Portfolio optimization for t and skewed-t returns

It is well-established that equity returns are not Normally distributed, but what should the portfolio manager do about this, and is it worth the effort? As we describe, there are now some good choices for multivariate modeling distributions that capture heavy tails and skewness in the data; we argue that among the best are the (Student) t and skewed t distributions. These can be efficiently calibrated to data, and show a much better fit to real data than the Normal distribution. By examining efficient frontiers computed using different distributional assumptions, we show, using for illustration 5 stocks chosen from the Dow index, that the choice of distribution has a significant effect on how much available return can be captured by an optimal portfolio on the efficient frontier. Portfolio optimization requires balancing risk and return; for this purpose one needs to employ some precise concept of “risk”. Already in 1952, Markowitz used the standard deviation (StD) of portfolio return as a risk measure, and, thinking of returns as normally distributed, described the efficient frontier of fully invested portfolios having minimum risk among those with a specified return. This concept has been extremely valuable in portfolio management because a rational portfolio manager will always choose to invest on this frontier. The construction of an efficient frontier depends on two inputs: a choice of risk measure (such as StD, V aR, or ES, described below), and a probability distribution used to model returns. Using StD (or equivalently, variance) as the risk measure has the drawback that it is generally insensitive to extreme events, and sometimes these are of most interest to the investor. Value at Risk (V aR) better reflects extreme events, but it does not aggregate risk in the sense of being subadditive on portfolios. This is a well-known difficulty addressed by the concept of a “coherent risk measure” in the sense of Artzner, et. al. [1999]. A popular example of a coherent risk measure is expected shortfall (ES), though V aR is still more commonly seen in practice. Perhaps unexpectedly, the choice of risk measure has no effect on the actual efficient frontier when the underlying distribution of returns is Normal – or more generally any “elliptical” distribution. Embrechts, McNeil, and Straumann [2001] show that when returns are elliptically distributed, the minimum risk portfolio for a given return is the same whether the risk measure is standard deviation, V aR, ES, or any other positive, homogeneous, translation-invariant risk measure. This fact suggests that the portfolio manager should pay at least as much attention to the family of probability distributions chosen to model returns as to the choice of which risk measure to use. It is now commonly understood that the multivariate Normal distribution is a poor model of generally acknowledged “stylized facts” of equity returns: • return distributions are fat-tailed and skewed • volatility is time-varying and clustered