Safety-first portfolio selection
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SAFETY-FIRST PORTFOLIO SELECTION V. I. Norkina and S. V. Boykob
UDC 519.865.5
Abstract. A. D. Roy’s safety-first (SF) approach to financial portfolio optimization is improved. Safety first means the minimization of the probability of negative returns. The improvement concerns a better estimation of the negative return probabilities by means of mean excess return risk functions. The search for the optimal SF-portfolio is similar to Roy’s geometric method but the efficient frontier is different. In case of a finite number of scenarios, the SF-portfolio selection problem is reduced to a mixed linear Boolean programming problem. Keywords: financial portfolio, optimization, return, downside risk, Roy safety, probability estimate, efficient frontier. INTRODUCTION In the present paper, we will improve A. D. Roy’s approach [1] to the safety optimization of a financial portfolio, which minimizes the estimate of the probability that portfolio’s return falls critically, the average return being constrained from below. The improvement implies a more accurate estimate of the probability of negative returns by threshold risk functions. Searching for the optimal safety portfolio reduces to the development and analysis of a new efficient frontier of portfolios non-dominated in risk functions. For the case of a finite set of return scenarios, the problem of choosing an SF-portfolio reduces to a mixed Boolean linear programming problem. The Markowitz model (1952) [2] of the optimization of a financial portfolio by risk–return criteria, where the variance (or standard deviation) of return is used as a risk measure, is widely known. The essence of the Markowitz model is developing an efficient frontier of portfolios with the minimum variance of return for the given average return and choosing an efficient portfolio that maximizes some utility function [3]. A related study of A. D. Roy, a British economist [1], was published independently in 1952, where he proposed to optimize a portfolio based on the criterion that the probability of the portfolio’s return falling below a minimum desired threshold is minimized (the safety-first criterion or SF-criterion). Since optimizing a probability function was a difficult computational problem in the 1950s, Roy proposed to minimize its upper-bound estimate obtained from the Chebyshev inequality. For an approximate problem, Roy gave a simple and elegant geometrical solution lying on the efficient frontier of non-dominated portfolios in the average return–risk (variance of return) plane. Though this work had not become widely popular, Roy’s ideas continued to develop, for example, in [4] (average return is maximized under a constrained probability of deriving an income less than a prescribed threshold), [5] (some quantile of return is optimized), [6] (income is maximized and the probability that income is less than a prescribed thereshold is minimized), [7] (lexicographic SF-principle is used), [8] (multiperiod SF-optimization is considered), [9] (the estimate of the probability of receiving less
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