Data-driven portfolio management with quantile constraints

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.

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:

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

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

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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:

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

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The solution of this system of equations is as follows:

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

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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

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

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