Managing ambiguity in asset allocation

Abstract

This paper is about the issue of input parameter uncertainty in portfolio optimization in a discrete setting with finite states (such as the case in a world with different macroeconomic regimes). In such a setting, being unable to assign reliable point estimates to the probabilities (or frequencies) of the states creates the ambiguity. We first describe how this ambiguity can be modeled probabilistically. Then, we show how this added uncertainty can be dealt with in optimal asset allocation problems. In simple-yet-realistic example applications we demonstrate that without sacrificing much of the upside, ambiguity managed portfolios may enhance the uniformity of returns across different states when compared to portfolios constructed by traditional methods. We stress that a key conclusion to be taken from these methods builds the case for insurance-like and potentially negative-yielding investments such as bonds and commodities so as to hedge the unforeseeable macrouncertainties for a smoother portfolio performance. Finally, we offer a variety of problem domains in which ambiguity management can be nested including macroeconomic scenario-based asset allocation, investing with regime-switching models, momentum investing, and risk-based investing.

Appendix

Denoting by $\varvec{P}$ a $K \times K$ matrix of zeros except $\varvec{p}$ on the diagonal, ${\mathbb{E}}^{\varvec{p}} \left[ {{\mathcal{U}}\left( \varvec{w} \right)} \right]$ in (5) can equivalently be expressed as:

$${\mathbb{E}}^{\varvec{p}} \left[ {{\mathcal{U}}\left( \varvec{w} \right)} \right] = \varvec{p}^{\text{T}} \varvec{Rw} - \frac{\lambda }{2}\varvec{w}^{\text{T}} \varvec{R}^{\text{T}} \varvec{PRw}$$

(12)

Assuming nonconstant asset return vectors in different scenarios are linearly independent, their Gram matrix $\varvec{R}^{\text{T}} \varvec{R}$ is positive definite. That is, for any arbitrary $\varvec{w} \in {\mathbb{R}}^{N}$, $\varvec{w}^{\text{T}} \varvec{R}^{\text{T}} \varvec{Rw} > 0$. Because the scenario probabilities are nonnegative, $\varvec{p} \ge 0$, we also have $\varvec{w}^{\text{T}} \varvec{R}^{\text{T}} \varvec{PRw} > 0$ which shows that the $N \times N$ matrix $\varvec{R}^{\text{T}} \varvec{PR}$ is positive definite. We conclude that ${\mathbb{E}}^{\varvec{p}} \left[ {{\mathcal{U}}\left( \varvec{w} \right)} \right]$ is strictly concave in $\varvec{w}$. It is also clear that ${\mathbb{E}}^{\varvec{p}} \left[ {{\mathcal{U}}\left( \varvec{w} \right)} \right]$ is linear and hence convex in $\varvec{p}$. Needless to say, it is continuous in both arguments. Assuming that ${\mathcal{C} } \subseteq {\mathbb{R}}^{N}$ is a compact convex set (such as the case when weights sum to one and no short sales), and given that $B_{\varepsilon } \left( {\varvec{\gamma},\eta } \right) \subseteq {\mathbb{R}}^{K}$ is also a compact convex set, we have by the Sion’s minimax theorem

$$\mathop {\text{Maximize}}\limits_{{\varvec{w} \in {\mathcal{C}}}} \; \mathop {\text{Minimize}}\limits_{{\varvec{p} \in B_{\varepsilon } \left( {\varvec{\gamma},\eta } \right)}}\,{\mathbb{E}}^{\varvec{p}} \left[ {{\mathcal{U}}\left( \varvec{w} \right)} \right] = \mathop {\text{Minimize}}\limits_{{\varvec{p} \in B_{\varepsilon } \left( {\varvec{\gamma},\eta } \right)}} \; \mathop {\text{Maximize}}\limits_{{\varvec{w} \in {\mathcal{C}}}}\,{\mathbb{E}}^{\varvec{p}} \left[ {{\mathcal{U}}\left( \varvec{w} \right)} \right]$$

(13)

and the solution $\left( {\varvec{w}^{*} ,\varvec{p}^{*} } \right)$ at which this identity attained is a saddle point. Because for any $\varvec{p} \in B_{\varepsilon } \left( {\varvec{\gamma},\eta } \right)$ the outer maximization is strictly concave, the optimum $\varvec{w}^{*}$ is unique. However, the optimum $\varvec{p}^{*}$ may take different values as we lack strict convexity.

