Global Minimum Variance Weights for Two Risky Assets

Summary

The document explains how to find the global minimum variance portfolio from two risky assets. It represents the portfolio with weights w and one minus w, then expresses portfolio variance using each asset’s variance and their covariance, represented through correlation and standard deviations. Differentiating this variance with respect to the first asset’s weight and setting the derivative to zero gives the minimizing weight in terms of the assets’ volatilities and correlation.

The portfolio’s expected return is the weighted average of the two expected returns. Substituting the minimizing weight into the variance and return formulas yields the portfolio’s variance, standard deviation, and expected return. The answer gives the core formulas but does not work through a numerical example or discuss constraints such as long-only weights; the stated minimum may therefore involve short positions when unrestricted weights are assumed.

Key ideas

• Portfolio variance depends on both asset variances and their covariance.
• The minimum variance weight is found by differentiating portfolio variance with respect to one asset’s weight.
• The second asset’s weight is one minus the first asset’s weight.
• Expected portfolio return is the weighted average of the two asset returns.
• The minimizing weight can imply a short position if no weight constraints are imposed.

Tags

# Computing the minimum variance portfolio for only two risky assets


# Computing the minimum variance portfolio for only two risky assets












Given two risky assets and their corresponding covariance matrix, how do I compute the global minimum variance portfolio, its standard deviation and its expected return?

## Answer by Brownian3 (score 4, accepted)

https://quant.stackexchange.com/a/16639

Assume the weights of the two assets are $w$,$1-w$ respectively;the expected returns and standard deviations are denoted by $\mu$,$\sigma$ with subscripts 1,2,p(for portfolio),i.e,we have $\mu_1$,$\mu_2$,$\mu_p$,$\sigma_1$,$\sigma_2$,$\sigma_p$.The correlation coefficent is $\rho$ Then

$$\sigma_p^2=w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\sigma_1\sigma_2\rho \,\,\,\,...(1)$$ $$\mu_p=w\mu_1+(1-w)\mu_2 \,\,\,\,\,\,\,\,\,\,\,\,...(2)$$ $$\frac{d_{\sigma_p}}{dw}=0$$ $$w=\frac{\sigma_2^2-\rho\sigma_1\sigma_2}{\sigma_2^2+\sigma_1^2-2\rho\sigma_1\sigma_2}\,\,\,\,\,\,...(3)$$

Substituting(3) into (1) and (2) and simplify them will lead to the answer.

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