Hi ahnuld,
Sorry I’m late to the party. You missed a test question about global minimum variance portfolio with two risky securities:
[ QUOTE ]
The global minimum variance portfolio formed from two risky securities will be riskless when the correlation coeffcient between the two securities is..
D) -1
[/ QUOTE ]
To understand how perfect negative correlation effects portfolio variance, we may examine the portfolio standard deviation formula:
Two securities with perfect negative correlation have the same standard deviation, but the direction of the deviations is perfectly opposite (as you know). Since I can’t type the rho character (for correlation) and sigma (for standard deviation), I will refer to them as p and s respectively.
Consider the three terms under the square root (we’ll ignore square root for now, and eliminate it later). By substituting –1 for p, the third term becomes:
- 2 w ( 1 – w ) s1 s2
When analyzing the three terms together, we can factor it using the binomial factoring property:
a^2 + b^2 – 2ab = ( a – b )^2
Therefore, the term under the square root become:
[w s1 – ( 1 – w ) s2]^2
Since this is a squared term and it is under a square root, we may cancel the power. In addition, the standard deviations are equal (since it is perfectly negatively correlated), therefore if the weights are equal, the equation will equal 0. When two securities are perfectly negatively correlated, using equal weights and factoring, all terms cancel each other in the portfolio variance formula.
While you were taking the test, you made incorrect assumptions about the effect on portfolio variance of short selling a security that is negatively correlated with another. I can see how you got confused. If we look at the two extremes (perfect positive and negative correlation) you may see why you missed the question. We assume equal weights (as you did while taking the test).
Consider two stocks that are perfectly positively correlated. For simplicity, imagine it is the same stock. You short 10 shares and long 10 shares. What would you expect to happen to the portfolio variance? It will be zero (coincidentally the return will be zero since it is the same security). A move in any direction of the long security will be offset by the move of the short security.
Now consider two perfectly negatively correlated securities. You short a security, hoping it will drop. The same drop will cause your long position value to rise by the same magnitude. This does not eliminate any variance. In fact, you are causing two securities that move in opposite directions to perform equally.
Hope my explanations and examples help you and make sense.
Clark