
:6
~
q
:5
~
q
:7
§
g(062
fs
n: Unused
FV: Unused
R3: S3
R7: c
DISPLAY
047,
45
4
048,
21
049,
1
050,
30
051,
36
052,
20
053,
10
054,
45
6
055,
10
056,
45
2
057,
45
1
058,
30
059,
20
060,
43 22
061,
44
5
062,
31
063,
45
6
064,
34
065,
21
066,
45
5
067,
34
068,
21
069,
45
7
070,
20
071,43,33,062
12c platinum
ALG KEYSTROKES
:6
q
:4
1
³
g’
1
:6
y
§
:8
z
~
³
g>
?5
t
:6
q
~
³
:5
q
~
§
:7
³
g(065
fs
REGISTERS
i: Unused
PV: Unused
R0: Unused
R1: S1
R4: n
R5: a
R8-R.0: Unused
85
DISPLAY
047,
45
6
048,
21
049,
45
4
050,
30
051,
1
052,
36
053,
43 20
054,
1
055,
30
056,
45
6
057,
22
058,
20
059,
45
8
060,
10
061,
34
062,
36
063,
43 22
064,
44
5
065,
31
066,
45
6
067,
21
068,
34
069,
36
070,
45
5
071,
21
072,
34
073,
20
074,
45
7
075,
36
076,43,33,065
PMT: Unused
R2: S2
R6: b
Program Instructions:
1. Key in the program and press fCLEARH.
2. Divide the data points to be input into 3 equal consecutive groups. Label them Groups
I, II and III for convenience.
86
Forecasting
3. Key in the first point of group I and press \(³)
4. Key in the first point of group II and press \(³).
5. Key in the first point of group III and press t.
6. Repeat steps 3, 4, and 5 for the balance of the data in each group. After executing step
5 the display shows how many sets of data have been entered.
7. To fit the data to a Gompertz curve, press g(013t . The resultant display is
the curve constant "a". Constants "b" and "c" may be obtained by pressing :6 and
:7 respectively. The display may need adjusting to see significant digits.
8. To calculate a projected value, key in the number of the period and press t.
9. Repeat step 8 for each period desired.
Example: The X-presso Company marked a revolutionary new coffee brewing machine
in 1990. Sales grew at a steady pace for several years, then began to slow. The sales
records for the first 9 years of the product's life were as follows.
Year
1
2
3
4
5
Sales($K)
18
41
49
151
188
Year
6
7
8
9
Sales($K)
260
282
322
340
What are the projected sales volumes for this product in its 10th and 12th year? What is
the maximum yearly sales volume for this product if the present trend continues? What
annual sales rate would the curve have predicted for the 5th year of the product's life?
(Arrange the data as follows:)
Group I
18
41
49
Group II
151
188
260
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
18\
18³
151\
151³
282t
282t
41\
41³
188\
188³
322t
322t
49\
49³
260\
260³
340t
340t
Group III
282
322
340
Display
0.00
18.00
151.00
1.00
41.00
188.00
2.00
49.00
260.00
3.00
Comments
Total number of entries.
Forecasting
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
g(013t
g(013t
:6
:6
:7
:7
10t
10t
12t
12t
100t
100t
0.004
0.65
373.92
349.09
363.36
373.92
5t
202.60
5t
Display
87
Comments
a
b
c
Sales in 10th year, (in $K).
Sales in 12th year, (in $K).
Maximum annual sales
(after very long product
life).
Sales in 5th year (actual
sales were $188K).
Forecasting with Exponential Smoothing
A common method for analyzing trends in sales, inventory and securities is the moving
average. Exponential smoothing is a version of the weighted moving average which is
readily adaptable to programmable calculator forecasting.
Exponential smoothing is often used for short term sales and inventory forecasts. Typical
forecast periods are monthly or quarterly. Unlike a moving average, exponential
smoothing does not require a great deal of historical data. However , it should not be used
with data which has more than a moderate amount of up or down trend.
When using exponential smoothing, a smoothing factor is chosen which affects the
sensitivity of the average much the same way as the length of the standard moving
average period. The correspondence between the two techniques can be represented by
the formula:
α=
2
n +1
where α is the exponential smoothing factor (with values from 0 to 1) and n is the length
of the standard moving average. As the equation shows, the longer the moving average
period, the smaller the equivalent and the less sensitive the average becomes to
fluctuations in current values.
88
Forecasting
Forecasting with exponential smoothing involves selecting the best smoothing factor
based on historical data and then using the factor for updating subsequent data and
forecasting. This procedure uses the following HP 12C Platinum program:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
\
\
:6
\
§
?+4
gF
t
d
d
:0
§
:2
:1
§
+
:2
Þ
~
?2
+
:0
§
:1
:3
§
+
?3
:1
§
:0
z
:2
+
?5
:3
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,44
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
028,
029,
030,
031,
032,
033,
034,
035,
036,
037,
45
40
43
45
45
45
45
44
45
45
45
44
45
45
45
44
45
36
36
6
30
36
20
4
40
31
33
33
0
20
2
1
20
40
2
16
34
2
40
0
20
1
3
20
40
3
1
20
0
10
2
40
5
3
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
?7
:6
³
t
g’
?+4
:7
§
:0
³
:2
§
:1
+
~
:2
§
?+2
:0
³
:3
§
:1
+
~
z
?3
:0
+
:2
?6
:3
³
?5
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,44
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,44
021,
022,
023,
024,
025,
026,
027,
028,
029,
030,
031,
032,
033,
034,
035,
036,
037,
44
45
43
40
45
45
45
45
45
40
45
45
45
44
45
45
44
45
44
7
30
6
36
31
20
4
7
20
0
36
2
20
1
40
34
30
2
20
2
0
36
3
20
1
40
34
10
3
0
40
2
30
6
3
36
5
Forecasting
12c platinum / 12C
RPN KEYSTROKES
:0
z
:2
+
?6
g(000
fs
89
12c platinum
DISPLAY
ALG KEYSTROKES
038,
45
0 :6
038,
45
6
039,
10 g(000
039,43,33,000
040,
45
2 fs
041,
40
042,
44
6
043,43,33,000
DISPLAY
n: Unused
FV: Unused
i: Unused
R0: α
R3: Tt-1
R7-R.4: Unused
R4: Σe2
REGISTERS
PV: Unused
R1: 1-α
R5: Dt
PMT: Unused
R2: St-1
R6: Dˆ t+1
Program Instructions:
Selecting the "best" smoothing constant (α ):
1. Key in the program and press fCLEARH.
2.
RPN Mode:
Key in the number 1 and press \.
3.
Key in the "trial " and press ?0-?1.
2.
ALG Mode:
Key in the number 1 and press -.
3.
Key in the "trial " and press ?0³?1.
4.
Key in the first historical value (X1) and press ?2.
5.
Key in the second historical value (X2) and press ?6t. The result is the error
between the forecast value ( Dˆ t+1) and the true value (Xt+1).
6.
Press t; the display shows the next forecast ( Dˆ t+2).
7.
Optional: Press :5 to display the smoothed estimate of current demand.
Continue steps 5 and 6 for X3, X4, ... Xn until all historical values have been entered.
When doing step 5 merely key in the value and press t (do not press ?6).
8.
9.
Press :4. This value represents the cumulative forecasting error (∑e2). Record the
value and the following additional values; press :0 (α), :2 (smoothed average
St-1), :3 (trend Tt-1) and :6 (forecast Dˆ t+1).
10. Press fCLEARH.
11. Repeat steps 2 through 10 until a "best" α is selected based on the lowest cumulative
forecasting error (Register 4).
90
Forecasting
Forecasting:
RPN Mode:
1. Key in the number 1 and press \.
2. Key in the selected α and press ?0-?1.
ALG Mode:
1. Key in the number 1 and press -.
2. Key in the selected α and press ?0³?1.
3. From the selection routine or from a previous forecast:
Key in the smoothed average St-1 and press ?2.
Key in the trend Tt-1 and press ?3.
Key in the forecast Dˆ t+1 and press ?6.
4. Key in the current data value and press t. The output is the error in forecasting the
value just entered.
5. Press t. The displayed value represents the forecast for the next period.
6. Record the following values: :0 (α), :2 (St-1), :3 (Tt-1) and :6 ( Dˆ t+1) for
use as initial values in the next forecast. You may also wish to record :5 (Dt).
7. Repeat steps 4, 5, and 6 for the next forecast if available.
Example: Select the best smoothing constant based on sales (in thousands of dollars) of
22, 23, 23, 25, 23, 27, 25. Given the current sales in month 8 of 26, forecast the following
month.
Select the smoothing constant (α ):
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
1\
1.5?0.5?0³
?1
?1
22?2
22?2
23?6t
23?6t
t
t
23tt
23tt
25tt
25tt
23tt
23tt
27tt
27tt
25tt
25tt
:4
:4
:0
:0
Display
0.00
1.00
0.50
0.50
22.00
0.00
23.00
23.25
25.25
23.69
27.13
25.95
23.61
0.50
Comments
Cumulative error (∑e2).
Smoothing constant (α ).
Forecasting
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
25.11
:2
:2
0.42
:3
:3
:6
:6
91
Comments
Smoothing average (St-1).
Trend (Tt-1).
Last forecast ( Dˆ t+1).
25.95
The procedure is repeated for several α's.
Smoothing Constant (α)
Cumulative Error (Σe2)
For the selected α = .25
.5
23.61
St+1 =
Tt-1 =
Dˆ t+1=
.1
25.14
24.28
0.34
25.64
.25
17.01
.2
18.03
Forecasting:
12c platinum / 12C
RPN Keystrokes
fCLEARH
1\
.25?0?1
24.28?2
.34?3
25.64?6
26t
t
:5
12c platinum
ALG Keystrokes
fCLEARH
1.25?0³
?1
24.28?2
.34?3
25.64?6
26t
t
:5
0.00
1.00
0.75
0.75
24.28
0.34
25.64
0.36
26.16
25.80
:0
:2
:3
:6
:0
:2
:3
:6
0.25
24.71
0.36
26.16
Display
Comments
Forecast for month 9, ( Dˆ t+1).
Expected usage for current
(month 8) period, (Smoothed
Dt).
α
Record for initial values when
month 9 actual figures become
available.
Note: At least 4 periods of current data should be entered before forecasting is attempted.
Pricing Calculations
Markup and Margin Calculations
Sales work often involves calculating the various relations between markup, margin,
selling price and costs. Markup is defined as the difference between selling price and cost,
divided by the cost. Margin is defined as the difference between selling price and cost,
divided by selling price. In other words, markup is based on cost and margin is based on
selling price.
The following keystroke sequences are given to readily make these calculations on the
HP 12C Platinum.
CALCULATE
GIVEN
Selling Price Cost & Markup
Selling Price
Cost
Cost
Markup
Markup
Margin
Margin
RPN KEYSTROKES
Key in cost, \,
key in markup
(in %), b+.
Cost & Margin
Key in cost,
\1\
key in margin (in %),
b-z.
Selling Price & Markup Key in selling price,
\1\,
key in markup (in %),
b+z.
Selling Price & Margin Key in selling price,
\,
key in margin
(in %), b-.
Cost and Selling Price Key in cost, \,
key in selling price,
à.
Margin
Key in margin,
\\1
~b-z.
Selling Price & Cost
Key in selling price,
\, key in cost,
àÞ.
Markup
Key in markup,
\\1
~b+z.
92
ALG KEYSTROKES
Key in cost, +,
key in markup
(in %), b³.
Key in cost,
y-,
key in margin (in %),
b³y.
Key in selling price,
y+,
key in markup (in %),
b³y.
Key in selling price,
-,
key in margin
(in %), b³.
Key in cost, ³,
key in selling price,
à.
Key in margin,
-y
~b³y.
Key in selling price,
³, key in cost,
àÞ.
Key in markup,
+y
~b³y.
Pricing Calculations
93
Example 1: If the cost of an item is $160 and the margin is 20%, what is the selling
price? What is the markup?
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
160\1\20
160y-20
b-z
b³y
20\\
201~b-z
y~b³y
Display
20.
200.00
20.00
25.00
Comments
Margin(%).
Selling price.
Margin(%)
Markup (%).
Example 2: If an item sells for $21.00 and has a markup of 50%, what is its cost? What is
the margin?
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
21\1\50
21y+50
b+z
b³y
50\\
50+
1~b+z
y~b³y
Display
50.00
14.00
50.00
33.33
Comments
Markup (%).
Cost.
Markup(%)
Margin (%).
The following HP 12C Platinum program may be helpful for repetitive calculations of
selling price and costs as well as conversions between markup and margin.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
\
g(004
Þ
1
~
b
+
z
t
gF
§
gF
§
g(000
fs
DISPLAY
000,
001,
36
002,43,33,004
003,
16
004,
1
005,
34
006,
25
007,
40
008,
10
009,
31
010,
43 40
011,
20
012,
43 40
013,
20
014,43,33,000
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
³
g(004
Þ
~
+
~
b
³
gF
g’
z
~
³
t
gF
g(000
fs
DISPLAY
000,
001,
36
002,43,33,004
003,
16
004,
34
005,
40
006,
34
007,
25
008,
36
009,
43 40
010,
43 20
011,
10
012,
34
013,
36
014,
31
015,
43 40
016,43,33,000
94
Pricing Calculations
n: Unused
FV: Unused
REGISTERS
i: Unused
PV: Unused
R0-R.8: Unused
PMT: Unused
Program Instructions:
1. Key in the program.
2. To calculate selling price, given the markup, key in the cost, press \(³), key in
the markup and press g(000tt.
3. To calculate cost, given the markup, key in the selling price, press \(³), key in
the markup and press g(000t.
4, To calculate selling price, given the margin, key in the cost, press \(³), key in
the margin and press g(003t.
5. To calculate cost, given the margin, key in the selling price, press \(³), key in
the margin and press g(003tt.
RPN Mode:
6. To calculate markup from the margin, key in the margin and press
\g(003t.
7. To calculate margin from the markup, key in the markup and press
\g(000t.
ALG Mode:
6. To calculate markup from the margin, key in the margin and press ³, re-key margin,
g(003t.
