Introduction This Solutions Handbook has been designed to supplement the HP-12C 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.
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.
9.5 200000 0 133190 1,979.56 Monthly payment on new mortgage. 899.23 Net monthly payment (to lender). -66,810.00 Net amount of cash advanced (by lender). -80,425.02 Present value of net -13,615.02 NPV to lender of net cash advanced 14.83 % nominal yield (IRR). -65,376.72 Present value of net monthly payment at 15.25%. 1,433.28 NPV to borrower. 0 11.5 0 0 12 15.
CLEAR 20 8 6.75 0.56 Monthly interest rate (into i). 200132.06 200,132.06 Loan amount (into PV). -2,031.55 Monthly payment on existing mortgage (calculated). 10 0.83 Monthly interest on wrap-around. 300000 -300,000.00 Amount of wrap-around (into PV). 0 Monthly payment on wrap-around (calculated). Net monthly payment received (into PMT). 3,585.23 0 1,553.69 -99,867.94 Net cash advanced (into PV). 15.85 Nominal yield (IRR) to lender (calculated). 200132.
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 1/2% 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 $129963.35 240 mos. $1681.71 $1681.71 $-1051.61 $-1051.61 $1681.71 200mos. $-200000 Keystrokes CLEAR Display -74,990.
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 and press CLEAR 2. Key in the remaining periods of the original mortgage and press 3. Key in the desired annual yield and press 4.
12 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.
Keystrokes CLEAR Display 180,000.00 Potential Gross Income. 9,000.00 Vacancy Loss. 171,000.00 Effective Gross Income. 94,145.00 Net Operating Income. -89,580.09 Annual Debt Service. 4,564.91 Cash Throw-Off. 60 250 12 5 76855 20 11.5 700000 12 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.
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? Keystrokes Display CLEAR 240.00 Mortgage term. 0.96 Mortgage rate. 20 11.5 Property value. 700000 10 -7,465.01 Monthly payment. 120.00 Projection period. -530,956.57 Mortgage balance in 10 years. Estimated resale. 800000 7 56,000.
KEYSTROKES CLEAR 0 DISPLAY 0001- 0 02- 11 1 03- 44 1 7 04- 45 7 2 05- 26 06- 2 07- 10 08- 7 1 44 091 1 2 10-44 7 1 40 1112- 1 1 2 13- 42 11 0 14- 44 0 5 15- 45 5 16- 6 6 11 17- 45 12 18- 45 6 19- 12 20- 33 21- 44 22- 4 6 33 23- 45 13 24- 45 4 25- 13 26- 33 10
27- 4 44 282936 1 0 0 33 43 35 30-43, 33 36 31- 45 1 32- 42 25 33-44 30 0 3417 4 0 35-43, 33 17 36- 11 2 37- 45 2 8 38- 45 8 392 40-44 25 40 410 42-45 2 33 48 0 43- 25 44- 30 3 45- 45 3 9 46- 45 9 473 1 7 0 48-44 25 40 3 49- 33 50- 30 51- 1 52- 45 7 53-44 20 0 54- 11 30
55- 20 561 2 45 5758- 1 2 59- 20 60- 40 61- 0 14 45 0 621 09 30 63- 45 1 64- 43 31 65- 34 66- 31 67-43, 33 09 REGISTERS i: Annual % PMT: Monthly R0: Used R2: PGI R4: Dep. value R6: Factor (DB) R8: % gr. (PGI) R.0: Vacancy rt. n: Used PV: Used FV: 0 R1: Counter R3: Oper. cost R5: Dep. life R7: Tax Rate R9: % gr. (op) 1. Press and press 2. Key in loan values: CLEAR .
5. Key in depreciable value and press 6. Key in depreciable life and press 7. Key in factor (for declining balance only) and press 8. Key in the Marginal Tax Rate (as a percentage) and press 9. Key in the growth rate in Potential Gross Income ( 0 for no growth) and press 4. 5. 6. 7. 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.
125 6 35 7 6 8 2.5 9 5 .0 125.00 Decline in balance factor. 35.00 Marginal Tax Rate. 6.00 Potential Gross Income growth rate. 2.50 Operating cost growth. 5.00 Vacancy rate. 1.00 -1,020.88 2.00 -822.59 3.00 -598.85 4.00 -72.16 5.00 232.35 6.00 565.48 7.00 928.23 8.00 1,321.62 9.00 1,746.81 10.00 -1,020.
25 175200 2 40296 3 1200000 4 40,296.00 1st year operating cost. 1,200,000.00 Depreciable value. 35 5 35.00 Depreciable life. 50 7 50.00 Marginal tax rate. 3.5 8 3.50 Potential Gross Income 1.6 9 1.60 Operating cost growth rate. 0 7.00 Vacancy rate. 31 7.00 Go to dep. step. 3242 23 1.00 18,021.07 2.00 20,014.26 3.00 22,048.90 4.00 24,123.14 5.00 26,234.69 Change to SL.
The user may change to a different depreciation method by keying in the desired function at line 35 in place of .
252 2 6 1 2 12 26- 45 2 27- 42 23 28- 45 2 29- 20 30- 48 31- 6 32- 20 33-44 40 1 34- 45 2 35- 42 25 3637- 34 45 381 6 2 30 39-44 40 1 40- 45 6 41- 26 42- 2 43- 10 44- 1 45 4546- 0 n: Used PV: Used FV: Used R1: Used 48-43 45 0 40 33 REGISTERS i: Used PMT: Used R0: Used R2: Desired yr.
R3: Dep. value R5: Factor R7-R.3: Unused R4: Dep. life R6: MTR 1. Key in the program and press 2. Key in the loan values: CLEAR . • Key in annual interest rate and press • Key in mortgage amount and press • Key in monthly payment and press . . . (If any of the values are unknown, they should be solved for.) 3. Key in depreciable value and press 4. Key in depreciable life in years and press 5. Key in accelerated depreciation factor for the declining balance method and press 3. 4. 5. 6.
700000 700,000.00 Mortgage. 9.5 0.79 Monthly interest. 20 240.00 Number of payments. -6,524.92 Monthly payment. 750,000.00 Depreciable value. 25.00 Depreciable life. 125.00 Factor. 48.00 Marginal Tax Rate. 900000 900,000.00 Purchase price. 1750000 1,750,000.00 Sale price. 8 8.00 Commission rate. 10 911,372.04 ATNCPR.
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. Therefor each periodic payment has a constant amount applied toward the principle and a varying amount of interest.
0 0 0 4 0 0 0 0 0 55,000.00 Remaining balance. 2,750.00 Second payment's interest. 7,750.00 Total second payment. 50,000.00 Remaining balance after the first year. 1,500.00 Seventh payment's interest. 6,500.00 Total seventh payment. 25,000.00 Remaining balance. 1,250.00 Eighth payment's interest. 6,250.00 Total eighth payment. 20,000.00 Remaining balance after fourth year.
