1. Effective versus Nominal Rates of Interest
Dr. Craig Ruff
Department of Finance
J. Mack Robinson College of Business
Georgia State University
© 2014 Craig Ruff

2. (3) Effective vs. Nominal Rates of Interest
It is difficult to compare rates with different compounding periods.
For instance, which is a better rate? 10%, compounded annually, or 9.8%, compounded monthly?
The effective rate of interest translates all rates to a ‘compounded annually’ basis, which allows an apples-to-apples comparison of rates.
Indeed, we have already seen applications of the concept behind the effective rate of interest. Recall that $100 deposited for 1 year at 10%, compounded semi-annually, will grow to $ at the end of one year. This implies that 10%, compounded semi-annually, is really earning a 10.25% annual rate. In this case, 10% is the nominal rate and 10.25% is the effective rate.
© 2014 Craig Ruff

3. (3) Effective vs. Nominal Rates of Interest
As you may recall, we found the 10.25% by investing $100 at 5% for six months and then rolling that $105 over for another six months. At the end of the second six months (one year), our money had grown to $ This is a 10.25% effective return (on an annual compounding basis).
What we want is a general formula for finding this effective rate.
The set-up for finding this general formula is to imagine we are putting $1 in for N years at an effective rate, compounded annually, and that will give us the same thing as putting $1 in for N years at the nominal rate, compounded non-annually. Now, we want to re-arrange this formula to get a formula for the effective rate.
© 2014 Craig Ruff

4. (3) Effective vs. Nominal Rates of Interest
With a little algebra (and recognizing that the $1 and N on each side of the equation kill each other), we can re-arrange this relationship to isolate the effective rate as:
Plugging in the numbers for 10%, compounded semi annually:
10% is the nominal rate.
10.25% is the effective rate.
One way to think of this is that 10%, compounded semi annually, is equivalent to 10.25%, compounded annually.
© 2014 Craig Ruff

5. (3) Effective vs. Nominal Rates of Interest
What is the effective rate of 10%, compounded monthly? The nominal rate is 10% and the value for m is 12 (since there are 12 months in a year). Plugging into the formula, we get:
One way to think of this is that 10%, compounded monthly, is equivalent to %, compounded annually.
What is the effective rate of 10%, compounded hourly? Again, the nominal rate is 10% and the value for m is 8760 (since there are 8760 hours in a year). Plugging into the formula, we get:
Or, 10%, compounded hourly, is equivalent to %, compounded
annually.
In using your calculator, watch out for parentheses…
© 2014 Craig Ruff

6. (3) Effective vs. Nominal Rates of Interest
Using 10%, we get the following effective rates for different compounding frequencies.
Compounding Rule:
Effective Rate
10%, compounded annually %
10%, compounded semi-annually %
10%, compounded monthly %
10%, compounded weekly (52 weeks/year) %
10%, compounded daily (365 days/year) %
10%, compounded hourly %
10%, compounded continuously (using the continuous compounding formula) %
© 2014 Craig Ruff

7. Recall the opening slide asked: Which is a better rate
Recall the opening slide asked: Which is a better rate? 10%, compounded annually, or 9.8%, compounded monthly? Calculating the effective rate can answer that question:
Now, we can compare the two accounts on an apples-to-apples basis. An account paying the equivalent of 10.25%, compounded annually, is paying a higher rate than the account only paying 10%, compounded annually
© 2014 Craig Ruff

8. Examples
© 2014 Craig Ruff

9. Example: Effective vs. Nominal Rates of Interest
Suppose you want to know the balance in your bank account of depositing $1000 in a bank account that pays 3%, compounded semi-annually, at the end of five years. We can solve this one as a standard FV of a single sum with non-annual compounding…
But we could also have used the effective rate of to solve this:
Buttons
Numbers to Enter PV
-1000 FV
???? 1,
I/Y
1.5 N 10
PMT
Note that because we converted 3%, compounded semi-annually, to an equivalent annual rate, we use an N of 5.
Buttons
Numbers to Enter PV
-1000 FV
???? 1,
I/Y
3.0225 N 5
PMT
© 2014 Craig Ruff

