Modern Portfolio Theory: Foundations, Analysis, and New Developments, + Website

Chapter 9

Non-Normal Distributions of Returns

Many portfolio-based finance theories utilize a mean-variance framework in which investors make decisions on the basis of means and variances of the rates of return. For example, the CAPM assumes all asset returns are normally distributed or investors have mean-variance preferences. Portfolio performance measures such as the Sharpe measure, the Treynor measure, and the Jensen measure are based on the two parameters of the normal distribution. However, there has been empirical evidence that asset returns are not normally distributed since Mandelbrot (1963), Fama (1965), and Blattberg and Gonedes (1974).¹ Researchers observe nonsymmetric, highly peaked, and longer-tailed (leptokurtic) characteristics in the empirical unconditional distribution of asset returns. Table 9.1 shows statistical results from normality tests of international stock market return distributions from 14 developed countries. Most of the countries show significantly positive skewness. In particular, skewness and kurtosis coefficients using weekly returns and monthly returns are quite different, even though skewness and kurtosis coefficients are pure index numbers that should remain invariant to the return measurement interval.² This evidence might be indicating that the returns are generated from several nonidentical distributions, such as a mixture of normal distributions.

Table 9.1 Summary Statistics of International Stock Market Return Distributions

In a dynamic ...