7. To calculate margin from the markup, key in the markup and press ³, re-key
markup, g(000t.
Example: Find the cost of an item selling for $38.00 with a margin of 30%. What is the
markup on the item? If the markup is raised to 50%, what will the selling price be?
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
38\
38³
30g(003
30g(003
tt
tt
30\
30³
g(003t
30g(003t
26.6\
26.6³
50g(000
50g(000
tt
tt
Display
38.00
30.00
26.60
30.00
42.86
26.60
50.00
39.90
Comments
Selling price.
Margin (%).
Cost.
Margin (%).
Markup (%).
Cost.
New markup.
New selling price.
Pricing Calculations
95
Calculations of List and Net prices With
Discounts
It is often useful to be able to quickly calculate a list or net price when the other price and
a series of discount rates are known. Alternatively, if the list and net price and several
discounts are known it may be desirable to calculate a missing discount. The following
series of keystrokes may be used:
RPN Mode:
1. Key in 1, press \\?1.
2. Key in the first discount (as a percentage) and press b-?§1d.
3. Repeat step 2 for each of the remaining known discount rates.
4. To calculate the list price, key in the net price and press :1z.
5. To calculate the net price, key in the list price and press :1§.
6. To calculate an unknown discount rate, immediately after step 3 (display should show
1.00), key in the net price, press \ and key in the list price.
7. Press :1§z-100§.
ALG Mode:
1. Key in 1, press ?1.
2. Key in 1-, key in the first discount (as a percentage) and press b³?§1.
3. Repeat step 2 for each of the remaining known discount rates.
4. To calculate the list price, key in the net price and press z:1³.
5. To calculate the net price, key in the list price and press §:1³.
6. To calculate an unknown discount rate, immediately after doing step 3 , key in the list
price, press §:1³, then key in the net price.
7. Press àÞ.
Example: The list price of an item is $3.28 and the net price is $1.45. Two of the discount
rates are 48% and 5%. What is the third discount rate?
12c platinum / 12C
12c platinum
RPN Keystrokes
ALG Keystrokes
1
1\\
?1
?1
48b-?§1 1-48b³?§1
d 5b-?§1 1-5b³?§1
d1.45\
3.28:1
3.28§:1
§z-100§
³1.45àÞ
Display
Comments
1.00
0.52
0.95
0.49
10.51
3rd discount rate (%).
96
Pricing Calculations
The following program for the HP 12C Platinum will be helpful in performing the
calculations:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
1
~
b
?§1
g(000
:1
§
z
1
~
Æ
2
§
g(000
fs
n: Unused
FV: Unused
DISPLAY
000,
001,
1
002,
34
003,
25
004,
30
005,44 20
1
006,43,33,000
007,
45
1
008,
20
009,
10
010,
1
011,
34
012,
30
013,
26
014,
2
015,
20
016,43,33,000
i: Unused
R0: Unused
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
1
~
b
³
?§1
g(000
:1
§
~
³
~
à
Þ
g(000
fs
REGISTERS
PV: Unused
R1: D’1xD’2...
DISPLAY
000,
001,
1
002,
30
003,
34
004,
25
005,
36
006,44 20
1
007,43,33,000
008,
45
1
009,
20
010,
34
011,
36
012,
34
013,
24
014,
16
015,43,33,000
PMT: Unused
R2-R7: Unused
Program Instructions:
1. Key in the program.
2. Key in 1 and press ?1.
3. Key in the first discount rate (as a percentage) and press t.
4. Repeat step 2 for each of the remaining discount rates.
RPN Mode:
5. To calculate the list price, key in the net price and press :1z.
6. To calculate the net price, key in the list price and press :1§.
7. To calculate the unknown discount rate, key in the net price, press \, key in the list
price and press g(007t.
Pricing Calculations
97
ALG Mode:
5. To calculate the list price, key in the net price and press z:1³.
6. To calculate the net price, key in the list price and press §:1³.
7. To calculate the unknown discount rate, key in the net price, press ³, key in the list
price and press g(008t.
Example: Calculate the unknown discount rate for the previous example. If the list price
is now raised to $3.75 what is the new net price?
12c platinum / 12C
12c platinum
RPN Keystrokes ALG Keystrokes
1?1
1?1
48t
48t
5t
5t
1.45\
1.45³
3.28g(007t 3.28g(008t
t
t
3.75:1§
3.75§:1³
Display
Comments
1.00
0.52
0.95
10.51
0.89
1.66
3rd discount rate (%).
Include 3rd discount rate in
calculation.
New net price.
Statistics
Curve Fitting
Exponential Curve Fit
Using the ° function of the HP 12C Platinum, a least squares exponential curve fit may
be easily calculated according to the equation y=AeBx. The exponential curve fitting
technique is often used to determine the growth rate of a variable such as a stock's value
over time, when it is suspected that the performance is non-linear. The value for B is the
decimal value of the continuous growth rate. For instance, assume after keying in several
end-of-month price quotes for a particular stock it is determined that the value of B is
0.10. This means that over the measured growth period the stock has experienced a 10%
continuous growth rate.
If B>0, you will have a growth curve. If B<0, you will have a decay curve.
Examples of these are given below.
DECAY CURVE
(B<0)
Quantity
Quantity
GROWTH CURVE
(B>0)
Time
Time
The procedure is as follows:
1. Press fCLEARH .
2. For each input pair of values, key in the y-value and press g°, key in the
corresponding x-value and press _ .
3. After all data pairs are input, press gR~ to obtain the correlation coefficient
(between ln y and x).
98
Statistics
99
4. Press 1 gRg>0gRg> to obtain A in the equation above.
RPN Mode:
5. Press ~dzg° to obtain B.
6. Press g>1- to obtain the effective growth rate (as a decimal).
ALG Mode:
5. Press z~d³g° to obtain B.
6. Press g>-1³ to obtain the effective growth rate (as a decimal).
7. To make a y-estimate, key in the x-value and press gRg>.
Example 1: A stock's price in history is listed below. What effective growth rate does this
represent? If the stock continues this growth rate, what is the price projected to be at the
end of 2004 (year 7)?
End of Year
1998(1)
1999(2)
2000(3)
2001(4)
2002(5)
2003(6)
2004(7)
Price
45
51.5
53.75
80
122.5
210
?
12c platinum / 12C
12c platinum
RPN Keystrokes
ALG Keystrokes
fCLEARH
fCLEARH
45g°1_
45g°1_
51.5g°2_
51.5g°2_
53.75g°3_
53.75g°3_
80g°4_
80g°4_
122.5g°5_
122.5g°5_
210g°6_
210g°6_
gR~
gR~
1.00
2.00
3.00
4.00
5.00
6.00
0.95
First data pair input.
Second data pair input.
Third data pair input.
Fourth data pair input.
Fifth data pair input.
Sixth data pair input.
Correlation coefficient
(between ln y and x).
1gRg>
0gRg>
~dzg°
g>17gRg>
27.34
0.31
0.36
232.35
A
B
Effective growth rate.
Projected price at end of year
7 (2004).
1gRg>
0gRg>
z~d³g°
g>-1³
7gRg>
Display
Comments
100
Statistics
For repeated use of this routine, the following HP 12C Platinum program will be useful.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
~
g°
~
_
g(000
gR
~
t
1
gR
g>
0
gR
g>
t
~
d
z
g°
t
g>
1
t
gR
g>
g(000
fs
n: Unused
FV: Unused
R3: Σx2
R7-R.6: Unused
DISPLAY
000,
001,
34
002,
43 23
003,
34
004,
49
005,43,33,000
006,
43
2
007,
34
008,
31
009,
1
010,
43
2
011,
43 22
012,
0
013,
43
2
014,
43 22
015,
31
016,
34
017,
33
018,
10
019,
43 23
020,
31
021,
43 22
022,
1
023,
30
024,
31
025,
43
2
026,
43 22
027,43,33,000
i: Unused
R0: Unused
R4: Σy
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
~
g°
~
_
g(000
gR
~
t
0
gR
g>
t
1
gR
g>
~
d
z
~
³
g°
t
g>
1
³
t
gR
g>
g(000
fs
REGISTERS
PV: Unused
R1: n
R5: Σy2
DISPLAY
000,
001,
34
002,
43 23
003,
34
004,
49
005,43,33,000
006,
43
2
007,
34
008,
31
009,
0
010,
43
2
011,
43 22
012,
31
013,
1
014,
43
2
015,
43 22
016,
34
017,
33
018,
10
019,
34
020,
36
021,
43 23
022,
31
023,
43 22
024,
30
025,
1
026,
36
027,
31
028,
43
2
029,
43 22
030,43,33,000
PMT: Unused
R2: Σx
R6: Σxy
Statistics 101
Program Instructions:
1. Key in the program and press fCLEARH.
2. For each input pair of values, key in the y-value and press \(³), key in the
corresponding x- value and press t.
3. After all data pairs are input, press g(006t to obtain the correlation
coefficient (between ln y and x).
4. Press t to obtain A.
5. Press t to obtain B.
6. Press t to obtain the effective growth rate as a decimal.
7. RPN: To make a y-estimate, key in the x-value and press t. For subsequent
estimates, key in the x-value and press g(025t.
7. ALG: To make a y-estimate, key in the x-value and press t. For subsequent
estimates, key in the x-value and press g(028t.
8. For a different set of data, press fCLEARH and go to step 2.
Example 2: Repeat example 1 using the program.
12c platinum /12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
45\1t
45³1t
51.5\2t
51.5³2t
53.75\3t
53.75³3t
80\4t
80³4t
122.5\5t
122.5³5t
210\6t
210³6t
g(006t
g(006t
1.00
2.00
3.00
4.00
5.00
6.00
0.95
t
t
t
7t
27.34
0.31
0.36
232.35
t
t
t
7t
Display
Comments
First data pair input.
Second data pair input.
Third data pair input.
Fourth data pair input.
Fifth data pair input.
Sixth data pair input.
Correlation coefficient
(between ln y and x).
A
B
Effective growth rate.
Projected price at the end of
year 7 (2004).
102
Statistics
Logarithmic Curve Fit
If your data does not fit a line or an exponential curve, try the following logarithmic curve
fit. This is calculated according to the equation y = A + B (ln x), and all x values must be
positive.
A typical logarithmic curve is shown below.
Y
X
The procedure is as follows:
1. Press fCLEARH.
2. Key in the first y-value and press \(³). Key in the first x-value and press
g°_. Repeat this step for each data pair.
3. After all data pairs are input, press gR~ to obtain the correlation coefficient
(between y and ln x).
4. Press 1gR0gR to obtain A in the equation above.
5. RPN: Press ~d- to obtain B.
5. ALG: Press -~d³ to obtain B.
6. To make a y-estimate, key in the x-value and press g°gR.
Statistics 103
Example 1: A manufacturer observes declining sales of a soon-to-be obsolete product, of
which there were originally 10,000 units in inventory. The cumulative sales figures over a
number of months, given below, may be approximated by a logarithmic curve of the form
y = A + B (ln x), where y represents cumulative sales in units and x the number of months
since the beginning. How many units will be sold by the end of the eighth month?
Month
Cumulative
Sales (units)
1
2
3
4
5
6
1431
3506
5177
6658
7810
8592
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
1431\
1431³
1g°_
1g°_
3506\
3506³
2g°_
2g°_
5177\
5177³
3g°_
3g°_
6658\
6658³
4g°_
4g°_
7810\
7810³
5g°_
5g°_
8592\
8592³
6g°_
6g°_
gR~
gR~
1gR0gR
~d8g°gR
1,066.15
4,069.93
9,529.34
1gR0gR
-~d³
8g°gR
Display
Comments
1.00
First pair data input.
2.00
Second pair data input.
3.00
Third pair data input.
4.00
Fourth pair data input.
5.00
Fifth pair data input.
6.00
0.99
Sixth pair data input.
Correlation coefficient
(between y and ln x).
Value of A.
Value of B.
Total units sold by end of
eighth month.
104
Statistics
Power Curve Fit
Another method of analysis is the power curve or geometric curve. The equation of the
power curve is y = AxB, and the values for A and B are computed by calculations similar to
linear regression. Some examples of power curves are shown below.
Y
Y
B>1
0 < B <1
B<0
0
X
X
0
The following keystrokes fit a power curve according to the equation ln y = ln A + B(ln x):
1. Press fCLEARH.
2. Key in the first y-value and press g°. Key in the first x-value and press
g°_. Repeat this step for all data pairs.
3. Press gR~, to obtain the correlation coefficient (between ln y and ln x).
4. Press 0gRg> to obtain A in the above equation.
5. RPN: Press 1gR0gR~d- to obtain B.
5. ALG: Press 1gR0gR-~d³ to obtain B.
6. To make a y-estimate, key in the x-value and press g°gRg>.
Example: If Galileo had wished to investigate quantitatively the relationship between the
time (t) for a falling object to hit the ground and the height (h) it has fallen, he might have
released a rock from various levels of the Tower of Pisa (which was leaning even then)
and timed its descent by counting his pulse. The following data are measurements Galileo
might have made.
t (pulses)
h (feet)
2
30
2.5
50
3.5
90
4
130
4.5
150
Statistics 105
Find the power curve formulas that best expresses h as a function of t (h = AtB).
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
30g°
30g°
2g°_
2g°_
50g°
50g°
2.5g°_
2.5g°_
90g°
90g°
3.5g°_
3.5g°_
130g°
130g°
4g°_
4g°_
150g°
150g°
4.5g°_
4.5g°_
gR~
gR~
Display
Comments
1.00
First pair data input.
2.00
Second pair data input.
3.00
Third pair data input.
4.00
Fourth pair data input.
5.00
1.00
Fifth pair data input.
Correlation coefficient
(between In y and ln x).
Value of A.
7.72
0gRg>
0gRg>
1gR0gR
1gR0gR
1.99
~d-~d³
The formula that best expresses h as a function of t is
Value of B.
h = 7.72t1.99
We know, as Galileo did not, that in fact h is proportional to t2.
