HP Hewlett Packard Calculator HP 12C User Manual

Download from Www.Somanuals.com. All Manuals Search And Download.

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 1: 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 abount 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?

Keystrokes

Display

CLEAR

Monthly payment on existing

mortgage received by lender.

-1,080.33

6.5

133190

Download from Www.Somanuals.com. All Manuals Search And Download.

9.5

1,979.56

Monthly payment on new mortgage.

Net monthly payment (to lender).

200000

899.23

Net amount of cash advanced (by

lender).

-66,810.00

133190

11.5

-80,425.02 Present value of net

-13,615.02 NPV to lender of net cash advanced

14.83

% nominal yield (IRR).

Present value of net monthly

payment at 15.25%.

-65,376.72

1,433.28

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?

Keystrokes

Display

144.00

Total number of months remaining in

original load (into n).

Download from Www.Somanuals.com. All Manuals Search And Download.

CLEAR

0.56

Monthly interest rate (into i).

Loan amount (into PV).

6.75

200,132.06

200132.06

Monthly payment on existing

mortgage (calculated).

-2,031.55

0.83

Monthly interest on wrap-around.

Amount of wrap-around (into PV).

-300,000.00

300000

Monthly payment on wrap-around

(calculated).

Net monthly payment received (into

PMT).

3,585.23

1,553.69

-99,867.94

15.85

Net cash advanced (into PV).

200132.06

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.

where:

n = number of years remaining in original mortgage

PMT = yearly payment of original mortgage

PV = remaining balance of original mortgage

n = number of years in wrap-around mortgage

PMT = yearly payment of wrap-around mortgage

PV = total amount of wrap-around mortgage

BAL = balloon payment

Download from Www.Somanuals.com. All Manuals Search And Download.

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

$-1051.61

200mos.

$-1051.61

$-200000

Keystrokes

Display

CLEAR

200000

-74,990.00

Net investment.

125010

1051.61

1681.71

630.10

Net cash flow received by lender.

The above cash flow occurs 200

times.

1,681.71

39.00

Next cash flow received by lender.

Cash flow occurs 39 times.

129963.35

131,645.06

11.84

Final cash flow.

Rate of return to lender.

Download from Www.Somanuals.com. All Manuals Search And Download.

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. Key in the monthly payment to be made by the lender on the original

mortgage and press

5. Press

6. Key in the net amount of cash advanced and press

7. Key in the total term of the wrap-around mortgage and press

8. If a balloon payment exists, key it in and press

9. Press

desired yield.

to obtain the payment amount necessary to achieve the

10. Key in the amount of the wrap-around mortgage and press

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?

Keystrokes

Display

Number of periods and monthly

interest rate.

CLEAR

200

1051.61

Present value of payments plus cash

advanced.

-165,776.92

74990

240

1,693.97

9.58

Monthly payment received by lender

Annual interest rate paid by borrower.

129963.35

2000

Download from Www.Somanuals.com. All Manuals Search And Download.

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 payments 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.

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?

Download from Www.Somanuals.com. All Manuals Search And Download.

Keystrokes

Display

CLEAR

180,000.00

Potential Gross Income.

250

9,000.00

Vacancy Loss.

171,000.00

94,145.00

Effective Gross Income.

Net Operating Income.

76855

11.5

-89,580.09

Annual Debt Service.

700000

4,564.91

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 are as follows:

1. Forecast or estimate Sales Price. Deduct sales and Transaction Costs.

The result is the Proceeds of Resale.

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

Download from Www.Somanuals.com. All Manuals Search And Download.

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

240.00

Mortgage term.

CLEAR

11.5

0.96

Mortgage rate.

Property value.

700000

-7,465.01

120.00

Monthly payment.

Projection period.

-530,956.57

Mortgage balance in 10 years.

Estimated resale.

800000

56,000.00

744,000.00

213,043.43

Transaction costs.

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.

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 program. This program calculates the

Net Operating Income using the Potential Gross Income, operational cost

and vacancy rate. The Net Operating 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

line 32 of the program in place of

Download from Www.Somanuals.com. All Manuals Search And Download.

KEYSTROKES

DISPLAY

00-

01-

02-

CLEAR

03-

04-

05-

06-

07-

08-

09-

10-44 40

11-

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-

23-

24-

25-

26-

42 11

45 12

45 13

_{Download from Www.Somanuals.}1_co0_{m. All Manuals Search And Download.}

27-

28-

29-

43 35

30-43, 33 36

31-

32-

42 25

33-44 30

34-

35-43, 33 17

36-

37-

38-

39-

40-44 40

41-

42-45 48

43-

44-

45-

46-

47-

48-44 40

49-

50-

51-

52-

53-44 20

54-

_{Download from Www.Somanuals.}1_co1_{m. All Manuals Search And Download.}

55-

56-

45 14

57-

58-

59-

60-

61-

62-

63-

64-

65-

66-

43 31

67-43, 33 09

REGISTERS

n: Used

PV: Used

FV: 0

i: Annual %

PMT: Monthly

R : Used

R : Counter

R : PGI

R : Oper. cost

R : Dep. value

R : Dep. life

R : Factor (DB)

R : Tax Rate

R : % gr. (PGI)

R : % gr. (op)

R. : Vacancy rt.

1. Press

and press

CLEAR

2. Key in loan values:

•

Key in annual interest rate and press

Key in principal to be paid and press

Key in monthly payment and press

(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

_{Download from Www.Somanuals.}1_co2_{m. All Manuals Search And Download.}

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 8.

10. Key in the growth rate in operational cost (0 if no growth) and press

11. Key in the vacancy rate (0 for no vacancy rate) and press

12. Key in the desired depreciation function at line 32 in the program.

13. Press

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

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%?

Keystrokes

Display

CLEAR

100000

80,000.00

Mortgage amount.

1.02

Monthly interest rate.

Mortgage term.

12.25

360

-838.32

9,900.00

3,291.75

75,000.00

20.00

Monthly payment.

Potential Gross Income.

1st year operating cost.

Depreciable value.

Useful life.

9900

3291.75

75000

_{Download from Www.Somanuals.}1_co3_{m. All Manuals Search And Download.}

125.00

35.00

6.00

Decline in balance factor.

Marginal Tax Rate.

125

Potential Gross Income growth rate.

Operating cost growth.

2.50

2.5

5.00

Vacancy rate.

Year 1

1.00

ATCF

Year 2

-1,020.88

2.00

ATCF

Year 3

-822.59

3.00

ATCF

Year 4

-598.85

4.00

ATCF

Year 5

-72.16

5.00

ATCF

Year 6

232.35

6.00

ATCF

Year 7

565.48

7.00

ATCF

Year 8

928.23

8.00

ATCF

Year 9

1,321.62

9.00

ATCF

Year 10

1,746.81

10.00

-1,020.88

ATCF

Example 2: An office building was purchased for $1,400,000. The value

of depreciable improvements is $1,200,000.00 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 $175,2000 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?

Keystrokes

Display

CLEAR

1050000

175,200.00

Potential Gross Income.

9173.81

9.5

_{Download from Www.Somanuals.}1_co4_{m. All Manuals Search And Download.}

175200

40296

1200000

40,296.00

1st year operating cost.

1,200,000.00 Depreciable value.

35.00

50.00

3.50

1.60

7.00

Depreciable life.

Marginal tax rate.

Potential Gross Income

Operating cost growth rate.

Vacancy rate.

3.5

1.6

Go to dep. step.

32- 42

1.00

Change to SL.

Year 1

ATCF

18,021.07

Year 2

2.00

ATCF

Year 3

20,014.26

3.00

ATCF

22,048.90

Year 4

4.00

ATCF

Year 5

24,123.14

5.00

ATCF

26,234.69

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

program which follows. In calculating the owner's income tax liability on

resale, this program assumes that the owner elects to have his capital

gain taxed at 40% of his Marginal Tax Rate. This assumption is in

accordance with a 1978 Federal tax ruling.* (*Federal Taxes, code sec.

1202 (32,036))

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).

_{Download from Www.Somanuals.}1_co5_{m. All Manuals Search And Download.}

The user may change to a different depreciation method by keying in the

desired function at line 35 in place of

KEYSTROKES

DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

CLEAR

08-

09-

10-

11-

12-

13-

14-

15-

16-

17-

45 14

42 14

43 11

18-44 40

CLEAR

19-

20-

21-

22-

23-

24-

42 34

_{Download from Www.Somanuals.}1_co6_{m. All Manuals Search And Download.}

25-

26-

27-

28-

29-

42 23

30-

31-

32-

33-44 40

34-

35-

36-

37-

38-

42 25

45 13

39-44 40

40-

41-

42-

43-

44-

45-

46-

47-

48-43 33 00

REGISTERS

n: Used

i: Used

PV: Used

FV: Used

PMT: Used

R : Used

R : Desired yr.

_{Download from Www.Somanuals.}1_co7_{m. All Manuals Search And Download.}

R : Dep. value

R : Dep. life

R : Factor

R : MTR

R -R : Unused

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 5.

6. Key in your Marginal Tax Rate as a percentage and press

7. Key in the purchase price and press

8. Key in the sale price and press

9. Key in the % commission charged on the sale and press

*If a dollar value is desired instead of a commission rate, key in

which does not affect the register values, at line 04 of the program.

10. Key in the number of years after purchase and press

Example 1: An apartment complex, purchased for $900,000 ten years

ago, is sold for $1,750,000. The closing cost are 8% of the sale price and

the income tax rate is 48%.

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?

Keystrokes

Display

0.00

CLEAR

_{Download from Www.Somanuals.}1_co8_{m. All Manuals Search And Download.}

700,000.00

0.79

Mortgage.

700000

9.5

Monthly interest.

Number of payments.

Monthly payment.

Depreciable value.

Depreciable life.

Factor.

240.00

-6,524.92

750,000.00

25.00

750000

125.00

125

48.00

Marginal Tax Rate.

Purchase price.

900,000.00

900000

1750000

1,750,000.00 Sale price.

8.00

Commission rate.

ATNCPR.

911,372.04

_{Download from Www.Somanuals.}1_co9_{m. All Manuals Search And Download.}

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.

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. Key in the constant periodic payment to principal and press

2. Key in periodic interest rate and press

3. 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

4. Press

5. Press

6. Press

to obtain the interest portion of the payment.

to obtain the total payment.

to obtain the remaining balance of the loan.

7. Return to step 4 for each successive payment.

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)?

Keystrokes

Display

5000

5.00

Semi-annual interest rate.

3,000.00

8,000.00

First payment's interest.

Total first payment.

60000

_{Download from Www.Somanuals.}2_co0_{m. All Manuals Search And Download.}

55,000.00

2,750.00

7,750.00

Remaining balance.

Second payment's interest.

Total second payment.

Remaining balance after the first

year.

50,000.00

1,500.00

Seventh payment's interest.

6,500.00

25,000.00

1,250.00

6,250.00

Total seventh payment.

Remaining balance.

Eighth payment's interest.

Total eighth payment.

Remaining balance after fourth

year.

20,000.00

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.

1. Press

and press

CLEAR

2. Key in the number of months in loan and press

3. Key in the add-on rate and press

4. Key in the amount of the loan and press

received; negative for cash paid out.)

* (*Positive for cash

5. Press

6. Press

to obtain the APR.

_{Download from Www.Somanuals.}2_co1_{m. All Manuals Search And Download.}

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

1,180.00

Amount of loan.

1000

-65.56

21.64

Monthly payment.

Annual Percentage Rate.

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:

a. Key in number of months of loan and press

and press

CLEAR

b. Key in APR and press

c. Key in 100 and press

3. Press

rate.

to obtain the add-on

Example 1: What is the equivalent add-on rate for an 18 month loan with

an APR of 14%.

Keystrokes

Display

CLEAR

7.63

Add-On Interest Rate.

