To log or not to log: The distribution of asset returns

In the context of the measurement of market risk, the random variable is taken as the rate of return of a financial asset. One may define the return in different ways, the two most common are arithmetic and geometric returns. The distinction between these two types of returns is not well understood. They are frequently assumed to be approximately equal. Moreover they both are assumed to be normally distributed. In this paper we explain the difference. We show that both types cannot be normally distributed, and that the difference grows larger as the volatility of the financial asset increases and the time resolution decreases.