Standard Error of the Mean
The standard error of the mean is a measure of how reliable the mean of a sample (X) is
as an estimator of the mean of the population from which the sample was drawn.
To calculate the standard error of the mean:
1. Press fCLEARH.
2. If you are summing one set of numbers, key in the first number and press _.
Continue until you have entered all of the values.
3. If you are summing two sets of numbers, key in the y-value and press \(³), key
in the x-value and press _. Continue until you have entered all of the values.
4. Press gÖ to obtain the mean of the x-values.
RPN Mode:
5. Press gv:1grz to obtain the standard error of the mean of the x-values.
6. Alternatively, press gv~:1grz to obtain the standard error for the
mean of the y-values.
106
Statistics
ALG Mode:
5. Press gvz:1gr³ to obtain the standard error of the mean of the xvalues.
6. Alternatively, press gv~z:1gr³ to obtain the standard error for the
mean of the y-values.
Example: A sample of 6 one-bedroom apartment rentals reveals that one rents for $190
per month unfurnished, one rents for $200 per month, two rent for $205 per month, one
rents for $216 per month, and one rents for $220 per month. What are the mean monthly
rental and the standard deviation? What is the standard error of the mean?
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
190_200_
190_200_
205_205_
205_205_
216_220_
216_220_
gÖ
gÖ
gv
gv
:1grz
z:1gr³
Display
6.00
206.00
10.86
4.43
Comments
Total number of inputs.
Average monthly rent.
Standard deviation.
Standard error of the mean.
Mean, Standard Deviation, Standard Error for
Grouped Data
Grouped data are presented in frequency distributions to save time and effort in writing
down (or entering) each observation individually. Given a set of data points
x1, x2, ... , xn
with respective frequencies
f1, f2, ... , fn
this procedure computes the mean, standard deviation, and standard error of the mean.
1. Press fCLEARH.
RPN Mode:
2. Key in the first value and press \\.
3. Key in the respective frequency and press ?+0§_. The display shows the
number of data points entered.
ALG Mode:
2. Key in the first value and press ³.
3. Key in the respective frequency and press ?+0§~³gF~_. The
display shows the number of data points entered.
4. Repeat steps 2 and 3 for each data point.
Statistics 107
5. To calculate the mean (average) press :0?1:6?3gÖ.
6. Press gv to find the standard deviation.
7. RPN: Press :0grz to find the standard error of the mean.
7. ALG: Press z:0gr³ to find the standard error of the mean.
Example 1: A survey of 266 one-bedroom apartment rentals reveals that 54 rent for $190
a month unfurnished, 32 rent for $195 per month, 88 rent for $200 per month, and 92 rent
for 206 per month. What are the average monthly rental, the standard deviation, and the
standard error of the mean?
12c platinum / 12C
RPN Keystrokes
fCLEARH
190\\
54?+0§_
195\\
32?+0§_
200\\
88?+0§_
206\\
92?+0§_
:0?1:6
?3gÖ
gv
:0grz
12c platinum
ALG Keystrokes
fCLEARH
190³54?+0
§~³gF~_
195³32?+0
§~³gF~_
200³88?+0
§~³gF~_
206³92?+0
§~³gF~_
:0?1:6
?3gÖ
gv
z:0gr³
Display
Comments
1.00
First data pair entered.
2.00
Second data pair entered.
3.00
Third data pair entered.
4.00
Fourth data pair entered.
199.44
5.97
0.37
Average monthly rent.
Standard deviation.
Standard error of the
mean.
Use the following HP 12C Platinum program for the previous example:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
?+0
§
_
g(000
:0
?1
:6
?3
gÖ
t
DISPLAY
000,
001,44 40
0
002,
20
003,
49
004,43,33,000
005,
45
0
006,
44
1
007,
45
6
008,
44
3
009,
43
0
010,
31
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
?+0
§
~
³
gF
~
_
g(000
:0
?1
DISPLAY
000,
001,44 40
0
002,
20
003,
34
004,
36
005,
43 40
006,
34
007,
49
008,43,33,000
009,
45
0
010,
44
1
108
Statistics
12c platinum / 12C
RPN KEYSTROKES
gv
t
:0
gr
z
g(000
fs
DISPLAY
011,
43 48
012,
31
013,
45
0
014,
43 21
015,
10
016,43,33,000
n: Unused
FV: Unused
R3: Σfixi2
R7-R.7: Unused
i: Unused
R0: Σfi
R4: Σxi
12c platinum
ALG KEYSTROKES
:6
?3
gÖ
t
gv
t
z
:0
gr
³
g(000
fs
REGISTERS
PV: Unused
R1: Σfi
R5: Σxi2
DISPLAY
011,
45
6
012,
44
3
013,
43
0
014,
31
015,
43 48
016,
31
017,
10
018,
45
0
019,
43 21
020,
36
021,43,33,000
PMT: Unused
R2: Σfixi
R6: Σfixi2
Program Instructions:
1. Key in the program.
2. Press fCLEARH.
3. RPN: Key in the first value and press \\.
3. ALG: Key in the first value and press ³.
4. Key in the respective frequency and press t. The display shows the number of data
points entered.
5. Repeat steps 3 and 4 for each data point.
6. RPN: To calculate the mean, press g(005t.
6. ALG: To calculate the mean, press g(009t.
7. Press t to find the standard deviation.
8. Press t to find the standard error of the mean.
9. For a new case, go to step 2.
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
190\\
190³
1.00
54t
54t
Comments
First data pair.
Statistics 109
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
195\\
195³
32t
32t
200\\
200³
88t
88t
206\\
206³
92t
92t
g(005t
g(009t
t
t
t
t
Display
Comments
2.00
Second data pair.
3.00
Third data pair.
4.00
199.44
5.97
0.37
Total number of data sets.
Average monthly rent (mean).
Standard deviation.
Standard error of the mean.
Chi-Square Statistics
The chi-square statistic is a measure of the goodness of fit between two sets of
frequencies. It is used to test whether a set of observed frequencies differs from a set of
expected frequencies sufficiently to reject the hypothesis under which the expected
frequencies were obtained.
In other words, you are testing whether discrepancies between the observed frequencies
(Oi) and the expected frequencies (Ei) are significant, or whether they may reasonable be
attributed to chance. The formula generally used is:
n
(Oi − Ei )2
i =1
Ei
χ2 = ∑
If there is a close agreement between the observed and expected frequencies, χ2 will be
small. If the agreement is poor, χ2 will be large.
The following keystrokes calculate the χ2 statistic:
1. Press fCLEARH .
RPN Mode:
2. Key in the first Oi value and press \ .
3. Key in the first Ei value and press ?0-\§:0z+ .
ALG Mode:
2. Key in the first Oi value and press - .
3. Key in the first Ei value and press ?0³g’z:0+~³.
4. Repeat steps 2 and 3 for all data pairs. The χ2 value is displayed.
110
Statistics
Example 1: A suspect die from a Las Vegas casino is brought to an independent testing
firm to determine its bias, if any. The die is tossed 120 times and the following results
obtained.
Number
Observed Frequency
1
25
2
17
3
15
4
23
5
24
6
16
The expected frequency = 120 throws / 6 sides, or E = 20 for each number, 1 thru 6.
(Since E is a constant in this example, there is no need to store it in R0 each time.)
12c platinum / 12C
12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
25\
2520?0-\
20?0³g’
§:0z+
z:0+~³
17\2017-20³g’
\§:0z+ z:0+~³
15\2015-20³g’
\§:0z+ z:0+~³
23\2023-20³g’
\§:0z+ z:0+~³
24\2024-20³g’
\§:0z+ z:0+~³
16\2016-20³g’
\§:0z+ z:0+~³
Display
Comments
1.25
1.70
2.95
3.40
4.20
5.00
χ2
The number of degrees of freedom is (n-1). Since n = 6, the degrees of freedom = 5.
Consulting statistical tables, you look up χ2 to a 0.05 significance level with 5 degrees of
freedom, and see that χ20.05,5 = 11.07. Since χ2 = 5 is within 11.07, we may conclude that
to a 0.05 significance level (probability = .95), the die is fair.
Try the following HP 12C Platinum program with the same example.
Statistics 111
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
?0
\
§
:0
z
+
g(000
fs
n: Unused
FV: Unused
12c platinum
ALG KEYSTROKES
fs
000,
fCLEARÎ
001, 44 0
~
002, 30
à
003, 36
b
004, 20
g’
005, 45 0
§
006, 10
~
007, 40
+
008,43,33,000 ~
³
g(000
fs
DISPLAY
i: Unused
R0: Ei
REGISTERS
PV: Unused
R1-R.9: Unused
DISPLAY
000,
001,
34
002,
24
003,
25
004,
43 20
005,
20
006,
34
007,
40
008,
34
009,
36
010,43,33,000
PMT: Unused
Program Instructions:
1. Key in the program.
2. Press fCLEARH.
3. Key in the first Oi value and press \(³).
4. Key in the first Ei value and press t.
5. Repeat steps 3 and 4 for all data pairs. The χ2 value is displayed.
6. For a new case, go to step 2.
12c platinum / 12C
RPN Keystrokes
fCLEARH
25\
20t
17\
20t
15\
20t
23\
20t
24\
20t
16\
20t
12c platinum
ALG Keystrokes
fCLEARH
25³
20t
17³
20t
15³
20t
23³
20t
24³
20t
16³
20t
Display
Comments
1.25
1.70
2.95
3.40
4.20
5.00
χ2
112
Statistics
Normal Distribution
The normal (or Gaussian) distribution is an important tool in statistics and business
analysis. The following HP 12C Platinum program gives an approximation to the upper
tail area Q under a standardized normal distribution curve, given x. The upper tail area
signifies the probability of occurrence of all values ≥ x.
f(x)
Q(x)
0
Q( x) ≅
x
1
(83 x + 351) + 562
EXP −
2
703 / x + 165
Relative error less than 0.042% over the range 0 < x < 5.5
Reference:
Stephen E. Derenzo, "Approximations for Hand Calculators Using Small Integer
Coefficients," Mathematics of Computation, Vol. 31, No. 137, page 2014,225; Jan 1977.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
?0
8
3
§
3
5
1
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
44
0
8
3
20
3
5
1
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
?0
7
0
3
z
~
+
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
44
0
7
0
3
10
34
40
Statistics 113
12c platinum / 12C
RPN KEYSTROKES
+
:0
§
5
6
2
+
7
0
3
:0
z
1
6
5
+
z
Þ
g>
2
z
g(000
fs
n: Unused
FV: Unused
DISPLAY
008,
40
009,
45
0
010,
20
011,
5
012,
6
013,
2
014,
40
015,
7
016,
0
017,
3
018,
45
0
019,
10
020,
1
021,
6
022,
5
023,
40
024,
10
025,
16
026,
43 22
027,
2
028,
10
029,43,33,000
i: Unused
R0: x
12c platinum
ALG KEYSTROKES
1
6
5
³
:0
§
8
3
+
3
5
1
§
:0
+
5
6
2
z
~
³
g>
+
³
y
g(000
fs
REGISTERS
PV: Unused
R1-R.6: Unused
DISPLAY
008,
1
009,
6
010,
5
011,
36
012,
45
0
013,
20
014,
8
015,
3
016,
40
017,
3
018,
5
019,
1
020,
20
021,
45
0
022,
40
023,
5
024,
6
025,
2
026,
10
027,
34
028,
36
029,
43 22
030,
40
031,
36
032,
22
033,43,33,000
PMT: Unused
Program Instructions:
1. Key in the program.
2. Key in x and press t to compute Q(x).
3. Repeat step 2 for each new case.
Example: Find Q(x) for x = 1.18 and x = 2.1.
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
0.12
1.18t
1.18t
0.02
2.1t
2.1t
Comments
Q(1.18)
Q(2.1)
114
Statistics
Covariance
Covariance is a measure of the interdependence between paired variables (x and y). Like
standard deviation, covariance may be defined for either a sample (Sxy) or a population
(S'xy) as follows:
Sxy = r · sx · sy
S'xy = r · s'x · s'y
The following procedure finds the covariance of a sample (Sxy) and of a population (S'xy):
1. Press fCLEARH.
2. Key in the y-values and press \(³).
3. Key in the x-values and press _.
4. Repeat steps 2 and 3 for all data pairs.
RPN Mode:
5. Press gv§\gRd§ to obtain the value of Sxy.
6. Press :1 1-:1z§ to obtain S'xy.
ALG Mode:
5. Press gv1gRd§~§~³ to obtain the value of Sxy.
6. Press 1-:1y§~³ to obtain S'xy.
Example 1: Find the sample covariance (Sxy) and population covariance (S'xy) for the
following paired variables:
xi
yi
26
92
30
85
44
78
50
81
62
54
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
92\26_
92³26_
85\30_
85³30_
78\44_
78³44_
81\50_
81³50_
54\62_
54³62_
51\68_
51³68_
7.00
40\74_
40³74_
gv1
gv§\
gRd§
gR
-354.14
d§
~§~³
:1 1-:1
1-:1y
-303.55
z§
§~³
68
51
74
40
Comments
Total number of entries.
Sxy
S'xy
Statistics 115
Try the previous example using the following HP 12C Platinum program:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
_
g(000
gv
§
\
gR
d
§
t
:1
1
:
z
§
g(000
fs
n: Unused
FV: Unused
R3: Σx2
R7-R.7: Unused
DISPLAY
000,
001,
49
002,43,33,000
003,
43 48
004,
20
005,
36
006,
43
2
007,
33
008,
20
009,
31
010,
45
1
011,
1
012,
30
013,
45
1
014,
10
015,
20
016,43,33,000
REGISTERS
i: Unused
R0: Unused
R4: Σy
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
_
g(000
gv
1
gR
d
§
~
§
~
³
t
1
:1
y
§
~
³
g(000
fs
PV: Unused
R1: n
R5: Σy2
DISPLAY
000,
001,
49
002,43,33,000
003,
43 48
004,
1
005,
43
2
006,
33
007,
20
008,
34
009,
20
010,
34
011,
36
012,
31
013,
1
014,
30
015,
45
1
016,
22
017,
20
018,
34
019,
36
020,43,33,000
PMT: Unused
R2: Σx
R6: Σxy
Program Instructions:
1. Key in the program.
2. Press fCLEARH.
3. Key in the y-value and press \(³).
4. Key in the x-value and press t. Repeat steps 3 and 4 for all data pairs.
5. Press g(003t to obtain the value of Sxy.
6. Press t to obtain S'xy.
7. For a new case, go to step 2.
116
Statistics
12c platinum / 12C
12c platinum
Display
Comments
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
92\26t
92³26t
85\30t
85³30t
78\44t
78³44t
81\50t
81³50t
54\62t
54³62t
51\68t
51³68t
Total number of entries.
