Finding Optimal Weights in a Three-Asset Portfolio

Summary

The document presents a portfolio allocation problem involving three assets, with expected returns, volatilities, and a correlation specified for two of them. One asset is described as having zero volatility. The questioner writes a portfolio variance expression and notes that the weights must sum to one, but cannot determine optimal allocations from those constraints alone.

The response points to the standard mean-variance approach: define an objective, such as maximizing expected return for a chosen risk level or optimizing a risk-adjusted measure, then use constrained optimization with the weights’ sum constraint. In matrix form, expected returns and the covariance matrix can support a direct calculation for common portfolio objectives. The discussion does not state which objective or practical constraints should define “optimal,” nor does it calculate weights. Its useful lesson is that a variance equation and a budget constraint do not uniquely determine an allocation; the optimization target and relevant covariance inputs are also needed.

Key ideas

• A variance constraint and weights summing to one do not by themselves identify optimal portfolio weights.
• Mean-variance allocation can be framed as constrained optimization over asset weights.
• The objective must specify what “optimal” means, such as a risk-return tradeoff.
• A covariance matrix provides the inputs for a matrix-based allocation calculation.

Tags

# Optimal asset allocation in three-asset portfolio


# Optimal asset allocation in three-asset portfolio












I apologize if similar question has been already asked.

I have to find optimal weights $w_F,_I,_M$ for the assets $F, I, M$ in the portfolio.

$E(r_F) = 0.03$

$σ_F = 0$

$E(r_I) = 0.2325$

$σ_I = 0.5$

$E(r_M) = 0.12$

$σ_M = 0.2$

$ρ_I,_M = 0.9$

After computing the $σ_p^2$, I get: $0.0225 =0.25w_I^2 + 0.04w_M^2 + 0.18w_I,_M $

So I also know that $ w_I + w_M + w_F = 1 $, but I can't get optimal values nevertheless. Any ideas?

## Answer by Xenarc (score 2, accepted)

https://quant.stackexchange.com/a/47387

I recommend having a look at this pdf, it outlines how to determine optimal weightings for n risky assets with linear algebra methods. The crux of the problem is that you must solve for the partial derivatives with respect to each weight, for 0 with the constraint of the sum of the weights = 1. This can be solved as a constrained optimisation problem / linear programming problem. Or, if you're looking for a quick answer, the pdf outlines a simple matrix operation to find the optimal weightings.

Hope this helps

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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