Note on simple and logarithmic return

Abstract

In this paper we describe and clarify the definitions and the usage of the simple and logarithmic returns for financial assets like stocks or portfolios. It can be proven that the distributions of the simple and logarithmic returns are really close to each other. Because of this fact we investigate the question whether the calculated financial risk depends on the use of simple or log returns. To show the effect of the return-type on the calculations, we consider and compare the riskiness order of stocks and portfolios. For our purposes, in the empirical study we use seven Hungarian daily stock prices and for the risk calculation we focus on the following risk measures: standard deviation, semivariance, Value at Risk and Expected Shortfall. The results clearly show that the riskiness order can depend on the use of the return type (i.e. log or simple return). Generally, often – due to missing data or the nature of the analysis – one has to use approximations. We also examine the effect of these approximations on the riskiness order of stocks and of portfolios. We found differences in the riskiness order using exact or approximated values. Therefore, we believe, if this is possible, exact values instead of approximated ones should be used for calculations. Additionally, it is important that one uses the same type of return within one study and one has to be aware of the possible instabilities when comparing return results. JEL Code: C18