100

_{Download from Www.Somanuals.}2_co2_{m. All Manuals Search And Download.}

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.

KEYSTROKES

DISPLAY

00-

CLEAR

01-

02-

03-

04-

05-

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

43 36

_{Download from Www.Somanuals.}2_co3_{m. All Manuals Search And Download.}

22-

23-

24-

25-

26-

27-

28-

29-

30-

31-

32-

33-

34-

35-

36-

37-

38-

39-

40-

41-

42-

43-

42 14

45 14

45 13

44-

45-

46-

47-

48-

49-

50-

_{Download from Www.Somanuals.}2_co4_{m. All Manuals Search And Download.}

51-

52-

43 35

53-43, 33 61

54-

55-

56-

57-

58-

59-

60-

61-

62-

63-

64-

65-

66-

67-

68-

69-

70-

71-

72-

73-

74-

75-

76-

42 14

45 13

77-45, 43 12

78-43, 33 00

_{Download from Www.Somanuals.}2_co5_{m. All Manuals Search And Download.}

REGISTERS

n: N

i: i

PV: Used

FV: 0

PMT: PMT

R : N

R : AIR

R : CL (%)

R : Loan

R : N/1200

R : Used

R -R : Unused

1. Key in the program.

2. Press

CLEAR

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 3.

7. Press

8. Press

9. Press

10. Press

to find the monthly payment amount.

to obtain the amount of credit life.

to calculate the total finance charge.

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?

Keystrokes

Display

CLEAR

36.00

Months.

6.75

Add-on interest rate.

Credit life (%).

Loan.

6.75

1.00

3100.00

-107.42

116.02

3100

Monthly payment.

Credit life.

_{Download from Www.Somanuals.}2_co6_{m. All Manuals Search And Download.}

-651.10

12.39

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:

1. Key in number of months in the loan and press

2. Key in payment number when prepayment occurs and press

3. Key in total finance charge and press

to obtain the unearned interest (rebate).

4. Key in periodic payment amount and press

obtain the amount of principal outstanding.

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?

Keystrokes

Display

180

5.81

Rebate.

39.33

190.84

Outstanding principal.

The following HP-12C program can be used to evaluate the previous example.

KEYSTROKES DISPLAY

00-

CLEAR

_{Download from Www.Somanuals.}2_co7_{m. All Manuals Search And Download.}

01-

02-

03-

04-

05-

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-

23-

24-

25-

26-43, 33 00

REGISTERS

i: Unused

n: Unused

_{Download from Www.Somanuals.}2_co8_{m. All Manuals Search And Download.}

PV: Unused

FV: Unused

PMT: Unused

R : Fin. charge

R : Payment#

R : # moths

R -R : Unused

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).

to obtain the unearned

5. Key in the periodic payment amount and press

principal outstanding.

to find the amount of

6. For a new case return to step 2.

Keystrokes

Display

5.81

Rebate.

180

39.33

190.84

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 on 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.

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 program

determines the monthly payments and remaining balance for each year

until the level payment is reached.

_{Download from Www.Somanuals.}2_co9_{m. All Manuals Search And Download.}

KEYSTROKES

DISPLAY

00-

01-

02-

CLEAR

03-

04-

05-

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

45 11

43 11

45 12

43 12

45 13

16-

17-

18-

19-

20-

21-

22-

23-

24-

43 11

45 14

25-

26-

27-

_{Download from Www.Somanuals.}3_co0_{m. All Manuals Search And Download.}

28-

29-

30-

31-

32-

33-

34-44 40

35-

36-

37-

43 35

38-43, 33 40

39-43, 33 25

40-

41-

42-

43-

44-

45 13

45-

46-

47-

48-

49-

50-

51-

52-

53-

54-

55-

_{Download from Www.Somanuals.}3_co1_{m. All Manuals Search And Download.}

56-

57-

58-

59-

60-

61-

62-

63-

64-

65-

42 14

66-

67-

68-44 40

69-44 30

70-

71-

43 35

72-43, 33 74

73-43, 33 48

74-

75-

76-

77-43, 33 76

REGISTERS

n: Used

i: i/12

PV: Used

FV: Used

PMT: Used

R : Used

R : Level Pmt.

R -R : Unused

1. Key in the program.

_{Download from Www.Somanuals.}3_co2_{m. All Manuals Search And Download.}

2. Press

CLEAR

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. The current year.

Then press

b. The monthly payment for the current year.

Then press to continue.

c. The remaining balance to be paid on the loan at the end of the cur-

rent year. Then press to return to step a. unless the level

to continue.

payment is reached. If the level payment has been reached, the

program will stop, displaying the monthly payment over the remain-

ing term of the loan.

8. For a new case press

00 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?

Keystrokes

Display

0.00

CLEAR

30.00

Term

12.50

Annual interest rate

Loan amount

12.5

50,000.00

5.00

50000

Rate of graduation

Year 1

1.00

-448.88

-50,194.67

2.00

1st year monthly payment.

Remaining balance after 1st year.

Year 2

_{Download from Www.Somanuals.}3_co3_{m. All Manuals Search And Download.}

-471.33

-51,665.07

3.00

2nd year monthly payment.

Remaining balance after 2nd year.

Year 3

-494.89

-52,215.34

4.00

3rd year monthly payment.

Remaining balance after 3rd year.

Year 4

-519.64

-52.523.34

5.00

4th year monthly payment.

Remaining balance after 4th year.

Year 5

-545.62

-52,542.97

5th year monthly payment.

Remaining balance after 5th year.

Monthly payment for remainder of

term.

-572.90

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 with the

following keystroke procedure. The original terms of the mortgage are

assumed to be known.

1. Press

and press

CLEAR

_{Download from Www.Somanuals.}3_co4_{m. All Manuals Search And Download.}

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 percent-

age) and press

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

d. Key in the monthly payment over this period and press

e. Press

to find the remaining balance, then press

CLEAR

3. Key in the adjusted annual interest rate (as a percentage) and press

. To calculate the new monthly payment:

a. Key in the remaining life of the mortgage (years) and press

b. Press

to find the new monthly payment.

To calculate the revised remaining term of the mortgage:

c. Key in the present monthly payment and press

d. Press

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.

Keystrokes

Display

50,000.00

Original amount of loan.

CLEAR

50000

0.96

Original monthly interest rate.

Period.

11.5

36.00

-495.15

Previous monthly payment.

495.15

_{Download from Www.Somanuals.}3_co5_{m. All Manuals Search And Download.}

-49,316.74

49,316.74

CLEAR

Remaining balance.

0.98

Adjusted monthly interest.

Remaining life of mortgage.

11.75

27.00

324.00

-504.35

-495.15

31.67

New monthly payment.

Previous monthly payment.

New remaining term (years).

495.15

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

2. Key in the number of the last payment period before payments close the

first time and press

and press

CLEAR

3. Key in the annual interest rate as a percentage and press

4. Press

5. Key in the number of payments which are skipped and press

6. Press 0

100

CLEAR

7. Key in the total number of years in the loan and press

_{Download from Www.Somanuals.}3_co6_{m. All Manuals Search And Download.}

8. Key in the loan amount and press

to obtain the

monthly payment amount when the payment is made at the end of the

month.

9. Press

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.

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?

Keystrokes

Display

Number of payment made before a

group of payments is skipped.

3.00

CLEAR

3,119.98

Monthly payment in arrears.

100

CLEAR

100000

_{Download from Www.Somanuals.}3_co7_{m. All Manuals Search And Download.}

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

2. Key in the initial investment and press

3. Key in the number of additional periodic deposits and press

and press

CLEAR

4. Key in the periodic interest rate and press

5. Key in the periodic deposit and press

6. Press

period.

to determine the value of the account at the end of the time

Example: You have just opened a savings account with a $200 deposit. If

you deposit $50 a month, and the account earns 5 1/4 % compounded

monthly, how much will you have in 3 years?

Keystrokes

Display

CLEAR

200

2,178.94

Value of the account.

5.25

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.

_{Download from Www.Somanuals.}3_co8_{m. All Manuals Search And Download.}

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

2. Key in the value of the savings account and press

3. Key in the periodic interest rate and press

and press

CLEAR

4. Key in the amount of the periodic withdrawal and press

5. Key in the amount remaining in the account and press

may be omitted if the account is depleted (FV=0).

. This step

6. Press

balance.

to determine the number of periods to reach the specified

Example: Your savings account presently contains $18,000 and earns 5

1/4% 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 $5,000, how many withdrawals can you make?

Keystrokes

Display

CLEAR

18000

71.00

Months to deplete account.

5.25

300

Months to reduce the account to

$5,000

53.00

5000

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 1/2% compounded monthly. Your current

balance is $1023.25. How much will you have accumulated in 5 months?

_{Download from Www.Somanuals.}3_co9_{m. All Manuals Search And Download.}

The cash flow diagram looks like this:

FV = ?

-50

PV = - 1023.25

Keystrokes

Display

CLEAR

1,299.22

Amount in account.

5.5

1023.25

Now suppose that at the beginning of the 6th month you withdrew $80.

What is the new balance?

Keystrokes

Display

1,219.22

New balance.

You increase your monthly deposit to $65. How much will you have in 3

months?

The cash flow diagram looks like this:

_{Download from Www.Somanuals.}4_co0_{m. All Manuals Search And Download.}

FV = ?

-65

-1219.22

Keystrokes

Display

1,431.95

Account balance.

Suppose that for 2 months you decide not to make a periodic deposit.

What is the balance in the account?

FV = ?

-1431.95

Keystrokes

Display

1,455.11

Account balance.

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 program determines the value of a savings account when

interest is compounded daily, based on a 365 day year. The user is able to

_{Download from Www.Somanuals.}4_co1_{m. All Manuals Search And Download.}

calculate the total amount remaining in the account after a series of

transactions on specified dates.

KEYSTROKES

DISPLAY

00-

01-

02-

03-

CLEAR

04-

05-

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-

23-

24-

25-

15 13

43 26

42 14

45 13

_{Download from Www.Somanuals.}4_co2_{m. All Manuals Search And Download.}

26-

27-44 40

28-

29-

30-

31-

32-

33-

34-

35-

36-

45 15

45 13

37-43, 33 13

REGISTERS

n: ∆days

PV: Used

FV: Used

i: i/365

PMT: 0

R : Initial date

R : Next date

R : $ amount

R : Interest

R -R : Unused

1. Key in the program

2. Press

CLEAR

and 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

5. Key in the amount of the initial deposit and press

6. Key in the date of the next transaction and press

7. Key in the amount of the transaction (positive for money deposited,

negative for cash withdrawn) and press

the account.

to determine the amount in

8. Repeat steps 6 and 7 for subsequent transactions.

9. To see the total interest to date, press 3.

_{Download from Www.Somanuals.}4_co3_{m. All Manuals Search And Download.}

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

125.00

Initial Deposit.

1.191981

5.25

125

2.241981

Balance in account, February 24,

1981.

185.65

256.18

206.95

383.62

3.161981

Balance in account, March 16, 1981.

Balance in account, April 6 1981.

Balance in account, June 1, 1981.

4.061981

6.0111981

175

7.061981

100

285.56

5.56

Balance in account, July 6, 1981.

Total interest.

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.

_{Download from Www.Somanuals.}4_co4_{m. All Manuals Search And Download.}

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. 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.

To calculate the equivalent payment period interest rate, information is

entered as follows:

1. Press

and press

CLEAR

2. Key in the annual interest rate (as a percent) and press

3. Key in the number of compounding periods per year and press

4. Key in 100 and press

5. Key in the number of payments (deposits) per year and press

CLEAR

The interest rate which corresponds to the payment period is now in

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?

Keystrokes

Display

CLEAR

0.42

Equivalent periodic interest rate.

365

100

CLEAR

_{Download from Www.Somanuals.}4_co5_{m. All Manuals Search And Download.}

2,519.61

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

0.11

Equivalent periodic interest rate.

