Deriving Unconstrained Mean–Variance Optimal Portfolio Weights

Summary

The document states a quadratic mean–variance utility objective, with expected portfolio return as the reward and return variance penalized by a risk-aversion coefficient. It asks for a textbook reference supporting the familiar expression for optimal weights.

The response derives that expression by differentiating the objective with respect to the weights, setting the gradient to zero, and solving the resulting linear system using the inverse covariance matrix. This derivation applies to the unconstrained optimization problem shown. It does not impose a budget or weights-sum-to-one constraint, nor does it discuss whether the covariance matrix is invertible or how additional constraints would change the solution. The exchange gives an algebraic justification but no textbook citation.

Key ideas

• The stated objective balances expected portfolio return against a variance penalty.
• Setting the gradient of the unconstrained objective to zero yields a linear system for optimal weights.
• The closed-form solution assumes no sum-to-one or other portfolio constraints.
• The derivation requires an invertible return covariance matrix.

Tags

# On a source for a mean-variance portfolio optimization result


# On a source for a mean-variance portfolio optimization result












In the context of a mean_variance framework consider an optimizing investor who chooses at time $T$ portfolio weights $w$ so as to maximize the quadratic objective function:

$$U(w) = E[R_p] - \frac{\gamma}{2}Var[R_p]= w'\mu - \frac{\gamma}{2}w'Vw$$

Where $E$ and $Var$ denote the mean and variance of the uncertain portfolio rate of return $R_p = w'R_{T+1}$ to be realized in time $T + 1$ and $\gamma$ is the relative risk aversion coefficient. The optimal portfolio weights will be:

$$w^* = \frac{1}{\gamma}V^{-1}\mu $$

Could I have a reference that proves this result? preferably a textbook that builds up to it.

## Answer by Stefan Voigt (score 3)

https://quant.stackexchange.com/a/17988

You do note require a sum up constraint that gives you that the weights sum up to 1? Then the problem is equivalent to a maximization without constraints: $$Z(\omega)=w'\mu - \frac{\gamma}{2}w'Vw$$ then it holds that $$\frac{dZ}{d\omega}=\mu-\gamma V\omega\overset{!}{=}0\\ \Leftrightarrow \frac{1}{\gamma}\mu=V\omega^*\\ \Leftrightarrow\omega^* = \frac{1}{\gamma}V^{-1}\mu $$

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