Computing Unconstrained Mean–Variance Portfolio Weights

The mean-variance utility problem

Let:

• $\mathbf{r}$ be the vector of risky-asset returns over one period;
• $r_f$ be the risk-free return over that period;
• ${\mathbf{r}}^e=\mathbf{r}-r_{f}1$ be the vector of excess returns;
• $\mathbf{\mu }=\mathbb{E}[{\mathbf{r}}^e]$ be the vector of expected excess returns;
• $\Sigma =\mathrm{Cov}({\mathbf{r}}^e)$ be the covariance matrix;
• $\mathbf{w}$ be the vector of allocations to risky assets.

The portfolio’s expected excess return and variance are

The standard mean-variance utility objective is

where $\gamma >0$ is the risk-aversion coefficient.

The two terms have a direct interpretation:

• ${\mathbf{w}}^{\mathsf{T}}\mathbf{\mu }$ rewards expected excess return.
• $\frac{\gamma }{2}{\mathbf{w}}^{\mathsf{T}}\Sigma \mathbf{w}$ penalizes variance.
• Larger $\gamma$ means variance is penalized more strongly, so the optimizer takes less risky exposure.

Here, “unconstrained” has a precise meaning: we impose no full-investment constraint such as

and no long-only or position-limit constraints. The risky weights can sum to less than, equal to, or greater than one.

When a risk-free asset is available, the residual cash allocation is

Thus, if the risky weights sum to $0.7$, the investor puts $30\%$ in the risk-free asset. If they sum to $1.2$, the investor borrows $20\%$ at the risk-free rate to lever the risky portfolio.

This is fundamentally different from the GMVP problem. The GMVP fixed total risky investment at $100\%$; this utility problem lets risk aversion determine the amount of capital exposed to the risky assets.

Deriving the unconstrained optimal weights

Assume $\Sigma$ is symmetric positive definite. Then the variance term is strictly convex, so the utility function is strictly concave and has a unique maximizer.

Start from

Differentiate with respect to the weight vector:

At the optimum, the gradient is zero:

Therefore,

Multiplying by ${\Sigma }^{-1}$ gives

This is the unconstrained mean-variance portfolio.

The first-order condition can also be read economically:

At the optimum, each asset’s expected excess return is balanced against its covariance contribution to portfolio risk, scaled by risk aversion. A high expected return alone does not guarantee a high allocation: the asset’s covariance with the whole portfolio matters.

Why this stationary point is the maximum

Define

Completing the square yields

Because $\Sigma$ is positive definite,

with equality only when $\mathbf{w}={\mathbf{w}}_{\star }$. The first term is therefore never positive, establishing that ${\mathbf{w}}_{\star }$ uniquely maximizes utility.

Risk aversion changes scale, not the risky direction

Define the unscaled risky direction

Then

For fixed $\mathbf{\mu }$ and $\Sigma$, changing $\gamma$ multiplies every risky position by the same scalar. It does not alter the relative composition of the risky sleeve.

If $\gamma$ doubles, then

Consequently:

while

Expected excess return therefore declines in proportion to $1/\gamma$, but variance declines in proportion to $1/{\gamma }^2$.

A useful implication is that the Sharpe ratio of this risky direction does not depend on $\gamma$:

Risk aversion selects how much of this efficient risky portfolio to hold, with the remainder allocated to the risk-free asset.

One boundary case deserves care. With $\gamma =0$, the problem becomes

If any positive-expected-excess-return direction exists and there are no constraints, the objective is unbounded: the optimizer would take arbitrarily large exposure. Thus $\gamma >0$, or practical portfolio constraints, are necessary for a finite solution.

Also, do not take the unconstrained weights and normalize them to sum to one as a general fix. Normalization changes the optimization problem: it removes the risk-free allocation decision and no longer corresponds to the specified value of $\gamma$.

Worked two-asset calculation

Use the covariance matrix from the previous lesson, now together with annual expected excess returns:

Suppose the investor has

The optimal risky allocation is

So the investor allocates approximately:

The risky weights sum to

The negative risk-free weight means borrowing at the risk-free rate to finance a $118.5\%$ allocation to the risky sleeve.

