Optimal portfolio choice with benchmarks

Abstract

We construct an algorithm that makes it possible to numerically obtain an investor’s optimal portfolio under general preferences. In particular, the objective function and risks constraints may be driven by benchmarks (reflecting state-dependent preferences). We apply the algorithm to various classic optimal portfolio problems for which explicit solutions are available and show that our numerical solutions are compatible with them. This observation allows us to conclude that the algorithm can be trusted as a viable way to deal with portfolio optimisation problems for which explicit solutions are not in reach.

Keywords:

• Optimal portfolio
• algorithm
• law-invariant
• GOP
• cost-efficiency
• state-dependent preferences

Acknowledgements

C. Bernard gratefully acknowledges support from the project on Systemic Risk funded by the GRI in financial services, the Louis Bachelier Institute, and from the Odysseus research grant funded by Flanders Research Foundation. R. H. De Staelen acknowledges the support of the Flanders Research Foundation (FWO15/PDO/076). S. Vanduffel acknowledges the financial support of the Stewardship of Finance Chair and of the Flanders Research Foundation.

Notes

No potential conflict of interest was reported by the authors.

1 Bernard et al. (2015) show that this is equivalent to having preferences that satisfy first-order stochastic dominance (FSD). Interestingly, many economists consider a violation of this property as grounds for refuting a particular theory; see e.g., Birnbaum et al. (1997), Birnbaum and Navarrette (1998), Levy (2008) for further discussions and empirical evidence of FSD violations. To illustrate the importance of FSD consistency in the literature, note for instance that Kahneman and Tversky (1979) have developed the cumulative prospect theory Tversky and Kahneman (1992) in order to address the FSD violation of their original prospect theory.

2 In the literature, a significant number of papers solve portfolio problems in a rather ad-hoc fashion and aim at obtaining explicit formulae. In this regard, we refer to Merton’s expected utility problem (Merton, 1969, 1971), Merton’s problem with the Basak–Shapiro Value-at-Risk constraint (Basak & Shapiro, 2001), Browne’s target probability optimisation problem (Browne, 1999), the optimal portfolio problem for a loss-averse investor as in Berkelaar et al. (2004) and optimal choice under Yaari’s dual theory ((He & Zhou, 2011); Yaari, 1987.

3 The state-price ${\xi }_t(\omega )$ is the price per unit of probability $\mathbb{P}$ of the "atomic” time and state-contingent claim (Arrow-Debreu security) that delivers one unit of a specific consumption good if a specific uncertain state $\omega$ realises at a specific future date t. For more information we refer to (Eeckhoudt et al., 2011).

4 Here, increasing refers to “non-decreasing”. Specifically, it does not mean that the portfolio is strictly increasing in $S_T^{\star }$.

5 A standard example of a law-invariant objective function is the expected utility functional, in which $V(X)=\mathbb{E}[u(X)].$ For the discretised version we would then obtain that $f(x_1,x_2,\dots ,x_n)=\frac{1}{n}{\sum }_{i=1}^{n}u(x_i).$

6 An example of risk constraint is a Value-at-Risk constraint, i.e., $\mathbb{P}(X_T⩾\underline{W})⩾1-\alpha$, so that assuming $x_1⩽x_2⩽\cdots ⩽x_n$, and $\frac{k}{n}=\alpha$, it amounts to constraining $(x_1,x_2,\dots ,x_n)$ in that $\underline{W}⩽x_k$ must hold.

7 The function implements an active-set algorithm that solves the Karush-Kuhn-Tucker (KKT) equations; see Nocedal and Wright (2006) and Floudas and Pardalos (2009). When the objective and constraints are twice differentiable and have Lipschitz continuous second derivatives in a neighbourhood of the optimum, the algorithm converges to the optimum when one starts close enough to it; see Chapter 18 of Nocedal and Wright (2006) and Hanson (1981), Hanson (1999).

8 Given a distribution function $F_X(x)$ we denote its survival function by ${\overline{F}}_X(x)=1-F_X(x)$.

9 In fact, if one would define the payoff-wise distance as

${\tilde{\delta }}_k^K(X):=\frac{\mathbb{E}[\left|X_{k,T}^{\ast }-X_T^{\star }\right|]}{\mathbb{E}[\left|X_T^{\star }\right|]},$

one would observe that its magnitude in all examples becomes much smaller.

10 When $X⩽Y$, almost surely, then i) $V_A(X)⩽V_A(Y)$ and ii) if X meets the risk constraints, then Y also meets the risk constraints.

11 If $F_{X|A}=F_{Y|A}$, then i) $V_A(X)=V_A(Y)$ and ii) if X meets the risk constraints, then Y also meets the risk constraints.

12 A natural extension is to also increase m.

13 In the case of state-independent preferences, there is a constant risk aversion $\eta$, and the results in Table 4 are obtained explicitly. Indeed, recall that for the CRRA utility, the optimal wealth obtained with an initial budget $W_0$ is given by$$\begin{array}{c}\hfill X_T^{\star }=W_0{e}^{(1-\frac{1}{\eta })M_T}{e}^{-\frac{1}{2}{(1-\frac{1}{\eta })}^2{V}_T}{\left(S_T^{\star }\right)}^{\frac{1}{\eta }},\end{array}$$

from which it follows that $\text{Var}(X_T^{\star })=W_0^2{e}^{2M_T-(1-\frac{2}{\eta })V_T}(e^{\frac{1}{{\eta }^2}{V}_T}-1)$. In the case of state-dependent preferences – that is, a varying risk aversion $\eta (·)$ – the results in Table 4 are obtained numerically.

Additional information

Funding

This work was supported by the Flanders Research Foundation [grant number FWO15/PDO/076].