7.00
40\74t
40³74t
-354.14 Sxy
g(003t
g(003t
-303.55 S'xy
t
t
Permutations
A permutation is an ordered subset of a set of distinct objects. The number of possible
permutations, each containing n objects, that can be formed from a collection of m distinct
objects is given by:
m Pn
=
m!
( m − n)!
where m, n are integers and 69 ≥ m ≥ n ≥ 0.
Use the following HP 12C Platinum program to calculate the number of possible
permutations.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
?0
~
ge
gF
:0
ge
z
g(000
fs
DISPLAY
000,
001,
44
0
002,
34
003,
43
3
004,
43 40
005,
45
0
006,
30
007,
43
3
008,
10
009,43,33,000
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
gF
~
³
~
ge
z
~
ge
³
g(000
fs
DISPLAY
000,
001,
43 40
002,
30
003,
34
004,
36
005,
34
006,
43
3
007,
10
008,
34
009,
43
3
010,
36
011,43,33,000
Statistics 117
n: Unused
FV: Unused
i: Unused
R0: n
REGISTERS
PV: Unused
R1-R.8: Unused
PMT: Unused
Program Instructions:
1. Key in the program.
2. Key in m and press \(³).
3. Key in n and press t to calculate mPn.
4. For a new case go to step 2.
Example: How many ways can 10 people be seated on a bench if only 4 seats are
available?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
10\
10³
5,040.00
4t
4t
Comments
10P4.
Combinations
A combination is a selection of one or more of a set of distinct objects without regard to
order. The number of possible combinations, each containing n objects, that can be
formed from a collection of m distinct objects is given by:
m Cn
=
m!
(m − n )!n!
Where m, n are integers and 69 ≥ m ≥ n ≥ 0.
Use the following HP 12C Platinum to calculate the number of possible combinations.
118
Statistics
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
?0
~
ge
gF
:0
ge
:0
ge
§
z
g(000
fs
n: Unused
FV: Unused
DISPLAY
000,
001,
44
0
002,
34
003,
43
3
004,
43 40
005,
45
0
006,
30
007,
43
3
008,
45
0
009,
43
3
010,
20
011,
10
012,43,33,000
i: Unused
R0: n
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
gF
~
³
~
ge
z
~
ge
z
gF
ge
³
g(000
fs
REGISTERS
PV: Unused
R1-R.8: Unused
DISPLAY
000,
001,
43 40
002,
30
003,
34
004,
36
005,
34
006,
43
3
007,
10
008,
34
009,
43
3
010,
10
011,
43 40
012,
43
3
013,
36
014,43,33,000
PMT: Unused
Program Instructions:
1. Key in the program.
2. Key in m and press \(³).
3. Key in n and press t to calculate mCn.
4. For a new case, go to step 2.
Example: A manager wants to choose a committee of three people from the seven
engineers working for him. In how many different ways can the committee be selected?
12c platinum / 12C
RPN Keystrokes
7\
3t
12c platinum
ALG
Keystrokes
7³
3t
Display
35.00
Comments
7C3.
Statistics 119
Random Number Generator
This HP 12C Platinum program calculates uniformly distributed pseudo-random numbers
ui in the range
0 < ui < 1
The following method is used:
•
ui+1 = fractional part of (997 ui)
•
where i = 0, 1, 2, ...
•
u0 = 0.5284163* (seed)
The period of this generator has a length of 500,000 numbers and the generator passes the
frequency test (chi square) for uniformity, the serial test and the run test. The most
significant digits (the left hand digits) are the most random digits. The right most digits
are significantly less random.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
.
5
2
8
4
1
6
3
?0
9
9
7
§
gT
?0
t
g(010
fs
n: Unused
FV: Unused
*
DISPLAY
000,
001,
48
002,
5
003,
2
004,
8
005,
4
006,
1
007,
6
008,
3
009,
44
0
010,
9
011,
9
012,
7
013,
20
014,
43 24
015,
44
0
016,
31
017,43,33,010
i: Unused
R0: Ui
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
.
5
2
8
4
1
6
3
§
9
9
7
³
gT
?0
t
g(009
fs
REGISTERS
PV: Unused
R1-R.7: Unused
DISPLAY
000,
001,
48
002,
5
003,
2
004,
8
005,
4
006,
1
007,
6
008,
3
009,
20
010,
9
011,
9
012,
7
013,
36
014,
43 24
015,
44
0
016,
31
017,43,33,009
PMT: Unused
Other seeds may be selected but the quotient of (seed x 107) divided by two or five must not be an
integer. Also, it would be wise to statistically test other seeds before using them.
120
Statistics
Program Instructions:
1. Key in the program.
2. To generate a random number, press t.
3. Repeat step 2 as many times as desired.
Example: Generate a sequence of 5 random numbers.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
t
t
t
t
t
t
t
t
t
t
Display
0.83
0.56
0.27
0.04
0.20
Comments
Random number.
Personal Finance
Homeowners Monthly Payment Estimator
It is often useful, when comparison shopping for a mortgage or determining the
appropriate price range of houses to consider, to be able to quickly estimate the monthly
payment given the purchase price, tax rate per $1000, percent down, interest rate and term
of the loan.
The calculation assumes that the assessed value is 100% of the sales price and does not
take into account financing of the closing costs.
A simple keystroke procedure may be used to calculate the monthly payment:
1.
Press g and press fCLEARG.
2.
Key in the annual interest rate and press gC.
3.
Key in the term of the loan (in years) and press gA.
4.
Key in the purchase prices and press ?1.
5.
6.
5.
6.
RPN Mode:
Key in the percent down and press b-$.
Key in the tax rate in dollars per thousand and press
:1§12000zÞ\PP+ .
(A negative sign is the convention for cash paid out).
ALG Mode:
Press -, key in the percent down and press
b$P:1§.
Key in the tax rate in dollars per thousand and press
Þz12000+~³.
(A negative sign is the convention for cash paid out).
Example: What would your monthly payments be on a $65,000 house in a neighborhood
with a $25 per thousand tax rate and a 10 ¾ % interest rate on a 35 year loan with 10%
down?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
gÂ
gÂ
fCLEARG
fCLEARG
0.90
10.75gC
10.75gC
420.00
35gA
35gA
65,000.00
65000?1
65000?1
121
Comments
Monthly interest rate.
Months of loan.
Purchase price.
122
Personal Finance
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
58,500.00
10b-$
-10b$
25:1§
P:1§
12000zÞ
25Þz12000+ -135.42
\PP
~
-672.16
+
³
Comments
Mortgage balance.
Approximate monthly taxes.
Approximate monthly
payment.
The following HP 12C Platinum program may be used instead of the above.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
gÂ
:1
:2
b
$
\
gF
+
:3
§
1
2
Æ
3
z
Þ
\
P
P
+
g(000
fs
n: Term
FV: 0
R3: Tax rate
DISPLAY
000,
001,
43
8
002,
45
1
003,
45
2
004,
25
005,
30
006,
13
007,
36
008,
43 40
009,
40
010,
45
3
011,
20
012,
1
013,
2
014,
26
015,
3
016,
10
017,
16
018,
36
019,
14
020,
14
021,
40
022,43,33,000
i: Interest
R0: Unused
R4-R.7: Unused
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
gÂ
:1
:2
b
$
P
:1
§
:3
z
1
2
Æ
3
~
~
³
g(000
fs
REGISTERS
PV: Loan
R1: Purch. Price
DISPLAY
000,
001,
43
8
002,
45
1
003,
30
004,
45
2
005,
25
006,
13
007,
14
008,
45
1
009,
20
010,
45
3
011,
10
012,
1
013,
2
014,
26
015,
3
016,
30
017,
34
018,
34
019,
36
020,43,33,000
PMT: Loan PMT
R2: % Down
Personal Finance
123
Program Instructions:
1. Key in the program.
2. Press fCLEARG.
3. Key in the annual interest rate and press gC .
4. Key in the term of the loan in years and press gA .
5. Key in the purchase price and press ?1.
6. Key in the percent down and press ?2.
7. Key in the tax rate in dollars per thousand and press ?3.
8. To calculate the approximate monthly payment, press t.
9. For a new case, store only the new variables by performing steps 3 thru 7 as needed.
Press t for the new approximate monthly payment.
Example: Solve the previous example using the HP 12C Platinum program.
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARG
fCLEARG
10.75gC
10.75gC
35gA
35gA
65000?1
65000?1
10?2
10?2
25?3
25?3
t
t
Display
Comments
0.90
420.00
65,000.00
10.00
25.00
-672.16
Monthly interest.
Months of loan.
Purchase price.
Percent down.
Tax rate per thousand.
Approximate monthly
payment.
What would the approximate payment be if the loan was at 10% interest?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
-638.33
10gCt
10gCt
Comments
Approximate monthly
payment.
What if the down payment is increased to 20%?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
-582.45
20?2t
20?2t
Comments
Approximate monthly
payment.
124
Personal Finance
Tax-Free Individual Retirement (IRA) or
Keogh Plan
The advent of tax-free retirement accounts (IRA or Keogh) has resulted in considerable
benefits for many persons who are not able to participate in group profit sharing or
retirement plans. The savings due to the tax-free status are often considerable, but
complex to calculate. Required data are: the years to retirement, the total annual
investment, the compound annual interest rate of the investment, and an assumed tax rate
(the dividend tax rate) which would be paid on a similar but taxable investment. This
program calculates:
1.
2.
3.
4.
The future cash value of the tax-free investment (the dividend tax rate does not apply).
The total cash paid in.
The total dividends paid (the tax-free status means these dividends are tax-free).
The future value of the investment at retirement, assuming that after retirement you
make withdrawals at a rate which causes the money to be taxed at the withdrawal tax
rate. This rate is often assumed to be one half of the dividend tax rate.
5. The diminished purchasing power assuming a given annual inflation rate.
6. The future value of a comparable taxable investment (the dividend tax rate applies).
7. The diminished purchasing power of a comparable taxable investment.
Notes:
•
The calculations run from the beginning of the first year to the end of the last
year.
•
The interest (annual yield), i, should be entered to as many significant figures as
possible for maximum accuracy.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:n
:P
§
t
+
t
:M
:2
b
t
1
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
45
45
45
45
11
14
20
31
40
31
15
2
25
30
31
1
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
:n
§
:P
+
t
~
³
t
:M
:2
b
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
45
45
45
45
11
20
14
40
31
34
36
31
15
30
2
25
Personal Finance
12c platinum / 12C
RPN KEYSTROKES
:3
b
+
:n
q
z
t
:¼
:1
b
¼
M
t
g(012
fs
n: Years
FV: Used
R3: Inflation %
DISPLAY
013,
45
3
014,
25
015,
40
016,
45 11
017,
21
018,
10
019,
31
020,
45 12
021,
45
1
022,
25
023,
30
024,
12
025,
15
026,
31
027,43,33,012
12c platinum
ALG KEYSTROKES
³
t
1
+
:3
b
³
q
:n
z
~
~
³
t
:¼
:1
b
¼
M
t
g(015
fs
REGISTERS
i: Used
PV: 0
R0: Unused
R1: Dividend Tax %
R4-R.5: Unused
3.
Key in the dividend tax rate as a percentage and press ?1.
3.
Key in the withdrawal tax rate as a percentage and press ?2.
3.
Key in the inflation rate as a percentage and press ?3.
6.
Key in years to retirement and press n.
7.
Key in the interest rates as a percentage and press ¼.
Key in the annual payment and press ÞP.
9.
DISPLAY
013,
36
014,
31
015,
1
016,
40
017,
45
3
018,
25
019,
36
020,
21
021,
45 11
022,
10
023,
34
024,
34
025,
36
026,
31
027,
45 12
028,
30
029,
45
1
030,
25
031,
12
032,
15
033,
31
034,43,33,015
PMT: Yearly Pmt
R2: Withdrawal Tax %
Program Instructions:
1. Key in the program.
2. Press fCLEARH and press g× .
8.
125
Press M to calculate the future value of the tax free investment.
10. Press t to compute the total cash paid in.
126
Personal Finance
11. Press t to compute the total dividends paid.
12. Press t to compute the future value when, after retirement, money is withdrawn at
a rate causing the tax rate to equal ½ the rate paid during the pay in period.
13. Press t to compute the diminished purchasing power, in terms of today's dollars,
of the future value assuming a 10% annual inflation rate.
14. Press t to compute the future value of an ordinary tax investment.
15. Press t to compute the diminished purchasing power of the ordinary tax
investment.
Example: Assuming a 35 year investment period with a dividend rate of 8.175% and an
income tax rate of 40%:
1.
2.
3.
4.
5.
6.
7.
If you invest $1500 each year in a tax free account, what will its value be at
retirement?
How much cash will be paid in?
What will be the value of the earned dividends?
After retirement, if you withdraw cash form the account at a rate such that it will be
taxed at a rate equal to one-half the rate paid during the pay-in period, what will be
the after-tax value?
What is the diminished purchasing power of that amount, in today's dollars, assuming
10% annual inflation?
If you invest the same amount ($1500 after taxes for a non-Keogh or non-IRA
account) each year with dividends taxed as ordinary income, what will be the total
tax-paid cash at retirement?
What is the purchasing power of that figure in terms of today's dollars?
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
g×
g×
40?1
40?1
20?2
20?2
10?3
10?3
35n
35n
8.175¼
8.175¼
1500ÞP
1500ÞP
M
M
t
t
t
t
t
t
t
t
t
t
t
t
Display
Comments
40.00
20.00
10.00
35.00
8.18
-1,500.00
290,730.34
-52,500.00
238,230.34
232,584.27
8,276.30
139,360.09
4,959.00
Dividend tax rate.
