StevenClark.com.au Time Value of Money: Compound Interest (redux)

The discussion in Time Value of Money: Present & Future about fixed amounts led us to Time Value of Money: Financial Tables and then onto Time Value of Money: Ordinary Annuities and Time Value of Money: Annuities Due. The last article took us into Time Value of Money: Mixed Streams. And in this article we revisit compound interest.

Semi-Annual and Quarterly Compounding

You will find that interest is going to be compounded by financial institutions in all sorts of increments – annually, monthly, weekly or even daily. So we need a fast and effective formula to calculate the effect of that compounding; where i is the interest rate, m is the number of terms in the year and n indicates the number of years:

Future Valuen = Present Value * (1 + (i / m))m*n

Solving for a quarterly compounding of \$2,000 at an interest rate of 8 per cent over 2 years:

• Future Value2 = \$2,000 * (1 + (0.08 / 4))4 * 2
• Future Value2 = \$2,000 * (1 + 0.02)8
• Future Value2 = \$2,000 * (1.02)8
• Future Value2 = \$2,000 * 1.1717
• Future Value2 = \$2,343.40

Skinning the Future Value Cat another way

The example I chose was specifically taken to show the relationship between the Future Value result of compounding over those eight time periods (\$2,343,40) with the equivalent result provided by the Financial Tables for Future Value of a fixed amount (\$2,332).

• Future Value = PMT * FVIVi,n
• Future Value = \$2,000 * FVIV0.08,2
• Future Value = \$2,000 * 1.166
• Future Value = \$2,332

Had the interest rate been 7 per cent and the quarterly interest over 3 years this would have been more difficult to prove on the Financial Tables because the term for each period’s interest rate would have reduced to 0.0175 per cent across 12 compounding periods. Understanding this relationship, it would be advisable to stick with the formula created specifically to calculate compound interest more frequently than a year.

Continuous Compounding

Continuous compounding is where interest is immediately and continuously earned and compounds interest on itself. The compounding periods become extremely small. This is the highest rate of interest that can be earned as compound interest.

To calculate a continuous compounding of interest the Future Value Interest Factor formula becomes:

FVIFi,n = ei*n

This alters the formula for calculating the Future Value and e can be taken as 2.7183:

Future Valuen(continuous compounding) = Present Value * (2.7183i*n)

To follow this continuous compounding calculation through with \$2,000 at 8 per cent over 2 years:

• Future Valuen = \$2,000 * (ei*n)
• Future Value2 = \$2,000 * (2.71830.08*2)
• Future Value2 = \$2,000 * (2.718316)
• Future Value2 = 2,347.02

The Next Step: Nominal & Effective Rates of Interest

In the next Time Value of Money article we will discuss the difference between nominal and effective annual rates of interest.

I would also advise you to pick up any decent copy of a managerial finance textbook to underpin these articles and to complete the exercises that will cement this understanding at the end of each chapter. The textbook will also provide more precise context important to your understanding.

Time Value of Money 101 Series  