Example 1: Calculate the APR and monthly payment of a 12% $1000 add-on loan which has a life of 18 months. Keystrokes Display CLEAR 18 1,180.00 Amount of loan. -65.56 Monthly payment. 21.64 Annual Percentage Rate. 12 1000 12 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 2. Enter the following information: 3. and press CLEAR . a.
Add-On Rate Loan with Credit Life. This HP-12C 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.
1 3 22- 1 23- 40 24- 34 25- 10 26- 45 270 28- 20 45 2930- 0 2 1 2 5 2 0 10 42 14 31- 16 32- 14 33- 31 34- 45 14 35- 45 0 36- 20 37- 16 38- 13 39- 45 13 40- 45 2 410 3 42- 25 45 0 43- 20 4445- 1 2 46- 10 47- 44 5 48- 26 49- 2 50- 20 24
61 51- 43 35 52- 43 35 53-43, 33 61 54- 5 0 1 45 5 55- 48 5657- 0 1 58- 40 59- 42 14 5 60- 44 5 5 61- 45 5 6263- 45 13 64- 34 65- 30 66- 3 31 45 3 67- 30 68- 16 69- 31 5 70- 45 5 3 71- 45 3 72- 40 73- 13 74- 0 00 45 0 75- 11 76- 12 77-45, 43 12 78-43, 33 00 25
REGISTERS i: i PMT: PMT R0: N R2: CL (%) R4: N/1200 R6-R9: Unused n: N PV: Used FV: 0 R1: AIR R3: Loan R5: Used 1. Key in the program. 2. Press 3. Key in the number of monthly payments in the loan and press 4. Key in the annual add-on interest rate as a percentage and press 5. Key in the credit life as a percentage and press 6. Key in the loan amount and press 7. Press to find the monthly payment amount. 8. Press to obtain the amount of credit life. 9.
-651.10 Total finance charge. 12.39 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: 1. Key in number of months in the loan and press 2.
01- 0 44 0203- 2 0 33 44 04- 2 33 1 05- 44 1 2 06- 45 2 0708- 2 1 44 1 10- 40 45 1213- 1 00 n: Unused 45 1 15- 20 45 1 17- 40 18- 10 45 2 20- 20 21- 31 22- 2 20 36 19- 2 0 14- 16- 1 2 09- 11- 0 30 45 2 23- 20 24- 34 25- 30 26-43, 33 00 REGISTERS i: Unused 28
PV: Unused FV: Unused R1: Payment# R3-R.6: Unused PMT: Unused R0: Fin. charge R2: # moths 1. Key in the program. 2. Key in the number of months in the loan and press 3. Key in the payment number when prepayment occurs and press . . 4. Key in the total finance charge and press interest (rebate). 5. Key in the periodic payment amount and press principal outstanding. 6. For a new case return to step 2. Keystrokes to obtain the unearned to find the amount of Display 30 25 5.81 Rebate. 190.
KEYSTROKES CLEAR 2 1 1 0 2 DISPLAY 0001- 43 8 02- 44 2 03- 34 04- 1 05- 25 06- 1 07- 40 08- 44 0 09- 45 11 10- 45 2 11- 3 1 1 30 12- 43 11 13- 45 12 14- 43 12 15- 45 13 16- 44 3 17- 1 18- 16 19- 14 20- 13 21- 16 22- 15 23- 0 1 24- 43 11 25- 45 14 26- 45 0 27- 30 10
1 1 28- 14 29- 13 30- 16 31- 15 32- 1 33- 1 1 34-44 40 1 2 35- 45 2 3637- 3 30 43 35 40 38-43, 33 40 25 39-43, 33 25 40- 45 3 41- 45 13 42- 10 4 43- 44 4 3 44- 45 3 451 13 46- 1 3 47- 44 3 3 48- 45 3 494 1 50- 31 45 51- 4 1 0 52- 45 0 1 53- 45 1 54- 21 55- 10 31
56- 20 57- 16 58- 14 60- 31 61- 15 62- 15 42 14 64- 31 65- 16 66- 13 67- 1 3 68-44 40 3 1 69-44 30 1 70- 45 1 71- 43 35 74 72-43, 33 74 48 73-43, 33 48 1 74- 4 76 n: Used PV: Used FV: Used R1: Used R3: Used R5-R9: Unused 1. 14 59- 63- 1 42 45 4 75- 16 76- 31 77-43, 33 76 REGISTERS i: i/12 PMT: Used R0: Used R2: Used R4: Level Pmt. Key in the program.
2. Press 3. Key in the term of the loan and press 4. Key in the annual interest rate and press . 5. Key in the total loan amount and press . 6. Key in the rate of graduation (as a percent) and press 7. Key in the number of years for which the loan graduates and press The following information will be displayed for each year until a level payment is reached. a. CLEAR . . to continue. The monthly payment for the current year. Then press c. . The current year. Then press b. .
-471.33 2nd year monthly payment. -51,665.07 Remaining balance after 2nd year. 3.00 Year 3 -494.89 3rd year monthly payment. -52,215.34 Remaining balance after 3rd year. 4.00 Year 4 -519.64 4th year monthly payment. -52.523.34 Remaining balance after 4th year. 5.00 Year 5 -545.62 5th year monthly payment. -52,542.97 Remaining balance after 5th year. -572.90 Monthly payment for remainder of term.
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 c. . .
-49,316.74 CLEAR 11.75 30 3 49,316.74 Remaining balance. 0.98 Adjusted monthly interest. 27.00 Remaining life of mortgage. 324.00 495.15 12 -504.35 New monthly payment. -495.15 Previous monthly payment. 31.67 New remaining term (years). 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.
8. Key in the loan amount and press 0 to obtain the monthly payment amount when the payment is made at the end of the month. 9. Press 0 1 . 10. Key in the annual interest rate as a percent and press find the monthly payment amount when the payment is made at the beginning of the month. to 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.
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 and press CLEAR . 2. Key in the initial investment and press 3. Key in the number of additional periodic deposits and press 4. Key in the periodic interest rate and press 5. Key in the periodic deposit and press 6. Press . . . .
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 and press CLEAR . 2. Key in the value of the savings account and press 3. Key in the periodic interest rate and press 4.
The cash flow diagram looks like this: FV = ? 1 2 -50 3 -50 4 -50 5 -50 -50 PV = - 1023.25 Keystrokes Display CLEAR 50 1,299.22 Amount in account. 5.5 1023.25 5 Now suppose that at the beginning of the 6th month you withdrew $80. What is the new balance? Keystrokes 80 Display 1,219.22 New balance. You increase your monthly deposit to $65.
FV = ? 1 2 -65 3 -65 -65 PV = -1219.22 Keystrokes Display 1,431.95 65 Account balance. 3 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 Keystrokes 2 Display 1,455.11 Account balance. 0 This type of procedure may be continued for any length of time, and may be modified to meet the user's particular needs.
calculate the total amount remaining in the account after a series of transactions on specified dates.
