Mean-Risk Optimization for Normal Mixture Distributions

This section develops the mean-risk portfolio optimization framework for normal mixture distributions. The key insight is that the normal mixture structure (1) enables a dimension reduction from \(d\) assets to a two-dimensional problem.

Coherent Risk Measures

Definition. Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space and \(\mathcal{L}(\Omega, \mathcal{F})\) the set of real-valued random variables. A coherent risk measure is a function \(\rho : \mathcal{L} \to \mathbb{R}\) satisfying [Artzner1999]:

1. Monotonicity: If \(X \leq Y\), then \(\rho(X) \geq \rho(Y)\).

2. Translation invariance: \(\rho(X + c) = \rho(X) - c\) for all \(c \in \mathbb{R}\).

3. Positive homogeneity: \(\rho(\lambda X) = \lambda \rho(X)\) for all \(\lambda \geq 0\).

4. Subadditivity: \(\rho(X + Y) \leq \rho(X) + \rho(Y)\).

Definition. For a continuous random variable \(X\) and \(\alpha \in (0, 1)\):

\[\begin{split}\operatorname{VaR}_\alpha(X) &:= -\inf\{x \in \mathbb{R} : \mathbb{P}(X \leq x) > \alpha\}, \\ \operatorname{CVaR}_\alpha(X) &:= -E[X \mid X \leq -\operatorname{VaR}_\alpha(X)].\end{split}\]

VaR is widely used but is not coherent (it lacks subadditivity). CVaR is coherent.

Risk Monotonicity for Normal Mixtures

Recall that a normal mixture random vector can be written as:

(1)\[X \stackrel{d}{=} \mu + \gamma Y + \sqrt{Y} Z,\]

where \(\mu, \gamma \in \mathbb{R}^d\), \(Y \geq 0\) is a univariate random variable, and \(Z \sim N(0, \Sigma)\) is independent of \(Y\).

Consider the univariate case \(d = 1\) with \(Z \sim N(0, \sigma^2)\), \(\sigma > 0\).

Theorem 1. Let \(\rho\) be a coherent risk measure that depends only on the distribution. If \(X\) follows (1), then:

1. \(\mu \mapsto \rho(X)\) is decreasing.

2. \(\gamma \mapsto \rho(X)\) is non-increasing.

3. \(\sigma \mapsto \rho(X)\) is non-decreasing on \(\mathbb{R}^+\).

Proof.

(i) By translation invariance: \(\rho(X) = \rho(\gamma Y + \sqrt{Y} \sigma Z) - \mu\), which is decreasing in \(\mu\).

2. For any \(\Delta\gamma \geq 0\):

\[\begin{split}\rho((\gamma + \Delta\gamma) Y + \sqrt{Y} \sigma Z) &\leq \rho(\gamma Y + \sqrt{Y} \sigma Z) + \rho(\Delta\gamma \, Y) \\ &\leq \rho(\gamma Y + \sqrt{Y} \sigma Z),\end{split}\]

since \(\rho(\Delta\gamma \, Y) \leq \rho(0) = 0\) by monotonicity (\(\Delta\gamma \, Y \geq 0\)).

(iii) The map \(\sigma \mapsto \rho(\gamma Y + \sigma \sqrt{Y} Z)\) is convex:

\[\rho(\gamma Y + (a \sigma_1 + (1-a) \sigma_2) \sqrt{Y} Z) \leq a \, \rho(\gamma Y + \sigma_1 \sqrt{Y} Z) + (1-a) \, \rho(\gamma Y + \sigma_2 \sqrt{Y} Z),\]

and symmetric about zero (since \(\gamma Y + \sigma \sqrt{Y} Z \stackrel{d}{=} \gamma Y - \sigma \sqrt{Y} Z\)). A convex symmetric function is non-decreasing on \(\mathbb{R}^+\). \(\square\)

Intuitively, (1) can be viewed as a portfolio with a risk-free component \(\mu\), a non-negative-return asset with weight \(\gamma\), and a risky asset with weight \(\sigma\). Any coherent risk measure prefers large \(\mu\) and \(\gamma\) and small \(\sigma\).

