The difference between arithmetic and geometric returns

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.

Screen shot 2013-08-13 at 7.56.04 PM

So the two returns are calculated somewhat differently.  However, the difference between Ra and Rg depends on the size of the return.  Suppose that Ra 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 Rg as a Taylor Series Expansion:

Screen shot 2013-08-13 at 7.56.21 PM

Since we said Ra is close to 0, R2, R3, …, Rn 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 Ra gets infinitely smaller.  Therefore, we can ignore the sequence of terms after Rand conclude that when Ra is small, Rg is approximately equal to  Ra.

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.

About schapshow

Math & Statistics graduate who likes gymnastics, 90s alternative music, and statistical modeling. View all posts by schapshow

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