The Mathematics of Money

Copyright © 2008, The McGraw-Hill Companies, Inc.

3.5 Solving for Rates and Times (Optional) 131

c. Which answer, (a) or (b), gives the actual amount of interest he will earn?

8. Oswayobank offers a 7.52% rate compounded monthly for a 7-year deposit account.

a. Find the APY for this account.

b. Use the APY to determine how much someone would need to deposit in order to end up with a $5,000 account

value at maturity.

c. Rework (b) using the nominal rate.

B. Additional Exercises

9. Parul deposited $4,000 in a CD paying 6.29% compounded continuously. Find the effective interest rate and use it to
   fi nd the future value after 4 years. Also, fi nd the future value after 4 years by using the nominal rate.


10. Howard invested $67,945.16 for 3 years, 8 months, and 17 days in an account paying 8.19% compounded daily
    using bankers’ rule. Find the total interest he will earn, both directly by using the nominal rate, and then by fi nding the
    effective rate and using it.

3.5 Solving for Rates and Times (Optional)

So far, we have dodged the question of how to use the compound interest formula to find

interest rates and terms. While the Rule of 72 provides a way to approximate these answers,

in this section we will see how to find exact answers. This will, however, require more

advanced algebraic techniques than we have been using so far, and this section assumes

that the reader has more than basic algebra skills. Solution methods are presented in the

following by means of examples.

Solving for the Interest Rate (Annual Compounding)

Example 3.5.1 Suppose that an investment grows from $2,500 to $3,000 in 5 years.

What is the effective interest rate earned by this investment?

We begin by substituting the values we have into the compound interest formula.

FV 
 PV(1  i)n

$3,000 
 $2,500(1  i)^5

We need to whittle away at this equation to isolate the i. A fi rst step toward this goal would

be to eliminate the $2,500 by dividing both sides.

1.2 
 (1  i)^5

This helps, but it leaves the diffi culty of getting rid of the exponent. To do this, we will use the

fact that when something raised to a power is itself raised to a power, you can multiply the

exponents. We’ve previously noted that exponents do not have to be whole numbers, and so

if we raise both sides of this equation to the^1 ⁄ 5 : power we get:

(1.2)^1 ⁄^5 
 (1  i)^5 

(^1) ⁄ 5
(1.2)^1 ⁄^5
(1  i)^1
1.0371372893
1  i