Effective Annual Rates of Return and Present Values

Effective Annual Rates of Return and Present Values

I just finished teaching a finance course to graduate students in an MBA program. Two of the subjects covered in this course were rates of return and present value. The students were able to handle the calculations but were unaware of some of the concepts.

What happens when you compound interest rates? Does the effective rate rise as the number of times that the rate is compounded increases? I am sure you all remember the offers by many banks. “Deposit with us and you will get a free toaster and interest will be compounded continuously.” There is a general belief that the more times that the interest is compounded, the higher the annual rate. This is an erroneous concept. As the number of times the interest is compounded the effective annual rate increase but there is an upper limit.

If the interest is 6% and it is compounded twice a year the effective annual rate is 6.09%. If it is compounded 12 times per year the effective annual rate is 6.168%. Compounding it 24 times per year results in an annual rate of 6.170% and compounding it 100 times per year results in an annual rate of 6.181%.

Note how the effective annual rate increases as the number of times it is compounded increases but the size of the increase becomes smaller and smaller. In fact, if you compounded 6% an infinite number of times in a year, the annual rate would be 6.184%. This is the maximum annual rate. Note how close this figure is to the effective rate for compounding 24 times per year.

If the annual interest rate was 12%, compounding it 4 times per year would result in an effective annual rate of 12.551%. However, compounding it 100 times per year results in an annual rate of 12.742% and compounding it an infinite number of times per year results in an annual rate of 12.750%; not much greater than the result from compounding it 24 times per year. The table below shows the effective annual rate for the nominal interest rates between 5% and 18% based upon an increased number of compounding periods. As can be seen, the increase due to compounding becomes very small as the number of compounding periods increases. This is a concept that is not always understood and thus it is often used to infer that increased compounding will yield greater rates of returns.

Tags: annual rates of return, interest rate, mba, profit

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