263 2 40 27-44 40 3 28- 45 15 29- 45 2 30- 40 31- 16 32- 13 1 33- 45 1 0 34- 44 0 35- 45 13 13 n: ∆days PV: Used FV: Used R1: Next date R3: Interest 36- 16 37-43, 33 13 REGISTERS i: i/365 PMT: 0 R0: Initial date R2: $ amount R4-R.4: Unused 1. Key in the program 2. Press 3. Key in the date (MM.DDYYYY) of the first transaction and press 4. Key in the annual nominal interest rate as a percentage and press CLEAR and press . . 5.
10. For a new case press and go to step 2. Example: Compute the amount remaining in this 5.25% account after the following transactions: 1. January 19, 1981 deposit $125.00 2. February 24, 1981 deposit $60.00 3. March 16, 1981 deposit $70.00 4. April 6, 1981 withdraw $50.00 5. June 1, 1981 deposit $175.00 6. July 6, 1981 withdraw $100.00 Keystrokes Display CLEAR 1.191981 125.00 Initial Deposit. 185.65 Balance in account, February 24, 1981. 256.18 Balance in account, March 16, 1981.
I savings plans however, money may become available for deposit or investment at a frequency different from the compounding frequencies offered. The HP 12C 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.
7 2,519.61 25 Future value. 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. Keystrokes Display CLEAR 5.5 4 0.11 Equivalent periodic interest rate. -11.49 Periodic payment. 100 52 CLEAR 8 52 6000 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).
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 program assumes that the purchase is financed with a loan and that the loan is made for the term of the lease.
1 2 5 15- 45 11 16-44 48 1 17- 45 12 18-44 48 2 19- 45 5 2021- 6 13 45 2223- 7 11 45 240 1 9 25- 45 0 26- 42 24 27-44 40 1 28- 45 9 30-45 13 48 311 32-45 34-45 0 14 48 332 7 12 290 6 1 11 48 35- 2 12 1 36- 45 1 3 37- 45 3 3839- 20 45 408 4142- 48 14 30 45 8 30
4 43- 45 4 0 44- 45 0 45- 21 46- 10 2 47-44 40 2 00 48-43, 33 00 REGISTERS i: Used PMT: Used R0: Used R2: Purch. Adv. R4: Discount R6: Dep. life R8: Used R.0: Used R.2: Used n: Used PV: Used FV: 0 R1: Used R3: Tax R5: Dep. Value R7: Factor (DB) R9: Used R.1: Used R.3: Unused Instructions: 1. Key in the program. -Select the depreciation function and key in at line 26. 2. Press 3. Input the following information for the purchase of the loan: and press CLEAR .
8. For declining balance depreciation, key in the depreciation factor (as a percentage) and press 9. 7. Key in the total first lease payment (including any advance payments) and press 1 3 2. 10. Key in the first year's maintenance expense that would be anticipated if the asset was owned and press . 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. 11. Key in the next lease payment and press .
2 3 4 5 6 7 8 9 10 200 200 200 1500 300 300 300 300 300 Keystrokes 1700 1700 1700 1700 1700 1700 1700 0 0 1000 750 Display 0.00 CLEAR 10 12 -10,000.00 Always use negative loan amount. 1,769.84 Purchase payment. 0.48 Marginal tax rate. 1.05 Discounting factor. 8,500.00 Depreciable value. 10.00 Depreciable life. 3,400.00 1st lease payment. 2 1,768.00 After-tax expense. 10000 .48 3 .05 1 4 10000 1500 5 10 6 1700 1 3 200 312.36 Present value of 1st year's net purchase.
200 -628.09 5th year. -226.44 6th year. -309.48 7th year. -388.81 8th year. 1700 200 1700 200 1700 200 1700 300 0 -1,034.72 9th year. 300 0 -1,080.88 10th year. 750.00 Buy back. 390.00 After tax buy back expense. 239.43 Present value. -150.49 Net lease advantage. 750 1 3 43 2 Break-Even Analysis Break-even analysis is basically a technique for analyzing the relationships among fixed costs, variable costs, and income.
Sa l es R ev u en Profit e To Co tal sts Variable Costs $ Break-Even Point Lo ss Fixed Costs 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: 1. Key in the fixed costs and press .
To calculate the sales volume needed to achieve a specified gross profit: 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 . . . . to calculate the sales volume. To calculate the required sales price to achieve a given gross profit at a specified sales volume: 1. Key in the fixed costs and press . 2. Key in the gross desired and press 3.
12000 12,000.00 4500 16,500.00 2500 6.60 6.75 13.35 Fixed cost. Sales price per unit to achieve desired gross profit. For repeated calculation the following HP-12C program can be used.
21- 2 00 2. 2 22- 40 23-43, 33 00 REGISTERS i: Unused PMT: Unused R0: Unused R2: V R4: U R6-R.6: Unused n: Unused PV: Unused FV: Unused R1: F R3: P R5: GP 1. 45 Key in the program and store the know variables as follows: a. Key in the fixed costs, F and press 1. b. Key in the variable costs per unit, V and press c. Key in the unit price, P (if known) and press d. Key in the sales volume, U, in units (if known) and press e. Key in the gross profit, GP, (if known) and press 2. 3. 4. 5.
Example 2: A manufacturer of automotive accessories produces rear view mirrors. A new line of mirrors will require fixed costs of $35,00 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? Keystrokes 35000 1 Display 35,000.00 Fixed cost. 8.25 2 8.25 Variable cost. 12.5 3 12.50 Sales price. 0 0.00 5 10 Break-even volume is between 8,235 and 8,236 units. 8,235.
3. Key in the number of units and press 4. Key in the fixed cost and press . 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 Keystrokes Display 13 13.00 Price per copy. 6.75 6.25 Profit per copy. 25.00 Close to break-even point. 13 13.00 Price per copy. 6.75 6.25 Profit per copy. 1.
REGISTERS i: Unused PMT: Unused R0: Unused R2: V R4-R.8: Unused n: Unused PV: Unused FV: Unused R1: F R3: P 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 to calculate the operating leverage. 4.
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%.
1 23- 16 24- 1 25- 40 26- 0 45 2700 20 28-43, 33 10 30- 16 45 3233- 1 35- 00 1 40 45 340 0 29- 31- 1 0 1 10 45 0 36- 20 37-43, 33 00 5 38- 45 5 6 39- 45 6 00 40- 10 41- 30 42-43, 33 00 43- 4 45 4445- 6 00 n: Unused PV: Unused 5 30 45 6 46- 20 47-43, 33 00 REGISTERS i: Unused PMT: Unused 61
R0: 100 R2: % discount R4: % op. exp. R6: 1-% tax FV: Unused R1: list price R3: mfg. cost R5: % net profit R7-R.3: Unused 1. Key in the program and press press 2. CLEAR , then key in 100 and 0. Key in 1 and press , then key in your appropriate tax rate as a decimal and press 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 c. Key in the manufacturing cost in dollars (if known) and press 3. d.
b. Press 12 43 . 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 $2.50? The standard company operating expense is 32% of net shipping (sales) price and tax rate is 48%. Keystrokes CLEAR Display 100.00 100 0 1 .48 11.98 1 35 2 2.50 3 32 4 6 0.52 48% tax rate. 11.98 List price ($). 35.00 Discount (%). 2.50 Manufacturing cost ($). 32.00 Operating expenses (%). 12 67.90 43 18.