Rustem and Howe (2002) and Žaković et al (2000) describe an interior point algorithm for solving the constrained saddle point problem. In what follows, we will utilize the KKT conditions in (Berç & Howe, 2002) chapter 3.2.3 and provide the reader with a system of equations. This system can be solved with readily available software such as Matlab’s cone solver.

For brevity in notation, let us introduce functions $f, \;g\; {\text{and}}\; h$. $f\left( {\varvec{w},\varvec{p}} \right): = - {\mathbb{E}}^{\varvec{p}} \left[ {{\mathcal{U}}\left( \varvec{w} \right)} \right]$. $g\left( \varvec{w} \right) \le 0$ is a general vector function defining the constraint set and $h\left( \varvec{p} \right) \le 0$ is a general vector function defining the constraints on the scenario probabilities. In other words, ${\mathcal{C}} = \left\{ {\varvec{w} \in {\mathbb{R}}^{N} |g\left( \varvec{w} \right) \le 0} \right\}$ and $B_{\varepsilon } \left( {\varvec{\gamma},\eta } \right) = \left\{ {\varvec{p} \in {\mathbb{R}}^{K} |h\left( \varvec{p} \right) \le 0} \right\}$. In detail, $g_{1} \left( \varvec{w} \right) = \varvec{w}^{\text{T}} 1 - 1$, $g_{2} \left( \varvec{w} \right) = - \varvec{w}^{\text{T}} 1 + 1$, $g_{3:3 + N} \left( \varvec{w} \right) = - \varvec{w}$, $h_{1} \left( \varvec{p} \right) = \varvec{p}^{\text{T}} 1 - 1$, $h_{2} \left( \varvec{p} \right) = - \varvec{p}^{\text{T}} 1 + 1$, and $h_{3} \left( \varvec{p} \right) = {\varvec{\upgamma}}^{\text{T}} {\text{ln}}\left( {\varvec{\gamma}/\varvec{p}} \right) - \varepsilon /\eta$. Further, let us assume $N_{{\mathcal{C}}}$ and $N_{B}$ denote the number of constraints in ${\mathcal{C}}$ and $B_{\varepsilon } \left( {\varvec{\gamma},\eta } \right),$ respectively.

Second, let us introduce slack variables to transform inequality constraints to equality constraints as shown in Exhibit A1.

Exhibit A1 Dual and slack variables for ${\mathcal{C}}$ and $B_{\varepsilon } \left( {\varvec{\gamma},\eta } \right)$

Now we are ready to introduce the log-barrier function for the minmax problem with barrier penalty $\nu > 0$:

$$\ell \left( {\varvec{w},\varvec{p},\nu } \right) = f\left( {\varvec{w},\varvec{p}} \right) - \nu \left[ {\mathop \sum \limits_{i = 1}^{{N_{{\mathcal{C}}} }} \ln \left( {s_{{\varvec{w},i}} } \right) - \mathop \sum \limits_{j = 1}^{{N_{B} }} \ln \left( {s_{{\varvec{p},j}} } \right)} \right]$$

(14)

Therefore our original MaxMin problem can be solved using the following minmax problem:

$$\mathop {\text{Minimize}}\limits_{{\varvec{w},\varvec{ s}_{\varvec{w}} }} \; \mathop {\text{Maximize}}\limits_{{\varvec{p},\varvec{ s}_{\varvec{p}} }} \left\{ {\ell \left( {\varvec{w},\varvec{p},\nu } \right) |g\left( \varvec{w} \right) +\,\varvec{s}_{\varvec{w}}\,=\,0, h\left( \varvec{p} \right) + \varvec{s}_{\varvec{p}} = 0,\; \varvec{s}_{\varvec{w}} > 0,\; s_{p} > 0} \right\}$$

