1. The Minimum Variance Frontier and Related Concepts
Dr. Himanshu Joshi

2. The Mean Variance Frontier
An investor’s objective in using mean variance approach to portfolio selection is to chose an efficient portfolio.
An efficient portfolio is one offering the highest expected returns for a given level of risk as measured by variance or S.D of returns.
Thus if an investor quantifies her risk tolerance using S.D, she seeks the portfolio that she expect will deliver her the greatest return for S.D of return consistent with her risk tolerance.

3. Assumed Expected Returns, Variances, and Correlation: Two Asset CaseC:\Documents and Settings\himanshu\Desktop\SRPM June -Sept 2014\Concavity of Efficient Set.xlsx
Asset 1 (Large Cap Stocks)
Asset 2 (Government Bonds)
Expected Return
15% 5%
Variance
225
100
S.D
10%
Correlation
0.5

4. Expected Return and S.D of Portfolio
E(RP) = w1 E(R1) + w2 E(R2)
σp2 = w12 σ12 + w22 σ w1 w2 σ1 σ2 ρ1,2

5. SD and Return for Portfolio

6. Minimum Variance Frontier (Using Variance)

7. Minimum Variance Frontier (using SD)

8. Portfolio Possibilities Curve/Minimum Variance Frontier
Figure 1 graph the possible combinations of risk and return for a portfolio composed of government bonds and large cap stocks. On Y-Axis: Returns, on X-Axis: Variance/SD.
The two asset case is a special because all two asset portfolios plot on the curve illustrated (there is a unique combination of two assets that provides a given level of expected return).
This is Portfolio Possibility Curve. A curve plotting the expected return and risk of the portfolios that can be formed using two assets.

9. Minimum Variance Frontier
We can also call the curve as Minimum Variance Frontier. Because it shows the minimum variance that can be achieved for a given level of expected return.
Minimum Variance Frontier is more useful concept as it can be applied to more than two assets.

10. Minimum Variance Frontier
In general case of more than two assets, any portfolios plotting on imaginary horizontal line at any expected return level have the same expected return, and as we move left on that line, we have less variance of return.
The attainable portfolio farthest to the left on such line is minimum-variance portfolio for the level of expected return and one point on the minimum-variance frontier.

11. Three or More Assets
With three or more assets, the minimum variance frontier is a true frontier. It is the border of a region representing all combinations of expected return and risk that are possible (border of feasible region).
The region results from the fact that with three or more assets, an unlimited number of portfolios can provide a given level of return.
EX. If we have three assets with expected return of 5%, 12% and 20% and we want 11% return on portfolio, we would use the following equation to solve for the portfolio weights:
11% = (5%x w1) +(12%x w2) +{20%x (1-w1-w2)}
This single equation with two unknowns, w1 and w2, has an unlimited number of possible solutions, each solution representing a portfolio.

12. Minimum Variance Frontier (Using Variance)

13. Global Minimum Variance Portfolio (GMVP)
In above figure, note that the variance of global minimum-variance portfolio (the one with smallest variance) appears to be close to when expected return of the portfolio is 6.43.
Weights of GMVP: This global minimum-variance portfolio has 14.3 percent of assets in large cap stocks and 85.7 percent of assets in government bonds.
Given these assumed returns, standard deviations, and correlations, a portfolio manager should not choose a portfolio with less than 14.3% of assets in large cap stocks because any such portfolio will have both :
A higher variance and a lower expected return than the GMVP.
All of the points below Point A are inferior to the global minimum-variance portfolio, and they should be avoided.

14. Efficient Frontier
Portfolios located below the global minimum variance portfolio (point A) are dominated by others that have same variance but higher returns.
Because these dominated portfolios use risk inefficiently, they are inefficient portfolios.
The portion of the minimum variance frontier beginning with the global minimum-variance portfolio and continuing above it is called Efficient Frontier.

15. Efficient Frontier
Portfolios lying on the efficient frontier offer the maximum expected return for their level of variance of return.
Efficient portfolios use risk efficiently: investors making portfolio choices in terms of mean return and variance of return can restrict their selections to portfolios lying on the efficient frontier.
This simplifies the selection process.
If an investor can quantify his risk tolerance in terms of variance or SD of return, the efficient portfolio for that level of variance will represent Optimal-mean variance choice.