What reduction in manufacturing cost would achieve the same result without necessitating an increase in list price above $11.98? 13 7.79 01 2.30 Manufacturing cost ($).
Securities After-Tax Yield The following HP-12C program calculate the after tax yield to maturity of a bond held for more than one year. The calculations assumes an actual/ actual day basis. For after-tax computations, the interest or coupon payments are considered income, while the difference between the bond or note face value and its purchase price is considered capital gains.
19- 25 20- 30 21- 0 45 0 22- 10 23- 14 1 24- 45 1 0 25- 45 0 26- 10 27- 13 6 28- 45 6 7 29- 45 7 30- 42 22 31-43, 33 00 00 n: Unused PV: Used FV: 0 R1: Purchase price R3: Coupon rate R5: Income rate R7: Used REGISTERS i: Yield PMT: Used R0: Used R2: Sales price R4: Capital rate R6: Used R8-R.5: Unused 1. Key in the program. 2. Key in the purchase price and press 3. Key in the sales price and press 4. Key in the annual coupon rate (as a percentage) and press 5.
8. Key in the purchase date (MM.DDYYYY) and press . 9. Key in the assumed sell date (MM.DDYYYY) and press 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, 1981 for $70 and expect to sell it in 5 years for $90.
2 3 02- 45 2 03- 43 26 04- 45 3 0506- 5 1 45 25 08- 1 09- 34 10- 30 45 1213- 5 5 07- 11- 4 10 4 20 44 14- 5 31 1 15- 45 1 2 16- 45 2 17- 43 26 18- 45 3 3 19- 34 20- 10 4 21- 45 4 5 22- 45 5 1 2 00 23- 10 24- 1 25- 30 26- 20 27- 26 28- 2 29- 20 30-43, 33 00 68
REGISTERS i: Unused PMT: Unused R0: Unused R2: Mat. date R4: redemp. value R6-R.5: Unused n: Unused PV: Unused FV: Unused R1: Settl. date R3: 360 or 360 R5: dis./price 1. Key in the program. 2. Press 3. Key in the settlement date (MM.DDYYYY) and press 4. Key in the maturity date (MM.DDYYYY) and press 5. Key in the number of days in a year (360 or 365) and press 6. Key in the redemption value per $100 and press 7. To calculate the purchase price: 8. . 1. 2. 3. 4. a.
3.211981 2 3.21 Maturity dtae. 360 3 360.00 360 day basis. 100 4 100.00 Redemption value per $100. 7.8 5 7.80 Discount rate. 96.45 Price. 8.09 Yield. Example 2: Determine the yield of this security; settlement date June 25, 1980; maturity date September 10, 1980; price $99.45; redemption value $101.33. Assume 360 day basis. Keystrokes Display 6.251980 1 6.25 Settlement date. 9.101980 2 9.10 Maturity dtae. 360.00 360 day basis. 101.33 Redemption value per $100. 99.45 Price.
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.
0.00 CLEAR 211570 1.00 112550 2.00 190060 3.00 171,393.33 211570 2.00 131760 3.00 3-month average for March. 144,790.00 112550 2.00 300500 3.00 3-month average for April. 207,440.00 190060 2.00 271120 3.00 3-month average for May. 234,460.00 3-month average for June.
10- 4 40 5 11- 45 5 4 12- 44 4 13- 5 40 6 14- 45 6 5 15- 44 5 16- 6 40 7 17- 45 7 6 18- 44 6 19- 7 40 8 20- 45 8 7 21- 44 7 22- 8 9 40 23- 45 9 24- 44 8 25- 9 0 9 40 26-45 48 0 27- 44 9 28- 10 40 1 29-45 48 1 0 30-44 48 0 31- 11 40 2 32-45 48 2 1 33-44 48 1 34- 12 0 3536- 73 40 45 0 10
3738- m* 01 n: Unused PV: Unused FV: Unused R1: X1 R3: X3 R5: X5 R7: X7 R9: X9 R.1: X11 R.3-R.4: Unused 31 44 -- 39-43, 33 01 REGISTERS i: Unused PMT: Unused R0: m R2: X2 R4: X4 R6: X6 R8: X8 R.0: X.0 R.2: X12 *At step 38, m=number of elements in the moving average, i.e. fir a 5 element moving average line 38 would be average line 38 would be 5 and for a 12-element 2 This program can be used for a moving average of 2 to 12 elements.
5. Continue as above, keying in and storing each data point in its appropriate register until m data points have been stored. 6. Press 7. Key in the next data point and press 00 to calculate the first moving average. 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.
171,393.33 3-month average for March. 131760 144,790.00 3-month average for April. 300500 207,440.00 3-month average for May. 271120 234,460.00 3-month average for June. 00 Seasonal Variation Factors Based on Centered Moving Averages. Seasonal variation factors are useful concepts in many types of forecasting.
08- 2 44 09- 2 40 4 10- 45 4 3 11- 44 3 12- 40 5 13- 45 5 4 14- 44 4 2 4 15- 2 16- 10 17- 40 18- 4 19- 10 20- 31 21- 2 01 2 22- 23 23- 31 24- 5 45 44 5 25-43, 33 01 REGISTERS i: Unused PMT: Unused R0: n R2: X2 R4: X4 R6-R.6: Unused n: Unused PV: Unused FV: Unused R1: X1 R3: X3 R5: X5 1. Key in the program. 2. Press 3. Key in the quarterly sales figures starting with the first quarter: a. CLEAR . Key in 1st quarter sales and press 77 1.
4. 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 Press 00 5. to calculate the centered moving average for the 3rd quarter of the first year. 5. Press to calculate the seasonal variation for this quarter. 6. Key in the next quarter's sales and press to calculate the moving average for the next quarter. 7. Press to calculate the seasonal variation.
449.75 390 4th quarter, 1978. 111.40 460.25 530 1st quarter, 1979. 98.86 476.38 560 2nd quarter, 1979. 81.87 490.00 513 3rd quarter, 1979. 107.94 503.75 434 4th quarter, 1979. 111.17 513.25 562 1st quarter, 1979. 99.95 521.38 593 2nd quarter, 1980. 83.24 Now average each quarter's seasonal variation for the two years? Keystrokes CLEAR Display 0.00 98.86 1.00 99.95 2.00 1st quarter average seasonal variation, %. 99.41 CLEAR 0.00 81.87 1.00 83.24 2.
