Modern Portfolio Theory and Mean-Variance Optimization › Multi-Asset Portfolio Optimization

With more than two assets, we need the full matrix formulation. The principles remain the same, but the algebra becomes more complex, requiring linear algebra tools to express and solve the optimization problem efficiently. Let $\mathbf{w}$ be the $n \times 1$ vector of weights, $\boldsymbol{\mu}$ the $n \times 1$ vector of expected returns, and $\Sigma$ the $n \times n$ covariance matrix.

The optimization problem has two related formulations, each emphasizing a different perspective on the objective:

Formulation 1: Minimize risk for a target return

This formulation starts with a desired level of expected return and asks: what is the least risky way to achieve it?

$$\begin{aligned} \min_{\mathbf{w}} \quad & \frac{1}{2}\mathbf{w}^\top \Sigma \mathbf{w} \\ \text{subject to} \quad & \mathbf{w}^\top \boldsymbol{\mu} = \mu_{\text{target}} \\ & \mathbf{w}^\top \mathbf{1} = 1 \end{aligned}$$

where:

• $\mathbf{w}$: vector of portfolio weights
• $\Sigma$: covariance matrix of asset returns
• $\boldsymbol{\mu}$: vector of expected returns
• $\mu_{\text{target}}$: required level of expected return
• $\mathbf{1}$: vector of ones (sum constraint)

The factor of $\frac{1}{2}$ in the objective is a mathematical convenience that simplifies the derivatives. It does not change the optimal solution because multiplying an objective by a positive constant does not alter which portfolio minimizes it.

Formulation 2: Maximize utility (mean-variance utility)

This formulation takes a different approach, directly modeling investor preferences through a utility function that trades off return against risk:

$$\begin{aligned} \max_{\mathbf{w}} \quad & \mathbf{w}^\top \boldsymbol{\mu} - \frac{\gamma}{2}\mathbf{w}^\top \Sigma \mathbf{w} \\ \text{subject to} \quad & \mathbf{w}^\top \mathbf{1} = 1 \end{aligned}$$

where:

• $\gamma$: risk aversion parameter ($\gamma > 0$)
• $\mathbf{w}$: vector of portfolio weights
• $\boldsymbol{\mu}$: vector of expected returns
• $\Sigma$: covariance matrix of asset returns
• $\mathbf{1}$: vector of ones

The parameter $\gamma$ captures how much you dislike risk. If you are highly risk-averse (large $\gamma$), you penalize variance heavily and will choose a conservative, low-volatility portfolio. If you are risk-tolerant (small $\gamma$), you care more about expected return and will accept higher volatility to chase higher gains. The target-return formulation traces the minimum-variance frontier, including its inefficient lower branch. For targets above the GMV return, its solutions also maximize mean-variance utility for a corresponding $\gamma>0$. Varying positive $\gamma$ traces the efficient upper branch and approaches the GMV portfolio as $\gamma$ tends to infinity.

For the target-return closed forms below, assume that $\Sigma$ is positive definite and that $\boldsymbol{\mu}$ is not proportional to $\mathbf{1}$. These conditions make $\Sigma$ invertible and $AC-B^2>0$. If the covariance matrix is singular, the inverse formulas do not apply as written; the constrained system must be treated directly.

For the first formulation, we form the Lagrangian using the techniques from Part I. Its equality multipliers are free variables, not fixed penalties. With differentiable functions and linearly independent constraint gradients, a constrained optimum satisfies stationarity together with the original constraints. We solve that system rather than minimize jointly over weights and multipliers:

$$\mathcal{L} = \frac{1}{2}\mathbf{w}^\top \Sigma \mathbf{w} - \lambda_1(\mathbf{w}^\top \boldsymbol{\mu} - \mu_{\text{target}}) - \lambda_2(\mathbf{w}^\top \mathbf{1} - 1)$$

where:

• $\mathcal{L}$: the Lagrangian function
• $\mathbf{w}$: vector of portfolio weights
• $\Sigma$: covariance matrix of asset returns
• $\lambda_1, \lambda_2$: Lagrange multipliers enforcing the return and budget constraints
• $\boldsymbol{\mu}$: vector of expected returns
• $\mu_{\text{target}}$: target portfolio return
• $\mathbf{1}$: vector of ones

When the optimal value varies differentiably with a constraint's right-hand side, the multiplier gives its signed local rate of change. With the sign convention used here, $\partial V/\partial\mu_{\text{target}}=\lambda_1$ for the optimal half-variance $V$. A finite change is generally only approximated by this rate times the change. Taking the derivative with respect to $\mathbf{w}$ and setting it to zero:

$$\frac{\partial \mathcal{L}}{\partial \mathbf{w}} = \Sigma \mathbf{w} - \lambda_1 \boldsymbol{\mu} - \lambda_2 \mathbf{1} = 0$$

where:

