I’ve mentioned before that geometric and arithmetic returns are computed differently and not understanding the difference between the two can lead to problems. Here I’ll show you why the continuously compounded arithmetic return is sometimes pretty close to the geometric return and sometimes far off and misleading.

So the two returns are calculated somewhat differently. However, the difference between R_{a} and R_{g} depends on the size of the return. Suppose that R_{a} is very small, almost 0. This is a reasonable assumption if the return period is very short; perhaps one day or even hours. Then we can rewrite R_{g} as a **Taylor Series Expansion:**

Since we said R_{a} is close to 0, R^{2}, R^{3}, …, R^{n} are all even closer to zero. Also, the denominator gets bigger as n increases, which makes each fraction even smaller. So the numerator gets arbitrarily smaller as the denominator gets arbitrarily larger, which means that each term beyond R_{a} gets infinitely smaller. Therefore, we can ignore the sequence of terms after R_{a }and conclude that when R_{a} is small, R_{g is approximately equal to }_{ }R_{a}.

As an example, suppose a stock price $90 at t = 1, $90.5 at t = 2, and $110 at t = 3. The arithmetic return for the first period is .00556, which is close to the geometric return of .00554. This is because the return is small. However, the arithmetic return does not approximate the geometric return as well for the second return. The arithmetic return is .21547, while the geometric return is .195131. Not an awful estimate, but not good either. The error gets larger as the return gets bigger.

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