Multi-Dimensional Risk and Mean-Kurtosis Portfolio Optimization

The mean-variance portfolio optimization theory of Markowitz assumes that stock returns are distributed according to normal probability density functions (pdfs). In reality, stock returns are more accurately described by leptokurtic pdfs which have kurtosis greater than zero. Stocks with leptokurtic distributions of returns are conventionally considered to be inherently more risky than stocks with normal pdfs. This paper examines portfolio optimization using the kurtosis as a risk measure. Maximizing the kurtosis as a function of portfolio weights is equivalent to maximizing the probability of large fluctuations from the mean which is counter-intuitive and contrary to normal practice. It is argued that risk is multidimensional and that the kurtosis is an ambiguous multi-dimensional risk measure