0.00 CLEAR 111.4 1.00 111.17 2.00 4th quarter average seasonal variation, %. 111.
20- 6 44 21- 6 40 8 22- 45 8 7 23- 44 7 24- 40 9 25- 45 9 8 26- 44 8 270 9 40 28-45 48 0 29- 44 9 30- 40 1 31-45 48 1 0 32-44 48 0 33- 40 2 34-45 48 2 1 35-44 48 1 36- 2 0 6 40 3 37-45 48 3 2 38-44 48 2 39- 2 40- 10 41- 40 42- 45 0 43- 10 44- 31 45- 45 6 46- 23 47- 31 81
3 48-44 48 3 01 48-43, 33 01 REGISTERS i: Unused PMT: Unused R0: n R2: X2 R4: X4 R6: X6 R8: X8 R.0: X10 R.2: X12 n: Unused PV: Unused FV: Unused R1: X1 R3: X3 R5: X5 R7: X7 R9: X9 R.1: X11 R.3: X13 1. Key in the program. 2. Press 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. CLEAR . Key in the 1st month and press 1. Key in the 2nd month and press 5. 2, etc., Key in the 10th month and press 0, etc.
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.
4 18- 30 19- 10 20- 45 21- 22 22- 21 6 23- 44 6 1 24- 45 1 3 25- 45 3 262 27- 20 45 2 28- 36 29- 20 30- 30 1 31- 45 1 3 32- 45 3 332 2 4 34- 40 45 2 35- 2 36- 20 37- 30 38- 10 39- 45 40- 1 4 4 10 41- 43 22 7 42- 44 7 6 43- 45 6 44- 1 45- 30 84
6 46- 45 6 4 47- 45 4 1 48- 21 49- 1 50- 30 51- 36 52- 20 53- 10 54- 6 45 55- 6 10 2 56- 45 2 1 57- 45 1 5 58- 30 59- 20 60- 43 22 61- 44 5 6263- 62 n: Unused 6 34 65- 21 45 5 67- 34 68- 21 69- 7 45 64- 66- 5 31 45 7 70- 20 71-43, 33 62 REGISTERS i: Unused 85
PV: Unused FV: Unused R1: S1 R3: S3 R5: a R7: c PMT: Unused R0: Unused R2: S2 R4: n R6: b R8-R.0: Unused 1. Key in the program and press CLEAR . 2. Divide the data points to be input into 3 equal consecutive groups. Label them Groups I, II and III for convenience. 3. Key in the first point of group I and press 4. Key in the first point of group II and press 5. Key in the first point of group III and press 6. Repeat steps 3, 4, and 5 for the balance of the data in each group.
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 Keystrokes Group Group II III 151 282 188 322 260 340 Display 0.00 CLEAR 18 18.00 151 151.00 282 1.00 41 41.00 188 188.00 322 2.00 49 49.00 260 260.00 340 3.00 Total number of entries. 0.004 a 6 0.65 b 7 373.92 c 10 349.09 Sales in 10th year, (in $K). 12 363.36 Sales in 12th year, (in $K). 100 373.
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.
2 14- 45 2 1 15- 45 1 2 2 16- 20 17- 40 18- 45 19- 16 20- 34 21- 44 220 2 23- 2 40 45 24- 0 20 1 25- 45 1 3 26- 45 3 27- 20 28- 40 3 29- 44 3 1 30- 45 1 310 32- 20 45 332 34- 0 10 45 35- 2 40 36- 44 5 3 37- 45 3 0 38- 45 0 392 4041- 89 10 45 2 40
42- 6 00 44 6 43-43, 33 00 REGISTERS i: Unused PMT: Unused R0: α R2: St-1 n: Unused PV: Unused FV: Unused R1: 1-α R3: Tt-1 R4: Σe2 R5: Dt R6: t+1 R7-R.4: Unused Selecting the "best" smoothing constant (α): 1. Key in the program and press CLEAR . 2. Key in the number 1 and press 3. Key in the "trial " and press 4. Key in the first historical value (X1) and press 5. Key in the second historical value (X2) and press . 0 1. 2. result is the error between the forecast value ( (Xt+1) 6.
1. Key in the number 1 and press 2. Key in the selected and press 3. From the selection routing or from a previous forecast: 4. . 0 1. o Key in the smoothed average St-1 and press o Key in the trend Tt-1 and press o Key in the forecast t+1 and press Key in the current data value and press 2. 3. 6. . The output is the error in forecasting the value just entered. 5. Press . The displayed value represents the forecast for the next period. 6.
4 23.61 Cumulative error (Σe2). 0 0.50 Smoothing constant (a). 2 25.11 Smoothing average (St-1). 3 0.42 Trend (Tt-1). 6 25.95 Last forecast (Dt+1). The procedure is repeated for several α's. Smoothing Constant (α) Cumulative Error .5 (Σe2) .1 .25 .2 23.61 25.14 17.01 18.03 For the selected α = .25 St+1= 24.28 Tt-1 = 0.34 Dt+1= 25.64 Forecasting: Keystrokes Display 0.00 CLEAR 1.00 1 .25 0.75 0 0.75 1 24.28 2 .34 0.34 3 25.64 24.28 6 25.64 0.36 26 26.16 5 25.
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.
Example 2: If an item sells for $21.00 and has a markup of 50%, what is its cost? What is the margin? Keystrokes Display 21 1 50 21.00 Selling price. 50.00 Markup (%). 14.00 Cost. 50 50.00 1 33.33 Margin (%). The following HP 12C program may be helpful for repetitive calculations of selling price and costs as well as conversions between markup and margin.
R0-R.8: Unused 1. FV: Unused Key in program. 2. To calculate selling price, given the markup, key in the cost, press , key in the markup and press 3. 00 To calculate cost, given the markup, key in the selling price, press , key in the markup and press 4. 03 . 03 . To calculate markup from the margin, key in the margin and press 03 7. . To calculate cost given the margin, key in the selling price, press , key in the margin and press 6.
list and new and several discounts are known it may be desirable to calculate a missing discount. The following series of keystrokes may be used: 1. Key in 1, press 1. 2. Key in the first discount (as a percentage) and press 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 1 . 5. To calculate the net price, key in the list price and press 1 . 6.
04- 30 1 05-44 20 1 00 06-43, 33 00 07- 1 1 2 00 n: Unused PV: Unused FV: Unused R1: R1D1xD2...D 45 1 08- 20 09- 10 10- 1 11- 34 12- 30 13- 26 14- 2 15- 20 16-43, 33 00 REGISTERS i: Unused PMT: Unused R0: Unused R2-R7: Unused 1. Key in the program. 2. Key in 1 and press 3. Key in the first discount rate (as a percentage) and press 4. Repeat step 2 for each of the remaining discount rates. 5. To calculate the list price, key in the net price and press 1 . 6.
1 1.00 1 48 0.52 5 0.95 1.45 3.28 3.75 10.51 3rd discount rate (%). 0.89 Include 3rd discount rate in calculation. 1.66 New net price.
Statistics Curve Fitting Exponential Curve Fit Using the function of the HP-12C, 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.
5. Press to obtain B. 6. Press 7. To make a y-estimate, key in the x-value and press 1 to obtain the effective growth rate (as a decimal). . 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 1982 (year 7)? End of Year 1976(1) 1977(2) 1978(3) 1979(4) 1980(5) 1981(6) 1982(7) Keystrokes CLEAR 45 Price 45 51.5 53.75 80 122.5 210 ? Display 1.