100

CLEAR

-11.49

Periodic payment.

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). How long will it take you to

accumulate $1000?

Keystrokes

Display

CLEAR

5.25

0.10

Equivalent periodic interest rate.

365

100

CLEAR

96.00

Weeks.

1000

_{Download from Www.Somanuals.}4_co6_{m. All Manuals Search And Download.}

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. 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.

KEYSTROKES

DISPLAY

00-

CLEAR

01-

02-

03-44 40

04-

05-

06-

07-

08-

09-

42 11

10-

11-

12-

13-

45 13

45 14

14-44 48

_{Download from Www.Somanuals.}4_co7_{m. All Manuals Search And Download.}

15-

16-44 48

17- 45 12

18-44 48

45 11

19-

20-

21-

22-

23-

24-

25-

26-

42 24

27-44 40

28-

29-

30-45 48

31-

32-45 48

33-

34-45 48

35-

36-

37-

38-

39-

40-

41-

42-

45 14

_{Download from Www.Somanuals.}4_co8_{m. All Manuals Search And Download.}

43-

44-

45-

46-

47-44 40

48-43, 33 00

REGISTERS

n: Used

i: Used

PV: Used

FV: 0

PMT: Used

R : Used

R : Purch. Adv.

R : Tax

R : Discount

R : Dep. Value

R : Dep. life

R : Factor (DB)

R : Used

R : Unused

Instructions:

1. Key in the program.

-Select the depreciation function and key in at line 26.

2. Press and press CLEAR

3. Input the following information for the purchase of the loan:

-Key in the number of years for amortization and press

-Key in the annual interest rate and press

-Key in the loan amount (purchase price) and press

-Press to find the annual payment.

4. Key in the marginal effective tax rate (as a decimal) and press

5. Key in the discount rate (as a decimal) or cost of capital and press

6. Key in the depreciable value and press

7. Key in the depreciable live and press

_{Download from Www.Somanuals.}4_co9_{m. All Manuals Search And Download.}

8. For declining balance depreciation, key in the depreciation factor (as a

percentage) and press 7.

9. Key in the total first lease payment (including any advance payments) and

press 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

. 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.

12. 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

steps 10 and 11 for the remainder of the term.

. Continue

13. After all the lease payments and expenses have been entered (steps 10 and

11), key in the lease buy back option and press

43 . If no buy back option exists, use the estimated salvage

value of the purchased equipment at the end of the term.

14. To find the net advantage of owning press

represents a net lease advantage.

2. A negative value

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 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 Maintenance Lease Payment Tax Credit Buy Back

1700+700

1700

200

_{Download from Www.Somanuals.}5_co0_{m. All Manuals Search And Download.}

200

1500

300

1700

1000

750

Keystrokes

Display

0.00

CLEAR

-10,000.00

Always use negative loan amount.

10000

1,769.84

0.48

Purchase payment.

Marginal tax rate.

.48

.05

1.05

Discounting factor.

Depreciable value.

10000

1500

8,500.00

10.00

Depreciable life.

1st lease payment.

After-tax expense.

3,400.00

1,768.00

1700

200

1700

200

1700

1000

Present value of 1st year's net

purchase.

312.36

200.43

2nd year's advantage.

1,000.00

907.03

Tax credit.

Present value of tax credit.

200

95.05

-4.38

3rd year.

4th year.

1700

200

1700

_{Download from Www.Somanuals.}5_co1_{m. All Manuals Search And Download.}

200

1700

200

1700

200

1700

200

1700

300

750

-628.09

-226.44

-309.48

-388.81

5th year.

6th year.

7th year.

8th year.

-1,034.72

-1,080.88

750.00

9th year.

10th year.

Buy back.

390.00

After tax buy back expense.

Present value.

Net lease advantage.

239.43

-150.49

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 income 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.

_{Download from Www.Somanuals.}5_co2_{m. All Manuals Search And Download.}

Profit

Sales Revenue

Variable

Costs

Break-Even Point

Fixed Costs

Loss

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

2. Key in the unit price and press

3. Key in the variable cost per unit and press

4. Press

To calculate the gross profit at a given volume:

1. Key in the unit price and press

to calculate the break-even volume.

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.

_{Download from Www.Somanuals.}5_co3_{m. All Manuals Search And Download.}

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. Key in the specified sales volume in units and press

4. Key in the variable cost per unit and press

sales price per unit.

to calculate the required

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?

Keystrokes

Display

12,000.00

Fixed cost.

12000

13.00

Sales price.

1,920.00

Break-even volume.

6.75

Find the gross profit if 2500 units are sold.

13.00

Sales price.

6.25

Profit per unit.

6.75

2500

12000

15,625.00

3,625.00

Gross profit.

If a gross profit of $4,500 is desired at a sales volume of 2500 units, what

should the sales price be?

_{Download from Www.Somanuals.}5_co4_{m. All Manuals Search And Download.}

12,000.00

16,500.00

6.60

Fixed cost.

12000

4500

2500

Sales price per unit to achieve

desired gross profit.

13.35

6.75

For repeated calculation the following HP-12C program can be used.

KEYSTROKES DISPLAY

00-

01-

02-

03-

CLEAR

04-43, 33 00

05-

06-

07-

08-

09-43, 33 00

10-

11-

12-

13-

14-

15-43, 33 00

16-

17-

18-

19-

20-

_{Download from Www.Somanuals.}5_co5_{m. All Manuals Search And Download.}

21-

22-

23-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : Unused

R : F

R : V

R : P

R : U

R : GP

R -R : Unused

1. 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. 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

c. Press

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. Press

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. Press

to calculate the required sales price.

_{Download from Www.Somanuals.}5_co6_{m. All Manuals Search And Download.}

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

Display

35,000.00

Fixed cost.

35000

8.25

12.5

8.25

12.50

0.00

Variable cost.

Sales price.

Break-even volume is between 8,235

and 8,236 units.

8,235.29

What would be the gross profit if the price is raised to $14.00 and the

sales volume is 10,000 units?

Keystrokes

Display

14.00

Sales price.

F and V are already stored.

Volume.

10,000.00

22,500.00

10000

Gross Profit.

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:

1. Key in the sales price per unit and press

2. Key in the variable cost per unit and press

_{Download from Www.Somanuals.}5_co7_{m. All Manuals Search And Download.}

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

13.00

Price per copy.

Profit per copy.

6.25

6.75

2000

25.00

Close to break-even point.

12000

13.00

6.25

Price per copy.

Profit per copy.

6.75

5000

Operating further from the breakeven

point and lesssensitive to changes in

sales volume.

1.62

12000

For repeated calculations the following HP-12C program can be used:

KEYSTROKES DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

08-

09-

CLEAR

10-43, 33 00

_{Download from Www.Somanuals.}5_co8_{m. All Manuals Search And Download.}

REGISTERS

i: Unused

n: Unused

PV: Unused

FV: Unused

PMT: Unused

R : Unused

R : F

R : V

R : P

R -R : Unused

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

leverage.

to calculate the operating

4. To calculate a new operating leverage at a different sales volume, key in

the new sales volume and press

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.

Keystrokes

Display

35,000.00

Fixed costs.

35000

8.25

Variable cost.

12.50

11.77

Sales price.

12.5

Operating leverage near break-even.

9000

Operating leverage further from

break-even.

1.70

20000

Profit and Loss Analysis

The HP-12C 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.

_{Download from Www.Somanuals.}5_co9_{m. All Manuals Search And Download.}

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%.

KEYSTROKES

DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

08-

09-

10-

CLEAR

11-43, 33 00

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-

_{Download from Www.Somanuals.}6_co0_{m. All Manuals Search And Download.}

23-

24-

25-

26-

27-

28-43, 33

29-

30-

31-

32-

33-

34-

35-

36-

37-43, 33 00

38-

39-

40-

41-

42-43, 33 00

43-

44-

45-

46-

47-43, 33 00

REGISTERS

n: Unused

PV: Unused

i: Unused

PMT: Unused

_{Download from Www.Somanuals.}6_co1_{m. All Manuals Search And Download.}

R : 100

FV: Unused

R : list price

R : % discount

R : mfg. cost

R : % op. exp.

R : % net profit

R : 1-% tax

R -R : Unused

1. Key in the program and press

press 0.

CLEAR

, then key in 100 and

2. Key in 1 and press

decimal and press

, then key in your appropriate tax rate as a

a. Key in the list price in dollars (if known) and press

b. Key in the discount in percent (if known) and press

c. Key in the manufacturing cost in dollars (if known) and press

d. Key in the operating expense in percent (if known) and press

e. Key in the net profit after tax in percent (if known) and press

4. To calculate list price:

a. Do steps 2 and 3b, c, d, e above.

b. Press

5. To calculate discount:

a. Do steps 2 and 3a, c, d, e above.

b. Press

6. To calculate manufacturing cost:

a. Do steps 2 and 3a, b, d, e, above.

b. Press 13

00.

7. To calculate operating expense:

a. Do steps 2 and 3a, b, c, e, above.

b. Press

8. To calculate net profit after tax:

a. Do steps 2 and 3a, b, c, d, above.

_{Download from Www.Somanuals.}6_co2_{m. All Manuals Search And Download.}

b. Press

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

Display

CLEAR

100.00

100

0.52

48% tax rate.

.48

11.98

35.00

2.50

List price ($).

11.98

Discount (%).

Manufacturing cost ($).

Operating expenses (%).

2.50

32.00

67.90

18.67

Net profit (%).

If manufacturing expenses increase to $3.25, what is the effect on net

profit?

3.25

Manufacturing cost.

3.25

58.26

13.66

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?

0.00

58.26

Maximum operating expense (%).

At 32% operating expense and $3.25 manufacturing cost, what should the

list price be to generate 20% net profit?

20.00

11.00

16.93

List price ($).

_{Download from Www.Somanuals.}6_co3_{m. All Manuals Search And Download.}

What reduction in manufacturing cost would achieve the same result

without necessitating an increase in list price above $11.98?

7.79

2.30

Manufacturing cost ($).

_{Download from Www.Somanuals.}6_co4_{m. All Manuals Search And Download.}

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.

KEYSTROKES

DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

08-

09-

10-

11-

12-

CLEAR

42 34

CLEAR

13-

14-

15-

16-

17-

18-

_{Download from Www.Somanuals.}6_co5_{m. All Manuals Search And Download.}

19-

20-

21-

22-

23-

24-

25-

26-

27-

28-

29-

30-

42 22

31-43, 33 00

REGISTERS

n: Unused

PV: Used

FV: 0

i: Yield

PMT: Used

R : Used

R : Sales price

R : Purchase price

R : Coupon rate

R : Capital rate

R : Income rate

R : Used

R -R : 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. Key in capital gains tax rate (as a percentage) and press

6. Key in the income tax rate (as a percentage) and press

7. Press

_{Download from Www.Somanuals.}6_co6_{m. All Manuals Search And Download.}

8. Key in the purchase date (MM.DDYYYY) and press

9. Key in the assumed sell date (MM.DDYYYY) and press

after-tax yield (as a percentage).

to find the

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. 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%)

Keystrokes

Display

10.00

Purchase price.

90.00

7.00

Selling price.

Annual coupon rate.

Capital gains tax rate.

Income tax rate.

25.00

50.00

10.01

8.53

Purchase Date.

% after tax yield.

10.011981

10.011986

Discounted Notes

A note is a written agreement to pay a sum of money plus interest at a

certain rate. Notes to not have periodic coupons, since all interest is paid

at maturity.

A discounted note is a note that is purchase below its face value. The

following HP 12C program finds the price and/or yield* (*The yield is a

reflection of the return on an investment) of a discounted note.

