Minimum Variance Frontier

Posted By: Advisor Analyzer Team

The Mean Variance framework is a strategic asset allocation approach that uses static data, as oppose to dynamic.  Like the Sharpe ratio, CML, Treynor, etc… the approach focuses on risk adjusted returns.  More specifically, the mean variance examines return on an investment per unit of risk as measured by standard deviation.

 

In the world of investments, there is a broad range of returns and risk profiles per asset class. When we combine all the highest expected returns per unit of risk profiles, it creates a frontier with risk on the x-axis and returns on the y-axis.  The portfolio’s that lie on the outer edge of frontier are the most efficient meaning they provide the highest level of returns per risk level, or the lowest risk for a give point of returns.  Any portfolio’s that lie below the frontier are less efficient and sub optimal relative to those on the outer edge. 

Efficient Portfolios

Essentially, portfolios with varying allocations can share the same returns and risk profiles.  It depends on the weighted average return of the portfolio, and the weighted average standard deviations after adjusting for correlations between assets.  Once we have determined the composition of any two portfolios on the frontier, we can determine any portfolio on the frontier by calculating:

 

·     R = (Weight A) (Return A) + (Weight B = 1 -  Weight A) (Return B)

 

By using the frontier, we can effectively determine the expected returns of a portfolio without knowledge of the asset allocation of that portfolio. 

 

The process of using adjacent portfolios to determine a third portfolio’s standard deviation characteristics is known as corner portfolios method.  This method however, is an estimate for portfolio risk as oppose to an accurate measure due to the fact that it does take into consideration the correlation of assets in the adjacent portfolios.  As the correlation decreases, the aggregate risk of the portfolio is reduced, and vice versa.  Intuitively speaking, two assets that are perfectly correlated will move identically providing no diversification benefit.  Conversely, assets that are negatively correlated will provide higher risk reduction since an increase in one asset will be offset by the decrease of another.

Resampled Efficient Frontier

While in theory, the efficient frontier is a set of viable portfolios with known expected returns, standard deviations and correlations, application of the process proves challenging.  First, the frontier is a representation of all possible investable assets, which is hardly an easy task to list.  More importantly, estimating the expected return, standard deviation, and correlation is a daunting challenge.  Historical returns (actual returns) are used to generate a mean and variance frontier and countless calculations are needed.  Since there are potentially infinite combinations of weightings to obtain the same risk and return profiles, computer testing is used to resample another set of portfolio weights with the same mean and variance.  Each run will generate another frontier with differing weights, and this process is repeated numerous times for each asset.  Finally, the portfolio manager averages the weights for each asset and uses the weights to determine the outer most portfolios. 

Conclusion

There are many performance measures beyond the mean variance frontier used in portfolio management today however the efficient frontier is arguably the cornerstone behind many of these methods.  The Sharpe ratio, CML, SML, Treynor, and MM can all be linked back to frontier’s fundamental concepts.