Withdrawal tax rate.
Inflation rate.
Years to retirement.
Dividend rate.
Annual payment.
Future value at retirement.
Cash paid in.
Earned dividends (untaxed).
After-tax value.
Diminished purchasing power.
Tax-paid cash at retirement.
Purchasing power of tax-paid
cash at retirement.
Personal Finance
127
Stock Portfolio Evaluation and Analysis
This program evaluates a portfolio of stocks given the current market price per share and
the annual dividend. The user inputs the initial purchase price of a stock, the number of
shares, the beta coefficient*, the annual dividend, and the current market price for a
portfolio of any size.
The program returns the percent change in value of each stock and the valuation and beta
coefficient* of the entire portfolio. Output includes the original portfolio value, the new
portfolio value, the percent change in the value and the annual dividend and yield as a
percent of the current market value. The overall beta coefficient of the portfolio is also
calculated.
Note:
The beta coefficient analysis is optional. Key in 1.00 if beta is not to be analyzed.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
:4
gm
g(021
?-4
§
:7
§
?+0
~
:7
§
?+1
d
§
?+3
:5
gF
à
t
*
DISPLAY
000,
001,
45
4
002,
43 35
003,43,33,021
004,44 30
4
005,
20
006,
45
7
007,
20
008,44 40
0
009,
34
010,
45
7
011,
20
012,44 40
1
013,
33
014,
20
015,44 40
3
016,
45
5
017,
43 40
018,
24
019,
31
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
?6
:4
gm
g(026
?-4
d
§
:7
³
?+0
~
§
:7
³
?+1
d
§
~
³
DISPLAY
000,
001,
44
6
002,
45
4
003,
43 35
004,43,33,026
005,44 30
4
006,
33
007,
20
008,
45
7
009,
36
010,44 40
0
011,
34
012,
20
013,
45
7
014,
36
015,44 40
1
016,
33
017,
20
018,
34
019,
36
The beta coefficient is a measure of a stock variability (risk) compared to the market in general.
Beta values for individual stocks can be acquired from brokers, investment publications or the local
business library.
128
Personal Finance
12c platinum / 12C
RPN KEYSTROKES
g(001
+
~
?7
§
?5
?+2
1
?4
t
g(001
:2
t
:0
t
à
t
:0
:1
t
Z
t
:3
:0
z
g(000
fs
n: Unused
FV: Unused
R3: ΣPiSiβi
R7: Si
DISPLAY
020,43,33,001
021,
40
022,
34
023,
44
7
024,
20
025,
44
5
026,44 40
2
027,
1
028,
44
4
029,
31
030,43,33,001
031,
45
2
032,
31
033,
45
0
034,
31
035,
24
036,
31
037,
45
0
038,
45
1
039,
31
040,
23
041,
31
042,
45
3
043,
45
0
044,
10
045,43,33,000
12c platinum
ALG KEYSTROKES
?+3
:5
:6
à
t
g(001
d
?5
§
~
?7
³
?+2
1
?4
t
g(001
:2
t
:0
t
à
t
:0
:1
t
Z
t
:3
z
:0
³
g(000
fs
REGISTERS
i: Unused
PV: Unused
R0: ΣPV
R1: ΣDIV
R4: Flag
R5: Pi ni
R8-R.1: Unused
DISPLAY
020,44 40
3
021,
45
5
022,
45
6
023,
24
024,
31
025,43,33,001
026,
33
027,
44
5
028,
20
029,
34
030,
44
7
031,
36
032,44 40
2
033,
1
034,
44
4
035,
31
036,43,33,001
037,
45
2
038,
31
039,
45
0
040,
31
041,
24
042,
31
043,
45
0
044,
45
1
045,
31
046,
23
047,
31
048,
45
3
049,
10
050,
45
0
051,
36
052,43,33,000
PMT: Unused
R2: ΣOrig. Val.
R6: Pi
Personal Finance
129
Program Instructions:
1. Key in the program.
2. Initialize the program by pressing fCLEARH.
3.
Key in the number of shares of a stock and press \(³).
4.
Key in the initial purchase of the stock and press t.
5.
Key in the beta coefficient of the stock and press \(³).
6.
Key in the annual dividend of the stock and press \(³).
7.
Key in the present price of the stock and press t. The display will show the
percent change in the stock value.
Repeat steps 3 through 7 until all the stocks are entered.
Next, to evaluate the entire portfolio, press :
8.
9.
RPN: g(031
ALG: g(037
10. Press t to see the initial portfolio value.
11. Press t to see the present portfolio value.
12. Press t to see the percent change in value.
13. Press t to see the total yearly dividend.
14. Press t to see the annual dividend yield as a percent of the current market value.
15. Press t to see the beta coefficient of the portfolio.
16. For a new case return to step 2.
Example: Evaluate the following portfolio:
Number of
Shares Held
100
200
50
500
Initial
Purchase
Price
25.63
30.25
89.88
65.25
Beta
Coefficient
Annual
Dividend
Present
Market Price
Stock
.8
1.2
1.3
.6
$1.70
$2.10
$4.55
$3.50
27.25
33.50
96.13
64.38
Int'l Heartburn
P. D. Q.
Datacrunch
N.W. Sundial
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
fCLEARH
fCLEARH
100\
100³
25.63t
25.63t
.8\
.8³
1.70\
1.70³
27.25t
27.25t
Display
0.00
100.00
1.00
0.80
1.70
6.32
Comments
Int'l Heartburn
Percent change in Stock's
value.
130
Personal Finance
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
200\
200³
30.25t
30.25t
1.2\
1.2³
2.10\
2.10³
33.5t
33.5t
200.00
1.00
1.20
2.10
10.74
50\
89.88t
1.3\
4.55\
96.13t
50³
89.88t
1.3³
4.55³
96.13t
50.00
1.00
1.30
4.55
6.95
500\
65.25t
.6\
3.50\
64.38t
500³
65.25t
.6³
3.50³
64.38t
500.00
1.00
0.60
3.50
-1.33
g(031
t
t
t
t
t
t
g(037
t
t
t
t
t
t
45,732.00
46,421.50
1.51
2,567.50
5.53
0.77
Display
Comments
P. D. Q.
Percent change in Stock's
value.
Datacrunch
Percent change in Stock's
value.
N. W. Sundial
Percent change in Stock's
value.
Original value.
Present value.
Percent change in value.
Total yearly dividend.
Annual dividend yield.
Portfolio beta coefficient.
Canadian Mortgages
In Canada, interest is compounded semi-annually with payments made monthly. This
results in a different monthly mortgage factor than is used in the United States and
preprogrammed into the HP 12C Platinum. This difference can be easily handled by the
addition of a few keystrokes. For any problem requiring an input for ¼, the Canadian
mortgage factor is calculated first and then this value is entered in for ¼ in the
calculation to give the answer for Canada.
The keystrokes to calculate the monthly Canadian mortgage factor are:
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
RPN Mode:
Press fCLEARGgÂ.
Key in 6 and press n.
Key in 200 and press \$.
Key in the annual interest rate as a percentage and press +ÞM.
Press ¼.
ALG Mode:
Press fCLEARGgÂ.
Key in 2 and press n.
Key in the annual interest rate as a percentage and press z2¼.
Key in 1 and press $M.
Press 12n¼.
The Canadian mortgage factor is now stored in ¼ for future use. The examples below
show how this factor is used for ¼ in Canadian mortgage problems. Example 3 shows
how to reverse this procedure and obtain the annual Canadian interest rate from the
monthly Canadian mortgage factor.
Periodic Payment Amount
Example 1: What is the monthly payment required to fully amortize a 30-year, $30,000
Canadian mortgage if the interest rate is 9%?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
fCLEARG
fCLEARG
gÂ
gÂ
6n200\$
2n9z2¼
9+ÞM¼
1$M12n¼ 0.74
360.00
30gA
30gA
30000$0M
P
30000$0M
P
-237.85
131
Comments
Canadian mortgage factor.
Total monthly periods in
mortgage life.
Monthly payment.
132
Canadian Mortgages
Number of Periodic Payments to Fully
Amortize a Mortgage
Example 2: An investor can afford to pay $440 per month on a $56,000 Canadian
mortgage. If the annual interest rate is 9 ¼ %, how long will it take to completely
amortize this mortgage?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
fCLEARG
fCLEARG
gÂ
gÂ
6n200\$
2n9.25z2¼
9.25+ÞM¼ 1$M12n¼ 0.76
-440.00
440ÞP
440ÞP
56000$0Mn 56000$0Mn 437.00
Comments
Canadian mortgage factor.
Monthly payment.
Total number of monthly
payments.
Effective Interest Rate (Yield)
Example 3: A Canadian mortgage has monthly payments of $612.77 with a maturity of
25 years. The principal amount is $75,500. What is the annual interest rate?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
fCLEARG
fCLEARG
gÂ
gÂ
25gA
25gA
612.77ÞP
612.77ÞP
0.72
75500$¼
75500$¼
6n0P
0P
200Þ$
12nM
8.75
2n¼§2³
M:$+
Comments
Canadian mortgage factor.
Annual interest rate.
Balance Remaining at End of Specified
Period
Example 4: A Canadian mortgage has monthly payments of $612.77 at 8.75% interest.
The principal amount is $75,500. What will be the outstanding balance remaining at the
end of 10 years?
Canadian Mortgages
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
fCLEARG
fCLEARG
gÂ
gÂ
6n200\$
2n8.75z2¼
8.75+ÞM¼ 1$M12n¼ 0.72
612.77ÞP
612.77ÞP
10gA
10gA
-61,877.18
75500$M
75500$M
133
Comments
Canadian mortgage factor.
Outstanding balance remaining
at the end of 10 years.
Miscellaneous
Learning Curve for Manufacturing Costs
Many production process costs vary with output according to the "learning curve"
equation. The production team becomes more proficient in manufacturing a given item as
more and more of them are fabricated and costs may be expected to decrease by a
predictable amount. The learning factor, r, characterizes the learning curve. For instance,
if r =.80 the curve is called an 80% learning curve.
It is readily apparent that the learning, or experience curve, has many uses in setting
production standards, forecasting costs, setting prices, etc. Note, however, that the
learning factor may change, especially after large numbers have been produced.
It the cost of the first unit of a run, C1, and the learning curve factor, r, are known, the
following procedure can be used to calculate the cost of the nth item:
1. Key in the cost of the first item, C1 and press \(³).
2. Key in the number of units produced, n, and press \(³).
RPN Mode:
3. Key in the learning factor, r, and press g°2g°z.
4. Then press q§ to calculate the cost of the nth unit, Cn.
ALG Mode:
3. Key in the learning factor, r, and press g°z2g°q.
4. Then press ~~§~³to calculate the cost of the nth unit, Cn.
Example 1: An electronic manufacturer begins a pilot run on a new instrument. From
past experience he expects the process to have a learning factor, r, or 0.90. If the first unit
costs $875 to produce, what is the expected cost of the 100th unit?
12c platinum / 12C
12c platinum
Display
RPN Keystrokes
ALG Keystrokes
875.00
875\
875³
100.00
100\
100³
.9g°2g°z .9g°z2g°q -0.15
434.51
q§
~~§~³
Comments
Cost of the 100th unit.
If the cost of the first unit, C1, and the nth unit, Cn, are known the learning factor may be
calculated. In addition, it is possible to calculate Cij, the average cost of the ith thru jth
134
Miscellaneous
135
unit. These calculations may be rapidly done with the following HP 12C Platinum
program:
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
g°
2
g°
z
?2
d
~
?1
z
g°
:2
z
g>
?2
g(000
:2
g°
2
g°
z
q
:1
§
g(000
?3
~
?4
:2
g°
2
g°
z
1
+
?0
q
:3
:0
DISPLAY
000,
001,
43 23
002,
2
003,
43 23
004,
10
005,
44
2
006,
33
007,
34
008,
44
1
009,
10
010,
43 23
011,
45
2
012,
10
013,
43 22
014,
44
2
015,43,33,000
016,
45
2
017,
43 23
018,
2
019,
43 23
020,
10
021,
21
022,
45
1
023,
20
024,43,33,000
025,
44
3
026,
34
027,
44
4
028,
45
2
029,
43 23
030,
2
031,
43 23
032,
10
033,
1
034,
40
035,
44
0
036,
21
037,
45
3
038,
45
0
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
g°
z
2
g°
³
?2
d
z
~
?1
³
g°
z
:2
³
g>
?2
g(000
:2
g°
z
2
g°
³
~
q
~
§
:1
³
g(000
?3
~
?4
:2
g°
z
2
DISPLAY
000,
001,
43 23
002,
10
003,
2
004,
43 23
005,
36
006,
44
2
007,
33
008,
10
009,
34
010,
44
1
011,
36
012,
43 23
013,
10
014,
45
2
015,
36
016,
43 22
017,
44
2
018,43,33,000
019,
45
2
020,
43 23
021,
10
022,
2
023,
43 23
024,
36
025,
34
026,
21
027,
34
028,
20
029,
45
1
030,
36
031,43,33,000
032,
44
3
033,
34
034,
44
4
035,
45
2
036,
43 23
037,
10
038,
2
136
Miscellaneous
12c platinum / 12C
RPN KEYSTROKES
q
:0
z
:4
:3
z
:1
§
g(000
fs
n: Unused
FV: Unused
R3: i
DISPLAY
039,
21
040,
30
041,
45
0
042,
10
043,
45
4
044,
45
3
045,
30
046,
10
047,
45
1
048,
20
049,43,33,000
i: Unused
R0: K+1
R4: j
12c platinum
ALG KEYSTROKES
g°
+
1
³
?0
~
q
~
³
:3
q
:0
³
~
~
z
:0
³
:4
:3
³
~
z
~
§
:1
³
fs
REGISTERS
PV: Unused
R1: C1
R5-R.3: Unused
DISPLAY
039,
040,
041,
042,
043,
044,
045,
046,
047,
048,
049,
050,
051,
052,
053,
054,
055,
056,
057,
058,
059,
060,
061,
062,
063,
064,
065,
066,
067,
43
44
45
45
45
45
45
45
PMT: Unused
R2: r
23
40
1
36
0
34
21
34
36
3
21
0
36
34
30
34
10
0
36
4
30
3
36
34
10
34
20
1
36
Miscellaneous
137
Program Instructions:
1. Key in the program.
Note: If the average costs are not going to be calculated:
RPN: lines 25 through 48 need not be keyed in
ALG: lines 32 through 67 need not be keyed in
2. To calculate r, the learning factor, if C1 and Cn are known:
a. Key in C1, the cost of the first unit and press \(³).
b. Key in Cn, the cost of the nth unit and press \(³).
c. Key in n, the number of units and press t to calculate r, the learning factor.