KEYSTROKES

DISPLAY

00-

01-

CLEAR

_{Download from Www.Somanuals.}6_co7_{m. All Manuals Search And Download.}

02-

03-

04-

05-

06-

43 26

07-

08-

09-

10-

11-

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-

43 26

23-

24-

25-

26-

27-

28-

29-

30-43, 33 00

_{Download from Www.Somanuals.}6_co8_{m. All Manuals Search And Download.}

REGISTERS

i: Unused

n: Unused

PV: Unused

FV: Unused

PMT: Unused

R : Unused

R : Mat. date

R : Settl. date

R : 360 or 360

R : redemp. value

R : dis./price

R -R : Unused

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:

a. Key in the discount rate and press

b. Press

c. Press

to calculate the purchase price.

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

b. Press 15

to calculate the yield.

c. For a new case, go to step 3.

Example 1: Calculate the price and yield on this U.S. Treasury Bill:

settlement date October 8, 1980; maturity date March 21, 1981; discount

rate 7.80%. Compute on a 360 day basis.

Keystrokes

Display

10.08

Settlement date.

10.081920

_{Download from Www.Somanuals.}6_co9_{m. All Manuals Search And Download.}

3.21

Maturity dtae.

360 day basis.

Redemption value per $100.

Discount rate.

Price.

3.211981

360

360.00

100.00

7.80

100

7.8

96.45

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

6.25

Settlement date.

Maturity dtae.

360 day basis.

Redemption value per $100.

Price.

6.251980

9.10

9.101980

360

360.00

101.33

99.45

8.84

101.33

99.45

Yield.

_{Download from Www.Somanuals.}7_co0_{m. All Manuals Search And Download.}

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 as follows.

1. Press

2. Key in the first m data points (where m is the number of data points in the

average) and press after each entry.

CLEAR

3. Press

4. Key in the oldest (first value) entered in step 2 and press

5. Key in the newest data point (m + 1) and press

to obtain the first average.

6. Press

to obtain the next value of the moving average.

7. Repeat steps 4 through 5 for the remaining data.

Example: An electronics sales firm wished to calculate a 3-month moving

average for the dollar volume of components sole each month. Sales for

the first six months of this year were:

January

February

March

April

$211,570

112,550

190,060

131,760

300,500

271,120

May

June

Keystrokes

Display

_{Download from Www.Somanuals.}7_co1_{m. All Manuals Search And Download.}

0.00

CLEAR

211570

1.00

2.00

112550

190060

3.00

171,393.33

2.00

3-month average for March.

3-month average for April.

3-month average for May.

3-month average for June.

211570

131760

3.00

144,790.00

2.00

112550

300500

3.00

207,440.00

2.00

190060

271120

3.00

234,460.00

For repeated calculations the following HP 12C program can be used for

up to a 12 element moving average:

KEYSTROKES

DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

08-

09-

CLEAR

_{Download from Www.Somanuals.}7_co2_{m. All Manuals Search And Download.}

10-

11-

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-

23-

24-

25-

26-45 48

27-

28-

29-45 48

30-44 48

31-

32-45 48

33-44 48

34-

35-

36-

_{Download from Www.Somanuals.}7_co3_{m. All Manuals Search And Download.}

37-

38-

44 --

39-43, 33 01

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : m

R : X

.2 12

R -R : Unused

.3 .4

*At step 38, m=number of elements in the moving average, i.e. fir a 5

element moving average line 38 would be 5 and for a 12-element

average line 38 would be

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, form line 01 until you

reach a

superscripted with the number of elements you desire. Key in

this line, then skip the reset of the program down to line 35. Then key in

lines 35 through 39, being sure to specify the register number at line 38,

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,

5).

m becomes the displayed line 17,

To run the program:

1. Key in the program.

2. Press

CLEAR

. Key in the number of elements, m, and press 0.

3. Key in the second data point and press

4. Key in the second data point and press

_{Download from Www.Somanuals.}7_co4_{m. All Manuals Search And Download.}

5. Continue as above, keying in and storing each data point in its appropriate

6. Press

to calculate the first moving average.

7. Key in the next data point and press

average.

to calculate the next moving

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:

KEYSTROKES

DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

08-

09-

10-

11-

CLEAR

12-43, 33 01

Keystrokes

CLEAR

Display

0.00

3.00

211,570.00

112,550.00

190,060.00

211570

112550

190060

_{Download from Www.Somanuals.}7_co5_{m. All Manuals Search And Download.}

171,393.33

144,790.00

207,440.00

234,460.00

3-month average for March.

3-month average for April.

3-month average for May.

3-month average for June.

131760

300500

271120

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

ration 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 ration 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 can

also be programmed to calculate monthly seasonal variations using a

centered 12 month moving averages. Programs for both of these

calculations are represented here:

An HP 12C program to calculate the quarterly seasonal variations based

on a centered 4-point moving average is:

KEYSTROKES

DISPLAY

00-

CLEAR

01-

02-

03-

04-

05-

06-

07-

_{Download from Www.Somanuals.}7_co6_{m. All Manuals Search And Download.}

08-

09-

10-

11-

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-

23-

24-

25-43, 33 01

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : n

R : X

R -R : Unused

1. Key in the program.

2. Press CLEAR

3. Key in the quarterly sales figures starting with the first quarter:

a. Key in 1st quarter sales and press 1.

_{Download from Www.Somanuals.}7_co7_{m. All Manuals Search And Download.}

b. Key in 2nd quarter sales and press

c. Key in 3rd quarter sales and press

d. Key in 4th quarter sales and press

e. Key in the 1st quarter sales for the next year and press

4. Press

the 3rd quarter of the first year.

5. Press to calculate the seasonal variation for this quarter.

to calculate the centered moving average for

6. Key in the next quarter's sales and press

average for the next quarter.

to calculate the moving

7. Press

to calculate the seasonal variation.

8. Repeat steps 6 and 7 for the balance of the data.

Example: Econo-Wise Home Appliance Company had quarterly sales for

the years 1978 thru 1980 as follows:

Sales (IN $K)

Quarterly

1978

1st

397

2nd

376

3rd

460

4th

501

1979

1980

455

513

390

434

530

562

560

593

Find the centered 4-quarter moving average and seasonal variation factor

for each quarter.

Keystrokes

Display

0.00

CLEAR

397.00

376.00

460.00

501.00

455.00

397

376

460

501

455

Centered 4-element average for

3rd quarter, 1978 seasonal

variation factor.

440.75

104.37

_{Download from Www.Somanuals.}7_co8_{m. All Manuals Search And Download.}

449.75

111.40

460.25

98.86

4th quarter, 1978.

1st quarter, 1979.

2nd quarter, 1979.

3rd quarter, 1979.

4th quarter, 1979.

1st quarter, 1979.

2nd quarter, 1980.

390

530

560

513

434

562

593

476.38

81.87

490.00

107.94

503.75

111.17

513.25

99.95

521.38

83.24

Now average each quarter's seasonal variation for the two years?

Keystrokes Display

0.00

CLEAR

98.86

99.95

1.00

2.00

1st quarter average seasonal

variation, %.

99.41

0.00

1.00

2.00

CLEAR

81.87

83.24

2nd quarter average seasonal

variation, %.

82.56

0.00

1.00

2.00

CLEAR

104.37

107.94

3rd quarter average seasonal

variation, %.

106.16

_{Download from Www.Somanuals.}7_co9_{m. All Manuals Search And Download.}

0.00

1.00

2.00

CLEAR

111.4

111.17

4th quarter average seasonal

variation, %.

111.29

An HP-12C program to calculate a centered 12-month moving average

and seasonal variation factor is as follows:

KEYSTROKES

DISPLAY

00-

CLEAR

01-

02-2

03-

04-

05-

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

16-

17-

18-

19-

_{Download from Www.Somanuals.}8_co0_{m. All Manuals Search And Download.}

20-

21-

22-

23-

24-

25-

26-

27-

28-45 48

29-

30-

31-45 48

32-44 48

33-

34-45 48

35-44 48

36-

37-45 48

38-44 48

39-

40-

41-

42-

43-

44-

45-

46-

47-

_{Download from Www.Somanuals.}8_co1_{m. All Manuals Search And Download.}

48-44 48

48-43, 33 01

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : n

R : X

1. Key in the program.

2. Press CLEAR

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

Key in the 2nd month and press

Key in the 10th month and press

Key in the 13th month and press

2, etc.,

0, etc.,

5. Press

the 7th month.

6. Press to calculate the seasonal variation for that month.

to calculate the centered moving average for

7. Key in the value for the next month (14th) and press

moving average for the next month (8th).

to calculate the

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

_{Download from Www.Somanuals.}8_co2_{m. All Manuals Search And Download.}

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 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.

KEYSTROKES

DISPLAY

00-

01-

CLEAR

43 23

02-44 40

03-

04-

43 23

05-44 40

06-

07-

43 23

08-44 40

09-

10-44 40

11-

12-43, 33 00

13-

14-

15-

16-

17-

_{Download from Www.Somanuals.}8_co3_{m. All Manuals Search And Download.}

18-

19-

20-

21-

22-

23-

24-

25-

26-

27-

28-

29-

30-

31-

32-

33-

34-

35-

36-

37-

38-

39-

40-

41-

42-

43 22

43-

44-

45-

_{Download from Www.Somanuals.}8_co4_{m. All Manuals Search And Download.}

46-

47-

48-

49-

50-

51-

52-

53-

54-

55-

56-

57-

58-

59-

60-

61-

62-

63-

64-

65-

66-

67-

68-

69-

70-

43 22

71-43, 33 62

REGISTERS

i: Unused

n: Unused

_{Download from Www.Somanuals.}8_co5_{m. All Manuals Search And Download.}

PV: Unused

FV: Unused

PMT: Unused

R : Unused

R : S

R : n

R : a

R : b

R : c

R -R : 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. 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

resultant display is the curve constant "a". Constants "b" and "c" may be

obtained by pressing 6 and 7 respectively.

. The

8. To calculate a projected value, key in the number of the period and press

9. Repeat step 8 for each period desired.

Example: The X-presso Company marked a revolutionary new coffee

brewing machine in 1968. 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 Sales($K)

151

188

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

_{Download from Www.Somanuals.}8_co6_{m. All Manuals Search And Download.}

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 Group Group

III

151

282

188

260

322

340

Keystrokes

CLEAR

Display

0.00

18.00

151.00

1.00

151

282

41.00

188.00

2.00

188

322

49.00

260.00

3.00

260

340

Total number of entries.

0.004

0.65

373.92

349.09

363.36

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

373.92

202.60

100

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.

_{Download from Www.Somanuals.}8_co7_{m. All Manuals Search And Download.}

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 =

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.

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 program:

KEYSTROKES

DISPLAY

00-

CLEAR

01-

02-

03-

04-

05-

06-

07-44 40

08-

09-

10-

11-

12-

13-

43 36

_{Download from Www.Somanuals.}8_co8_{m. All Manuals Search And Download.}

14-

15-

16-

17-

18-

19-

20-

21-

22-

23-

24-

25-

26-

27-

28-

29-

30-

31-

32-

33-

34-

35-

36-

37-

38-

39-

40-

41-

_{Download from Www.Somanuals.}8_co9_{m. All Manuals Search And Download.}

42-

43-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : α

R : 1-α

R : S

t-1

R : T

R : Σe

t-1

R : D

R :

t+1

R -R : Unused

Selecting the "best" smoothing constant (α):

1. Key in the program and press

2. Key in the number 1 and press

3. Key in the "trial " and press

CLEAR

4. Key in the first historical value (X ) and press

5. Key in the second historical value (X ) and press

. The

result is the error between the forecast value (

) and the true value

t+1

)

t+1

6. Press

; the display shows the next forecast (

t+2

7. Optional: Press

demand.

5 to display the smoothed estimate of current

8. Continue steps 5 and 6 for X , X , ... X until all historical values have

been entered. When doing step 5 merely key in the value and press

(do not press

9. Press

6).