16. Intervention of Correlations..
The trade-off between risk and return for a portfolio depends not only on the expected asset returns and variances but also on the correlation of asset returns.
We can illustrate the Minimum Variance frontiers with different correlation between the two assets: -1,0,0.5,and 1.

17. Correlation of (-1)

18. Correlation (+1)

19. Correlation (0)

20. Correlation of (+0.5)

21. Inferring the Results with different Correlations
The correlations illustrated in figures are -1, 0,0.5,+1.
These figures illustrates a number of interesting characteristics about the minimum-variance frontiers and diversification:
1. The endpoints of all of the frontiers are the same. This fact should not be surprising, because at one endpoint all of the assets are in government bonds and at other end point of the assets are in large cap stocks. At each endpoint, the expected return and standard deviations are simply the return and standard deviation of the relevant asset. (stocks or bonds)

22. Inferring the Results with different Correlations
2. When correlation is +1, the minimum-variance frontier is an upward-slopping straight line. If we start at any point on the line, for each one percentage point increase standard deviation we achieve the same constant increment in expected return. With a correlation of +1, the return on one asset is an exact positive linear function of the return on the other asset. Because fluctuations in the returns on the two assets track each other in this way, the returns on one asset can not dampen or smooth out the fluctuations in the returns on the other asset. For correlation of +1, diversification has not potential benefit.

23. Inferring the Results with different Correlations
When we move from a correlation of +1 to a correlation of 0.5, the minimum variance frontier bows out to the left, in the direction of smaller standard deviation.
With any correlation less than +1, we can achieve any feasible level of expected return with a smaller standard deviation of return than of +1 correlation case.
As we move from a correlation of 0.5 to each smaller value of correlation, the minimum-variance frontier bows out farther to the left.

24. Inferring the Results with different Correlations
The frontiers for correlations of 0.5, 0 and -1 have negatively sloped part.
This means that if we start at the lowest point (100% in government bonds) and shift money into stocks until we reach the global minimum-variance portfolio, we can get more expected return with less risk.
Therefore, relative to an initial position of fully invested in government bonds, there are diversification benefits in each of these correlation cases.

25. Inferring the Results with different Correlations
A diversification benefit is a reduction in portfolio standard deviation of return through diversification without accompanying decrease in expected return.
Because the minimum-variance frontier lows out further to the left as we lower correlation, we can also conclude that as we lower correlation, holding all other values constant, there are increasingly large potential benefit to diversification.

26. Inferring the Results with different Correlations
When the correlation is -1, the minimum-variance frontier has two linear segments. The two segments join at the global minimum-variance portfolio, which has standard deviation of 0. with a correlation of -1, portfolio risk can be reduced to zero, if desired.

27. Inferring the Results with different Correlations
The efficient frontier is the positively sloped part of the minimum-variance frontier. Holding all other values constant, as we lower correlation, the efficient frontier improves in the sense of offering a higher expected return for a given feasible level of standard deviation of return.

28. Extension to the Three Asset Case..
Asset 1 Large Cap Stocks
Asset 2 Government Bonds
Asset 3
Small Cap Stocks
Expected Return
15% 5%
Variance
225
100
Standard Deviation
10%
Correlations:
Large Cap and Bonds
0.5
Large Cap and Small Cap
0.8
Bonds and Small Cap

29. Extension to the Three Asset Case
Would adding another asset to the possible investment choices improve the available trade-offs between risk and return?

30. Extension to the Three Asset Case
A new Asset permit us to move to a superior minimum-variance frontier.

31. Weights? Large Cap Govt Bonds Small Cap 1 0.05 0.9 0.1 0.8 0.10 0.151
0.05
0.9
0.1
0.8
0.10
0.15
0.7
0.2
0.6
0.25
0.5
0.3
0.4
Quadratic Programming

32. Diversification and Portfolio Size
σp2 = σ {(1-ρ)/n + ρ}