Minimizing Portfolio Variance for Two Correlated Assets

Summary

The document explains how to choose weights for two risky assets when their returns are highly correlated and their volatilities differ. It gives the variance of a weighted return portfolio in terms of the asset weights, individual variances, and cross-asset correlation and standard deviations. With those inputs, an investor can differentiate the variance expression and solve for weights that minimize portfolio risk.

The example supplies a correlation of 0.9 and daily volatilities of 2.5% and 5%, but it does not work through the resulting weights or quantify a return improvement. The method addresses variance minimization, not maximizing expected return; that would require expected return estimates and additional constraints. Leverage or shorting may change the feasible weights, but the answer does not discuss those choices or their risks.

Key ideas

• Portfolio variance depends on both the assets’ individual variances and their covariance.
• The covariance term is determined by correlation and each asset’s standard deviation.
• Substituting estimated inputs into the variance expression allows optimization over portfolio weights.
• A minimum-variance allocation alone does not establish that expected returns will increase.

Tags

# How to optimize two highly correlated risky assets?


# How to optimize two highly correlated risky assets?












Suppose you have two highly correlated risky assets.

Correlation coefficient: 0.9

Volatility: Asset 1 price varies 2.5% /day Asset 2 price varies 5% / day

What can be done to do reduce the risk and increase the return?

Consider how to weigh each asset (ex. 50/50, 60/40, 80/20) and the possibility of leverage or shorting (if necessary).

## Answer by mark leeds (score 1, accepted)

https://quant.stackexchange.com/a/70723

The variance of the linear combination of returns $(\omega_{1} \times ret_1 + \omega_2 \times ret_2) = $

$\omega_{1}^2 \times \sigma^2(ret_{1}) + \omega_2^2 \times \sigma^2(ret_{2}) + \omega_1 \times \omega_2 \times \rho_{1,2} \times \sigma(ret_{1}) \times \sigma(ret_{2})$.

You have the variances and the correlations and the standard deviations. So, you can put those in and take the derivative with respect to $\omega_1$ and $\omega_2$ to hopefully find the minimum.

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.