Skewed Target Range Strategy for Multiperiod Portfolio Optimization Using a Two-Stage Least Squares Monte Carlo Method

In this paper, we propose a novel investment strategy for portfolio optimization problems. The proposed strategy maximizes the expected portfolio value bounded within a targeted range, composed of a conservative lower target representing a need for capital protection and a desired upper target representing an investment goal. This strategy favorably shapes the entire probability distribution of returns, as it simultaneously seeks a desired expected return, cuts off downside risk and implicitly caps volatility and higher moments. To illustrate the effectiveness of this investment strategy, we study a multiperiod portfolio optimization problem with transaction costs and develop a two-stage regression approach that improves the classical least squares Monte Carlo (LSMC) algorithm when dealing with difficult payoffs, such as highly concave, abruptly changing or discontinuous functions. Our numerical results show substantial improvements over the classical LSMC algorithm for both the constant relative risk-aversion (CRRA) utility approach and the proposed skewed target range strategy (STRS). Our numerical results illustrate the ability of the STRS to contain the portfolio value within the targeted range. When compared with the CRRA utility approach, the STRS achieves a similar mean-variance efficient frontier while delivering a better downside risk-return trade-off.

[1]  Duan Li,et al.  Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation , 2000 .

[2]  R. Tyrrell Rockafellar,et al.  Coherent Approaches to Risk in Optimization Under Uncertainty , 2007 .

[3]  D. Morton,et al.  Efficient Fund of Hedge Funds Construction Under Downside Risk Measures , 2006 .

[4]  Huyên Pham,et al.  A numerical algorithm for fully nonlinear HJB equations: An approach by control randomization , 2013, Monte Carlo Methods Appl..

[5]  Sid Browne,et al.  The Risk and Rewards of Minimizing Shortfall Probability , 1999 .

[6]  D. Bernoulli Exposition of a New Theory on the Measurement of Risk , 1954 .

[7]  J. Tobin Estimation of Relationships for Limited Dependent Variables , 1958 .

[8]  X. Zhou,et al.  Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework , 2000 .

[9]  Vijay S. Bawa,et al.  Safety-First, Stochastic Dominance, and Optimal Portfolio Choice , 1978, Journal of Financial and Quantitative Analysis.

[10]  Alexandre M. Baptista,et al.  Economic implications of using a mean-VaR model for portfolio selection: A comparison with mean-variance analysis , 2002 .

[11]  Jyrki Wallenius,et al.  Multiple criteria decision making: Foundations and some approaches , 2012 .

[12]  James O. Williams Maximizing the Probability of Achieving Investment Goals , 1997 .

[13]  Lucie Teplá,et al.  Optimal investment with minimum performance constraints , 2001 .

[14]  Sid Browne,et al.  Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark , 1999, Finance Stochastics.

[15]  Ramesh K. S. Rao,et al.  Asset Pricing in a Generalized Mean-Lower Partial Moment Framework: Theory and Evidence , 1989, Journal of Financial and Quantitative Analysis.

[16]  Edward Carr Franks,et al.  Targeting Excess-of-Benchmark Returns , 1992 .

[17]  P. Shevchenko,et al.  Bias-Corrected Least-Squares Monte Carlo for Utility Based Optimal Stochastic Control Problems , 2019 .

[18]  Huyên Pham,et al.  A large deviations approach to optimal long term investment , 2003, Finance Stochastics.

[19]  Hideo Nagai Downside risk minimization via a large deviations approach , 2012 .

[20]  Tien Foo,et al.  Asset Allocation in a Downside Risk Framework , 2000 .

[21]  C. Oosterlee,et al.  On Pre-Commitment Aspects of a Time-Consistent Strategy for a Mean-Variance Investor , 2016 .

[22]  U. Makov,et al.  Optimal portfolios with downside risk , 2017 .

[23]  P. A. Forsyth,et al.  The 4% strategy revisited: a pre-commitment mean-variance optimal approach to wealth management , 2017 .

[24]  Lorenzo Garlappi,et al.  Numerical Solutions to Dynamic Portfolio Problems: The Case for Value Function Iteration using Taylor Approximation , 2009 .

[25]  Nico van der Wijst,et al.  Optimal portfolio selection and dynamic benchmark tracking , 2005, Eur. J. Oper. Res..

[26]  Robert Jarrow,et al.  Downside Loss Aversion and Portfolio Management , 2006, Manag. Sci..

