Minimum variance portfolio

Minimum variance portfolio

In a market with multiple risky assets, the minimum variance portfolio is a particular combination of those assets that yields the minimum volatility. To be more specific, consider the market depicted in Figure 1. Here, the blue curve represents the efficient frontier. That is, all portfolios that lie on it are efficient portfolios (e.g., D and E) whereas those that lie below it are inefficient portfolios (e.g., A, B, and C). In this context, the leftmost portfolio on the efficient frontier is the minimum variance portfolio (MVP). This is because it is the efficient portfolio that has the lowest level of risk. The MVP appeals to investors that are particularly risk-averse.

efficient vs inefficient portfolios
Figure 1: Efficient frontier stretching from the global minimum variance portfolio to the maximum return portfolio

Minimum variance portfolio formula

Let’s suppose we have two risky assets: A and B. Then, we can write the portfolio variance as:

portfolio variance with two assets

where σA and σB are the standard deviations of returns (i.e., return volatility) for these two assets, σAB is the covariance of returns, and ωA and ωB are the investment weights. Given that ωA + ωB = 1 (i.e., we’re fully invested in these two assets), we have ωB = 1 − ωA. So, we can rewrite the portfolio variance as follows:

return volatility investment weight

We’d like to find the investment weight ωA* that yields the minimum variance portfolio:

portfolio variance minimization problem

We can find that by taking the derivative of portfolio variance with respect to ωA and setting it equal to zero:

investment weight first order condition

Solving for the optimal investment weight ωA* yields the minimum variance portfolio formula:

minimum variance portfolio formula

Remember that the covariance between the returns of two assets σAB can also be written as ρABσAσB, where ρAB is the correlation coefficient. So, we can write the formula using the correlation coefficient instead of covariance as well:

formula with the correlation coefficient

Minimum variance portfolio calculator

minimum variance portfolio calculator

Three inputs are needed to use our minimum variance portfolio calculator:

  • The standard deviation of returns on the 1st asset in percentages (σA ≥ 0).
  • The standard deviation of returns on the 2nd asset in percentages (σB ≥ 0).
  • The correlation coefficient between the returns of the two assets (ρAB), which needs to be between −1 and +1.

Note that if the covariance between the returns of the two assets (σAB) is known/given instead of the correlation, the latter can be easily calculated as: ρAB = σAB / (σAσB).

Once the inputs are entered, the calculator gives the optimal investment weight for each asset (ωA* and ωB*), and the variance (σMVP2) and standard deviation (σMVP) of the portfolio that has the minimum variance. It is worth highlighting that either ωA* or ωB* can be negative, in which case short selling is implied.

To illustrate the use of the calculator, try the following example. If asset A has a volatility of 20%, asset B has a volatility of 10%, and the correlation coefficient between the returns of A and B is −0.25, then investing 25% in A and 75% in B yields the minimum variance portfolio, which has a variance of 62.5 and volatility of 7.9%.

Summary

One of the most significant insights of modern portfolio theory is that investors can reduce risk through diversification. In this lesson, we explained the concept of the (global) minimum variance portfolio. For any given number of assets, this portfolio has the lowest risk where risk is measured in terms of variance.

Further reading:

Clarke, de Silva, and Thorley (2006), “Minimum-Variance Portfolios in the U.S. Equity Market“, The Journal of Portfolio Management, Vol. 33(1), pp. 10-24.

what is next?

This lesson is part of our free course on investments.

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