00- CLEAR 0102- 00 43 34 04- 49 05-43, 33 00 43 34 08- 31 1 10- 43 2 11- 43 22 12- 0 13- 43 2 14- 43 22 15- 31 16- 34 17- 33 18- 10 19- 43 20211 00 2 07- 09- 0 23 03- 06- 1 34 23 31 43 22 22- 1 23- 30 24- 31 25- 43 2 26- 43 22 27-43, 33 00 101
REGISTERS i: Unused PMT: Unused R0: Unused R2: Σx n: Unused PV: Unused FV: Unused R1: n R3: Σx2 R4: Σy Σy2 R6: Σxy R5: R7-R.6: Unused 1. Key in the program and press CLEAR 2. For each input pair of values, key in the y-value and press in the corresponding x-value and press 3. After all data pairs are input, press . , key . 06 to obtain the correlation coefficient (between ln y and x). 4. Press to obtain A. 5. Press to obtain B. 6.
7 27.34 A 0.31 B 0.36 Effective growth rate. 232.35 Projected price at the end of year 7 (1982). 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.
1. Press CLEAR 2. Key in the first y-value and press press 3. . . Key in the first x-value and . Repeat this step for each data pair. After all data pairs are input, press to obtain the correlation coefficient (between y and ln x). 4. Press 1 0 to obtain A in the equation above. 5. Press 6. To make a y-estimate, key in the x-value and press to obtain B. .
1 0 8 0.99 Correlation coefficient (between y and ln x). 1,066.15 Value of A. 4,069.93 Value of B. 9,529.34 Total units sold by end of eighth month. 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.
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) 2 30 h (feet) 2.5 50 3.5 90 4 130 4.5 150 Find the power curve formulas that best expresses h as a function of t (h = AtB). Keystrokes Display CLEAR 30 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. 1.
1. Press CLEAR . 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 , key in the x-value and press . Continue until you have entered all of the values. 4. Press 5. Press to obtain the mean of the x-values. 1 to obtain the standard error of the mean of the x-values. 6.
this procedure computes the mean, standard deviation, and standard error of the mean. 1. Press CLEAR . 2. Key in the first value and press 3. Key in the respective frequency and press 4. Repeat steps 2 and 3 for each data point. 5. To calculate the mean (average) press . 0 . The display shows the number of data points entered. 0 1 6 3 . 6. Press to find the standard deviation. 7. Press 0 to find the standard error of the mean.
00- CLEAR 01-44 0 00 40 0 02- 20 03- 49 04-43, 33 00 0 05- 45 0 1 06- 44 1 6 07- 45 6 3 08- 44 3 09- 43 0 1011- 31 43 120 00 48 31 13- 45 0 14- 43 21 15- 10 16-43, 33 00 REGISTERS i: Unused PMT: Unused R0: Σfi R2: Σfixi n: Unused PV: Unused FV: Unused R1: Σfi R3: Σfixi2 R4: Σxi R5: Σxi2 R6: Σfixi2 R7-R.7: Unused 1. Key in the program. 2. Press 3. Key in the first value and press CLEAR . .
4. Key in the respective frequency and press . The display shows the number of data points entered. 5. Repeat steps 3 and 4 for each data point. 6. To calculate the mean, press 7. Press to find the standard deviation. 8. Press to find the standard error of the mean. 9. For a new case, go to step 2. Keystrokes 05 . Display CLEAR 190 1.00 First data pair. 2.00 Second data pair. 3.00 Third data pair. 4.00 Total number of data sets. 199.44 Average monthly rent (maen). 5.
n 2 x = ∑ ( Oi – Ei ) --------------------Ei i=1 If there is a close agreement between the observed and expected frequencies, x2 will be small. If the agreement is poor, x2 will be large. The following keystrokes calculate the x2 statistic: 1. Press CLEAR . 2. Key in the first Oi value and press 3. Key in the first Ei value and press . 0 0 . 4. Repeat steps 2 and 3 for all data pairs. The x2 value is displayed.
15 20 2.95 0 23 20 3.40 0 24 4.20 20 0 16 20 X2 5.00 0 The number of degrees of freedom is (n-1). Since n = 6, the degrees of freedom = 5. Consulting statistical tables, you look up x2 to a 0.05 significance level with 5 degrees of freedom, and see that x20.05,5 = 11.07. Since x2 = 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 program with the same example.
1. Key in the program. 2. Press 3. Key in the first Oi value and press 4. Key in the first Ei value and press 5. Repeat steps 3 and 4 for all data pairs. The x2 value is displayed. 6. For a new case, go to step 2. Keystrokes Display CLEAR . . . CLEAR 25 1.25 20 17 1.70 20 15 2.95 20 23 3.40 20 24 4.20 20 16 X2 5.00 20 Normal Distribution The normal (or Gaussian) distribution is an important tool in statistics and business analysis.
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 214-225; Jan 1977.
3 04- 20 05- 3 115
5 06- 5 1 07- 1 08- 40 09- 0 45 0 10- 20 5 11- 5 6 12- 6 2 13- 2 14- 40 7 15- 7 0 16- 0 3 17- 3 18- 0 45 0 19- 10 1 20- 1 6 21- 6 5 22- 5 23- 40 24- 10 25- 16 262 00 n: Unused PV: Unused FV: Unused 43 27- 2 28- 10 29-43, 33 00 REGISTERS i: Unused PMT: Unused R0: x R1-R.6: Unused 1. Key in program. 2. Key in x and press 22 to computed Q(x).
3. Repeat step 2 for each new case. Example: Find Q(x) for x = 1.18 and x = 2.1. Keystrokes Display 1.18 0.12 Q(1.18) 2.1 0.02 Q(2.1) 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 CLEAR . 2.
54 62 51 68 40 74 11 1 -354.14 Sxy -303.
n: Unused PV: Unused FV: Unused R1: n i: Unused PMT: Unused R0: Unused R2: Σx R3: Σx2 R4: Σy R5: Σy2 R7-R.7: Unused R6: Σxy 1. Key in the program. 2. Press 3. Key in the y-value and press 4. Key in the x-value and press 5. Press 03 6. Press to obtain S'xy. 7. For a new case, go to step 2. Keystrokes Display CLEAR . . . Repeat steps 3 and 4 for all data pairs. . to obtain the value of Sxy. CLEAR 92 26 85 30 78 44 81 50 54 62 51 68 40 74 03 7.00 Total number of entries.
m! mPn = -------------------( m – n )! where m, n are integers and 69 ≥ m ≥ n ≥ 0. Use the following HP-12C program to calculate the number of possible permutations. KEYSTROKES DISPLAY CLEAR 00- 0 01- 44 02- 0 34 03- 43 3 04- 43 36 05- 45 0 0607- 00 n: Unused PV: Unused FV: Unused R1-R.8: Unused 0 30 43 3 08- 10 09-43, 33 00 REGISTERS i: Unused PMT: Unused R0: n 1. Key in the program. 2. Key in m and press 3. Key in n and press 4. For a new case go to step 2. .
