Deriving Mean–Variance Portfolio Weights With Lagrange Multipliers

Mean-variance optimisation finds the weights with the smallest variance for a specified expected return. With a fully invested portfolio, unrestricted short selling, and a positive-definite covariance matrix, two Lagrange multipliers give the exact solution. One enforces the target return and the other enforces that the weights sum to one. Long-only limits turn the problem into a constrained quadratic programme.

Set up the two constraints and solve the multipliers

Let $w$ be the weight vector, $\mu$ the expected returns, $\Sigma$ the covariance matrix and $m$ the target return. Solve

\min_w \frac12 w^\top\Sigma w,\qquad \mathbf{1}^\top w=1,\quad \mu^\top w=m.

The Lagrangian is

\tfrac12w^\top\Sigma w-\lambda(\mu^\top w-m)-\gamma(\mathbf{1}^\top w-1)

. Stationarity gives

w=\Sigma^{-1}(\lambda\mu+\gamma\mathbf{1}).

Define $A=\mathbf{1}^\top\Sigma^{-1}\mathbf{1}$ , $B=\mathbf{1}^\top\Sigma^{-1}\mu$ , and $C=\mu^\top\Sigma^{-1}\mu$ . Substitution into the constraints yields $A\gamma+B\lambda=1$ and $B\gamma+C\lambda=m$ . With $D=AC-B^2>0$ ,

\gamma=\frac{C-Bm}{D},\quad \lambda=\frac{Am-B}{D},\qquad \sigma_p^2(m)=\frac{Am^2-2Bm+C}{D}.

The global minimum-variance portfolio has return

B/A

. The efficient branch consists of targets

m\geq B/A

. Below that return, moving towards the minimum-variance point increases expected return and decreases variance. When all assets have the same expected return, the return constraint is either redundant or infeasible and

D=0

Differentiating

\tfrac12 w^\top\Sigma w

gives

\Sigma w

because

\Sigma

is symmetric. The two linear constraints contribute multiples of

\mathbf 1

and

\mu

. Solving for

w

at fixed multipliers therefore gives a linear combination of

\Sigma^{-1}\mathbf 1

and

\Sigma^{-1}\mu

. Substituting that combination back into the budget and return equations produces a two-by-two system. Its determinant is positive when

\mu

is not proportional to

\mathbf 1

and

\Sigma

is positive definite. In computation, solve the linear systems rather than forming an explicit matrix inverse.

There is a useful boundary case before you invert the two-by-two matrix. If all assets have the same expected return, the return constraint is proportional to the budget constraint. Every fully invested portfolio then has the same expected return. A target at that return leaves only the global minimum-variance problem, while any other target is infeasible. This is why a zero determinant needs interpretation. Adding a small numerical constant to force the two-multiplier formula through would hide the fact that the two constraints are carrying the same information.

Check the solution on a two-asset target

Suppose equities have an expected return of 8% and volatility of 18%, bonds have an expected return of 4% and volatility of 8%, and their correlation is 0.20. The 6% return target forces an even split. The covariance is

0.20(0.18)(0.08)=0.00288

. The resulting variance is

0.01114

and volatility is about 10.55%. The global minimum-variance portfolio allocates about 10.65% to equities, giving an expected return of roughly 4.43%. The 6% target sits on the efficient branch.

Solve the two constraints in Python

Solve the linear systems directly to avoid forming a matrix inverse.

import numpy as np

mu = np.array([0.08, 0.04])
cov = np.array([[0.0324, 0.00288], [0.00288, 0.0064]])
constraints = np.column_stack([np.ones(mu.size), mu])
basis = np.linalg.solve(cov, constraints)
multipliers = np.linalg.solve(constraints.T @ basis, [1.0, 0.06])
weights = basis @ multipliers
print(weights, np.sqrt(weights @ cov @ weights))

Deriving Mean–Variance Portfolio Weights With Lagrange Multipliers

Set up the two constraints and solve the multipliers

Check the solution on a two-asset target

Solve the two constraints in Python

See how correlation changes two-asset portfolio risk

References and next steps