The Mathematics of Money

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

Then, dividing both sides by 100 we get:

1.0406 
 1  R

And then subtracting 1 from both sides gives us:

R 
 0.0406 or 4.06%.

This should come as no surprise, and in fact you may well have figured this out for yourself

without bothering with the algebra. The 3.98% compounded daily earned $4.06 interest

above and beyond the original $100, and figuring out what percent that represents of $100

is simple. In fact, this was actually the reason why we chose to use $100 and 1 year origi-

nally—because with these choices the effective interest rate is easy to see.

We can use this approach whenever we need to find the effective rate for a given nominal

rate.

“FORMULA” 3.3.1

Finding the Effective Interest Rate

To fi nd the effective rate for a given nominal rate and compounding frequency,

simply fi nd the FV of $100 in 1 year using the nominal rate and compounding. The

effective interest rate (rounded to two decimal places) will be the same number as

the amount of interest earned.

The following example will illustrate:

Example 3.3.4 Find the equivalent annual rate for 7.35% compounded quarterly.

Following the procedure given above:

FV
$100  1 + ___0.0735 4 

4


 $107.56.

The interest earned is $7.56, and so we conclude that the equivalent annual rate is 7.56%.

We can verify this by using this rate to fi nd the same future value:

FV 
 $100(1 + 0.0756)^1 
 $107.56.

Of course, the equivalent annual rate in Example 3.3.4 is not exactly 7.56%. The future

value was rounded to two decimal places, as is customary, and so the rate based on this

answer is also rounded to two decimal places. In many situations two decimal places is

good enough; however, if more precision is needed we can get it by carrying the future

value out to as many decimal places as we need.

Example 3.3.5 Rework Example 3.3.4, this time fi nding the rate to three decimal places.

FV
$100  1 + 0.0735___ 4 

4


 $107.555

As strange as it may seem to carry a dollar value out to three decimal places, this allows us

to conclude that the equivalent annual rate is 7.555% (to three decimal places.)

A Formula for Effective Rates (Optional)

The procedure given above is perfectly adequate to find effective rates in any circum-

stance. For those who desire a more traditional style of formula, though, we can develop

one. Suppose we let:

r 
 the nominal rate

c 
 the number of compoundings per year

R 
 the effective rate

Then using the fact that the nominal and effective rates will give the same future value, and

once again using $100 and 1 year, we get the equation:

$100  1  __cr (^) 
c

$100(1  R)^1
3.3 Effective Interest Rates 117