3. To calculate the cost of the nth unit when C1 and r are known:
a. Key in C1 and press ?1. Key in r and press ?2. (Note: This step may be
skipped if step 2 has just been done).
b. Key in the number of units, n and calculate Cn, the cost of the nth unit by pressing
RPN: g(016t.
ALG: g(019t.
4. To calculate the average cost per unit of the ith through jth unit, Cij, if C1 and r are
known.
a. Key in C1 and press ?1. Key in r and press ?2. (Note: This step may be
skipped if step 2 has just been done).
b. Key in the number of the last unit of the batch, j and press \(³).
c. Key in the number of the first unit of the batch, i, and calculate the average cost per
unit by pressing
RPN: g(025t.
ALG: g(032t.
Example 2: The electronic manufacturer cited in example 1 found that the 100th
instrument actually cost $395 to manufacture. Find the actual learning factor, r, the cost of
the 500th unit and the average cost of units 500 thru 1000. (Recall that C1 was $875).
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
875\
875³
395\
395³
100t
100t
500g(016t 500g(019t
1000\
1000³
500g(025t 500g(032t
Display
875.00
395.00
0.89
299.14
1000.00
280.00
Comments
Actual r.
Cost of the 500th unit.
Average cost of the 500th thru
1000th unit.
138
Miscellaneous
Queuing and Waiting Theory
Waiting lines, or queues, cause problems in many marketing situations. Customer
goodwill, business efficiency, labor and space considerations are only some of the
problems which may be minimized by proper application of queuing theory.
Although queuing theory can be complex and complicated subject, handheld calculators
can be used to arrive at helpful decisions.
One common situation that we can analyze involves the case of several identical stations
serving customers, where the customers arrive randomly in unlimited numbers. Suppose
there are n (1 or more) identical stations serving the customers. λ is the arrival rate
(Poisson input) and µ is the service rate (exponential service). We will assume that all
customers are served on a first-come, first-served basis and wait in a single line (queue)
then are directed to whichever station is available. We also will assume that no customers
are lost from the queue. This situation, for instance, would be closely approximated by
customers at some banking operations.
The formulas for calculating some of the necessary probabilities are too complex for
simple keystroke solution. However, tables listing these probabilities are available and
can be used to aid in quick solutions. Using the assumptions outlined above and a suitable
table giving mean waiting time as a multiple of mean service (see page 512 of the
Reference) the following keystroke solutions may be obtained:
RPN Mode:
1. Key in the arrival rate of customers, λ, and press \.
2. Key in the service rate, µ, and press z to calculate ρ, the intensity factor. (Note ρ
must be less than n for valid results, otherwise the queue will lengthen without limit).
3. Key in n, the number of servers and press z to calculate ρ/n.
4. For a given n and ρ/n find the mean waiting time as a multiple of mean service time
from the table. Key it in and press \.
5. Calculate the average waiting time in the queue by keying in the service rate, µ, and
pressing ?1z?2.
6. Calculate the average waiting time in the system by pressing :1y+.
7. Key in λ and press :2§ to calculate the average queue length.
8. Key in ρ, the intensity factor (from step 2 above) and press + to calculate the
average number of customers in the system.
Miscellaneous
139
ALG Mode:
1. Key in the arrival rate of customers, λ, and press z.
2. Key in the service rate, µ, and press z to calculate ρ, the intensity factor. (Note ρ
must be less than n for valid results, otherwise the queue will lengthen without limit).
3. Key in n, the number of servers and press ³ to calculate ρ/n.
4. For a given n and ρ/n find the mean waiting time as a multiple of mean service time
from the table. Key it in and press z.
5. Calculate the average waiting time in the queue by keying in the service rate, µ, and
pressing ?1+?2.
6. Calculate the average waiting time in the system by pressing :1y³.
7. Key in λ and press §:2+ to calculate the average queue length.
8. Key in ρ, the intensity factor (from step 2 above) and press ³ to calculate the
average number of customers in the system.
Reference:
Richard E Trueman, "An Introduction to Quantitative Methods for Decision Making,"
Holt, Rinehart and Winston, New York, 1977
Example 1: Bank customers arrive at a bank on an average of 1.2 customers per minute.
They join a common queue for three tellers. Each teller completes a transaction at the rate
of one customer every 2 minutes (0.5 customers per minute). What is the average waiting
time in the queue? In the system? What is the average number of customers in the queue?
In the system?
12c platinum / 12C 12c platinum
Display
RPN Keystrokes ALG Keystrokes
1.20
1.2\
1.2z
2.40
.5z
.5z
0.80
3z
3³
Comments
ρ, intensity factor.
ρ/n
From Table 12.2, page 512 of the reference, the mean waiting time as a multiple of mean
service time for n = 3, ρ/n = 0.8 is 1.079. (Note S is used instead of n in the reference's
notation).
12c platinum / 12C 12c platinum
RPN Keystrokes ALG Keystrokes
1.079\
1.079z
.5?1z?2
.5?1+?2
:1y+
:1y³
1.2:2§
1.2§:2+
2.4+
2.4³
Display
1.08
2.16
4.16
2.59
4.99
Comments
Average wait in queue (min).
Average wait in system (min).
Average queue length.
Average # of customers in
system.
140
Miscellaneous
If the number of servers is limited to one, with other conditions remaining the same
(unlimited queue, Poisson arrival, exponential service), the average queue length can be
readily calculated without reference to tables:
RPN Mode:
1. Key in the arrival rate, λ and press ?1.
2. Key in the service rate, µ and press ?2
z\\2q~ 1~-z to calculate the average number of customers
waiting in queue at any one time.
3. Press :1z to calculate the average waiting time.
4. Press :2y+ to calculate the average total time the customer spends in the
system.
5. Press :1§ to calculate the average number of customers in the system.
ALG Mode:
1. Key in the arrival rate, λ and press ?1z.
2. Key in the service rate, µ and press ?2³y-g’~³y to calculate
the average number of customers waiting in queue at any one time.
3. Press z:1+ to calculate the average waiting time.
4. Press :2y§ to calculate the average total time the customer spends in the
system.
5. Press :1³ to calculate the average number of customers in the system.
Example 2: A small grocery store has but a single check-out counter. Customers arrive at
a rate of 1 every 2 minutes (λ = .5) and, on the average, customers can be checked out at a
rate of .9 per minute (µ). What is the average number of customers in the waiting line at
any time? The average waiting time? What is the average total time for a customer to wait
and be checked out? The average number of customers in the system?
12c platinum / 12C
RPN Keystrokes
.5?1
.9?2z\
\2q~
1~
-z
12c platinum
Display
ALG Keystrokes
0.50
.5?1z
.9?2
0.56
³
y-g’
0.69
~³y
:1z
:2y+
z:1+
:2y§
1.39
2.50
:1§
:1³
1.25
Comments
(intensity factor, ρ)
Average # customers waiting in
queue.
Average waiting time.
Average total time in the
system.
Average # customers in system.
Miscellaneous
141
With an HP 12C Platinum program one can readily calculate the necessary probabilities
for this type of problem (dispensing with the use of tables) and perform additional
calculations as well.
12c platinum / 12C
RPN KEYSTROKES
fs
fCLEARÎ
1
?-0
:.0
:0
0
go
g(009
g(016
+
q
gF
ge
z
_
g(001
:.0
:7
q
1
:.0
:7
z
z
:7
ge
z
?6
:2
+
y
?1
:6
§
?2
:.0
DISPLAY
000,
001,
1
002,44 33
0
003,45 48
0
004,
45
0
005,
0
006,
43 34
007,43,33,009
008,43,33,016
009,
40
010,
21
011,
43 40
012,
43
3
013,
10
014,
49
015,43,33,001
016,45 48
0
017,
45
7
018,
21
019,
1
020,45 48
0
021,
45
7
022,
10
023,
30
024,
10
025,
45
7
026,
43
3
027,
10
028,
44
6
029,
45
2
030,
40
031,
22
032,
44
1
033,
45
6
034,
20
035,
44
2
036,45 48
0
12c platinum
ALG KEYSTROKES
fs
fCLEARÎ
1
?-0
:0
0
go
g(008
g(017
:.0
q
:0
z
:0
ge
³
_
g(001
:.0
z
:7
1
³
Þ
:.0
q
:7
z
~
z
:7
ge
³
?6
+
:2
³
DISPLAY
000,
001,
1
002,44 30
0
003,
45
0
004,
0
005,
43 34
006,43,33,008
007,43,33,017
008,45 48
0
009,
21
010,
45
0
011,
10
012,
45
0
013,
43
3
014,
36
015,
49
016,43,33,001
017,45 48
0
018,
10
019,
45
7
020,
30
021,
1
022,
36
023,
16
024,45 48
0
025,
21
026,
45
7
027,
10
028,
34
029,
10
030,
45
7
031,
43
3
032,
36
033,
44
6
034,
40
035,
45
2
036,
36
142
Miscellaneous
12c platinum / 12C
RPN KEYSTROKES
§
:7
:.0
z
?3
:.0
+
?4
:8
z
?5
:3
:8
z
?6
t
:8
:7
:9
§
§
g>
:2
§
g(053
fs
DISPLAY
037,
20
038,
45
7
039,45 48
0
040,
30
041,
10
042,
44
3
043,45 48
0
044,
40
045,
44
4
046,
45
8
047,
10
048,
44
5
049,
45
3
050,
45
8
051,
10
052,
44
6
053,
31
054,
45
8
055,
45
7
056,
45
9
057,
20
058,
30
059,
20
060,
43 22
061,
45
2
062,
20
063,43,33,053
12c platinum
ALG KEYSTROKES
y
?1
?2
:6
?§2
:7
:.0
³
:.0
z
~
§
:2
+
?3
:.0
z
?4
:8
³
?5
:3
z
:8
³
?6
t
:7
§
:9
:8
§
~
³
Þ
g>
§
:2
³
g(064
fs
DISPLAY
037,
22
038,
44
1
039,
44
2
040,
45
6
041,44 20
2
042,
45
7
043,
30
044,45 48
0
045,
36
046,45 48
0
047,
10
048,
34
049,
20
050,
45
2
051,
40
052,
44
3
053,45 48
0
054,
10
055,
44
4
056,
45
8
057,
36
058,
44
5
059,
45
3
060,
10
061,
45
8
062,
36
063,
44
6
064,
31
065,
45
7
066,
20
067,
45
9
068,
30
069,
45
8
070,
20
071,
34
072,
36
073,
16
074,
43 22
075,
20
076,
45
2
077,
36
078,43,33,064
Miscellaneous
n: Unused
FV: Unused
R3: Lq
R7: n
R.1: Unused
i: Unused
R0: K
R4: L
R8: λ
REGISTERS
PV: Unused
R1: P0
R5: T
R9: µ
143
PMT: Unused
R2: Pb
R6: Used, Tq
R.0: ρ
Program Instructions:
1. Key in the program and press fCLEARH.
2. Key in the number of servers, n and press ?0?7.
RPN Mode:
3. Key in the arrival rate of customers, λ and press ?8.
4. Key in the service rate of each server, µ and press ?9.
5. Press z?.0 to calculate and store ρ the intensity factor.
ALG Mode:
3. Key in the arrival rate of customers, λ and press ?8z.
4. Key in the service rate of each server, µ and press ?9.
5. Press ³?.0 to calculate and store ρ the intensity factor.
6. Press t to see Tq, the average waiting time in the queue. Display P0, probability
that all servers are idle, by pressing :1. Display Pb, probability that all servers are
busy by pressing :2. Display Lq, average number waiting in the queue by pressing
:3. Display L, the average number in the system (waiting and being served), by
pressing :4.
Display T, average total time through the system, by pressing :5. Tq, the average
waiting time in the queue, may again be displayed by pressing :6.
7. If desired, calculate P(t), the probability of waiting longer than a given time, by keying
in the time and pressing t.
8. Repeat step 7 for other times of interest.
Example 3: Using the data from example 1 of the keystroke solutions verify the data
obtained. In addition, obtain P0, the probability that none of the tellers are busy, and Pb
the probability that all the tellers are busy. What is the probability that a customer will
have to wait 2 minutes or more?
144
Miscellaneous
12c platinum / 12C
RPN Keystrokes
fCLEARH
3?0?7
1.2?8
.5?9
z?.0
t
:1
:2
12c platinum
ALG Keystrokes
fCLEARH
3?0?7
1.2?8z
.5?9
³?.0
t
:1
:2
0.00
3.00
1.20
0.50
2.40
2.16
0.06
0.65
:3
:4
:5
2t
:3
:4
:5
2t
2.59
4.99
4.16
0.36
Display
Comments
n
λ
µ
ρ
Tq average waiting time in queue.
P0 probability all servers are idle.
Pb probability all servers are
busy.
Lq average # waiting in queue.
L, average # waiting in system.
T, average total time in system.
Probability of having to wait 2
minutes or more.
Appendix A
Algebraic Versions of Programs from
Part III of the Owner's Handbook
About this Appendix
This appendix contains algebraic versions of the eleven programs found in Part III of the
HP12C Platinum Owner's Handbook and Problem-Solving Guide. It does not cover the
step-by-step examples given in Part III.
These programs are solely for the HP 12C Platinum, and work properly only when the
calculator is set to ALG mode.
