# The Compounding Frequency of Interest Rates and the Value of an Investment

The riskiness of investments can be expressed as an interest rate which represents the investment’s annual expected return. Although this is a convenient method for comparing the risk of multiple investments, the terms of the investment must be carefully evaluated to accurately value the asset.

A Certificate of Deposit (CD) is an investment normally issued by large banks as a means to bring money into the organization. Purchasers of a CD are paid interest in exchange for loaning the bank money. CDs rarely pay a high interest rate due in part to the fact the investment risk is low. In other words, the chances that a bank will default on a CD are so low that the required return or interest rate of the CD is low. Today, rates of about two to five percent are common with the higher rates going to longer term CDs. In previous decades, CD interest rates were sometimes as high as seven or eight percent in response to the economic conditions of the time.

Suppose that a bank offers a one-year CD at an interest rate of 5% with a minimum investment of $1,000. How much will the CD be worth in one year? Since the interest rate is expressed as an annual rate, the value of the CD in one year is:

1,000 * 1.05 = 1,050

So, the investment will be worth $1,050 in one year. However, this case assumes that the interest rate is compounded annually. Often, interest rates are compounded with other frequencies but the most common are annual, semi-annual, quarterly, and monthly.

To find the future value of an investment with compounding other than annual, some simple math is needed. Take the same example above but assume that the interest rate is compounded semi-annually (twice per year). First, divide the interest rate in half to get 2.5%. Then, apply the Future Value formula as usual.

FV = PV * (1 + r)2

1,050.63 = 1,000 * (1 + 0.025)2

So with semi-annual compounding, the investment is worth just a bit more than if the interest is compounded annually. Now suppose that the interest is compounded quarterly:

1,050.95 = 1,000 * (1 + 0.0125)4

Notice that since the interest is computed quarterly, we divide the interest rate by 4 (four quarters in a year) and raise the expression 1.0125 to the fourth power. Suppose the interest is compounded monthly:

1,051.16 = 1,000 * (1 + 0.004167)12

Although it isn’t common, some interest rates are compounded continuously. This means that interest is computed on an ongoing basis, not at regular intervals. To value the investment with continuous compounding, use the following formula:

FV = PV * _e_r

where *e* is approximately equal to 2.7183. *e* is commonly found in mathematics and represents the base for natural logarithms. On a scientific calculator, it is usually indicated as a key with the symbol _e_x. For our example here, the value of the investment with continuous compounding is:

1,051.27 = 1,000 * e0.05

In the example used in this article, the total future value of the investments didn’t change much when the number of compounding periods was increased. It ranged from a high of $1051.27 to a low of $1,050, a difference of only $1.27. However, remember that the interest rates on CDs are low because of their low risk. The number of periods in an investment’s life becomes more significant as both the amount invested and the interest rate increase. The same investment over the course of many years or with a larger interest rate would yield a much larger difference.