Data-driven portfolio management with quantile constraints

Elçin Çetinkaya; Aurélie Thiele

doi:10.1007/s00291-015-0396-9

OR Spectrum

July 2015, Volume 37, Issue 3, pp 761–786 | Cite as

Data-driven portfolio management with quantile constraints

Authors
Authors and affiliations

Elçin Çetinkaya
Aurélie ThieleEmail author

Regular Article

First Online: 14 March 2015

Received: 24 May 2013

Accepted: 27 February 2015

264 Downloads
5 Citations

Abstract

We investigate an iterative, data-driven approximation to a problem where the investor seeks to maximize the expected return of her portfolio subject to a quantile constraint, given historical realizations of the stock returns. The approach, which was developed independently from Calafiore (SIAM J Optim 20:3427–3464 2010) but uses a similar idea, involves solving a series of linear programming problems and thus can be solved quickly for problems of large scale. We compare its performance to that of methods commonly used in the finance literature, such as fitting a Gaussian distribution to the returns (Keisler, Decision Anal 1:177–189 2004; Rachev et al. Advanced stochastic models, risk assessment and portfolio optimization: the ideal risk, uncertainty and performance measures, Wiley, New York 2008). We also analyze the resulting efficient frontier and extend our approach to the case where portfolio risk is measured by the inter-quartile range of its return. Our main contribution is in the detail of the implementation, i.e., the choice of the constraints to be generated in the master problem, as well as the numerical simulations and empirical tests, and the application to the inter-quartile range as a risk measure.

Keywords

Data-driven optimization Quantile constraints Iterative algorithm

This is a preview of subscription content, to check access.

Notes

Acknowledgments

We would like to thank to the audience of our talk at the 21st International Symposium on Mathematical Programming (ISMP 2012) in Berlin, Germany, and two anonymous reviewers for their insightful comments that have substantially improved the clarity of the paper as well as its positioning with respect to existing literature.

Appendix 1

Denote $e ^{R_i^t}$ the return of stock $i$ during time period $t$. Then return of stock $i$ from time 1 to time $T$ is $e^ {\sum _{t=1}^T R_i^t}$. Therefore, the portfolio return over $T$ period can be formulated as:

$$\begin{aligned} W = \sum _{i=1}^n x_i e^ {\sum _{t=1}^T R_i^t}. \end{aligned}$$

Then, the first and the second moments of the portfolio return are calculated as:

$$\begin{aligned} E[W]= & {} \sum _{i=1}^T x_i E[e^ {\sum _{t=1}^T R_i^t}] = \sum _{i=1}^n e^ {\left( \tilde{\mu _i} T + \frac{ \tilde{ \sigma _i}^2 T}{2}\right) } \end{aligned}$$

(11)

$$\begin{aligned} E[W^2]= & {} E \left[ \left( \sum _{i=1}^T x_i e^ {\sum _{t=1}^T R_i^t} \right) ^2 \right] \nonumber \\= & {} \sum _{i=1}^n \left( x_i^2 e^{2 T\tilde{ \mu _i} + 2 T \tilde{\sigma _i}^2 } +\sum _{j=1, j \ne i}^n x_i x_j e^{ \left( (\tilde{\mu _i} + \tilde{\mu _j})T+ \frac{T}{2} (\tilde{ \sigma _i}^2 + \tilde{\sigma _j}^2 +2 \rho _{i,j} \tilde{\sigma _i} \tilde{\sigma _j}) \right) } \right) \nonumber \\ \end{aligned}$$

(12)

We define the vector $b \in \mathcal {R}^{n} $ such that

$$\begin{aligned} b_i = e^ {\left( \tilde{\mu _i} T + \frac{ \tilde{ \sigma _i}^2 T}{2}\right) }\quad \forall i, \end{aligned}$$

and the matrix A $ \in \mathcal {R}^{n x n} $ such that

$$\begin{aligned} A_{i,j}= e^{ \left( (\tilde{\mu _i} + \tilde{\mu _j})T+ \frac{T}{2} (\tilde{ \sigma _i}^2 +\tilde{ \sigma _j}^2 + 2 \rho _{i,j}\tilde{\sigma _i} \tilde{ \sigma _j}) \right) } \, \forall i, \quad \forall j, \, \text{ and } \, i \ne j \end{aligned}$$

$$\begin{aligned} A_{i,i}= e^{2 T \tilde{\mu _i} + 2 T \tilde{\sigma _i}^2 } \quad \forall i. \end{aligned}$$