They should be tested by running the corresponding program examples given in the
HP12C Platinum Owner's Handbook and Problem-Solving Guide. They work exactly the
same as the RPN versions. The instructions for running these programs are not included
here. They are simply listed with reference to the relevant section in the HP12C Platinum
Owner's Handbook and Problem-Solving Guide. The register usage is the same as in the
RPN versions.
In order to execute miscellaneous side calculations which are done during the program
examples, these additional steps are required in algebraic mode:
1. The first three depreciation programs specify a side calculation to calculate the total
depreciation through the current year - the algebraic version of this is:
:$+:3-~-:M³.
2. Section 14 - the Advance Payments with Residual-Solving for Payment program
Example 1 stores 15/12 in R1 and Example 2 stores 18/12 in R1. The algebraic
keystrokes are: 15z12³?1 and 18z12³?1.
3. At the end of the bond programs the total price is obtained by pressing: +~³.
4. Section 16 - the 30/360 Day Basis Bonds program Example 2 stores 93 ⅜ in R1. The
algebraic keystrokes are: 3z8+93³?1.
The program listings have been formatted in double columns and apart from one instance
the listings do not span pages, thus making key entry convenient.
145
146
Appendix A
Algebraic Mode Programs
Section 12: The Rent or Buy Decision
ALG KEYSTROKES
fs
fCLEARÎ
M
M
:n
?0
:2
gA
:3
gC
:$
:1
$
:M
0
M
P
d
:0
gA
f!
~
d
z
~
:4
§
:.0
b
³
:P
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
026,
027,
028,45
029,
030,
031,
45
44
45
43
45
43
45
45
45
45
43
42
45
48
45
15
15
11
0
2
11
3
12
13
30
1
13
15
0
15
14
33
0
11
11
34
33
10
34
30
4
20
0
25
36
14
ALG KEYSTROKES
:4
:5
+
:8
~
P
d
Þ
:7
b
:$
M
t
:0
gA
:1
Þ
:6
$
¼
:gC
t
:9
gC
M
M
fs
DISPLAY
032,
033,
45
034,
035,
45
036,
037,
45
038,
039,
040,
041,
042,
043,
044,
45
045,
046,
047,
45
048,
049,
050,
45
051,
43
052,
45
053,
054,
055,
45
056,
057,
058,45,43
059,
060,
45
061,
43
062,
063,
30
4
30
5
40
8
30
34
14
33
16
30
7
25
30
13
15
31
0
11
1
16
30
6
13
12
12
31
9
12
15
15
Appendix A
147
Section 13: Straight-Line Depreciation
ALG KEYSTROKES
fs
fCLEARÎ
z
1
2
³
?1
~
?2
1
³
?0
1
fV
§
:1
³
?3
:$
~
$
:n
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
44
44
44
42
45
44
45
45
10
1
2
36
1
34
2
30
1
36
0
1
23
20
1
36
3
13
30
34
13
11
ALG KEYSTROKES
:1
n
:0
gm
g(038
:2
gu
:0
fV
t
1
?+0
?+2
g(029
:2
gu
:$
:M
³
:3
g(033
fs
DISPLAY
023,
30
024,
45
1
025,
11
026,
45
0
027,
43 35
028,43,33,038
029,
45
2
030,
43 31
031,
45
0
032,
42 23
033,
31
034,
1
035,44 40
0
036,44 40
2
037,43,33,029
038,
45
2
039,
43 31
040,
45 13
041,
30
042,
45 15
043,
36
044,
45
3
045,43,33,033
148
Appendix A
Section 13: Declining-Balance Depreciation
ALG KEYSTROKES
fs
fCLEARÎ
z
1
2
³
?1
~
?2
1
³
?0
1
f#
§
:1
³
?3
:$
~
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
44
44
44
42
45
44
45
10
1
2
36
1
34
2
30
1
36
0
1
25
20
1
36
3
13
30
34
ALG KEYSTROKES
$
:0
gm
g(034
:2
gu
:0
f#
t
1
?+0
?+2
g(025
:2
gu
:$
:M
³
:3
g(029
fs
DISPLAY
021,
13
022,
45
0
023,
43 35
024,43,33,034
025,
45
2
026,
43 31
027,
45
0
028,
42 25
029,
31
030,
1
031,44 40
0
032,44 40
2
033,43,33,025
034,
45
2
035,
43 31
036,
45 13
037,
30
038,
45 15
039,
36
040,
45
3
041,43,33,029
Appendix A
149
Section 13: Sum-of-the-Years-Digits Depreciation
ALG KEYSTROKES
fs
fCLEARÎ
z
1
2
³
?1
~
?2
1
³
?0
1
fÝ
§
:1
³
?3
:$
~
$
:n
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
44
44
44
42
45
44
45
45
10
1
2
36
1
34
2
30
1
36
0
1
24
20
1
36
3
13
30
34
13
11
ALG KEYSTROKES
:1
n
:0
gm
g(038
:2
gu
:0
fÝ
t
1
?+0
?+2
g(029
:2
gu
:$
:M
³
:3
g(033
fs
DISPLAY
023,
30
024,
45
1
025,
11
026,
45
0
027,
43 35
028,43,33,038
029,
45
2
030,
43 31
031,
45
0
032,
42 24
033,
31
034,
1
035,44 40
0
036,44 40
2
037,43,33,029
038,
45
2
039,
43 31
040,
45 13
041,
30
042,
45 15
043,
36
044,
45
3
045,43,33,033
150
Appendix A
Section 13: Full- and Partial- Year Depreciation with Crossover
ALG KEYSTROKES
fs
fCLEARÎ
z
1
2
³
?6
:n
~
³
?4
d
?0
1
?-0
?2
?3
f#
§
:6
³
?1
:$
~
$
:1
:$
:M
³
~
:0
1
go
g(042
d
d
1
gu
DISPLAY
000,
001,
10
002,
1
003,
2
004,
36
005,
44
6
006,
45 11
007,
30
008,
34
009,
36
010,
44
4
011,
33
012,
44
0
013,
1
014,44 30
0
015,
44
2
016,
44
3
017,
42 25
018,
20
019,
45
6
020,
36
021,
44
1
022,
45 13
023,
30
024,
34
025,
13
026,
45
1
027,
45 13
028,
30
029,
45 15
030,
36
031,
34
032,
45
0
033,
1
034,
43 34
035,43,33,042
036,
33
037,
33
038,
1
039,
43 31
ALG KEYSTROKES
d
t
1
?+2
?-0
f#
?+1
?5
:$
:M
z
:4
³
go
g(057
g(069
d
0
:0
go
g(090
:$
:5
$
1
?-4
g(043
:4
n
0
?6
1
?-2
?+0
:5
?-1
:3
fV
?+1
DISPLAY
040,
33
041,
31
042,
1
043,44 40
2
044,44 30
0
045,
42 25
046,44 40
1
047,
44
5
048,
45 13
049,
30
050,
45 15
051,
10
052,
45
4
053,
36
054,
43 34
055,43,33,057
056,43,33,069
057,
33
058,
0
059,
45
0
060,
43 34
061,43,33,090
062,
45 13
063,
30
064,
45
5
065,
13
066,
1
067,44 30
4
068,43,33,043
069,
45
4
070,
11
071,
0
072,
44
6
073,
1
074,44 30
2
075,44 40
0
076,
45
5
077,44 30
1
078,
45
3
079,
42 23
080,44 40
1
Appendix A
151
Full- and partial-Year Depreciation with Crossover (continued)
ALG KEYSTROKES
1
?-0
?+2
?+3
d
:0
1
go
g(078
d
DISPLAY
081,
1
082,44 30
0
083,44 40
2
084,44 40
3
085,
33
086,
45
0
087,
1
088,
43 34
089,43,33,078
090,
33
ALG KEYSTROKES
d
:2
gu
d
t
:6
gm
g(078
g(062
fs
DISPLAY
091,
33
092,
45
2
093,
43 31
094,
33
095,
31
096,
45
6
097,
43 35
098,43,33,078
099,43,33,062
Section 14: Lease with Advance Payments - Solving For Payment
ALG KEYSTROKES
fs
fCLEARÎ
gÂ
fCLEARG
:0
:1
n
:2
¼
1
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
43
42
45
45
45
8
34
0
30
1
11
2
12
1
ALG KEYSTROKES
Þ
P
$
+
:1
z
:3
~
³
fs
DISPLAY
010,
011,
012,
013,
014,
45
015,
016,
45
017,
018,
16
14
13
40
1
10
3
34
36
Section 14: Lease with Advance Payments - Solving For Yield
ALG KEYSTROKES
fs
fCLEARÎ
gÂ
fCLEARG
:0
:1
n
:2
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
43
42
45
45
45
8
34
0
30
1
11
2
ALG KEYSTROKES
P
§
:1
:3
$
¼
:gC
fs
DISPLAY
008,
009,
010,
45
011,
012,
45
013,
014,
015,45,43
14
20
1
30
3
13
12
12
152
Appendix A
Section 14: Advance Payments With Residual - Solving for Payment
ALG KEYSTROKES
fs
fCLEARÎ
gÂ
fCLEARG
:0
n
:1
¼
:3
M
$
+
:2
³
?5
0
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
43
42
45
45
45
45
44
8
34
0
11
1
12
3
15
13
40
2
36
5
0
ALG KEYSTROKES
M
:n
:4
n
1
Þ
P
$
+
:4
z
:5
~
³
fs
DISPLAY
015,
016,
45
017,
018,
45
019,
020,
021,
022,
023,
024,
025,
45
026,
027,
45
028,
029,
15
11
30
4
11
1
16
14
13
40
4
10
5
34
36
Section 15: Nominal Rate Converted to Effective Rate
ALG KEYSTROKES
fs
fCLEARÎ
fCLEARG
n
~
z
~
¼
DISPLAY
000,
001,
002,
003,
004,
005,
006,
42
34
11
34
10
34
12
ALG KEYSTROKES
1
$
M
1
n
¼
fs
DISPLAY
007,
008,
009,
010,
011,
012,
1
13
15
1
11
12
Appendix A
153
Section 16: 30/360 Day Basis Bonds
ALG KEYSTROKES
fs
fCLEARÎ
fCLEARG
g×
:2
z
2
P
?6
+
:5
M
:3
:4
gÒ
d
z
1
8
0
n
gT
1
~
§
:6
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
020,
021,
022,
023,
024,
025,
42
43
45
44
45
45
45
43
43
45
34
7
2
10
2
14
6
40
5
15
3
4
26
33
10
1
8
0
11
24
1
30
34
20
6
ALG KEYSTROKES
DISPLAY
026,
36
³
027,
44
6
?6
028,
45
6
:6
029,
45
0
:0
030,
43 35
gm
031,43,33,041
g(041
032,
10
z
033,
2
2
034,
12
¼
035,
13
$
036,
16
Þ
037,
30
038,
34
~
039,
36
³
040,43,33,000
g(000
041,
33
d
042,
40
+
043,
45
1
:1
044,
36
³
045,
16
Þ
046,
13
$
047,
12
¼
048,
20
§
049,
2
2
050,
36
³
fs
154
Appendix A
Section 16: Annual Coupon Bonds
ALG KEYSTROKES
fs
fCLEARÎ
fCLEARG
gÂ
:0
n
:2
P
:1
¼
:3
M
$
:5
Æ
6
Þ
³
?6
:5
DISPLAY
000,
001,
002,
003,
004,
005,
006,
007,
008,
009,
010,
011,
012,
013,
014,
015,
016,
017,
018,
019,
42
43
45
45
45
45
45
44
45
34
8
0
11
2
14
1
12
3
15
13
5
30
26
6
16
36
6
5
ALG KEYSTROKES
gÒ
?7
:6
:4
gÒ
z
:7
n
0
P
M
:n
§
:2
Þ
+
t
~
³
Þ
fs
DISPLAY
020,
43
021,
44
022,
45
023,
45
024,
43
025,
026,
45
027,
028,
029,
030,
031,
45
032,
033,
45
034,
035,
036,
037,
038,
039,
26
7
6
4
26
10
7
11
0
14
15
11
20
2
16
40
31
34
36
16
Appendix B
Formulas Used
Real Estate
Wrap-Around Mortgage
n1 = number of years remaining in original mortgage.
PMT1 = yearly payment of original mortgage.
PV1 = remaining balance of original mortgage.
n2 = number of years in wrap-around mortgage.
PMT2 = yearly payment of wrap-around mortgage.
PV2 = total amount of wrap-around mortgage.
r = interest rate of wrap-around mortgage as a decimal.
FV = balloon payment.
PMT2 1 − (1 + r )− n2 PMT1 1 − (1 + r )− n1
−
PV2 − PV1 =
+ FV (1 + r )− n2
r
r
After-Tax Cash Flows
ATCFk = After-Tax Cash Flow for kth year.
Intk = interest for kth year.
Depk = depreciation for kth year.
r = appropriate tax rate.
NOI = Net Operating Income.
ATCFk = NOI (1 − r) − 12 x PMT + r x (Intk + Depk).
After-Tax Net Cash Proceeds of Resale
CO = capital purchase.
CPR = sales price − closing costs.
r = marginal tax rate.
c= capital gains tax rate.
NCPR = CPR − remaining balance of mortgage.
ATNCPR = NCPR − r x (Total Dep. − SL Dep.) − c x (CPR − CO + SL Dep.)
= NCPR − [c x (CPR − CO) + r x (Total Dep.) +(c − r) x (SL Dep.) ]
155
156
Appendix B
Lending
Loans With a Constant Amount Paid Towards Principal
BALk = remaining balance after time period k.
CPMT = Constant payment to principal.
BALk = PV − (k x CPMT)
kth payment to interest = i (BALk) = (PMTi)k
kth total payment = CPMT + (PMTi)k
Add-On Interest Rate to APR
r = add-on rate as a decimal.
n = number of monthly payments.
APR = 1200i, where i is the solution in the following equation:
n
1 − (1 + i )− n
=
n
i
1+ r
12
Add-On to APR with Credit Life
CL = credit life as decimal.
AMT = loan amount.
