The Mathematics of Money

118 Chapter 3 Compound Interest

Writing off the exponent of 1 and dividing both sides by $100 we get:

(^)  1  __rc (^) 
c

(1  R)
Then, subtracting 1 from both sides, we obtain the formula:

FORMULA 3.3.2

The Effective Rate Formula

Effective rate
(^)  1 + rc (^) 
c
 1
where
r
the NOMINAL INTEREST RATE
and
c
the number of COMPOUNDINGS PER YEAR
Example 3.3.6 Rework Example 3.3.5, this time using the formula
Effective rate
(^)  1  .____^07354 
4
 1
1.07555  1
. 07555
7.555%
Note that, since the formula gives the effective rate as a decimal, in order to have three
decimal places in the percent we needed to round to 5 decimal places, so that we would
have the correct number when we moved the decimal to the left.
Using Effective Rates for Comparisons
One of the main advantages of effective rates is that they allow us to readily compare
nominal rates with different compounding frequencies. By using equivalent annual rates,
we can make an “apples to apples” comparison, making any comparison of rates as simple
as the one in Example 3.3.1.
The following example will illustrate:
Example 3.3.7 Which of these interest rates is actually the highest?
Bank Rate Compounding
Tu lly Savings Bank 5.95% Annual
Bank of Manlius 5.85% Monthly
Cincinnatus Trust 5.75% Daily
To compare these nominal rates, we convert each to its equivalent annual (effective) rate.^6
For Tully Savings, no work is required since the rate is already annual.
For Bank of Manlius: FV
$100  1 + 0.0585 12 
12

$106.01
For Cincinnatus Trust: FV
$100  1  0.0575 365 
365

$105.92
This allows us to complete a new table using the effective rates:
Bank Nominal Rate Effective Rate
Tu lly Savings Bank 5.95% annually 5.95%
Bank of Manlius 5.85% monthly 6.01%
Cincinnatus Trust 5.75% daily 5.92%
By comparing the effective rates, we see that Bank of Manlius is actually offering the highest
interest rate, followed by Tully Savings Bank, with Cincinnatus bringing up the rear.
(^6) While in this example we use the $100 and 1-year procedure to determine the effective rates, the optional formula
can equally well be used.
cb