4. This value represents the cumulative forecasting error

(Σe ). Record the value and the following additional values; press

(α),

2 (smoothed average S ),

3 (trend T ) and

t-1

(forecast

t+1

10. Press

CLEAR

11. Repeat steps 2 through 10 until a "best" α is selected based on the lowest

cumulative forecasting error (Register 4).

Forecasting:

_{Download from Www.Somanuals.}9_co0_{m. All Manuals Search And Download.}

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:

Key in the smoothed average S and press

t-1

Key in the trend T and press

t-1

Key in the forecast

and press

. The output is the error in

t+1

4. Key in the current data value and press

forecasting the value just entered.

5. Press

period.

. The displayed value represents the forecast for the next

6. Record the following values:

and 6 (D ) for use as initial values in the next forecast. You may

also wish to record

0 (α),

2 (S ),

3 (T

)

t-1

t+1

5 (D ).

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):

Keystrokes

Display

0.00

CLEAR

1.00

0.50

22.00

0.00

23.00

23.25

25.25

23.69

27.13

25.95

_{Download from Www.Somanuals.}9_co1_{m. All Manuals Search And Download.}

23.61

0.50

Cumulative error (Σe ).

Smoothing constant (a).

Smoothing average (S ).

25.11

0.42

t-1

Trend (T ).

t-1

Last forecast (D ).

25.95

t+1

The procedure is repeated for several α's.

Smoothing Constant (α)

.5 .1 .25 .2

Cumulative Error (Σe )

23.61 25.14 17.01 18.03

For the selected α = .25

= 24.28

= 0.34

t+1

t-1

= 25.64

t+1

Forecasting:

Keystrokes

CLEAR

Display

0.00

1.00

0.75

.25

0.75

24.28

0.34

24.28

.34

25.64

0.36

25.64

26.16

Forecast for month 9, (

t+1

Expected usage for current (month

25.80

8) period, (Smoothed D ).

0.25

24.71

0.36

Record for initial values when

month 9 actual figures become

available.

26.16

Note: At least 4 periods of current data should be entered before

forecasting is attempted.

_{Download from Www.Somanuals.}9_co2_{m. All Manuals Search And Download.}

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.

CALCULATE

GIVEN

KEYSTROKES

Selling Price Cost & Markup

Key in cost,

(in %),

, key in markup (in %),

, key in margin

Selling Price Cost & Margin

Selling Price & Key in selling price,

, key in

Cost

Markup

markup (in %0,

Selling Price & Key in selling price,

Cost

Margin

margin (in %0,

Cost and

Selling Price

Markup

Margin

Key in cost,

, key in selling price,

Key in margin,

Margin

Selling Price &

Cost

Key in selling price,

Key in markup,

, key in cost,

Markup

Example 1: If the cost of an item is $160 and the margin is 20%, what is

the selling price? What is the markup?

Keystrokes

Display

160.00

Cost.

160

20.00

200.00

20.00

25.00

Margin (%).

Selling price.

Markup (%).

_{Download from Www.Somanuals.}9_co3_{m. All Manuals Search And Download.}

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

21.00

Selling price.

Markup (%).

Cost.

50.00

14.00

50.00

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.

KEYSTROKES

DISPLAY

00-

01-

CLEAR

02-43, 33 04

03-

04-

05-

06-

07-

08-

09-

10-

11-

12-

13-

43 36

14-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

PMT: Unused

_{Download from Www.Somanuals.}9_co4_{m. All Manuals Search And Download.}

R -R : Unused

FV: Unused

1. Key in program.

2. To calculate selling price, given the markup, key in the cost, press

, key in the markup and press 00

3. To calculate cost, given the markup, key in the selling price, press

, key in the markup and press 00

4. To calculate selling price, given the margin, key in the cost, press

, key in the margin and press 03

5. To calculate cost given the margin, key in the selling price, press

, key in the margin and press 03

6. To calculate markup from the margin, key in the margin and press

7. To calculate margin from the markup, key in the markup and press

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?

Keystrokes

Display

38.00

Selling price.

Markup (%).

Cost.

30.00

26.60

42.86

26.60

39.90

Markup (%).

Cost.

26.6

New selling price.

Calculations of List and Net prices With

Discounts

If it often useful to be able to quickly calculate list or net price when the

other price and a series of discount rates are known. Alternatively, if the

_{Download from Www.Somanuals.}9_co5_{m. All Manuals Search And Download.}

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

2. Key in the first discount (as a percentage) and press

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

5. To calculate the net price, key in the list price and press

6. To calculate an unknown discount rate, immediately after doing step 3

(display should show 1.00), key in the net price, press

in the list price.

and key

7. Press

100

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?

Keystrokes

Display

1.00

1.45

3.28

0.49

10.51

3rd discount rate (%).

100

The following program for the HP 12C will be helpful in performing the

calculations:

KEYSTROKES

DISPLAY

00-

01-

02-

CLEAR

03-

_{Download from Www.Somanuals.}9_co6_{m. All Manuals Search And Download.}

04-

05-44 20

06-43, 33 00

07-

08-

09-

10-

11-

12-

13-

14-

15-

16-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : Unused

R : R D xD ...D

R -R : 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

6. To calculate the net price, key in the list price and press

7. To calculate the unknown discount rate, key in the net price, press

, key in the list price and press 07

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?

Keystrokes

Display

_{Download from Www.Somanuals.}9_co7_{m. All Manuals Search And Download.}

1.00

0.52

0.95

1.45

3.28

10.51

3rd discount rate (%).

Include 3rd discount rate in

calculation.

0.89

1.66

New net price.

3.75

_{Download from Www.Somanuals.}9_co8_{m. All Manuals Search And Download.}

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=Ae . 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

Examples of these are given below.

The procedure is as follows:

1. Press

CLEAR

2. For each input pair of values, key in the y-value and press

, key in

the corresponding x-value and press

3. After all data pairs are input, press

coefficient (between ln y and x).

to obtain the correlation

4. Press 1

above.

to obtain A in the equation

_{Download from Www.Somanuals.}9_co9_{m. All Manuals Search And Download.}

5. Press

6. Press

to obtain B.

1 to obtain the effective growth rate (as a decimal).

7. To make a y-estimate, key in the x-value and press

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

Price

Year

1976(1)

1977(2)

1978(3)

1979(4)

1980(5)

1981(6)

1982(7)

51.5

53.75

122.5

210

Keystrokes

CLEAR

Display

1.00

First data pair input.

2.00

3.00

4.00

5.00

6.00

Second data pair input.

Third data pair input.

Fourth data pair input.

Fifth data pair input.

Sixth data pair input.

51.5

53.75

122.5

210

Correlation coefficient (between ln

y and x).

0.95

27.34

0.31

0.36

Effective growth rate.

Projected price at end of year 7

(1982).

232.35

For repeated use of this routine, the following HP-12C program will be

useful.

KEYSTROKES

DISPLAY

_{Download from Www.Somanuals}1_.c0_om0_{. All Manuals Search And Download.}

00-

01-

02-

03-

04-

CLEAR

43 23

05-43, 33 00

06-

07-

08-

09-

10-

11-

12-

13-

43 22

14-

15-

16-

17-

18-

19-

20-

43 22

43 23

21-

22-

23-

43 22

24-

25-

26-

43 22

27-43, 33 00

_{Download from Www.Somanuals}1_.c0_om1_{. All Manuals Search And Download.}

REGISTERS

i: Unused

n: Unused

PV: Unused

FV: Unused

PMT: Unused

R : Unused

R : n

R : Σx

R : Σy

R : Σx

R : Σxy

R : Σy

R -R : Unused

1. Key in the program and press

CLEAR

2. For each input pair of values, key in the y-value and press

, key

in the corresponding x-value and press

3. After all data pairs are input, press

to obtain the

correlation coefficient (between ln y and x).

4. Press

5. Press

6. Press

to obtain A.

to obtain B.

to obtain the effective growth rate as a decimal.

7. To make a y-estimate, key in the x-value and press

estimates, key in the x-value and press

. For subsequent

8. For a different set of data, press

Keystrokes Display

CLEAR

and go to step 2.

1.00

First data pair input.

2.00

3.00

4.00

5.00

6.00

Second data pair input.

Third data pair input.

Fourth data pair input.

Fifth data pair input.

Sixth data pair input.

51.5

53.75

122.5

210

Correlation coefficient (between ln

y and x).

0.95

_{Download from Www.Somanuals}1_.c0_om2_{. All Manuals Search And Download.}

27.34

0.31

0.36

Effective growth rate.

Projected price at the end of year 7

(1982).

232.35

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.

The procedure is as follows:

_{Download from Www.Somanuals}1_.c0_om3_{. All Manuals Search And Download.}

1. Press

CLEAR

2. Key in the first y-value and press

. Key in the first x-value and

press

. Repeat this step for each data pair.

3. After all data pairs are input, press

to obtain the correlation

coefficient (between y and ln x).

4. Press 1

5. Press

to obtain A in the equation above.

to obtain B.

6. To make a y-estimate, key in the x-value and press

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 fit by a logarithmic cure 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 eighth months?

Month

Cumulative Sales (units)

1431 3506 5177 6658 7810 8592

Keystrokes

Display

CLEAR

1.00

First pair data input.

1431

3506

2.00

3.00

4.00

5.00

6.00

Second pair data input.

Third pair data input.

Forth pair data input.

Fifth pair data input.

Sixth pair data input.

5177

6658

7810

8592

_{Download from Www.Somanuals}1_.c0_om4_{. All Manuals Search And Download.}

Correlation coefficient (between y

and ln x).

0.99

1,066.15

4,069.93

Value of A.

Value of B.

Total units sold by end of eighth

month.

9,529.34

Power Curve Fit

Another method of analysis is the power curve or geometric curve. The

equation of the power curve is y = Ax , and the values for A and B are

computed by calculations similar to linear regression. Some examples of

power curves are shown below.

The following keystrokes fit a power curve according to the equation ln y =

ln A + B(ln x):

1. Press

CLEAR

2. Key in the first y-value and press

. Key in the first x-value and

press

. Repeat this step for all data pairs.

3. Press

, to obtain the correlation coefficient (between ln y

and ln x).

4. Press 0

5. Press 1

to obtain A in the above equation.

to obtain B.

6. To make a y-estimate, key in the x-value and press

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 hasfallen, he might have released a rock from various

_{Download from Www.Somanuals}1_.c0_om5_{. All Manuals Search And Download.}

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.

2.5

3.5

4.5

t (pulses)

h (feet)

130

150

Find the power curve formulas that best expresses h as a function of t (h =

At ).

Keystrokes

Display

CLEAR

1.00

First pair data input.

2.5

3.5

130

2.00

3.00

4.00

5.00

Second pair data input.

Third pair data input.

Fourth pair data input.

Fifth pair data input.

150

4.5

Correlation coefficient (between In

y and ln x).

1.00

7.72

Value of A.

Value of B.

1.99

The formula that best expresses h as a function of t is

h = 7.72t^1.99

We know, as Galileo did not, that in fact h is proportional to t .

Standard Error of the Mean

The standard error of the mean is a measure of how reliable the mean of a

sample ( ) is as an estimator of the mean of the population from which the

sample was drawn.

To calculate the standard error of the mean:

_{Download from Www.Somanuals}1_.c0_om6_{. All Manuals Search And Download.}

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

entered all of the values.

. Continue until you have

4. Press

to obtain the mean of the x-values.

to obtain the standard error of the

5. Press

mean of the x-values.

6. Alternatively, press

standard error for the mean of the y-values.

to obtain the

Example: A sample of 6 one-bedroom apartment rentals reveals that one

rents for $190 per month unfurnished, one rents for $200 pre 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?

Keystrokes

Display

CLEAR

190

205

220

200

205

6.00

Total number of inputs.

