Whats the EAR

A bank is offering 12 percent compounded quarterly. If you put $100 in an account, how much will you have at the end of one year? What's the EAR? How much will you have at the end of two years?

The bank is effectively offering 12%/4 = 3% every quarter. If you invest $100 for four periods at 3 percent per period, the future value is:

Future value = $100 X 1.034 = $100 X 1.1255 = $112.55

The EAR is 12.55 percent: $100 X (1 + .1255) = $112.55.

We can determine what you would have at the end of two years in two different ways. One way is to recognize that two years is the same as eight quarters. At 3 percent per quarter, after eight quarters, you would have:

Alternatively, we could determine the value after two years by using an EAR of 12.55 percent; so after two years you would have:

Thus, the two calculations produce the same answer. This illustrates an important point. Anytime we do a present or future value calculation, the rate we use must be an actual or effective rate. In this case, the actual rate is 3 percent per quarter. The effective annual rate is 12.55 percent. It doesn't matter which one we use once we know the EAR.

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