Portfolio Optimization with Tail Risk Measures and Non-Normal Returns
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The traditional Markowitz mean-variance portfolio optimization theory uses volatility as the sole measure of risk. However, volatility is flawed both intuitively and theoretically: being symmetric it does not differentiate between gains and losses; it does not satisfy an expected utility maximization rationale except under unrealistic assumptions and is not a coherent risk measure. The past decade has seen considerable research on better risk measures, with the two tail risk measures Value-at-Risk (VaR) and Expected Tail Loss (ETL) being the main contenders, as well as research on modeling skewness and fat-tails that are prevalent in financial return distributions. There are two main approaches to the latter problem: (a) constructing modified VaR (MVaR) and modified ETL (METL) using Cornish-Fisher asymptotic expansions to provide non-parametric skewness and kurtosis corrections, and (b) fitting a skewed and fat-tailed multivariate parametric distribution to portfolio returns and optimizing the portfolio using ETL based on Monte Carlo simulations from the fitted distribution. It is an open question how MVaR and METL compare with one another and with empirical VaR and ETL, and also how much improvement can be obtained in fitting parametric distributions. In this dissertation, we first show that MVaR and METL are very sensitive to outliers, sometimes rendering complete failure of a portfolio. Then we propose new robust skewness and kurtosis estimates, study their statistical behavior and that of the resulting robust MVaR and METL through the use of influence functions, and show through extensive empirical studies that robust MVaR and METL can effectively curb the failure of the original estimates. We use the same experimental approach to show that the simple empirical ETL optimization yields portfolio performance essentially equivalent to that of the much more complex method of fitting multivariate fat-tailed skewed distributions. Finally we address the following important problem: VaR and ETL based portfolio optimization do not have expected utility maximization rationales. Thus we establish a method of designing coherent spectral risk measures based on non-satiated risk-averse utility functions. We show that the resulting risk measures satisfy second order stochastic dominance and their empirical portfolio performances are slightly improved over ETL.
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