216

206.00

10.86

4.43

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, ... , x_n

with respective frequencies

f₁, f₂, ... , f_n

_{Download from Www.Somanuals}1_.c0_om7_{. All Manuals Search And Download.}

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

display shows the number of data points entered.

4. Repeat steps 2 and 3 for each data point.

. The

5. To calculate the mean (average) press

6. Press

7. Press

to find the standard deviation.

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?

Keystrokes

Display

CLEAR

1.00

First data pair entered.

190

195

2.00

Second data pair entered.

Third data pair entered.

Fourth data pair entered.

Average monthly rent.

200

3.00

206

4.00

199.44

5.97

0.37

Standard deviation.

Standard error of the mean.

Use the following HP-12C program for the previous example:

KEYSTROKES DISPLAY

_{Download from Www.Somanuals}1_.c0_om8_{. All Manuals Search And Download.}

00-

CLEAR

01-44 40

02-

03-

04-43, 33 00

05-

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

43 48

43 21

16-43, 33 00

REGISTERS

n: Unused

i: Unused

PMT: Unused

R : Σf

PV: Unused

FV: Unused

R : Σf

R : Σf x

i i

R : Σx

R : Σf x

i i

R : Σx

R : Σf x

i i

R -R : Unused

1. Key in the program.

2. Press CLEAR

3. Key in the first value and press

_{Download from Www.Somanuals}1_.c0_om9_{. All Manuals Search And Download.}

4. Key in the respective frequency and press

number of data points entered.

. The display shows the

5. Repeat steps 3 and 4 for each data point.

6. To calculate the mean, press

to find the standard deviation.

to find the standard error of the mean.

7. Press

8. Press

9. For a new case, go to step 2.

Keystrokes Display

CLEAR

1.00

First data pair.

190

195

2.00

3.00

4.00

Second data pair.

200

Third data pair.

206

Total number of data sets.

199.44

5.97

Average monthly rent (maen).

Standard deviation.

0.37

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 (O_i) and the expected frequencies (E_i) are

significant, or whether they may reasonable be attributed to chance. The

formula generally used is:

_{Download from Www.Somanuals}1_.c1_om0_{. All Manuals Search And Download.}

(O_i– E_i)

---------------------

E_i

x²

∑

i = 1

If there is a close agreement between the observed and expected

frequencies, x will be small. If the agreement is poor, x will be large.

The following keystrokes calculate the x statistic:

1. Press

CLEAR

2. Key in the first O value and press

3. Key in the first E value and press

4. Repeat steps 2 and 3 for all data pairs. The x value is displayed.

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

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 R₀each time.)

Keystrokes

Display

CLEAR

1.25

1.70

_{Download from Www.Somanuals}1_.c1_om1_{. All Manuals Search And Download.}

2.95

3.40

4.20

5.00

The number of degrees of freedom is (n-1). Since n = 6, the degrees of

freedom = 5.

Consulting statistical tables, you look up x to a 0.05 significance level

with 5 degrees of freedom, and see that x _0.05,5= 11.07. Since x = 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.

KEYSTROKES

DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

CLEAR

08-43, 33 00

REGISTERS

n: Unused

PV: Unused

FV: Unused

i: Unused

PMT: Unused

R : E

R -R : Unused

_{Download from Www.Somanuals}1_.c1_om2_{. All Manuals Search And Download.}

1. Key in the program.

2. Press CLEAR

3. Key in the first O value and press

4. Key in the first E value and press

5. Repeat steps 3 and 4 for all data pairs. The x value is displayed.

6. For a new case, go to step 2.

Keystrokes

Display

CLEAR

1.25

1.70

2.95

3.40

4.20

5.00

Normal Distribution

The normal (or Gaussian) distribution is an important tool in statistics and

business analysis. The following HP-12C 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.

_{Download from Www.Somanuals}1_.c1_om3_{. All Manuals Search And Download.}

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.

KEYSTROKES

DISPLAY

00-

01-

CLEAR

02-

03-

_{Download from Www.Somanuals}1_.c1_om4_{. All Manuals Search And Download.}

04-

05-

_{Download from Www.Somanuals}1_.c1_om5_{. All Manuals Search And Download.}

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-

23-

24-

25-

26-

27-

28-

43 22

29-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : x

R -R : Unused

1. Key in program.

2. Key in x and press

to computed Q(x).

_{Download from Www.Somanuals}1_.c1_om6_{. All Manuals Search And Download.}

3. Repeat step 2 for each new case.

Example: Find Q(x) for x = 1.18 and x = 2.1.

Keystrokes

Display

0.12

0.02

Q(1.18)

Q(2.1)

1.18

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 (S_xy) or a population (S'_xy) as follows:

S_xy= r * s_x* s_y

S'_xy= r * s'_x* s'_y

The following procedure finds the covariance of a sample (S_xy) and of a

population (S' ):

1. Press

CLEAR

2. Key in the y-values and press

3. Key in the x-values and press

. Repeat steps 2 and 3 for all data

pairs.

4. Press

to obtain the value of S .

5. Press

1 1

to obtain S' .

Example 1: Find the sample covariance (S_xy) and population covariance

(S' ) for the following paired variables:

Keystrokes

CLEAR

Display

7.00

Total number of entries.

_{Download from Www.Somanuals}1_.c1_om7_{. All Manuals Search And Download.}

-354.14

-303.55

1 1

Try the previous example using the following HP-12C program:

KEYSTROKES

DISPLAY

00-

01-

CLEAR

02-43, 33 00

03-

04-

05-

06-

07-

08-

09-

43 48

10-

11-

12-

13-

14-

15-

16-43, 33 00

REGISTERS

_{Download from Www.Somanuals}1_.c1_om8_{. All Manuals Search And Download.}

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : Unused

R : n

R : Σx

R : Σy

R : Σx

R : Σxy

R : Σy

R -R : Unused

1. Key in the program.

2. Press CLEAR

3. Key in the y-value and press

4. Key in the x-value and press

. Repeat steps 3 and 4 for all data pairs.

5. Press

6. Press

to obtain S' .

. to obtain the value of S .

7. For a new case, go to step 2.

Keystrokes Display

CLEAR

7.00

Total number of entries.

-354.14

-303.55

Permutation

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:

_{Download from Www.Somanuals}1_.c1_om9_{. All Manuals Search And Download.}

(m – n)!

-------------------

mPn =

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

00-

01-

02-

03-

04-

05-

06-

07-

08-

CLEAR

43 36

09-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : n

R -R : Unused

1. Key in the program.

2. Key in m and press

3. Key in n and press

to calculate P .

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?

Keystrokes

Display

_{Download from Www.Somanuals}1_.c2_om0_{. All Manuals Search And Download.}

P .

5,040.00

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:

------------------------

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.

KEYSTROKES

DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

08-

09-

10-

11-

CLEAR

43 36

12-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

PMT: Unused

_{Download from Www.Somanuals}1_.c2_om1_{. All Manuals Search And Download.}

R : n

FV: Unused

R -R : Unused

1. Key in the program.

2. Key in m and press

3. Key in n and press

to calculate C .

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?

Keystrokes

Display

C .

35.00

Random Number Generator

This HP-12C program calculates uniformly distributed pseudo-random

numbers u in the range

0 < u < 1.

The following method is used:

•

u + 1 = fractional part of (997 u )

•

where i = 0, 1, 2, ...

u = 0.5284163* (seed), *Other seeds may be selected but the quotient of

(seed x 10 ) divided by two or five must not be an integer. Also, it would be

wise to statistically test other seeds before using them. )

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.

KEYSTROKES

DISPLAY

00-

CLEAR

01-

02-

_{Download from Www.Somanuals}1_.c2_om2_{. All Manuals Search And Download.}

03-

04-

05-

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

16-

43 24

17-43, 33 10

REGISTERS

n: Unused

i: Unused

PMT: Unused

R : U

PV: Unused

FV: Unused

R -R : 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

_{Download from Www.Somanuals}1_.c2_om3_{. All Manuals Search And Download.}

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

2. Key in the annual interest rate and press

3. Key in the term of the loan (in years) and press

and press

CLEAR

4. Key in the purchase prices and press

5. Key in the percent down and press

6. Key in the tax rate in dollars per thousand and press

12000

. ( 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 3/4 % interest

rate on a 35 year loan with 10% down?

Keystrokes

Display

0.90

Monthly interest rate.

CLEAR

10.75

420.00

Months of loan.

Purchase price.

Mortgage balance.

65,000.00

58,800.00

65000

-135.42

Approximate monthly taxes.

12000

_{Download from Www.Somanuals}1_.c2_om4_{. All Manuals Search And Download.}

-672.16

Approximate monthly payment.

The following HP-12C program may be used instead of the above.

KEYSTROKES DISPLAY

00-

01-

02-

03-

04-

05-

06-

07-

08-

09-

10-

11-

CLEAR

43 36

12-

13-

14-

15-

16-

17-

18-

19-

20-

21-

22-43, 33 00

REGISTERS

_{Download from Www.Somanuals}1_.c2_om5_{. All Manuals Search And Download.}

n: Term

PV: Loan

FV: 0

i: Interest

PMT: Loan PMT

R : Unused

R : Purch. Price

R : % Down

R : Tax rate

R -R : Unused

1. Key in the program.

2. Press

CLEAR

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

8. To calculate the approximate monthly payment, press

9. For a new case, store only the new variables by performing steps 3 thru 7

as needed. Press for the new approximate monthly payment.

Example: Solve the previous example using the HP-12C program..

Keystrokes

Display

CLEAR

10.75

0.90

Monthly interest.

420.00

65,000.00

10.00

Months of loan.

Purchase price.

65000

Percent down.

25.00

Tax rate per thousand.

Approximate monthly payment.

-672.16

What would the approximate payment be if the loan was at 10% interest?

10 -638.33 Approximate monthly payment.

What if the down payment is increased to 20%?

20 -582.45 Approximate monthly payment.

_{Download from Www.Somanuals}1_.c2_om6_{. All Manuals Search And Download.}

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. Required data are: the

years to retirement, the total annual investment, the compound annual

interest rate of the investment, and an assumed tax rate which would be

paid on a similar non taxfree investment. This program calculates:

1. The future cash value of the tax-free investment.

2. The total cash paid in.

3. The total dividends paid.

4. The future value of the investment at retirement, assuming that after

retirement you withdrew the money at a rate which causes the money to

be taxed at 1/2 the rate at which it would otherwise have been taxed during

the pay in period.

5. The diminished purchasing power assuming a given annual inflation rate.

6. The future value of a comparable taxable investment.

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 fig-

ures as possible for maximum accuracy.

The assumed 10% annual inflation rate may be changed by modifying the

program at lines 19 and 20.

The assumed tax rate used to calculate the after tax value of the tax-free

investment may be changed by modifying the program at line 9.

KEYSTROKES

DISPLAY

00-

01-

CLEAR

45 11

45 14

02-

03-

04-

05-

_{Download from Www.Somanuals}1_.c2_om7_{. All Manuals Search And Download.}

06-

07-

08-

09-

10-

11-

12-

13-

14-

15-

45 15

16-

17-

18-

19-

20-

21-

22-

23-

24-

45 11

25-

26-

27-

45 12

28-

29-

30-

31-

32-

33-

34-43, 33 17

_{Download from Www.Somanuals}1_.c2_om8_{. All Manuals Search And Download.}

REGISTERS

n: Years

PV: 0

i: Used

PMT: Yearly Pmt

R : Unused

FV: Used

R : Tax %

R -R : 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

CLEAR

and press

5. Key in the interest rates as a percentage and press

6. Key in the annual payment and press

7. Press

8. Press

9. Press

10. Press

to calculate the future value of the tax free investment.

to compute the total cash paid in.

to compute the total dividends paid.

to compute the future value when, after retirement, money is

withdrawn at a rate causing the tax rate to equal 1/2 the rate paid during

the pay in period.