10 10P4. 5,040.00 4 Combination 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! mCn = ------------------------( m – n )!n! Where m, n are integers and 69 ≥ m ≥ n ≥ 0. Use the following HP-12C to calculate the number of possible combinations.
R0: n FV: Unused R1-R.8: Unused 1. Key in the program. 2. Key in m and press 3. Key in n and press 4. For a new case, go to step 2. . to calculate mCn. 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? Keystrokes 7 Display 7C3. 35.00 3 Random Number Generator This HP-12C program calculates uniformly distributed pseudo-random numbers ui in the range 0 < ui < 1.
2 8 4 1 6 3 030405060708- 2 8 4 1 6 3 09- 0 44 0 9 9 1011- 9 9 7 12- 7 13- 20 0 10 14- 43 24 15- 44 0 16- 31 17-43, 33 10 REGISTERS i: Unused PMT: Unused R0: Ui n: Unused PV: Unused FV: Unused R1-R.7: Unused 1. Key in the program. 2. To generate a random number, press 3. Repeat step 2 as many times as desired. . Example: Generate a sequence of 5 random numbers. Keystrokes Display 0.83 0.83 0.83 0.83 0.
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.
-672.16 Approximate monthly payment. The following HP-12C program may be used instead of the above.
n: Term PV: Loan FV: 0 i: Interest PMT: Loan PMT R0: Unused R1: Purch. Price R2: % Down R3: Tax rate R4-R.7: Unused 1. Key in the program. 2. Press 3. Key in the annual interest rate and press 4. Key in the term of the loan in years and press 5. Key in the purchase price and press 6. Key in the percent down and press 7. Key in the tax rate in dollars per thousand and press 3. 8. To calculate the approximate monthly payment, press . 9.
Tax-Free Individual Retirement (IRA) of Keogh Plan. The advent of tax-free retirement accounts (IRA or Keogh) has resulted in considerable benefits for many person who are not able to participate in group profit sharing or retirement plans. The savings due to tax-free status are often considerable, but complex to calculate.
0607- 1 5 1 1 0 09- 5 10- 25 11- 16 12- 1 13- 40 45 20 16- 31 17- 1 18- 48 1920- 1 0 45 11 22- 21 23- 10 24- 31 45 2627- 17 15 15- 25- 1 1 48 21- 1 45 08- 14- 1 31 12 1 45 1 28- 25 29- 30 30- 20 31- 12 32- 15 33- 31 34-43, 33 17 128
REGISTERS i: Used PMT: Yearly Pmt R0: Unused n: Years PV: 0 FV: Used R1: Tax % R2-R.5: Unused 1. Key in the program. 2. Press 3. Key in the tax rate as a percentage and press 4. Key in years to retirement and press 5. Key in the interest rates as a percentage and press 6. Key in the annual payment and press 7. Press to calculate the future value of the tax free investment. 8. Press to compute the total cash paid in. 9. Press to compute the total dividends paid. 10.
6. If you invest the same amount ($1500, *after taxes for a not-Keogh or IRA account.) each year with dividends taxed as ordinary income, what will be the total tax-paid cash at retirement? 7. What is the purchasing power of that figure in terms of today's dollars? Keystrokes Display CLEAR 40.00 Tax rate. 35 35.00 Years to retirement. 8.175 8.18 Dividend rate. 1500 -1,500.00 Annual payment. 290,730.34 Future value at retirement. -52,500.00 Cash Paid in. 238,230.34 Earned dividends.
• Prices are input in the form XXX.ND where N is the numerator and D is the Denominator of the fractional portion of the price, e.g. 25 5/8 is input as 25.58. • The beta coefficient analysis is optional. Key in 1.00 if beta is not to be analyzed.
0 25-44 40 2627- 7 34 45 281 3 5 01 29-44 20 40 2 1 31- 20 32-44 40 3 33- 45 5 34- 43 36 35- 24 36- 31 37-43, 33 01 38- 40 39- 34 44 42- 44 5 43-44 40 2 45- 01 2 1 44 4 46- 31 47-43, 33 01 48- 45 490 7 20 444 1 33 415 7 30- 40- 7 0 50- 2 31 45 0 51- 31 52- 24 132
53- 31 0 54- 45 0 1 55- 45 1 56- 31 57- 23 58- 31 3 59- 45 3 0 60- 45 0 00 n: Unused PV: Unused FV: Unused R1: ΣDIV R3: ΣPiSiβi R5: Pi R7: Si 61- 10 62-43, 33 00 REGISTERS i: Unused PMT: Unused R0: ΣPV R2: ΣOrig. Val. R4: Flag R6: XXX.ND R8-R.1: Unused Instructions: 1. Key in the program. 2. Initialize the program by pressing 3. Key in the number of shares of a stock and press 4. Key in the initial purchase of the stock and press 5.
9. Next, to evaluate the entire portfolio, press 48. 10. Press to see the initial portfolio value. 11. Press to see the present portfolio value. 12. Press to see the percent change in value. 13. Press to see the total yearly dividend. 14. Press to see the annual dividend yield as a percent of the current market value. 15. Press to see the beta coefficient of the portfolio. 16. For a new case return to step 2.
89.78 1.00 1.3 1.30 4.55 4.55 96.18 6.95 500 500.00 65.14 1.00 .6 0.60 3.50 3.50 64.38 -1.34 Percent change in Stock's value. 45,731.25 Original value. 46,418.75 Present value. 1.50 Percent change in value. 2,567.50 Total yearly dividend. 5.53 Annual dividend yield. 0.77 Portfolio beta coefficient. 48 Percent change in Stock's value. N. W.
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. This difference can be easily handled by the addition of a few keystrokes. For any problem requiring an input for , the Canadian mortgage factor is calculated first and then this value is entered in for in the calculation to give the answer for Canada.
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 1/4 %, how long will it take to completely amortize this mortgage? Keystrokes Display CLEAR 6 0.76 Canadian mortgage factor. -440.00 Monthly payment. 437.00 Total number of monthly payments. 200 9.25 440 56000 0 Effective Interest Rate (Yield) Example 3: A Canadian mortgage has monthly payments of $612.
CLEAR 6 0.72 Canadian Mortgage factor. -61,877.18 Outstanding balance remaining at the end of 10 years. 200 8.75 612.
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.
00- CLEAR 012 43 0203- 2 43 0405- 2 10 44 07- 34 44 00 2 2 10- 43 23 11- 45 2 10 13- 43 22 14- 44 2 15-43, 33 00 16- 45 2 17- 43 23 1819- 00 3 2 43 23 20- 10 21- 21 22- 1 1 10 12- 2 2 33 09- 2 23 06- 08- 1 23 45 1 23- 20 24-43, 33 00 2526- 140 44 3 34
4 27- 44 4 2 28- 45 2 29- 43 23 2 3031- 1 43 23 32- 10 33- 1 34- 40 35- 0 2 44 36- 0 21 3 37- 45 3 0 38- 45 0 39- 21 40- 30 41- 0 45 42- 0 10 4 43- 45 4 3 44- 45 3 45- 30 46- 10 47- 1 00 n: Unused PV: Unused FV: Unused R1: C1 R3: i 45 1 48- 20 49-43, 33 00 REGISTERS i: Unused PMT: Unused R0: K+1 R2: r R4: j 141
R5-R.3: Unused 1. Key in the program, (Note: If the average cost are not going to be calculated, lines 25 through 48 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 . b. Key in Cn, the cost of the nth unit and press . c. Key in n, the number of units and press to calculate r the learning factor. 3. To calculate the cost of the nth unit when C1 and r are known: a. 1. Key in r and press 2.
