Appendix 4A

Continuous Compounding

Legend has it that many years ago, the world's bankers employed nearsighted men in green visors and armbands to compound interest continuously in the smoky back rooms of commercial banks. But no matter how fast they worked, it proved impossible for these unfortunate lackeys to compound interest on a continuous basis. One day, a particularly clever clerk discovered that holding period rates of return can be transformed into continuously compounded rates of return with a simple formula. Here's what he discovered.

Continuously Compounded Rates of Return

As the number of compounding intervals within a period approaches infinity, returns are said to be compounded continuously. At any instant, the rate of return is then called the instantaneous rate of return. Henceforth, let's denote continuously compounded rates of return with italics, so that c04a-math-0001 represents the continuously compounded version of a holding period interest rate c04a-math-0002.

Suppose you have an amount c04a-math-0003 today and you want to know how large this value will be after c04a-math-0004 periods if it earns a continuously ...

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