TY - JOUR
T1 - On the price of risk in a mean-risk optimization model
AU - Dentcheva, Darinka
AU - Stock, Gregory J.
N1 - Publisher Copyright:
© 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.
PY - 2018/10/3
Y1 - 2018/10/3
N2 - We investigate a mean-risk model for portfolio optimization where the risk quantifier is selected as a semi-deviation or as a standard deviation of the portfolio return. We analyse the existence of solutions to the problem under general assumptions. When the short positions are not constrained, we establish a lower bound on the cost of risk associated with optimizing the mean–standard deviation model and show that optimal solutions do not exist for any positive price of risk which is smaller than that bound. If the investment allocations are constrained, then we obtain a lower bound on the price of risk in terms of the shadow prices of said constraints and the data of the problem. A Value-at-Risk constraint in the model implies an upper bound on the price of risk for all feasible portfolios. Furthermore, we provide conditions under which using this upper bound as the cost of risk parameter in the model provides a non-dominated optimal portfolio with respect to the second-order stochastic dominance. Additionally, we study the relationship between minimizing the mean–standard deviation objective and maximizing the coefficient of variation and show that both problems are equivalent when the upper bound is used as the cost of risk. Additional relations between the Value-at-Risk constraint and the coefficient of variation are discussed as well. We illustrate the results numerically.
AB - We investigate a mean-risk model for portfolio optimization where the risk quantifier is selected as a semi-deviation or as a standard deviation of the portfolio return. We analyse the existence of solutions to the problem under general assumptions. When the short positions are not constrained, we establish a lower bound on the cost of risk associated with optimizing the mean–standard deviation model and show that optimal solutions do not exist for any positive price of risk which is smaller than that bound. If the investment allocations are constrained, then we obtain a lower bound on the price of risk in terms of the shadow prices of said constraints and the data of the problem. A Value-at-Risk constraint in the model implies an upper bound on the price of risk for all feasible portfolios. Furthermore, we provide conditions under which using this upper bound as the cost of risk parameter in the model provides a non-dominated optimal portfolio with respect to the second-order stochastic dominance. Additionally, we study the relationship between minimizing the mean–standard deviation objective and maximizing the coefficient of variation and show that both problems are equivalent when the upper bound is used as the cost of risk. Additional relations between the Value-at-Risk constraint and the coefficient of variation are discussed as well. We illustrate the results numerically.
KW - Coefficient of variation
KW - Mean-semi-deviation
KW - Portfolio optimization
KW - Stochastic orders
KW - Value-at-risk
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U2 - 10.1080/14697688.2018.1436765
DO - 10.1080/14697688.2018.1436765
M3 - Article
AN - SCOPUS:85044072357
SN - 1469-7688
VL - 18
SP - 1699
EP - 1713
JO - Quantitative Finance
JF - Quantitative Finance
IS - 10
ER -