Optimal Risky-Asset Allocation with a Risk-Free Asset

Summary

The document raises a portfolio-choice problem: how to allocate wealth between a risky market asset and a risk-free asset when utility rewards expected return and penalizes return variance. It gives a quadratic mean-variance utility function with a risk-aversion parameter and proposes expressions for portfolio expected return and variance. The central issue is how to derive the optimal weights from the first-order conditions.

The post itself contains no solution or numerical inputs for the assets’ expected returns, volatilities, or correlation, so it cannot establish an optimal allocation. Its useful lesson is that the risk-aversion parameter should influence the optimal risky exposure when the objective is correctly specified and differentiated. The proposed expected-return expression also uses portfolio volatility as an input, which may be inconsistent with treating asset weights as the decision variables; the setup needs careful formulation before solving. The document is therefore a problem statement rather than a completed allocation method.

Key ideas

• Mean-variance utility trades expected portfolio return against portfolio variance.
• Risk aversion should affect the optimal allocation to the risky asset.
• The portfolio variance formula includes each asset’s weighted variance and their covariance contribution.
• An allocation cannot be calculated without expected returns, volatilities, correlation, and a consistent return formulation.

Tags

# Optimal asset allocation


# Optimal asset allocation












I apologize if similar question has been already asked.

I have to compute optimal allocation of investment I between market asset and risk free asset. The investor's utility function is $U(w)=E(r_P)- γσ^2(r_P))/2$, $γ = 3$.

Then I use two formulas:

$E(r_P) = r_F + ((E(r_M) - r_F)σ_P)/σ_M$

$σ^2_P = w^2_1σ^2_1 + w^2_2σ^2_2 + 2w_1 w_2 σ_1 σ_2 ρ$

Which I plug in the utility function, before taking the derivatives wrt $w_1$ & $w_2$ . The result I obtained after equating both FOC seems wrong, because I can't use $γ = 3$ information anywhere to solve the problem:

$w_1 = ρ^2$

So where's the mistake?

Also I don't know the code for fraction line, so sorry about the messy notation.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.