• $\Sigma$: covariance matrix
• $\mathbf{w}$: vector of portfolio weights
• $\lambda_1, \lambda_2$: Lagrange multipliers
• $\boldsymbol{\mu}$: vector of expected returns
• $\mathbf{1}$: vector of ones

This first-order condition states that at the optimum, the marginal increase in risk from adjusting any weight must be exactly offset by the value of that weight in meeting the return and budget constraints. Rearranging terms to solve for $\mathbf{w}$ explicitly:

$$\begin{aligned} \Sigma \mathbf{w} &= \lambda_1 \boldsymbol{\mu} + \lambda_2 \mathbf{1} && \text{(isolate } \Sigma \mathbf{w} \text{)} \\ \mathbf{w} &= \Sigma^{-1}(\lambda_1 \boldsymbol{\mu} + \lambda_2 \mathbf{1}) && \text{(pre-multiply by } \Sigma^{-1} \text{)} \\ &= \lambda_1 \Sigma^{-1} \boldsymbol{\mu} + \lambda_2 \Sigma^{-1} \mathbf{1} && \text{(distribute)} \end{aligned}$$

where:

• $\mathbf{w}$: optimal portfolio weight vector
• $\Sigma^{-1}$: inverse of the covariance matrix
• $\boldsymbol{\mu}$: vector of expected returns
• $\mathbf{1}$: vector of ones
• $\lambda_1, \lambda_2$: Lagrange multipliers

The vector $\Sigma^{-1}\boldsymbol{\mu}$ is a covariance-adjusted return direction, while $\Sigma^{-1}\mathbf{1}$ is proportional to the GMV portfolio. Their coefficients and normalization matter: both terms jointly satisfy the target-return and budget constraints, and $\lambda_1$ can be negative on the lower branch.

The Lagrange multipliers $\lambda_1$ and $\lambda_2$ are determined by the constraints. Substituting back:

$$\begin{aligned} \mu_{\text{target}} &= \lambda_1 \boldsymbol{\mu}^\top \Sigma^{-1} \boldsymbol{\mu} + \lambda_2 \boldsymbol{\mu}^\top \Sigma^{-1} \mathbf{1} \\ 1 &= \lambda_1 \mathbf{1}^\top \Sigma^{-1} \boldsymbol{\mu} + \lambda_2 \mathbf{1}^\top \Sigma^{-1} \mathbf{1} \end{aligned}$$

where:

• $\mu_{\text{target}}$: target return
• $\lambda_1, \lambda_2$: Lagrange multipliers
• $\boldsymbol{\mu}$: vector of expected returns
• $\Sigma^{-1}$: inverse covariance matrix
• $\mathbf{1}$: vector of ones

These two equations form a linear system in the two unknowns $\lambda_1$ and $\lambda_2$. We can solve for $\lambda_1$ and $\lambda_2$ in terms of scalar constants derived from the problem parameters:

$$\lambda_1 = \frac{C\mu_{\text{target}} - B}{AC - B^2}, \quad \lambda_2 = \frac{A - B\mu_{\text{target}}}{AC - B^2}$$

where:

• $A = \boldsymbol{\mu}^\top \Sigma^{-1} \boldsymbol{\mu}$
• $B = \boldsymbol{\mu}^\top \Sigma^{-1} \mathbf{1} = \mathbf{1}^\top \Sigma^{-1} \boldsymbol{\mu}$
• $C = \mathbf{1}^\top \Sigma^{-1} \mathbf{1}$
• $\mu_{\text{target}}$: target portfolio return

The constants $A$, $B$, and $C$ depend only on the assets' expected returns and covariance structure, not on the target return. This means we can precompute them once and then quickly find optimal portfolios for any target return, making efficient frontier construction computationally efficient.

With positive-definite covariance, setting $\lambda_1 = 0$ (ignoring the return target) gives the global minimum variance (GMV) portfolio. This formula does not require distinct expected returns:

$$\mathbf{w}_{\text{GMV}} = \frac{\Sigma^{-1} \mathbf{1}}{\mathbf{1}^\top \Sigma^{-1} \mathbf{1}}$$

where:

• $\mathbf{w}_{\text{GMV}}$: weights of the global minimum variance portfolio
• $\Sigma^{-1}$: inverse of the covariance matrix
• $\mathbf{1}$: vector of ones

This formula represents risk minimization in its most basic form. The term $\Sigma^{-1}\mathbf{1}$ identifies the portfolio structure that minimizes variance without regard for expected returns, while the denominator normalizes the weights to sum to one. Each weight depends on the full covariance matrix through $\Sigma^{-1}\mathbf{1}$; lower standalone variance or lower correlations do not by themselves guarantee a higher weight. With short selling allowed, some GMV weights can be negative.

This portfolio minimizes variance within the stated asset universe subject to the budget constraint. If variance alone is the objective, it is optimal regardless of expected return. A short investment horizon does not by itself imply that objective; cash needs, liabilities, and available risk-free investments can also matter.