(15)

which has the following first order necessary and sufficient conditions given the convex–concavity assumptions:

$$\nabla_{\varvec{w}} f\left( {\varvec{w}^{\varvec{*}} \left( \nu \right),\varvec{p}^{\varvec{*}} \left( \nu \right)} \right) + \nabla_{\varvec{w}} g\left( {\varvec{w}^{\varvec{*}} \left( \nu \right)} \right)\varvec{\mu}_{{\varvec{w},1:2 + N}} = 0$$

(16)

$$\nabla_{\varvec{p}} f\left( {\varvec{w}^{\varvec{*}} \left( \nu \right),\varvec{p}^{\varvec{*}} \left( \nu \right)} \right) - \nabla_{\varvec{p}} h\left( {\varvec{p}^{\varvec{*}} \left( \nu \right)} \right)\varvec{\mu}_{{\varvec{p},1:3 }} = 0$$

(17)

$$\varvec{S}_{\varvec{w}}\varvec{\mu}_{{\varvec{w},1:2 + N}} - \nu 1 = 0,\quad \varvec{S}_{\varvec{p}}\varvec{\mu}_{{\varvec{p},1:3 + N}} - \nu 1 = 0$$

(18)

$$g\left( {\varvec{w}^{\varvec{*}} \left( \nu \right)} \right) + \varvec{s}_{\varvec{w}} = 0,\quad h\left( \varvec{p} \right) + \varvec{s}_{\varvec{p}} = 0$$

(19)

$$\varvec{s}_{{\varvec{w},1:2 + N}} > 0,\;\varvec{s}_{{\varvec{p},1:3}} > 0,\;\varvec{\mu}_{{\varvec{w},1:2 + N}} \,\ge\, 0,\;\varvec{\mu}_{{\varvec{p},1:3}} \ge 0,$$

(20)

where $\varvec{S}_{\varvec{a}}$ denotes a matrix of zeros except $\varvec{s}_{{\varvec{a},.:.}}$ on the diagonal and $1$ is a column vector of ones with the appropriate length. Let us use $\varvec{x} = \left[ {\varvec{w};\varvec{p};\varvec{s}_{{\varvec{w},1:2 + N}} ; \varvec{s}_{{\varvec{p},1:3}} ;\varvec{\mu}_{{\varvec{w},1:2 + N}} ;\varvec{\mu}_{{\varvec{p},1:3}} } \right]_{3N + K + 10 \times 1}$ as a placeholder for all the decision variables and define the following matrices:

$\varvec{A}_{\varvec{w}} = \left[ {\begin{array}{llll} &\quad{1^{\text{T}}}& \\ &\quad{ - 1^{\text{T}}}& \\ 1 &\quad \ldots &\quad 0 \\ 0 &\quad \ddots &\quad 0 \\ 0 &\quad \ldots &\quad 1 \\ \end{array}} \right]_{N + 2 \times N} ,\;\varvec{b}_{\varvec{w}} = \left[ {\begin{array}{cc} { - 1} \\ 1 \\ 0 \\ \vdots \\ 0 \\ \end{array} } \right]_{N + 2 \times 1} ,\;\varvec{A}_{\varvec{p}} = \left[ {\begin{array}{cc}{{\textbf{1}}^{\text{T}} } \\ {-{\textbf{1}}^{\text{T}}} \\ \end{array} } \right]_{2 \times K} ,\;\varvec{b}_{\varvec{p}} = \left[ {\begin{array}{cc} { - 1} \\ 1 \\ \end{array} } \right]_{2 \times 1}.$

Then the system of Eqs. (16)–(20) can be expressed as $\varvec{F}^{\varvec{\nu}} \left( \varvec{x} \right) = 0$ as follows:

$$\varvec{F}_{1:N}^{\nu } \left( \varvec{x} \right) = 0 = \left[ { - \varvec{R}^{\text{T}} \varvec{p} + \lambda \varvec{R}^{\text{T}} \varvec{PRw} + \varvec{A}_{\varvec{w}}^{\text{T}}\varvec{\mu}_{{\varvec{w},1:2 + N}} } \right]_{N \times 1}$$