The Log-Normal approximation of the portfolio return is represented as $e ^{Y}$ where $ Y \sim N(\mu ^*, \sigma ^*)$. Then the following equations hold:

$$\begin{aligned} E[W]= & {} b' x = E [ e^Y ]= e^ { \mu ^* + \frac{ {\sigma ^*}^2}{2}} \nonumber \\ E[W^2]= & {} x'Ax = e^ { 2\mu ^* + 2 {\sigma ^*}^2} \end{aligned}$$

(13)

The solution of this system of equations is as follows:

$$\begin{aligned} \mu ^*= & {} 2 ln(b'x)- \frac{1}{2}ln(x'Ax) \nonumber \\ {\sigma ^*}^2= & {} ln(x'Ax) - 2 ln(b'x) \end{aligned}$$

(14)

Then, the expected return maximization problem with quantile constraint is written as:

$$\begin{aligned} \begin{array}{rl} \max &{} {\left( e^{\mu ^* + \frac{{\sigma ^*}^2}{2}}\right) } \\ \text{ s.t. } &{} \mu ^* + \phi ^{-1} (\alpha ) \sigma ^* \ge ln(q_m), \\ &{} x \in X, \end{array} \end{aligned}$$

which is equivalent to

$$\begin{aligned} \begin{array}{rl} \max &{}b^T x \\ \text{ s.t. } &{} 2 \ln (b^Tx) - \frac{1}{2} \ln (x^TAx) + \phi ^{-1} (\alpha ) \sqrt{ \ln (b^Tx) - 2 \ln (x^TAx)} \ge \ln (q_m), \\ &{} x \in X. \end{array} \end{aligned}$$

(15)

