Annual Percentage Rate Calculations

Annual Percentage Rate Calculations

Thus far, we've been careful to use a discount rate that is consistent with the frequency of the cash flows--for example, 1% per month with monthly payments or 10% per year with annual payments. In practice, interest rates are typically stated in one of two ways, as an annual percentage rate calculations or as an annual percentage yield (APY), even though interest may be calculated and paid more often than annually.

Annual Percentage Rate (APR)

The annual percentage rate (APR) is the periodic rate times the number of periods in a year. The APR is a nominal rate, a rate "in name only". The true (effective) annual rate may be different from the APR because of the compounding frequency.

The Compounding frequency is how often interest is compounded. For example, the compounding frequency might be monthly (12 times per year), quarterly (4 times), or annually (once). The periodic rate is an effective rate, but recall that two periods of interest is more than double one. The second period's interest includes interest on the first period's interest.

With m compounding periods per year and a periodic rate of r, the APR is:

APR = (m)(r)

The Effect of Compounding Frequency on Future Value

How does compounding frequency affect future value? To answer this question, let's compare yearly, semiannually, quarterly, monthly and weekly compounding for saving $10,000 for a year at a 12% APR.

The future value of $10,000 in one year is shown in below table for all of these compounding frequencies. The APY equals the 12% APR for yearly compounding. But the table shows how the future value and APY increase as the compounding frequency increases.

Another way to understand an APY is to say that it's the total interest earned in a year (annual interest) divided by the principal. That is,

APY = annual interest/principal

For example, the annual interest for monthly compounding is $1268.25, which, divided by $10,000, gives the same 12.68%.

Table

Future Values and APYs for Various Compounding Frequencies

Continuous Compounding

If more frequent compounding increases the future value, what if we compound daily, hourly, or even every minute? These are all examples of discrete compounding, where interest is compounded a finite number of times per year. If interest is compounded an infinite number of times per year, we have continuous compounding.

The APR and APY with Continuous Compounding

When m, the compounding frequency, becomes large enough, compounding becomes essentially continuous, Without giving the proof, it turns out that with continuous compounding:

Where e is approximately 2.718². The function  e^x  is called an exponential function. It is usually found on a calculator with either an " " or "exp" on the key.