Subtle is the Lord, but malicious He is not: the calculation of abnormal stock returns in applied research

We use the expected logarithmic returns formula for the Geometric Brownian Motion (GBM) in conjunction with the expected logarithmic returns formula for the Feller diffusion to illustrate the nature and magnitude of errors which arise in computed abnormal returns when one applies an expected logarithmic returns formula which is incompatible with the stochastic process that generates a stock’s returns. Empirical analysis based on FTSE 100 stock price data for the five year period ending in 2017 shows that the scale of the errors in computed abnormal returns will hinge on the volatility of the returns generating process but will be particularly pronounced for relatively low stock prices. Although our principal focus is with comparing abnormal returns on the GBM and Feller diffusion, we also simulate logarithmic returns for the Uhlenbeck and Ornstein (1930) process, several interpretations of the Constant Elasticity of Variance (CEV) process and the scaled ‘t’ process of Praetz (1972) and Blattberg and Gonedes (1974). Taken in conjunction with the GBM and the Feller diffusion, these processes underpin virtually every equilibrium based asset pricing model which appears in the literature. However, computing abnormal returns for any of these processes using the expected logarithmic returns formula for the GBM inevitably leads to errors in the abnormal returns. Hence, an important principle which emerges from our analysis is that it is crucially important for researchers and others to test the compatibility of empirically observed returns with the distributional assumptions on which the empirical analysis is based if the complications arising from mis-specified modelling procedures are to be avoided.