Portfolio optimization under downside risk measures
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AbstractPortfolio optimization with respect to a risk measure that is coherent, easy to evaluate on large portfolios, and only penalizes low returns is of great value to practitioners and academics. In this thesis we consider risk measures defined by a-quantiles, and risk measures defined by tail means. We call these measures downside risk measures. We derive analytic expressions for all these risk measures, and investigate their characteristics. The particular quantile based risk measures we consider are: value at risk (VaR) , which is defined as the difference between the expected wealth and the corresponding a-quantile, capital at risk (GaR), defined as the difference between the wealth invested into the bond and the corresponding a-quantile, and relative value at risk (RVaR) , which is the ratio of the value at risk to the expected wealth. The tail mean based risk measures we investigate are: conditional value at risk (GVaR), defined as the difference between the riskless wealth and the tail mean, conditional capital at risk (GGaR) , defined as the difference between the expected wealth and the tail mean, and relative conditional value at risk (RGVaR) , which is the ratio of the conditional value at risk to the expected wealth. We show that only GGaR is a coherent risk measure, while GVaR and RGVaR are subadditive. The quantile based risk measures VaR, GaR and RVAR are not subadditive in general, so that none of these measures is coherent. We investigate continuous time portfolio selection problems under downside risk measures GaR, GGaR, VaR, RVaR, GVaR and RGVaR, in the Black Scholes setting, with time dependent parameters and deterministic, time dependent portfolios. Based on an idea introduced by Emmer at al., we introduce the fundamental dimension reduction procedure which transforms m-dimensional optimization problems into one-dimensional optimization problems. This idea leads to an optimal strategy which is a weighted average of the bond and Merton's portfolio, where the weights depend on the choice of the risk measure and the investor's risk tolerance. This result is an illustration of the two-fund separation theorem. The optimization results under GGaR and GaR favor investing into stocks over a longer time horizon, which is consistent with the common knowledge that stocks in the long run tend to outperform bonds. Under RVaR and RGVaR the portion of the wealth invested into the risky assets depends only on the investor's risk tolerance, regardless of the initial investment or the market setting. The optimization results under VaR and GVaR are counterintuitive in the sense that in better markets and during longer time horizons, we tend to invest less into the risky assets under these risk measures. We also investigate constrained portfolio selection problems where short-selling is not allowed, and optimization problems where the optimal solution is a constant portfolio. Finally, we provide several numerical examples which illustrate how the time dependency of the parameters can model the business cycle or the periodicity in the stocks' dynamics.
Bibliography: p. 213-216