# Programs

## Finance Programs

### Periodic Payments (PERPMT)

#### Summary Description

This program finds the present or future value of single payments, the present value of annuities, and the future value of sinking funds.

#### Detailed Description

Money has different values at different times. Most people understand that one dollar in your hand today is worth more than one dollar in your hand next year, when things will be more expensive. Another way of putting this is that one dollar a year from today is worth less than a dollar today. The worth today of an amount of money in the future is called the present value of that money; the worth at some time in the future of an amount of money today is called the future value of that money.

The interest rate or discount rate is the rate at which the value of money changes over time. Generally, one speaks of an interest rate when looking forward in time (when talking about the future value of present money), and a discount rate when looking backward in time (when talking about the present value of future money). For example, if your bank gives you 5% per year interest on a deposit, a \$100 deposit today will be worth \$105 in a year’s time; or \$100 in a year’s time is discounted to \$95.24 today.

For a single payment — one sum now turning into one sum in the future — the governing equation is:

FV = PV (1+r)n

where PV is the present value, FV is the future value, r is the periodic rate, and n is the number of periods. This equation can be turned around to be:

PV = FV (1+r)-n

with the same meanings.

An annuity is a series of equal payments made over time, looked at from the present. Many sweepstakes have prizes structured as annuities: a million dollar prize may be twenty yearly payments of \$50,000 rather than a single \$1,000,000 lump sum. As you might expect from the above, twenty yearly payments of \$50,000 is worth substantially less than a million dollars. At a 5% discount rate, twenty yearly payments of \$50,000 is worth only \$623,110.52. At a more aggressive 15% discount rate, the same twenty payments are worth only \$312,966.57. Annuities are traditionally viewed as being bought now, and giving payments at the end of the periods.

Finally, a sinking fund is a series of equal payments over time, looked at from the future. For example, you may want to accumulate the cost of a particular expense by making periodic deposits into a bank account. Twelve monthly payments of \$100 into a bank account with interest rate 5% (compounded monthly) will yield \$1,233.00 rather than the \$1,200 of the payments. Sinking funds are traditionally viewed as having payments at the beginning of the periods, and giving the accumulated value at the end of the last period.

A mortgage or loan is basically an annuity that a financial institution buys from you. They give you a lump sum, and you in return give them a number of payments over time. You can figure your payments by figuring the present value of an annuity with a \$1 payment, and dividing that into the amount you borrowed. The quotient is your monthly payment in dollars. For example, suppose you borrow \$35,000 at 14.5% with a four year payoff schedule. The payment period is one month; the periodic payment is 1.20833% (14.5% per year divided by 12 months per year); the number of periods is 48 (4 years times 12 months per year). An annuity of \$1.00 a month for 48 months at 1.20833% per month has a present value of \$36.26088; dividing this into \$35,000, the amount borrowed, gives 965.2276; so the mortgage payment would be 965.2276 times \$1, or \$965.23 per month.

#### Usage

Load and run the program PERPMT.

The program displays the prompt:

```1. FV single pmt
2. PV single pmt
3. Annuity
4. Sinking fund?
```

Enter the number of the analysis you want.

The program displays the prompts:

```Rate?
Periods?
Payment?
```

For each of these prompts, enter the appropriate value. “Rate” is the interest or discount rate per period, in percent. “Periods” is the number of periods. “Payment” is the single payment for options 1 and 2, or the periodic payment for options 3 and 4.

For FV single pmt and Sinking fund, the program will display:

```FV:
```

followed by the future value of the single payment or of the sinking fund. For PV single pmt and Annuity, the program will display:

```PV:
```

followed by the present value of the single payment or of the annuity.

#### Resources Used

 Resource Description Memory 283 bytes total program memory used. A The type of analysis being performed. This is 1 for future value of single payment, 2 for present value of single payment, 3 for annuity, or 4 for sinking fund. B The periodic rate. C The number of periods. D The amount of the single or periodic payment.

#### Program Source

The program is available as a text file with .CAT contents, or may be entered as shown below. A semicolon (“;”) marks the beginning of a comment, and is not to be entered into the calculator. Remember that these programs are copyrighted; see the copyright issues page for limitations on redistribution.

Program PERPMT:

```Lbl 0
"1. FV single pmt"
"2. PV single pmt"
"3. Annuity"
"4. Sinking fund"?->A           ; -> is assignment arrow
(A<>Int A) Or (A<1) Or (A>4)=>Goto 0
; <> is not equals => is conditional
"Rate"?->B
"Periods"?->C
"Payment"?->D
1+B/100->B
A=1=>Goto 1
A=2=>Goto 2
A=3=>Goto 3
A=4=>Goto 4
Lbl 1
"FV:"
DB^C_                           ; _ is display triangle
Goto 0
Lbl 2
"PV:"
DB^(-)C_                        ; (-) is change sign operator
Goto 0
Lbl 3
"PV:"
D(1-B^(-)C)(B-1)x^-1_           ; x^-1 is reciprocal operator
Goto 0
Lbl 4
"FV:"
DB(B^C-1)(B-1)x^-1_
Goto 0
```

Page last updated 5 July 2001.

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