The Mathematics of Money

116 Chapter 3 Compound Interest

Example 3.3.3 Leo has a life insurance policy with Trustworthy Mutual Life of

Nebraska. The company credits interest to his policy’s cash value^4 and offers Leo the

choice of two different options:

Option Rate Compounding

Daily Dividends 8.00% Daily (bankers’ rule)

Annual Advancement 8.33% Annual

Which option would give Leo the most interest?

Using the same approach as before, we fi nd the future value using any amount and any

term that we like. As before, we will make “nice” choices of PV 
 $100 and time 
 1 year.

Daily Dividends: FV
$100  1 + _____0.08 360 

360


 $108.33

Annual Advancement: FV 
 $100(1  0.0833)^1 
 $108.33

This one ends in a draw. Both options pay the same result, and so it makes no difference

ratewise which option Leo chooses.^5

When two rates and compoundings give the same result, we say that they are equivalent.

Since both options result in the same amount of interest being paid, we can use the two

interchangeably. Even if Leo chose the Daily Dividends option paying 8% compounded

daily, the company could just as well credit his interest using the 8.33% with annual

compounding.

For any given interest rate and compounding frequency, there is always an equivalent

annually compounded rate. Before taking this idea further, let’s introduce some terminology

to keep things straight.

Definitions 3.3.1

The annually compounded rate which produces the same results as a given interest rate

and compounding is called the equivalent annual rate (EAR) or the effective interest rate.

The original interest rate is called the nominal rate.

So, for example, we could say that if the nominal rate is 8.00% compounded daily (bank-

ers’ rule), then the equivalent annual rate (or the effective rate) is 8.33%.

How to Find the Effective Interest Rate for a Nominal Rate

Now let’s return to the question posed immediately before Example 3.3.3. Recall that the

Bank of Bolivar was offering an annually compounded interest rate that did not quite match

the daily compounded rate offered by Richburg Savings Bank. What annually compounded

interest rate should Bolivar offer to match 3.98% compounded daily? Or in other words,

what effective rate is equivalent to 3.98% compounded daily?

Suppose we call the effective rate R. Then, if we were to use R to find the future value

of $100 in 1 year, we would need to end up with $104.06, the same FV we got using 3.98%

compounded daily. So, plugging R into the future value formula, we would get:

FV 
 $100(1  R)^1

$104.06 
 $100(1  R)^1

Since an exponent of 1 doesn’t really do anything, it can be ignored, and so we then get:

$104.06 
 $100(1  R)

(^4) The cash value of a life insurance policy is an amount of money for which the policy can be “cashed in.” Not all
types of policies have cash values, but for those that do the cash value usually grows with time.
(^5) Actually, there is a very slight difference between the two. The future value using the annual rate is exactly
$108.33, while the future value with the daily rate actually comes out to be $1.0832774,399, which rounds to
$108.33. The difference is slight, but it is still there, and might not be lost in rounding if we use a higher PV.
However, since it amounts to less than a penny on a $100 deposit over the course of an entire year, in just about
any situation we would consider the difference far too small to matter.
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