The Mathematics of Money

174 Chapter 4 Annuities

For entry in the calculator, once again your best bet is to calculate the future value factor

however you have been, and then once you have that result on the screen, follow step

2 below:

Operation Result

Steps to evaluate s n (^) | (^) i 24.91152003
/(1  .09/4)^20
15.96371237
Using Formula 4.4.4:
Plugging in to the formula we get:
an (^) | (^) i

1  (1  i)n
____i

1  (1  0.0225)^20

__0.0225
15.96371237

On the calculator, this can be evaluated either step by step or by entering everything at

once.

Step by step (the fi rst steps):

Operation Result

.09/4
 0.0225

 1 
 1.0225

^ 20 
 0.640816472

Unfortunately, we want to subtract this value from 1, and with subtraction the order matters.

This forces us hold this result (either by writing it down or by storing it in the calculator’s

memory) and then use it in the next step:

Operation Result

1 0.640816472
 0.359183528

/(.09/4)
 15.96371237

Alternatively, entering it all at once we get:

Operation Result

(1(1.09/4)^20)/(.09/4)
 15.96371237

However you choose to enter these, be careful about the negative exponent. On many

popular calculator models, there are two “” keys, one for subtraction and one for negative

numbers. You want to make sure that you are using the “” key for negative numbers when

you enter the exponent. Using the wrong key will most likely lead to an error.

Annuity Present Values and Loans

One of the most common applications of present values is using them to find loan pay-

ments. Most loans with a fixed payment schedule can be mathematically analyzed by using

annuity present values (though there are a few exceptions, discussed in Chapter 10.) The

following examples will illustrate this:

Example 4.4.7 Cienna is thinking about buying a condominium. She fi gures that

she can afford monthly loan payments of $650, and that she would be taking out a

30-year loan with an interest rate of 8.4%. On the basis of these assumptions, what is

the most she can afford to borrow? If she has $7,500 to use as a down payment, what

is the most she can afford to pay for the condo?

To begin with, we can see that $650 per month is an annuity, and the money that she

receives to buy the condo is its present value.

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