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.

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 represents the continuously compounded version of a holding period interest rate .

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

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