[27]  V. Agarwal,et al.  Risks and Portfolio Decisions Involving Hedge Funds , 2004 .

[28]  Michael J. Stutzer,et al.  A Portfolio Performance Index , 2000 .

[29]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[30]  S. Sheu,et al.  Asymptotics of the probability minimizing a "down-side" risk , 2010, 1001.2131.

[31]  Alessandro Balata,et al.  Regress-Later Monte Carlo for optimal control of Markov processes , 2017, 1712.09705.

[32]  Andreas Winkelbauer,et al.  Moments and Absolute Moments of the Normal Distribution , 2012, ArXiv.

[33]  James C. T. Mao,et al.  SURVEY OF CAPITAL BUDGETING: THEORY AND PRACTICE , 1970 .

[34]  John N. Tsitsiklis,et al.  Regression methods for pricing complex American-style options , 2001, IEEE Trans. Neural Networks.

[35]  Jules H. van Binsbergen,et al.  Solving dynamic portfolio choice problems by recursing on optimized portfolio weights or on the value function? , 2007 .

[36]  Bengt Muthén,et al.  Moments of the censored and truncated bivariate normal distribution , 1990 .

[37]  Cornelis W. Oosterlee,et al.  Accurate and Robust Numerical Methods for the Dynamic Portfolio Management Problem , 2017 .

[38]  Lorenzo Garlappi,et al.  Solving Consumption and Portfolio Choice Problems: The State Variable Decomposition Method , 2010 .

[39]  J. Carriére Valuation of the early-exercise price for options using simulations and nonparametric regression , 1996 .

[40]  Nicholas Barberis,et al.  A Model of Casino Gambling , 2009, Manag. Sci..

[41]  Steen Koekebakker,et al.  Portfolio Performance Evaluation with Generalized Sharpe Ratios: Beyond the Mean and Variance , 2008 .

[42]  R Burr Porter,et al.  Semivariance and Stochastic Dominance: A Comparison , 1974 .

[43]  Francis A. Longstaff,et al.  Valuing American Options by Simulation: A Simple Least-Squares Approach , 2001 .

[44]  R. C. Merton,et al.  On Estimating the Expected Return on the Market: An Exploratory Investigation , 1980 .

[45]  Cornelis W. Oosterlee,et al.  The Stochastic Grid Bundling Method: Efficient Pricing of Bermudan Options and their Greeks , 2013, Appl. Math. Comput..

[46]  William W. Hogan,et al.  Computation of the Efficient Boundary in the E-S Portfolio Selection Model , 1972, Journal of Financial and Quantitative Analysis.

[47]  Tsong-Yue Lai Portfolio selection with skewness: A multiple-objective approach , 1991 .

[48]  James C. T. Mao,et al.  Models of Capital Budgeting, E-V VS E-S , 1970, Journal of Financial and Quantitative Analysis.

[49]  Michel Denault,et al.  Dynamic portfolio choices by simulation-and-regression: Revisiting the issue of value function vs portfolio weight recursions , 2015, Comput. Oper. Res..

[50]  U. Cherubini,et al.  RiskMetrics Technical Document , 2015 .

[51]  Hiroshi Konno,et al.  A mean-absolute deviation-skewness portfolio optimization model , 1993, Ann. Oper. Res..

[52]  M. Milevsky,et al.  ASSET ALLOCATION AND ANNUITY‐PURCHASE STRATEGIES TO MINIMIZE THE PROBABILITY OF FINANCIAL RUIN , 2006 .

[53]  P. Fishburn Mean-Risk Analysis with Risk Associated with Below-Target Returns , 1977 .

[54]  C. Bacon,et al.  Practical Risk‐Adjusted Performance Measurement , 2012 .

[55]  K. Hamza,et al.  Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach , 2016, Quantitative Finance.

[56]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[57]  Shaler Stidham,et al.  A note on separation in mean-lower-partial-moment portfolio optimization with fixed and moving targets , 2005 .

[58]  E. Rowland Theory of Games and Economic Behavior , 1946, Nature.

[59]  A. R. Norman,et al.  Portfolio Selection with Transaction Costs , 1990, Math. Oper. Res..

[60]  Cornelis W. Oosterlee,et al.  Multi-period mean–variance portfolio optimization based on Monte-Carlo simulation , 2016 .

[61]  A. Tversky,et al.  Advances in prospect theory: Cumulative representation of uncertainty , 1992 .