10. Example: Effective vs. Nominal Rates of Interest
Point:
We did this FV in two ways:
The nominal rate (3%) and ‘non-annual’ compounding, and
The effective rate (3.0225%) and ‘annual’ compounding.
Both approaches gave us the exact same outcome. This is not surprising, as 3%, compounded semi-annually, is an equivalent growth rate as %, compounded annually.
© 2014 Craig Ruff

11. Example: Effective vs. Nominal Rates of Interest
As another example, let’s try the same idea but with more difficult numbers. Suppose you deposit $1000 at 10%, compounded monthly. How much money will you have at the end of 19 quarters?
As before, we can solve this two different ways: (1) using the nominal rate and the non-annual formula and (2) using the effective rate and the annual formula. The point is that we will get the same answer.
Approach 1):
Since the rate is compounded monthly, the length of the period is a month. You need to tell your calculator that the periodic (monthly) rate is 10/12 = And since there are 3 months to a quarter and there are 19 quarters, there are a total of 57 (19*3) periods (months).
Buttons
Numbers to Enter PV
-1000 FV
???? IY
10/12 = N
19*3=57
PMT
© 2014 Craig Ruff

12. Example: Effective vs. Nominal Rates of Interest
Continuing with the same example…
Approach 2):
Here, we are using the effective rate. Since the effective rate is compounded annually, the length of the period is a year. Thus, you will tell your calculator that the periodic (yearly) rate is and the money is in the account for 4.75 periods (years). (As there are 4 quarters in a year and there are nineteen quarters, that means there are 4.75 years (19/4 = 4.75).)
Buttons
Numbers to Enter PV
-1000 FV
????
I/Y N
19/4=4.75
PMT
© 2014 Craig Ruff

13. Example: Effective vs. Nominal Rates of Interest
Again, the key point of the last example is that the two approaches yielded identical answers. That is not surprising as….
10%, compounded monthly, is an equivalent compounding rate as %, compounded annually.
Nominal
Effective
© 2014 Craig Ruff

14. Example: Effective vs. Nominal Rates of Interest
As another example…
I plan to invest $1000 for 10 months.
Bank A’s savings account is paying 10%, compounded quarterly.
Bank B’s savings account is paying %, compounded annually.
In which bank should I deposit my money?
The hard way to do this would be to calculate our wealth at the end using both banks’ rates and determine which bank makes us richer.
© 2014 Craig Ruff

15. Example: Effective vs. Nominal Rates of Interest
If I deposit the money in Bank A, I will earn the equivalent of 2.5% per quarter. Since I will have my money on deposit for 3 and 1/3 quarters, I can calculate the ending balance as:
Just for fun, notice that this approach also yields the same answer:
Now, I consider Bank B….
If I deposit the money in Bank B (which is paying %, compounded annually), at the end of 10 months, I will have:
The two banks give us the exact same ending wealth. Based on rates alone, it does not matter which bank we pick.
© 2014 Craig Ruff

16. Example: Effective vs. Nominal Rates of Interest
The easier way to determine the answer is to simply compare the two banks’ effective rates. That will provide an apples-to-apples comparison. Recall that Bank A’s savings account is paying 10%, compounded quarterly, and Bank B’s savings account is paying %, compounded annually. The effective rates are (although we really don’t need to calculate Bank B’s):
Since the effective rates as the same, it is no surprise that the bank’s yielded the same ending wealth.
Bank A’s savings account which is paying 10%, compounded quarterly, is really paying the same rate as Bank B’s savings account which is paying %, compounded annually.
Looking as both banks’ effective rates allows us to make an apples-to-apples comparison. In this case, the effective rates turned out to be identical.
© 2014 Craig Ruff

17. End