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.
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 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? Keystrokes .5 1 .9 2 Display 0.50 0.56 2 0.69 Average # customers waiting in queue. 1 1.39 Average waiting time. 2 2.50 Average total time in the system. 1 1.25 Average # customers in system.
14- 49 01 15-43, 33 01 0 16-45 48 0 17- 45 7 7 181 21 190 7 7 1 20-45 48 0 21- 45 7 22- 10 23- 30 24- 10 25- 45 7 26- 43 3 27- 10 6 28- 44 6 2 29- 45 2 30- 40 31- 22 1 32- 44 1 6 33- 45 6 342 0 20 35- 44 2 36-45 48 0 377 0 20 38- 45 7 39-45 48 0 40- 30 41- 10 146
3 0 42- 44 3 43-45 48 0 44- 40 4 45- 44 4 8 46- 45 8 47- 10 5 48- 44 5 3 49- 45 3 8 50- 45 8 5152- 6 10 44 53- 6 31 8 54- 45 8 7 55- 45 7 9 56- 45 9 2 53 n: Unused PV: Unused FV: Unused R1: P0 R3: Lq R5: T 57- 20 58- 30 59- 20 60- 43 22 61- 45 2 62- 20 63-43, 33 53 REGISTERS i: Unused PMT: Unused R0: K R2: Pb R4: L R6: Used, Tq 147
R8: λ R.0: ρ R7: n R9: µ R.1: Unused 1. Key in the program and press CLEAR . 2. Key in the number of servers, n and press 3. Key in the arrival rate of customers, λ, and press 4. Key in the service rate of each server, µ, and press 5. Press 6. Press 0 7. 8. 9. 0 to calculate and store ρ, the intensity factor. 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.
2 2 0.65 Pb probability all servers are busy. 3 2.59 Lq average # waiting in queue. 4 4.99 L, average # waiting in system. 5 4.16 T, average total time in system. 0.36 Probability of having to wait 2 minutes or more.
Appendix 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. • r = interest rate of wrap-around mortgage as a decimal. • FV = balloon payment.
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.
• FC = (G - AMT - CL) Rule of 78's Rebate • PV = finance charge. • Ik = interest charged at month k. • n = number of months in loan. • 2(n – k + 1) l k = ------------------------------PV n(n + 1) • ( n – k )l Rebate = --------------------k2 • BALk = (n - k) x PMT - Rebatek 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.
Compounding Periods Different From Payment Periods • C = number of compounding periods per year. • P = number of payments 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.
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. Securities 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.
Simple Moving Average • X = moving average. • m = number of elements in moving average. • X1 + X2 + X3 + X ...x m nm m X 1 = ------------------------------------------------------m • X 1 + X 2 + X 3 + X...xm + 1 nm m X 2 = ----------------------------------------------------------------m • etc. Seasonal Variation Factors Based on a Centered Moving Average • Xc = centered moving average • m = number of elements in the centered moving average. • Xm + 1 X ...
• 1 1 t + 1 = S t + --- T t α 2 1 S1 S3 – S2 c = exp --- ------------------------------------ n S 1 + S 3 – 2S 2 • ( b – 1 ) ( S2 – S1 ) a = exp ----------------------------------------2 n b(b – 1) • Where S1, S2, and S3 are: • n S1 = ∑ n b –1 Iny i = n ln c + b ( ln a ) --------------b–1 i=1 • 2n S2 = ∑ Iny i = n ln c + b n+1 n b –1 ( ln a ) --------------b–1 i = n+1 • • a, b and c are determined by solving the three equations above simultaneously.
3n S3 = ∑ Iny i = n ln c + b 2n + 1 i = 2n + 1 • Smoothed average St = αXt + (1 - α)St - 1 • Change, Ct = St - St - 1 • Trend, Tt = αCt + (1 - α)Tt - 1 • Current period expected usage, • Forecast of next period expected usage, • Error, et = t - Xt • Cumulative error = ∑ 2 et t=1 • Initial conditions: St-1 = Xt-1 Tt-1 = 0 Pricing Calculations Markup and Margin Calculations • Ma = margin(%). • Mu = markup(%). • S = selling price. • C = cost.
• S–C Ma = 100 -------------S • S–C Mu = 100 -------------C • C S = ------------------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 + ---------100 158
Calculations of List and Net Prices with Discounts • L = List price. • N = Net price. • D = Discount(%). • D D' = 1 – ---------100 • N L = -------------------------------------------------D' 1 × D' 2 × SSDDF ... D’x • N D x = 1 – -------------------------------------------------------------- ...
• Σ y i ln x i – 1 --- Σ ln x i Σ ln y i n B = -------------------------------------------------------2 1 2 Σ ( ln x i ) – --- ( Σ ln x ) n • 1 A = --- ( Σ y i – B Σ ln x i ) n • = A + B (ln x) Power Curve Fit • y = AxB (A>0) • ln y = ln A + Bln x • ( Σ ln x i ) ( Σ ln y i ) ------------------------------------n B = ------------------------------------------------2 2 ( Σ ln x i ) -------------------Σ ( ln x i ) – n • Σ ln y Σ ln x A = exp -------------i – B -------------i n n • = AxB Standard Er
• Σ fi xi mean X = ---------Σ fi • 2 standard deviation S x = 2 Σ f i x i – ( Σ f i )X ------------------------------------Σ fi – 1 • standard error S x = Σ fi Personal Finance Tax-Free Retirement Account (IRA) or Keogh Plan • n = the number of years to retirement. • i = the compunded annual interest. • PMT = the earnings used for investment (and taxes). • FV= future value. • tax= the percent tax expressed as a decimal.
Portfolio beta coefficient: • n β = ∑ Pi Si βi --------------T T Canadian Mortgages • r = annual interest rate expressed as a decimal. • monthly factor 1 -- r 6 = 1 + --- – 1 × 100 2 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 • Cn = C1nk Cij = the average cost of the ith through jth unit.
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.
• 1 PV = PMT 1 ( 1 + I ) A -------------- I B (1 + Q) - 1 -------------------- Q B (1 + C) ( n – AB ) ----------------- I + ---------------------------------------------- AB (1 + I) 1 (1 + I) where: • 1+C Q = ------------------- – 1 A (1 + I) • A = number of payments per year • B = number of years that payments increase • C = percentage increase in periodic payments (as a decimal) • PMT1 = amount
