If you reviewed the section on exponents, you may remember the following graph comparing the growth of compound interest and simple interest.

Compound interest causes an investment to grow quickly. This growth would accelerate if the compounding frequency became shorter. Given this, what would happen if the investment were compounded continuously? Would the investment continue to grow at a faster and faster rate, or does the pace of growth reach a limit as the compounding frequency becomes very small?

To answer this question, consider the formula to calculate the future value of an investment with compound interest.

Try making the compounding frequency smaller with an interest rate of 100 percent and an initial investment of \$1.

 Compounding Period Periods Per Year Future Value Annual 1 2 Semiannual 2 2.25 Bimonthly 6 2.521626 Monthly 12 2.613035 Weekly 52 2.692597 Daily 365 2.714567 Hourly 8760 2.718127 Minute 525600 2.718279

In terms of the future value of the investment, the effect of increasing the compounding frequency approaches a limit over time. Looking at the table above, you can see that at the end of a year there is only a penny of difference between an investment compounded daily and one compounded each minute. At the same time, there is a huge growth in the number of periods.

When compounding becomes continuous, the future value of the investment of a dollar at an interest rate of 100 percent approximates 2.718281828459045.... As you might recall, this is the value of e, the base of a natural logarithm. Therefore, the formula for calculating the value of an investment using continuous compounding includes e.

In this formula,

C0 = Initial investment
e = 2.718281828459045...
R = Annual rate of interest
T = Time period in years

From the information above, you can create a logarithm based on e, a fractional number. If both the natural exponential and natural logarithmic functions are plotted on the same grid, you may observe the natural logarithm approaching 2.718..., as you calculated above.

You may also note that the curve of the graph of e is similar in shape to the one above displaying compound interest. In addition, the graphs are reflection images of each other, as if you were holding a mirror along the line y = x.