zbMATH — the first resource for mathematics

Portfolio-optimization models for small investors. (English) Zbl 1271.91094
Summary: Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level.
Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends.
In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.

91G10 Portfolio theory
91G60 Numerical methods (including Monte Carlo methods)
Full Text: DOI
[1] Bonami, P; Lejeune, M, An exact solution approach for portfolio optimization problems under stochastic and integer constraints, Operat Res, 57, 650-670, (2009) · Zbl 1226.90049
[2] De Loera, J; Hemmecke, R; Köppe, M; Weismantel, R, FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension, Math Program, 115, 273-290, (2008) · Zbl 1151.90029
[3] De Loera, J; Hemmecke, R; Onn, S; Weismantel, R, N-fold integer programming, Discret Opt, 5, 231-241, (2008) · Zbl 1151.90025
[4] Konno, H; Wijayanayake, A, Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints, Math Program, 89, 233-250, (2001) · Zbl 1014.91053
[5] Konno, H; Yamamoto, R, Integer programming approaches in Mean-risk models, Comput Manag Sci, 2, 339-351, (2005) · Zbl 1128.91027
[6] Konno, H; Yamazaki, H, Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market, Manag Sci, 17, 519-531, (1991)
[7] Lüthi, HJ; Doege, J, Convex risk measures for portfolio optimization and concepts of flexibility, Math Program, 104, 541-559, (2005) · Zbl 1094.91033
[8] Lüthi, HJ; Studer, G; Zimmermann, U (ed.); Derigs, U (ed.); Gaul, W (ed.); Möhring, R (ed.); Schuster, K-P (ed.), Maximum loss for risk measurement of portfolios, 386-391, (1997), Berlin · Zbl 0916.90017
[9] Mansini, R; Speranza, G, An exact approach for portfolio selection with transaction costs and rounds, IIE Trans, 37, 919-929, (2005)
[10] Mansini, R; Speranza, MG, Heuristic algorithms for the portfolio selection problem with minimum transaction lots, Eur J Oper Res, 114, 219-233, (1999) · Zbl 0935.91022
[11] Markowitz, H, Portfolio selection, J Financ, 7, 77-91, (1952)
[12] Rockafellar, RT; Uryasev, S, Optimization of conditional value-at-risk, J Risk, 2, 21-41, (2000)
[13] Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: modeling and theory. SIAM, Philadelphia · Zbl 1094.91033
[14] Speranza, MG, A heuristic algorithm for a portfolio optimization model applied to the Milan stock market, Comput Oper Res, 23, 433-441, (1996) · Zbl 0849.90015
[15] Young, MR, A minimax portfolio selection rule with linear programming solution, Manag Sci, 44, 673-683, (1998) · Zbl 0999.91043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.