Portfolio Return as Normal Mixture

In the \(d\)-dimensional case, a portfolio with weight \(w \in \mathbb{R}^d\) (\(w^\top \mathbf{e} = 1\)) has return:

(2)\[w^\top X \stackrel{d}{=} w^\top \mu + w^\top \gamma \, Y + \sqrt{w^\top \Sigma w \, Y} \, Z,\]

where \(Z \sim N(0, 1)\). The expected return is:

\[E[w^\top X] = w^\top \mu + w^\top \gamma \, E[Y].\]

Mean-Risk Optimization

The generalized mean-risk optimization problem is:

(3)\[\min_w \; \rho(w^\top X) \quad \text{s.t.} \quad w^\top \mathbf{e} = 1, \quad E[w^\top X] \geq m,\]

where \(m \in \mathbb{R}\).

Dimension Reduction via the Efficient Surface

Proposition. The solution of (3) is:

\[w^* = \Sigma^{-1} [\mu \; \gamma \; \mathbf{e}] \, A^{-1} [\tilde{\mu}^* \; \tilde{\gamma}^* \; 1]^\top,\]

where \(\tilde{\mu}^*, \tilde{\gamma}^* \in \mathbb{R}\) solve the two-dimensional problem:

\[\min_{\tilde{\mu}, \tilde{\gamma}} \; \rho\!\left(\tilde{\mu} + \tilde{\gamma} \, Y + \sqrt{g(\tilde{\mu}, \tilde{\gamma}) \, Y} \, Z\right) \quad \text{s.t.} \quad \tilde{\mu} + \tilde{\gamma} \, E[Y] \geq m,\]

with

\[g(\tilde{\mu}, \tilde{\gamma}) = [\tilde{\mu}, \tilde{\gamma}, 1] \, A^{-1} [\tilde{\mu}, \tilde{\gamma}, 1]^\top, \qquad A = [\mu \; \gamma \; \mathbf{e}]^\top \Sigma^{-1} [\mu \; \gamma \; \mathbf{e}].\]

Proof. Define \(w^*(\tilde{\mu}, \tilde{\gamma})\) as the solution of:

(4)\[w^*(\tilde{\mu}, \tilde{\gamma}) := \arg\min_w \rho(w^\top X) \quad \text{s.t.} \quad w^\top \mathbf{e} = 1, \; w^\top \mu = \tilde{\mu}, \; w^\top \gamma = \tilde{\gamma}.\]

Then (3) is equivalent to optimizing over \((\tilde{\mu}, \tilde{\gamma})\) with constraint \(\tilde{\mu} + \tilde{\gamma} E[Y] \geq m\).

By Theorem 1 and (2), when \(w^\top \mu\) and \(w^\top \gamma\) are fixed, \(\rho(w^\top X)\) is non-decreasing in \(w^\top \Sigma w\). Therefore (4) reduces to:

\[w^*(\tilde{\mu}, \tilde{\gamma}) = \arg\min_w w^\top \Sigma w\]

with the same constraints. By Lagrange multipliers:

\[w^*(\tilde{\mu}, \tilde{\gamma}) = \Sigma^{-1} [\mu \; \gamma \; \mathbf{e}] \, A^{-1} [\tilde{\mu} \; \tilde{\gamma} \; 1]^\top.\]

Substituting into (2) completes the proof. \(\square\)

This reduces the \(d\)-dimensional problem to a two-dimensional one in \((\tilde{\mu}, \tilde{\gamma})\). The surface \((\tilde{\mu}, \tilde{\gamma}) \mapsto \rho\) is the efficient surface, generalizing the classical efficient frontier.

Worst-Case Risk Measures

Definition. Let \(\mathcal{P}\) be a set of probability distributions. The worst-case coherent risk measure is:

\[\rho^*(w) := \sup_{f \in \mathcal{P}} \rho(w^\top X).\]

For the box uncertainty set of normal mixture models:

\[\mathcal{P} = \{f_X(\cdot | \mu, \gamma, \Sigma) : \underline{\mu} \preceq \mu \preceq \overline{\mu}, \; \underline{\gamma} \preceq \gamma \preceq \overline{\gamma}, \; \underline{\Sigma} \preceq \Sigma \preceq \overline{\Sigma}, \; f_Y \text{ fixed}\},\]

where \(\preceq\) denotes element-wise inequality, the following holds:

Proposition. For any \(w \in \mathbb{R}^d_+\):

\[\rho^*(w) = \rho\!\left(w^\top \underline{\mu} + w^\top \underline{\gamma} \, Y + \sqrt{w^\top \overline{\Sigma} w \, Y} \, Z\right).\]

This follows directly from Theorem 1: the worst case uses the smallest \(\mu\), smallest \(\gamma\), and largest \(\Sigma\).

References

[Artzner1999]

Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228.