The portfolio’s expected excess return is

or about $8.89\%$ annually. Its variance is

so its volatility is approximately

If instead $\gamma =6$, all risky allocations halve:

The investor now allocates roughly $40.74\%$ to the risk-free asset. The risky sleeve has the same relative composition, but the total amount of risk taken is lower.

Reliable Python implementation

As with the GMVP, use a linear solve rather than explicitly constructing ${\Sigma }^{-1}$.

import numpy as np

def mean_variance_weights(excess_mu, cov, gamma):
    """
    Unconstrained mean-variance risky-asset weights.

    Parameters
    ----------
    excess_mu : array-like, shape (n_assets,)
        Expected excess returns over the same horizon as `cov`.
    cov : array-like, shape (n_assets, n_assets)
        Positive-definite covariance matrix.
    gamma : float
        Positive mean-variance risk-aversion coefficient.

    Returns
    -------
    w_risky : ndarray, shape (n_assets,)
        Risky-asset weights. Their sum need not equal one.
    """
    excess_mu = np.asarray(excess_mu, dtype=float)
    cov = np.asarray(cov, dtype=float)

    if gamma <= 0:
        raise ValueError("gamma must be strictly positive.")

    if cov.shape != (len(excess_mu), len(excess_mu)):
        raise ValueError("cov must be square and match excess_mu.")

    # Solves (gamma * Sigma) w = mu without explicitly inverting Sigma
    return np.linalg.solve(gamma * cov, excess_mu)


cov = np.array([
    [0.0225, 0.0075],
    [0.0075, 0.0625]
])

mu_excess = np.array([0.06, 0.10])
gamma = 3.0

w_risky = mean_variance_weights(mu_excess, cov, gamma)
w_cash = 1.0 - w_risky.sum()

expected_excess_return = w_risky @ mu_excess
variance = w_risky @ cov @ w_risky
volatility = np.sqrt(variance)

print("Risky weights:", w_risky)
print("Risky-weight sum:", w_risky.sum())
print("Implied cash weight:", w_cash)
print("Expected excess return:", expected_excess_return)
print("Volatility:", volatility)

# First-order-condition check: gamma * Sigma * w = mu
print("FOC holds:", np.allclose(gamma * cov @ w_risky, mu_excess))

Expected output, up to rounding:

Risky weights: [0.74074074 0.44444444]
Risky-weight sum: 1.1851851851851851
Implied cash weight: -0.18518518518518512
Expected excess return: 0.08888888888888889
Volatility: 0.17213259316477408
FOC holds: True

In a real implementation, keep several distinctions explicit:

• Estimate $\mathbf{\mu }$ as expected excess returns, not raw returns, when interpreting the residual as a risk-free allocation.
• Use a covariance matrix estimated over the same horizon as $\mathbf{\mu }$.
• Do not interpret $\gamma$ independently of the return horizon and units used in estimation.
• Treat extreme long-short positions as an output to diagnose, not necessarily a trade list to implement.

Why this matters for random-matrix portfolio methods

The mean-variance solution depends on two estimated quantities:

Thus it inherits instability from both estimated expected returns and the inverse sample covariance matrix.

The covariance component has the spectral representation

where ${\hat{\lambda }}_i$ and ${\hat{\mathbf{q}}}_i$ are sample eigenvalues and eigenvectors. Small sample eigenvalues enter through $1/{\hat{\lambda }}_i$, potentially producing large and unstable portfolio tilts.

Random matrix theory will help us address the covariance part of that issue. It cannot, by itself, make expected-return estimates reliable; mean estimation is often an even more severe practical problem.

Key takeaways

• The unconstrained mean-variance problem is

• With $\gamma >0$ and positive-definite $\Sigma$, its unique solution is

• The weights refer to the risky sleeve. Their sum need not equal one; the remaining allocation is cash or borrowing at the risk-free rate.
• Raising $\gamma$ proportionally reduces every risky position while preserving the risky sleeve’s direction.
• In code, compute weights with np.linalg.solve(gamma * cov, excess_mu), not an explicit matrix inverse.
• The same inverse-covariance sensitivity seen in the GMVP is present here, now combined with uncertainty in expected-return estimates.

Next, we will examine why covariance inversion specifically amplifies estimation error in low-variance sample-eigenvector directions.