11. Press

to compute the diminished purchasing power, in terms of

today's dollars, of the future value assuming a 10% annual inflation rate.

12. Press

to compute the future value of an ordinary tax investment.

to compute the diminished purchasing power of the ordinary

13. Press

tax investment.

Example: Assuming a 35 year investment period with a dividend rate of

8.175% and a tax rate of 40%.

1. If you invest $1500 each year in a tax free account, what will its value be at

retirement?

2. How much cash will be paid in?

3. What will be the value of the earned dividends?

4. After retirement, if you withdrew 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?

5. What is the diminished purchasing power of that amount, in today's

dollars, assuming 10% annual inflation?

_{Download from Www.Somanuals}1_.c2_om9_{. All Manuals Search And Download.}

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

CLEAR

Display

40.00

Tax rate.

35.00

Years to retirement.

Dividend rate.

8.18

8.175

1500

-1,500.00

290,730.34

-52,500.00

238,230.34

232,584.27

8,276.30

139,360.09

Annual payment.

Future value at retirement.

Cash Paid in.

Earned dividends.

After-tax value.

Diminished purchasing power.

Tax-paid cash at retirement.

Purchasing power of tax-paid cash

at retirement.

4,959.00

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.

*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.

Notes:

_{Download from Www.Somanuals}1_.c3_om0_{. All Manuals Search And Download.}

•

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.

KEYSTROKES

DISPLAY

00-

01-

02-

CLEAR

43 24

43 35

03-

04-43, 33 15

05-

06-

07-

08-

09-

10-

11-

43 25

43 36

43 24

12-

13-

14-

15-

16-

17-

18-

19-

43 25

43 35

20-43, 33 38

21-44 30

22-

23-

24-

_{Download from Www.Somanuals}1_.c3_om1_{. All Manuals Search And Download.}

25-44 40

26-

27-

28-

29-44 40

30-

31-

32-44 40

33-

34-

35-

36-

43 36

37-43, 33 01

38-

39-

40-

41-

42-

43-44 40

44-

45-

46-

47-43, 33 01

48-

49-

50-

51-

52-

_{Download from Www.Somanuals}1_.c3_om2_{. All Manuals Search And Download.}

53-

54-

55-

56-

57-

58-

59-

60-

61-

62-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : ΣPV

R : ΣDIV

R : ΣOrig. Val.

R : ΣP S β

R : Flag

i i

R : P

R : XXX.ND

R : S

R -R : Unused

Instructions:

1. Key in the program.

2. Initialize the program by pressing

CLEAR

3. Key in the number of shares of a stock and press

4. Key in the initial purchase of the stock and press

5. Key in the beta coefficient of the stock and press

6. Key in the annual dividend of the stock and press

7. Key in the present price of the stock and press

show the percent change in the stock value.

. The display will

8. Repeat steps 3 through 7 until all the stocks are entered.

_{Download from Www.Somanuals}1_.c3_om3_{. All Manuals Search And Download.}

9. Next, to evaluate the entire portfolio, press

48.

10. Press

11. Press

12. Press

13. Press

14. Press

to see the initial portfolio value.

to see the present portfolio value.

to see the percent change in value.

to see the total yearly dividend.

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.

Example: Evaluate the following portfolio:

Number of

Held

Initial

Purchase

Price

Beta

Annual

Present

Stock

Coefficient Dividend Market Price

100

200

25 5/8

30 1/4

89 7/8

65 1/4

$1.70

$2.10

$4.55

$3.50

27 1/4

33 1/2

96 1/8

64 3/8

Int'l Heartburn

P. D. Q.

1.2

1.3

Datacrunch

N.W. Sundial

500

Keystrokes

CLEAR

100

Display

0.00

100.00

1.00

Int'l Heartburn

25.58

0.80

1.70

27.14

200

6.34

Percent change in Stock's value.

P. D. Q.

200.00

1.00

30.14

1.2

1.20

2.10

33.12

10.74

50.00

Percent change in Stock's value.

Datacrunch

_{Download from Www.Somanuals}1_.c3_om4_{. All Manuals Search And Download.}

1.00

1.30

4.55

6.95

500.00

1.00

0.60

3.50

-1.34

89.78

1.3

4.55

96.18

500

Percent change in Stock's value.

N. W. Sundial

65.14

3.50

64.38

Percent change in Stock's value.

Original value.

45,731.25

46,418.75

1.50

Present value.

Percent change in value.

Total yearly dividend.

Annual dividend yield.

Portfolio beta coefficient.

2,567.50

5.53

0.77

_{Download from Www.Somanuals}1_.c3_om5_{. All Manuals Search And Download.}

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. The keystrokes to calculate the Canadian

Mortgage factor are:

1. Press

CLEAR

2. Key in 6 and press

3. Key in 200 and press

4. Key in the annual interest rate as a percentage and press

5. Press

The Canadian mortgage factor is now stored in

for future use. The

in Canadian

examples below show how this factor is used for

mortgage problems.

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%?

Keystrokes

Display

CLEAR

0.74

Canadian mortgage factor

200

360.00

Total monthly periods in mortgage life

Monthly payment

30000

-237.85

_{Download from Www.Somanuals}1_.c3_om6_{. All Manuals Search And Download.}

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

0.76

Canadian mortgage factor.

200

9.25

440

-440.00

437.00

Monthly payment.

Total number of monthly payments.

56000

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?

Keystrokes

Display

CLEAR

0.72

Canadian mortgage factor.

612.77

75500

8.75

Annual interest rate.

200

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?

Keystrokes

Display

_{Download from Www.Somanuals}1_.c3_om7_{. All Manuals Search And Download.}

CLEAR

200

0.72

Canadian Mortgage factor.

8.75

612.77

Outstanding balance remaining at the

end of 10 years.

-61,877.18

75500

_{Download from Www.Somanuals}1_.c3_om8_{. All Manuals Search And Download.}

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, C and press

2. Key in the number of units produced, n, and press

3. Key in the learning factor, r, and press

4. Then press to calculate the cost of the nth unit, C .

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?

Keystrokes

Display

875.00

875

100

100.00

-0.15

434.51

Cost of the 100th unit.

If the cost of the first unit, C₁, and the nth unit, C_n, are known the learning

factor may be calculated. In addition, it is possible to calculate C_ij, the

average cost of the ith thru jth unit. These calculations may be rapidly

done with the following HP-12C program:

KEYSTROKES

DISPLAY

_{Download from Www.Somanuals}1_.c3_om9_{. All Manuals Search And Download.}

00-

CLEAR

01-

02-

03-

43 23

04-

05-

06-

07-

08-

09-

10-

11-

12-

13-

14-

43 23

43 22

15-43, 33 00

16-

17-

18-

19-

43 23

20-

21-

22-

23-

24-43, 33 00

25-

26-

_{Download from Www.Somanuals}1_.c4_om0_{. All Manuals Search And Download.}

27-

28-

29-

30-

31-

43 23

32-

33-

34-

35-

36-

37-

38-

39-

40-

41-

42-

43-

44-

45-

46-

47-

48-

49-43, 33 00

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : K+1

R : C

R : r

R : i

R : j

_{Download from Www.Somanuals}1_.c4_om1_{. All Manuals Search And Download.}

R -R : Unused

1. Key in the program, (Note: If the average cost are not going to be calcu-

lated, lines 25 through 48 need not be keyed in).

2. To calculate r, the learning factor, if C and C are known:

a. Key in C , the cost of the first unit and press

b. Key in C , the cost of the nth unit and press

c. Key in n, the number of units and press

learning factor.

to calculate r the

3. To calculate the cost of the nth unit when C and r are known:

a. Key in C and press

1. Key in r and press

This step may be skipped if step 2 has just been done).

b. Key in the number of units, n and calculate C , the cost of the nth

2. (Note:

unit by pressing

4. To calculate the average cost per unit of the ith through jth unit, C , if C

and r are known.

a. Key in C 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

c. Key in the number of the first unit of the batch, i, and calculate the

average cost per unit by pressing 25

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 C₁was $875).

Keystrokes

Display

875.00

875

395

100

500

1000

395.00

0.89

Actual r.

299.14

1000.00

Cost of the 500th unit.

Average cost of the 500th thru 1000th

unit.

280.00

500

_{Download from Www.Somanuals}1_.c4_om2_{. All Manuals Search And Download.}

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 firstcome, 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:

1. Key in the arrival rate of customers, λ, and press

2. Key in the service rate, µ, and press 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

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 2.

to calculate ρ/n.

6. Calculate the average waiting time in the system by pressing

7. Key in λ and press

to calculate the average queue length.

8. Key in ρ, then intensity factor (from step 2 above) and press

calculate the average number of customer in the system.

Reference:

_{Download from Www.Somanuals}1_.c4_om3_{. All Manuals Search And Download.}

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?

Keystrokes

Display

1.20

1.2

2.40

0.80

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).

1.08

1.079

2.16

4.16

2.59

4.99

Average wait in queue (min).

Average wait in system (min).

Average queue length.

1.2

2.4

Average # of customers in system.

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:

1. Key in the arrival rate, λ, and press

2. Key in the service rate, µ, and press

to calculate the average number of

customers waiting in queue at any one time.

3. Press

4. Press

to calculate the average waiting time.

to calculate the average total time the customer

spends in the system.

5. Press 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).

_{Download from Www.Somanuals}1_.c4_om4_{. All Manuals Search And Download.}

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

Display

0.50

0.56

0.69

Average # customers waiting in

queue.

1.39

2.50

1.25

Average waiting time.

Average total time in the system.

Average # customers in system.

With an HP-12C program on can readily calculate the necessary

probabilities for this type of problem (dispensing with the use of tables)

and perform additional calculations as well.

KEYSTROKES

DISPLAY

00-

01-

CLEAR

02-44 33

03-45 48

04-

05-

06-

43 34

07-43, 33 09

08-43, 33 16

09-

10-

11-

12-

13-

43 36

_{Download from Www.Somanuals}1_.c4_om5_{. All Manuals Search And Download.}

14-

15-43, 33 01

16-45 48

17-

18-

19-

20-45 48

21-

22-

23-

24-

25-

26-

27-

28-

29-

30-

31-

32-

33-

34-

35-

36-45 48

37-

38-

39-45 48

40-

41-

_{Download from Www.Somanuals}1_.c4_om6_{. All Manuals Search And Download.}

42-

43-45 48

44-

45-

46-

47-

48-

49-

50-

51-

52-

53-

54-

55-

56-

57-

58-

59-

60-

61-

62-

43 22

63-43, 33 53

REGISTERS

n: Unused

i: Unused

PV: Unused

FV: Unused

PMT: Unused

R : K

R : P

R : L

R : T

R : Used, T

_{Download from Www.Somanuals}1_.c4_om7_{. All Manuals Search And Download.}

R : n

R : λ

R : µ

R : ρ

R : 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 to calculate and store ρ, the intensity factor.

to see T , the average waiting time in the queue. Display P ,

probability that all servers are idle, by pressing

probability that all servers are busy by pressing

average number waiting in the queue by pressing

1. Display P ,

2. Display L ,

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. T , the average waiting time in the queue, may again be

displayed by pressing

7. If desired, calculate P(t), the probability of waiting longer than a given time,

by keying in the time and pressing

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 P_bthe probability that all the tellers are busy.

What is the probability that a customer will have to wait 2 minutes or

more?

Keystrokes

Display

0.00

CLEAR

3.00

1.20

0.50

2.40

2.16

0.06

1.2

T average waiting time in queue.

P probability all servers are idle.

_{Download from Www.Somanuals}1_.c4_om8_{. All Manuals Search And Download.}

P probability all servers are busy.

0.65

2.59

4.99

4.16

L average # waiting in queue.

L, average # waiting in system.

