Time Value of Money Concepts

Compound Growth

Compound interest simply means that interest is earned on interest. Compounding magnifies the impact that a given interest rate has on the growth of principal. Sometimes, as will be illustrated below, this impact can be fairly dramatic.

There are two 'levers' by which compounding exerts its influence:

  • the compounding frequency or interval (the number of compounding periods in a year); and
  • the term (the number of years over which the compounding takes place).

Compounding Frequency

The more frequent the interval of compounding, the greater the impact. This is illustrated in the examples below. However note that the compounding frequency "lever" is subject to the law of diminishing returns.

Example. What is the value of $100 invested for 1 year at 8%?

1. Annual Compounding. In this case there is no compounding effect because the term is only one year, the same as the compounding frequency. Thus, all we have is simple interest (i.e., the effective rate is equal to the nominal rate)

2. Monthly Compounding. In this case there are 12 compounding periods. Interest earned each month is added to the balance and is itself available to earn interest in each succeeding month. Thus, the future value is greater than the amount calculated using annual compounding.

3. Weekly Compounding. As should be expected, increasing the frequency of the compounding period increases the impact of the interest rate. That it does so should be intuitive: more interest is available sooner to earn more interest. Whereas before we had to wait until the end of the month before the interest was 'added back to the pot', now it is being credited each week.

4. Daily Compounding. Now instead of earning interest weekly, we earn it daily. As expected the, the impact of the interest rate is magnified. However, this time the impact is not as dramatic as might be expected.

5. Continuous Compounding. What if instead of days we used an hourly compounding period? Or minutes? Or seconds? Better yet, what if the compounding period were continuous?

It is possible to calculate such a compounding period using the formula below. (See Continuous Compounding for more information.) The result is the maximum effect that compounding frequency can exert on a given interest rate and term.

Although the compounding is continuous, the result is not much greater than what we got with daily compounding. Which was not a whole lot more than we got with weekly compounding. This demonstrates, as mentioned earlier, that while compounding frequency magnifies the effect of a given interest rate, it is subject to the law of diminishing returns.

Term

Compounding exerts its most dramatic effect (for a given interest rate) when the term is extended. In other words, the longer an amount is subject to compounding, the greater the effect. Unlike compounding frequency, the impact of term length is not subject to the law of diminishing returns.

Using the previous example of $100 invested at 8%, the following calculations show the future value with monthly compounding at 1, 5, 10, and 25 years.

After 1 year:

After 5 years:

After 10 years:

After 25 years:

The effect of term length is impressive: the original sum doubles in less than 10 years and increases more than seven fold in 25 years.

These results are even more impressive when you consider that this example illustrates the growth of a single sum; there is no additional principal being added, only the crediting of periodic interest earned.

Thus, when we speak of the miracle of compounding for a given interest rate, we are talking more in terms of the impact of the term (the length of time an amount is subject to compounding) than the frequency (the number of compounding periods per year). In fact, if we had used continuous in the last example where the term was 25, the future value would have been only 738.91, an increase of less than 1% over the 734.02 earned with monthly compounding.