FC = finance charge.
n
1 + r
× AMT = G
12
2
n
n
1 − × CL − × CL × r
12
12
G
= PMT
n
G × CL × n
= amount of credit life
12
FC = (G − AMT − CL)
Rule of 78's Rebate
PV = finance charge.
Ik = interest charged at month k.
n = number of months in loan.
Ik =
2(n − k + 1)
PV
n(n + 1)
Appendix B
Rebate =
(n − k )I k
2
BALk = (n − k) × PMT − Rebatek
Graduated Payment Mortgage
n = total number of payments in the loan
I = interest rate per payment period, as a decimal
A = number of payments per year
B = number of years that payments increase
C = percentage increase in periodic payments (as a decimal)
PMT1 = amount of the first payment
PV = amount of the loan.
1 − (1 + I )−( n − AB )
(
1 + C )B
−A
B
I
1 − (1 + I ) (1 + Q ) − 1
PV = PMT1
+
AB
I
Q
(1 + I )
where:
Q=
1+ C
(1 + I )A
−1
Skipped Payments
A = number of payments per year.
B = number of years.
C = annual percentage rate as decimal.
D = periodic payment amount.
E = loan amount.
K = number of last payment before payments close the first time.
L = number of skipped payment.
DEND =
DBEGIN
C A C
1 + − 1
A
A
E
×
C − AB C A C A− K C A− L − K
1 − 1 + 1 + − 1 +
− 1
+ 1 +
A
A
A
A
D
= END
C
1+
A
157
158
Appendix B
Savings
Compounding Periods Different From Payment Periods
C = number of compounding periods per year.
P = number of payment periods per year.
i = periodic interest rate, expressed as a percentage.
r = i / 100, periodic interest rate expressed as a decimal.
iPMT = ((1 + r / C)C/P − 1)100
Investment Analysis
Lease vs. Purchase
PMTp = loan payment for purchase.
PMTL = lease payment.
In = interest portion of PMTp for period n.
Dn = depreciation for period n.
Mn = maintenance for period n.
T = marginal tax rate.
Net purchase advantage =
k
∑
cost of leasing(n) − cost of owning (n)
n =1
(1 + i )n
Cost of leasing(n) = (1 − T) PMTL
Cost of owning(n) = PMTp − T(In + Dn) + (1 − T)Mn
Break-Even Analysis and Operating Leverage
GP = Gross Profit.
P = Price per unit.
V = Variable costs per unit.
F = Fixed costs.
U = number of Units.
OL = Operating Leverage.
GP = U(P − V) − F
OL =
U (P − V )
U (P − V ) − F
Profit and Loss Analysis
Net income = (1 − tax) (net sales price − manufacturing expense − operating expense)
Net sales price = list price(1 − discount rate)
where operating expense represents a percentage of net sales price.
Appendix B
159
Securities and Options
Discounted Notes
Price (given discount rate)
B = number of days in year (annual basis).
DR = discount rate (as a decimal).
DSM = number of days from settlement date to maturity date.
P = dollar price per $100 per value.
RV = redemption value per $100 par value.
DSM
P = [RV ] − DR × RV ×
B
Yield (given price)
B = number of days in year (annual basis).
DSM = number of days from settlement date to maturity date.
P = dollar price per $100 par value.
RV = redemption value per $100 par value.
Y = annual yield of investment with security held to maturity (as a decimal).
RV − P B
Y =
×
P DSM
Black-Scholes Formula for Valuing European Options
P = current asset price.
r%= risk-free rate (continuous, per time unit).
s% = volatility (continuous, per time unit).
T = term of option (same time unit as r% and s%).
X = exercise price of option.
N(z)= probability that a unit normal random variable is less than z.
Call Value = P × N (d1 ) − Q × N (d 2 )
Put Value = Call Value + Q − P
where:
d1 = LN (P / Q ) / v + v / 2 , d 2 = d1 − v
Q = Xe (−T × r % / 100 ) , v = s % / 100 × T
160
Appendix B
Forecasting
Simple Moving Average
x = moving average.
m = number of elements in moving average.
x1 =
x1 + x2 + x3 + ... + xm
m
x2 =
x2 + x3 + x4 + ... + xm −1
m
etc.
Seasonal Variation Factors Based on a Centered Moving Average
xc = centered moving average
m = number of elements in the centered moving average.
x
x1
+ (x2 + x3 + ... + xm ) + m +1
2
2
xc =
m
SV = Seasonal variation factor.
xi = value of the ith data point.
xi = centered moving average of the ith data point.
SV =
xi
xi
Gompertz Curve Trend Analysis
( x)
y = ca b
where x, y, a, b, and c are positive.
1/ n
S − S2
b = 3
S 2 − S1
1 S S − S 2
2
c = exp 1 3
n S1 + S3 − 2 S 2
(b − 1)(S − S )
2
1
a = exp
b bn − 1 2
(
)
where S1, S2, and S3 are:
Appendix B
n
S1 = ∑ ln yi = n ln c + b(ln a )
i =1
S2 =
S3 =
161
bn −1
b −1
2n
∑ ln yi = n ln c + b n+1 (ln a )
i =n +1
bn −1
b −1
3n
∑ ln yi = n ln c + b 2n+1 (ln a )
i =2 n +1
bn −1
b −1
a, b and c are determined by solving the three equations above simultaneously.
Forecasting With Exponential Smoothing
α = smoothing constant (0 < α < 1)
Xt = actual current period usage
Smoothed average, S t = αX t + (1 − α ) S t −1
Change, Ct = S t - St - 1
Trend, Tt = αΧ t + (1 − α )Τ t − 1
Current period expected usage, Dt = S t +
(1− α ) T
α
t
1
Forecast of next period expected usage, Dˆ t +1 = S t + Tt
α
Error, et = Dˆ t − X t
Cumulative error =
m
∑ et 2
t =1
Initial conditions : St −1 = X t −1 and Tt −1 = 0
Pricing Calculations
Markup and Margin Calculations
Ma = margin(%).
Mu = markup(%).
S = selling price.
C = cost.
Ma = 100
S −C
S
Mu = 100
S −C
C
162
Appendix B
S=
C
Ma
1−
100
Mu
S = C 1 +
100
Ma
C = S 1 −
100
S
C=
Mu
1+
100
Mu
Ma =
Mu
1+
100
Ma
Mu =
Ma
1−
100
Calculations of List and Net Prices with Discounts
L = List price.
N = Net price.
D = Discount(%).
D′ = 1 −
L=
D
100
N
D1′ × D2′ × ... × D ′x
N
D x = 1001 −
′
′
′
(
)
×
×
×
L
D
D
...
D
1
2
x −1
Statistics
Exponential Curve Fit
y = Ae Bx
ln y = ln A + Bx
1
B=
∑ xi ln yi − n (∑ xi )(∑ ln yi )
1
∑ xi 2 − n (∑ xi )
2
Appendix B
∑ ln yi
∑ xi
A = exp
−B
n
n
yˆ = Ae Bx
Logarithmic Curve Fit
y = A + B(ln x )
1
B=
A=
∑ yi ln xi − n (∑ yi )(∑ ln xi )
1
∑ (ln xi )2 − n (∑ ln xi )
2
1
(∑ yi − B∑ ln xi )
n
yˆ = A + B(ln x )
Power Curve Fit
y = AxB
(A>0)
ln y = ln A + B(ln x )
1
B=
∑ (ln xi )(ln yi ) − n (∑ ln xi )(∑ ln y i )
1
∑ (ln xi )2 − n (∑ ln xi )
2
∑ ln yi
∑ ln xi
A = exp
−B
n
n
yˆ = Ax B
Standard Error of the Mean
Sx =
Sx
Sy =
n
Sy
n
Mean, Standard Deviation, Standard Error for Grouped Data
mean, x =
∑ f i xi
∑ fi
standard deviation, S x =
standard error, S x =
Sx
∑ fi
∑ f i xi 2 − (∑ f i ) x 2
∑ fi − 1
163
164
Appendix B
Personal Finance
Tax-Free Retirement Account (IRA) or Keogh Plan
n = the number of years to retirement.
i = the compounded annual interest.
PMT = the total annual investment.
FV= future value, after applicable taxes.
r = the assumed tax rate on interest expressed as a decimal.
w = the withdrawal tax rate expressed as a decimal.
For ordinary taxable investment:
FV =
{
For tax-free investment:
FV =
}
PMT
[1 + i(1 − r )] [1 + i(1 − r )]n − 1
i (1 − r )
[
]
PMT × (1 − w)
(1 + i ) (1 + i )n − 1
i
Stock Portfolio Evaluation and Analysis
n = the number of issues held.
Pi = the current market price / share of a stock.
Si = the number of shares of a stock held.
βi = the beta coefficient of an individual stock.
T = the total present value of a portfolio.
Portfolio beta coefficient:
n
β=∑
i =1
Pi Si βi
T
Canadian Mortgages
r = annual interest rate expressed as a decimal.
r 1 / 6
monthly factor = 1 + − 1 × 100
2
Appendix B
165
Miscellaneous
Learning Curve for Manufacturing Cost
Cn = Cost of the nth unit, C1 = Cost of the first unit.
n = number of units, r = learning factor.
k = ln r / ln 2, C n = C1n k
Cij = the average cost of the ith through jth unit.
Cij =
C1 j k +1 − i k +1
j − i k + 1
This formula is only approximate and may give appreciable error at small i.
Queuing and Waiting Theory
n = number of servers.
λ = arrival rate of customers (Poisson input).
µ =service rate for each server (exponential service).
ρ = Intensity factor = λ / µ ( ρ < n for valid results).
P0 = Probability that all servers are idle.
Pb = Probability that all servers are busy.
Lq = Average number of customers in queue.
L = Average number of customers in the system (waiting and being served).
Tq = Average waiting time in queue.
T = Average total time through the system.
P(t) = Probability of waiting longer than time t.
n −1 k
n
ρ
ρ
+
P0 = ∑
k = 0 k!
ρ
−
n
!
1
n
Pb =
Lq =
−1
ρ n P0
ρ
n! 1 −
n
Lq
ρ Pb
, L = Lq + ρ , T = L / λ , Tq =
n−ρ
λ
P(t ) = Pb e −(nµ − λ ) t
Subject Index
Page numbers in bold type indicate primary references: page numbers in regular type
indicate secondary references.
A
About This Handbook
Add-On Interest Rate Converted to APR
Add-On Rate Loan with Credit Life
After-Tax Cash Flows
After-Tax Net Cash Proceeds of Resale
After-Tax Yield
Algebraic Versions of Programs from Part III of the Owner's Handbook
APR Converted to Add-On Interest Rate
B
2
25 , 156
27 , 156
14 , 155
19 , 155
65
145
26
Balance Remaining at End of Specified Period
Before-Tax Cash Flows
Before-Tax Reversions (Resale Proceeds)
Black-Scholes Formula for Valuing European Options
Bonds: 30/360 Day Basis Bonds
Bonds: Annual Coupon Bonds
Break-Even Analysis
C
132
12
13
69 , 159
153
154
53 , 158
Calculations of List and Net prices With Discounts
Canadian Mortgages
Chi-Square Statistics
Combinations
Compounding Periods Different From Payment Periods
Covariance
Curve Fitting
D
95 , 162
131 , 164
109
117
46 , 158
114
98
Depreciation: Declining-Balance Depreciation
Depreciation: Full- and Partial- Year Depreciation with Crossover
Depreciation: Straight-Line Depreciation
Depreciation: Sum-of-the-Years-Digits Depreciation
Discounted Notes
148
150
147
149
67 , 159
166
Subject Index
167
E
Effective Interest Rate (Yield)
Exponential Curve Fit
F
132
98 , 162
Forecasting
Forecasting with Exponential Smoothing
Formulas Used
G
74 , 160
87 , 161
155
Gompertz Curve Trend Analysis
Graduated Payment Mortgage
H
83 , 160
157 , 32
Homeowners Monthly Payment Estimator
I
121
Income Property Cash Flow Analysis
Initial Deposit with Periodic Deposits
Interest Rebate - Rule of 78's
Introduction
Investment Analysis
L
12
40
30 , 156
2
49 , 158
Learning Curve for Manufacturing Cost
Lease vs. Purchase
Leasing: Advance Payments With Residual - Solving for Payment
Leasing: Lease with Advance Payments - Solving For Payment
Leasing: Lease with Advance Payments - Solving For Yield
Lending
Loan With a Constant Amount Paid Towards Principal
Logarithmic Curve Fit
M
134 , 165
49 , 158
152
151
151
24 , 156
24 , 156
102 , 163
Markup and Margin Calculations
Mean, Standard Deviation, Standard Error for Grouped Data
Miscellaneous
N
92 , 161
106 , 163
134 , 165
Normal Distribution
112
Number of Periodic Payments to Fully Amortize a Mortgage
132
Number of Periods to Deplete a Savings Account or to Reach a Specified Balance 41
O
Operating Leverage
59 , 158
168
Subject Index
P
Periodic Deposits and Withdrawals
Periodic Payment Amount
Permutations
Personal Finance
Portfolio beta coefficient
Power Curve Fit
Presentation of Algebraic and RPN
Pricing Calculations
Profit and Loss Analysis
Q
42
131
116
121 , 164
164
104 , 163
2
92 , 161
61 , 158
Queuing and Waiting Theory
R
138 , 165
Random Number Generator
Real Estate
Real Estate: The Rent or Buy Decision
Refinancing
Rule of 78's Rebate
S
119
7 , 155
146
7
156
Savings
Savings Account Compounded Daily
Savings: Nominal Rate Converted to Effective Rate
Seasonal Variation Factors Based on a Centered Moving Average
Securities and Options
Simple Moving Average
Skipped Payments
Standard Error of the Mean
Statistics
Stock Portfolio Evaluation and Analysis
T-W
40 , 158
44 , 160
152
160
65 , 159
74 , 160
38 , 157
105 , 163
98 , 162
127 , 164
Tax-Free Individual Retirement (IRA) or Keogh Plan
Using the RPN Programs on the HP-12C
Variable Rate Mortgages
Wrap-Around Mortgage
124 , 164
3
36
8 , 155
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