• Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers

## 1.2.2 Compounding More Frequently Than Annually

Interest can be computed more frequently than annually. With this greater precision, banks (for example) can offer accounts that can be withdrawn in the middle of the year. But it can also be misleading if the interest rate on a consumer loan states the annual percentage rate (APR) when interest is actually compounded more frequently than annually. Let's see how this works.

Semiannual compounding

What is the effect if interest is paid twice a year, one-half of the stated interest rate after six months and another half of the state rate after 12 months. In one year, \$100 at 8% interest compounded semiannually will be:

 Month P x (1 + %) Value at end of period 6 \$100 x (1 + .04) \$104 12 \$104 x (1 + .04) \$108.16

Although the nominal rate was 8%, the effective rate was \$8.16. No wonder banks that offer compound interest advertise effective rates to borrowers!

Quarterly compounding

The more the compounding, the greater the effective rate. Consider \$100 at 8% interest compounded quarterly:

 Month P x (1 + %) Value at end of period 3 \$100 x (1 + .02) \$102 6 \$102 x (1 + .02) \$104.04 9 \$104.04 x (1 + .02) \$106.12 12 \$106.12 x (1 + .02) \$108.24

Now the effective rate is 8.24%.

We could have gotten the same result using a modified version of our future value formula:

 FV = PV (1 + i/m)nm FV future value at the end of period PV the present value (or initial principal) i the interest rate paid each period n the number of periods m the number of coumpouding periods

Continuous compounding

You can even compute future values assuming continuous compounding. Using the formula

FV = PV*(1 + i/m)n*m
(where m is the frequency of compounding)

it is possible to use some calculus to compute future values when interest is compounded continuously. The formula is:

 FV = PV*ei*n (where e is the exponential function, 2.7183) FV future value at the end of period PV the present value (or initial principal) i the interest rate paid each period n the number of periods

This means that if you invest \$100 at 8% compounded continually, your effective rate is approximately 8.33% (after one year, \$100 becomes \$108.3288).

 A note on APR "Truth in lending" laws require that lenders disclose the annual percentage rate (APR) when making certain consumer loans, such as for credit cards. This is the nominal rate computed by multiplying the periodic rate by the number of periods in one year. This means a bank credit card that advertises an 18% rate is charging nominally 1.5% per month, but this is then compounded continuously. This, of course,is misleading! The effective rate is more than 18% -- it's actually about 19.56%. That is if you took out a loan for \$10,000 you would have to pay \$11,9560 to repay it after a year, not the \$11,800 that the advertised rate suggested. See Regulation Z. How did the banking lobby convince lawmakers that disclosure of the nominal APR is appropriate for consumer loans, but that disclosure of the higher effective rate is appropriate for deposit savings accounts?

 1.2.1 Future Value of a Single Amount 1.2.3 Future Value of an Annuity