T, average total time in system.

Probability of having to wait 2

minutes or more.

0.36

_{Download from Www.Somanuals}1_.c4_om9_{. All Manuals Search And Download.}

Appendix

Real Estate

Wrap-Around Mortgage

•

n = number of years remaining in original mortgage.

PMT = yearly payment of original mortgage.

PV = remaining balance of original mortgage.

n = number of years in wrap-around mortgage.

PMT = yearly payment of wrap-around mortgage.

•

r = interest rate of wrap-around mortgage as a decimal.

FV = balloon payment.

PMT₁[1 – (1 + r)^–n1

-------------------------------------------------------

]

PMT₂[1 – (1 + r)^–n2

]

–n2

PV₂– PV₁= ------------------------------------------------------- –

+ FV(1 + r)

After-Tax Cash Flows

•

ATCF = After-Tax Cash Flow for kth year.

Int = interest for kth year.

Dep = depreciation for kth year.

•

r = appropriate tax rate.

NOI = Net Operating Income.

ATCF = NOI (1 - r) - 12 x PMT + r x (Int + Dep ).

After-Tax Net Cash Proceeds of Resale

•

CO = capital purchase.

CPR = sales price - closing costs.

r = marginal tax rate.

NCPR = CPR - remaining balance of mortgage.

ATNCPR = NCPR + r x [(.6 SL Dep. - Total Dep) + .4 x (CO - CPR)]

_{Download from Www.Somanuals}1_.c5_om0_{. All Manuals Search And Download.}

Lending

Loans with a constant amount paid towards Principal

•

BAL = remaining balance after time period k.

•

CPMT = Constant payment to principal.

BAL = PV - (k x CPMT)

•

Kth payment to interest = i (BAL ) = (PMT )

i k

Kth total payment = CPMT + (PMT)

i 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:

1 – (1 + i)^–n

------------------ = ------------------------------

1 + ------ r

Add-On to APR with Credit Life

•

CL = credit life as decimal.

AMT = loan amount.

FC = finance charge.









------

1 +

--------------------------------------------------------------









------

1–

CL–

CL × r









•

---- = P M T

G × CL × n

amount of credit life

---------------------------- =

_{Download from Www.Somanuals}1_.c5_om1_{. All Manuals Search And Download.}

•

FC = (G - AMT - CL)

Rule of 78's Rebate

•

PV = finance charge.

I = interest charged at month k.

•

n = number of months in loan.

2(n – k + 1)

l = ------------------------------ P V

n(n + 1)

•

(n – k)l_k

Rebate= --------------------

•

BAL = (n - k) x PMT - Rebate

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.







1 + --- – 1 ---



------------------------------------------ ---------------------------------------------------------------------------------------------------------

D_END

−

















−

1 – 1 + --- AB

1 + --- – 1 + --- K + 1 + ---

– 1

















D_END

D_BEGIN= --------------

1 + ---

Savings

_{Download from Www.Somanuals}1_.c5_om2_{. All Manuals Search And Download.}

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.

C/P

•

= ((1 + r / C)

- 1)100

PMT

Investment Analysis

Lease vs. Purchase

•

PMT = loan payment for purchase.

PMT = lease payment.

I = interest portion of PMT for period n.

D = depreciation for period n.

M = maintenance for period n.

•

T = marginal tax rate.

cost of leasing (n) - cost of owning (n)

costjfdisafsdakflsafsa(f, xk)FDSAFF

----------------------------------------------------------------------------------------------------------

Net purchasing advantage =

(1 + i)ⁿ

∑

n = 1

Cost of owning(n) = PMT - T(I + D ) + (1 - T)M

•

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

U(P – V)

OL = --------------------------------

U(P – V) – F

_{Download from Www.Somanuals}1_.c5_om3_{. All Manuals Search And Download.}

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.

RV = redemption value per $100 par value.

DSM

-------------

P = [RV] – DR × RV ×

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

DSM

------------------

-------------

Y =

Forecasting

_{Download from Www.Somanuals}1_.c5_om4_{. All Manuals Search And Download.}

Simple Moving Average

•

X = moving average.

m = number of elements in moving average.

X + X + X + X

...x_m

nmm

X₁= -------------------------------------------------------

•

X + X + X + X

+ 1

...x_m

nmm

X₂= ----------------------------------------------------------------

etc.

Seasonal Variation Factors Based on a Centered Moving Average

•

X = centered moving average

•

m = number of elements in the centered moving average.

X₁

------ + (X ₂+ X₃+ X

_m+ 1

...x

) + -----------------

X_c= --------------------------------------------------------------------------------------

•

SV = Seasonal variation factor.

x = value of the ith data point.

•

i = centered moving average of the ith data point.

X_i

SV = ----

X_i

Gompertz Curve Trend Analysis

(bx)

•

y = ca

where x, y, a, b, and c are positive.

(1 – a)

D = S + ---------------- T

S₃– S₂

-------------------

S₂– S₁









b =

_{Download from Www.Somanuals}1_.c5_om5_{. All Manuals Search And Download.}

•

 

 

---

= S +

t + 1

S₁S₃– S²₂

------------------------------------

S + S – 2S











c = exp --

₂

(b – 1)(S₂– S₁)

-----------------------------------------

a = exp

b(bⁿ– 1)²

•

Where S , S , and S are:

bⁿ– 1

b – 1

--------------

Iny_i= nlnc + b(lna)

S₁

∑

i = 1

•

bⁿ– 1

Iny_i= nlnc + b^{n + 1}(lna)

--------------

S₂

b – 1

∑

i = n + 1

•

a, b and c are determined by solving the three equations above simulta-

neously.

Forecasting With Exponential Smoothing

•

a = smoothing constant (0 < a < 1)

X = actual current period usage

_{Download from Www.Somanuals}1_.c5_om6_{. All Manuals Search And Download.}

bⁿ– 1

b – 1

S₃

Iny_i= nlnc + b^{2n + 1}(lna)

--------------

∑

i = 2n + 1

•

Smoothed average S = αX + (1 - α)S - 1

Change, C = S - S - 1

Trend, T = αC + (1 - α)T - 1

Current period expected usage,

•

Forecast of next period expected usage,

Error, e = t - X

•

Cumulative error =

∑

t = 1

Initial conditions: S = X

•

t-1

= 0

t-1

Pricing Calculations

Markup and Margin Calculations

•

Ma = margin(%).

Mu = markup(%).

S = selling price.

C = cost.

_{Download from Www.Somanuals}1_.c5_om7_{. All Manuals Search And Download.}

•

S – C

Ma = 100-------------

S – C

Mu = 100-------------

S = -------------------

100

---------

1 –

•

100







S = C 1 + ---------





C = S 1 –







---------

100

C = -------------------

1 + ---------

100

Ma = -------------------

1 + ---------

100

Mu = -------------------

1 + ---------

100

_{Download from Www.Somanuals}1_.c5_om8_{. All Manuals Search And Download.}

Calculations of List and Net Prices with Discounts

•

L = List price.

N = Net price.

D = Discount(%).

100

---------

D' = 1 –

•

L = --------------------------------------------------

D'₁× D'₂× S..S. D’DF







-------------------------------------------------------------

= 1 –



L(D₁× D₂× D...D × D_{X – 1}

)

Statistics

Exponential Curve Fit

•

y = Ae

Σx_ilny_i– --(Σx_i)(Σlny_i)

B = ------------------------------------------------------------

Σx_i²– --(Σx_i)

•

Σlny_i

Σx_i

A = exp ------------- – B -------

= -Ae

Logarithmic Curve Fit

y = A + B(ln x)

•

_{Download from Www.Somanuals}1_.c5_om9_{. All Manuals Search And Download.}

•

Σy_ilnx_i– --Σlnx_iΣlny_i

B = -------------------------------------------------------

Σ(lnx_i) – --(Σlnx)

•

A = --(Σy_i– BΣlnx_i)

= A + B (ln x)

Power Curve Fit

•

y = Ax (A>0)

ln y = ln A + Bln x

(Σlnx_i)(Σlny_i)

-------------------------------------

B = ------------------------------------------------

(Σlnx_i)²

--------------------

Σ(lnx_i) –

•

Σlny_i

Σlnx_i

A = exp ------------- – B -------------

= Ax

Standard Error of the Mean

•

= -------

= ------

Mean, Standard Deviation, Standard Error for Grouped Data

_{Download from Www.Somanuals}1_.c6_om0_{. All Manuals Search And Download.}

•

Σf_ix_i

X = ----------

Σf_i

mean

Σf_ix²_i– (Σf_i)X²

-------------------------------------

Σf_i– 1

standard deviation

Sx =

•

standard error S_x

Σf_i

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.

For ordinary taxable investment:

•

PMT

FV = ------------------------ [1 + i (1 – t a x )]{[1 + i(1 – tax)] – 1}

i(1 – tax)

For tax-free investment:

•

n – 1

PMT

FV = ------------(1 + i)[(1 + i)

]

Stock Portfolio Evaluation and Analysis

•

n = the number of issues held.

P = the current market price / share of a stock.

•

S = the number of shares of a stock held.

β = the beta coefficient of an individual stock.

T = the total present value of a portfolio.

_{Download from Www.Somanuals}1_.c6_om1_{. All Manuals Search And Download.}

Portfolio beta coefficient:

•

P_iS_iβ_i

---------------

β =

∑

Canadian Mortgages

•

r = annual interest rate expressed as a decimal.

•

















monthly factor

1 + -- – 1 × 100









Miscellaneous

Learning Curve for Manufacturing Cost

•

C = Cost of the nth unit.

C = Cost of the first unit.

•

n = number of units.

r = learning factor.

k = ln r / ln 2

•

C = C n

C = the average cost of the ith through jth unit.

•

j^{k + 1}– i^{k + 1}

C₁

----------------------------

C_ij= ----------

k + 1

j – 1

•

This formula is only approximate and may give appreciable error at small i.

_{Download from Www.Somanuals}1_.c6_om2_{. All Manuals Search And Download.}

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).

P = Probability that all servers are idle.

•

P = Probability that all servers are busy.

L = Average number of customers in queue.

L = Average number of customers in the system (waiting and being

served).

•

T = Average waiting time in queue.

•

T = Average total time through the sytem.

P(t) = Probability of waiting longer than time t.

–1

n – 1

ρ^k

ρⁿ

P₀

----- + -----------------------





∑

n! 1 –





k = 0

•

ρⁿP₀

P_b= -----------------------





n! 1 –





•

ρP_b

L_q= ------------

n – ρ

L_q

T_q= -----

L = L_q+ ρ

T = L / λ

-(n_{µ - λ})t

•

P(t) = P e

Graduated Payment Mortgage

_{Download from Www.Somanuals}1_.c6_om3_{. All Manuals Search And Download.}

•









(1 + C)^B

(1 + I)















(n – AB)



----------------- 



(1 + Q)^B



(1 + I)

PV = PMT

+ ---------------------------------------------

₁









(1 + I)^AB

--------------- ----------------------













where:

•

1 + C

Q = ------------------- – 1

(1 + I)^A

•

A = number of payments per year

B = number of years that payments increase

C = percentage increase in periodic payments (as a decimal)

PMT = amount of the first payment

_{Download from Www.Somanuals}1_.c6_om4_{. All Manuals Search And Download.}

Graham Field Pill Reminder Device 5717 1 EA User Manual
Haier Refrigerator BC 80E User Manual
Hamilton Beach Iron 14515R User Manual
Healthrider Treadmill HRTL16991 User Manual
Honeywell Humidifier HEV 312 User Manual
Honeywell Satellite Radio 5817 User Manual
HP Hewlett Packard Indoor Furnishings 5344S User Manual
HP Hewlett Packard Power Supply T2200 XR User Manual
Hubbell Automobile MRH User Manual
Image Treadmill IMTL19901 User Manual