(21)

$$\varvec{F}_{N + 1:N + K}^{\nu } \left( \varvec{x} \right) = 0 = \left[ { - \varvec{Rw} + (\lambda /2)\left( {\varvec{Rw}} \right)^{2} - \left[ {\varvec{A}_{\varvec{p}}^{\text{T}} , -\varvec{\gamma}/\varvec{p}} \right]\varvec{\mu}_{{\varvec{p},1:3}} } \right]_{K \times 1}$$

(22)

$$\varvec{F}_{N + K + 1:2N + K + 2}^{\nu } \left( \varvec{x} \right) = 0 = \left[ {\varvec{S}_{\varvec{w}}\varvec{\mu}_{{\varvec{w},1:2 + N}} - \nu 1} \right]_{N + 2 \times 1}$$

(23)

$$\varvec{F}_{2N + K + 3:2N + K + 5}^{\nu } \left( \varvec{x} \right) = 0 = \left[ {\varvec{S}_{\varvec{p}}\varvec{\mu}_{{\varvec{p},1:3}} - \nu 1} \right]_{3 \times 1}$$

(24)

$$\varvec{F}_{2N + K + 6:3N + K + 7}^{\nu } \left( \varvec{x} \right) = 0 = \left[ {\varvec{A}_{\varvec{w}} \varvec{w} - \varvec{b}_{\varvec{w}} + \varvec{s}_{{\varvec{w},1:2 + N}} } \right]_{N + 2 \times 1}$$

(25)

$$\varvec{F}_{3N + K + 8:3N + K + 9}^{\nu } \left( \varvec{x} \right) = 0 = \left[ {\varvec{A}_{\varvec{p}} \varvec{p} - \varvec{b}_{\varvec{p}} + \varvec{S}_{{\varvec{p},1:2}} } \right]_{2 \times 1}$$

(26)

$$\varvec{F}_{3N + K + 10}^{\nu } \left( \varvec{x} \right) = 0 = \left[ {{\varvec{\upgamma}}^{\text{T}} {\text{ln}}\left( {\varvec{\gamma}/\varvec{p}} \right) - \varepsilon /\eta \varvec{ } + s_{{\varvec{p},3}} } \right]_{1 \times 1}$$

(27)

with $\varvec{s}_{{\varvec{w},1:2 + N}} > 0$, $\varvec{s}_{{\varvec{p},1:3}} > 0$, $\varvec{\mu}_{{\varvec{w},1:2 + N}} \ge 0$, $\varvec{\mu}_{{\varvec{p},1:3}} \ge 0$.

The traditional method for solving $\varvec{F}^{\varvec{\nu}} \left( \varvec{x} \right) = 0$ is a Newton algorithm for a given $\nu$. Yet, we can offer two quick and dirty ways to solve the above system for small size problems. First, one can implement a nonlinear least squares solver with bounds on decision variables with the objective of minimizing the norm of $\varvec{F}\left( \varvec{x} \right)$. Second, one can solve a redundant feasibility problem with a nonlinear solver such as Matlab’s fmincon. To do that one simply solves the optimization problem $\mathop {\text{Minimize}}\limits_{{\varvec{x} \in \left[ {\varvec{l},\varvec{u}} \right]}} 0$ subject to $\varvec{F}^{\varvec{\nu}} \left( \varvec{x} \right) = 0,$ where $\varvec{l}$ and $\varvec{u}$ are lower and upper bounds on $\varvec{x}$ that handles the constraints on slack and dual variables. If speed is a concern, one can supply the analytical Jacobian and Hessian of the nonlinear constraint function $\varvec{F}^{\varvec{\nu}} \left( \varvec{x} \right) = 0$.

Lastly, on the issue of selecting $\nu$, interior point methods first start from a suitably large $\nu$ and reduce it towards 0 while checking the magnitudes of primal and dual feasibility as well as complementary slackness. If these are small enough, the algorithm assumes the solution is optimal enough. For our purposes, in small size problems, we will set $\nu$ at a relatively small number such as 1e−10 and solve the system $\varvec{F}^{\varvec{\nu}} \left( \varvec{x} \right) = 0$ as we can live with this level of infeasibility. For very large problems, it is recommended to implement an interior point method such as the one studied in Žaković et al 2000).