References

Artzner P, Delbaen F, Eber JM, Health D (1999) Coherent measures of risk. Math Finance 9(3):203–228CrossRefGoogle Scholar
Beasley J (2013) Portfolio optimization: models and solution approaches. Tutor Oper Res 1:201–221Google Scholar
Benati S, Rizzi R (2007) A mixed integer linear programming formulation of the optimal mean/value-at-risk portfolio problem. Eur J Oper Res 176:423–434CrossRefGoogle Scholar
Benninga S, Wiener Z (1998) Value-at-risk (var). Math Educ Res 7(4)Google Scholar
Calafiore GC (2010) Random convex programs. SIAM J Optim 20(6):3427–3464CrossRefGoogle Scholar
Calafiore GC (2013) Direct data-driven portfolio optimization with guaranteed shortfall probability. Automatica 49:370–380CrossRefGoogle Scholar
Calafiore GC, Monastero B (2012) Data-driven asset allocation with guaranteed short-fall probability. IEEE American Control Conference, pp 3687–3692Google Scholar
Cetinkaya E (2014) Essays in robust and data-driven risk management. Ph.D. thesis, Lehigh University, BethlehemGoogle Scholar
Colombo M (2007) Advances in interior point methods for large scale linear programming. Ph.D. thesis, Doctor of Philosophy University of EdinburghGoogle Scholar
Cornuejols G, Tutuncu R (2007) Optimization methods in finance. Cambridge University Press, New YorkGoogle Scholar
Dentcheva D, Ruszczynski A (2006) Portfolio optimization with stochastic dominance constraints. J Bank Finance 30(2):433–451CrossRefGoogle Scholar
El-Ghaoui L, Oks L, Oustry F (2000) Worst-case value-at-risk and robust asset allocation: a semidefinite programming approach. Tech. Rep. M00/59, University of California, BerkeleyGoogle Scholar
Fabozzi F, Kolm P, Pachamanova D, Focardi S (2007) Robust portfolio optimization and management. Wiley, New YorkGoogle Scholar
Fenton LF (1969) The sum of lognormal probability distributions in scatter transmission systems. IRE Trans Commun Syst CS 8(3):57–67Google Scholar
Gaivoronski A, Pflug G (2005) Value-at-risk in portfolio optimization: properties and computational approach. J Risk 7(2):1–31Google Scholar
Goh J, Lim K, Sim M, Zhang W (2012) Portfolio value-at-risk optimization for asymmetrically distributed asset returns. Eur J Oper Res 221(2):397–406CrossRefGoogle Scholar
Harlow WV (1991) Asset allocation in a downside risk framework. Financial Anal J 47(5):28–40CrossRefGoogle Scholar
Keisler J (2004) Value of information in portfolio decision analysis. Decision Anal 1(3):177–189CrossRefGoogle Scholar
Kim JH, Powell WB (2011) Quantile optimization for heavy-tailed distribution using asymmetric signum functions. Princeton UniversityGoogle Scholar
Larsen N, Mausser H, Uryasev S (2002) Algorithms for optimization of value-at-risk. In: Pardalos P, Tsitsiringos VK (eds) Financial Engineering, E-Commerce and Supply Chain. Applied Optimization, vol 126. Springer, pp 19–46Google Scholar
Linsmeier TJ, Pearson ND (2000) Value at risk. Financial Anal J 56(2):47–67CrossRefGoogle Scholar
Lobo M, Fazel M, Boyd S (2006) Portfolio optimization with linear and fixed transaction costs. Ann Oper Res 152(5)Google Scholar
Markovitz HM (1952) Portfolio selection. J Finance 7(1):77–91Google Scholar
Markovitz HM (1959) Portfolio selection. Wiley, New YorkGoogle Scholar
Naumov AV, Kibzun AI (1992) Quantile optimization techniques with application to chance constrained problem for water-supply system design. Tech. Rep. 92–5, Department of Industrial and Operations Engineering at University of Michigan and Department of Applied Mathematics Moscow Aviation Institute, Ann Arbor. Mi 48109 and Moskiw, 127080, RussiaGoogle Scholar
Oyama T (2007) Determinants of stock prices: the case of zimbabwe. A Working Paper of the International Monetary FundGoogle Scholar
Pankov AR, Platonov EN, Semenikhin KV (2002) Minimax optimization of investment portfolio by quantile criterion. Autom Remote control 64(7):1122–1137CrossRefGoogle Scholar
Pfaff B (2013) Financial risk modeling and portfolio optimization with R. Wiley, New YorkGoogle Scholar
Rachev S, Stoyanov S, Fabozzi F (2008) Advanced stochastic models, risk assessment and portfolio optimization: the ideal risk, uncertainty and performance measures. Wiley, New YorkGoogle Scholar
Rockafellar RT, Uryasev S (2000) Optimization of conditional value at risk. J Risk 2(3):21–41Google Scholar
Rodriguez GJL (1999) Portfolio optimization with quantile-based risk measures. Ph.D. thesis, Massachusetts Institute of Technology, MassachusettsGoogle Scholar
Roy A (1952) Safety first and the holding of assets. Econometrica 20(3):431–449CrossRefGoogle Scholar
Ruszczynski A, Vanderbei R (2003) Frontiers of stochastically nondominated portfolios. Econometrica 71(4):1287–1297CrossRefGoogle Scholar
Sharpe WF (1966) Mutual fund performance. J Bus 39:119–138CrossRefGoogle Scholar
Sharpe WF (1971) Mean absolute deviation characteristic lines for securities and portfolios. Manag Sci 18(2):B1–B13CrossRefGoogle Scholar
Sortino FA, Price LN (1994) Performance measurement in downside risk framework. J Invest 3:59–64CrossRefGoogle Scholar
Szegö G (2002) Measures of risk. J Bank Finance 26:1253–1272CrossRefGoogle Scholar
Uryasev S (ed) (2000) Probabilistic constrained optimization: methodology and applications. Springer, New YorkGoogle Scholar
Wozabal D (2012) Value-at-risk optimization using the difference of convex algorithm. OR Spectrum 34:681–683CrossRefGoogle Scholar
Yitzhaki S (1982) Stochastic dominance, mean variance, and gini’s mean difference. Am Econ Assoc 72(1):178–185Google Scholar
Zymler S, Kuhn D, Rustem B (2013) Worst-case value-at-risk of nonlinear portfolios. Manag Sci 59(1):172–188CrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Elçin Çetinkaya
- 1
Aurélie Thiele
- 2
Email author

1.Amazon.comSeattleUSA
2.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA

Data-driven portfolio management with quantile constraints

Abstract

Keywords

Notes

Acknowledgments

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article