The expectation of a random variable is its probability-weighted average, E[X]=∑ixipi. It is linear: E[aX+bY+c]=aE[X]+bE[Y]+c, always, whether or not X and Y are independent. The variance measures spread around the mean, Var(X)=E[(X−E[X])2]=E[X2]−E[X]2, and the standard deviation σ=Var(X) has the same units as X.
Cov(X,Y)=E[XY]−E[X]E[Y],ρ=σXσYCov(X,Y)(1)
Covariance measures co-movement: positive when X and Y tend to be above their means together. Correlation ρ rescales it to lie between −1 and 1. Variances add only when covariance is zero: in general Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y). For a portfolio with weights wi this is the quadratic form Var=∑i,jwiwjCov(Ri,Rj)=wTΣw.
A discrete return and a two-asset portfolioA return (in percent) is −2, 1 or 4 with probabilities 0.2, 0.5, 0.3. Then E[X]=−0.4+0.5+1.2=1.3, E[X2]=0.8+0.5+4.8=6.1 and Var(X)=6.1−1.32=4.41, so σ=2.1. Now split money 50/50 between assets with σ1=0.2, σ2=0.3 and ρ=0.5: Var=0.25(0.04)+0.25(0.09)+2(0.25)(0.5)(0.2)(0.3)=0.0475, so σp≈0.218, below the average of 0.25.
Expectation is linear, with or without independence.
Variance: E[X2]−E[X]2; Var(aX+b)=a2Var(X).
Covariance and correlation measure co-movement; Cov(X,X)=Var(X).
Zero correlation is not independenceIndependent variables have zero covariance, but not the other way round. If X is symmetric around 0 and Y=X2, then Cov(X,Y)=0 although Y is completely determined by X. Correlation only sees linear co-movement; a market that is calm until it crashes can have low correlation and strong dependence in the tails.
Lab · what's graded
The expected return15 pts
The variance15 pts
A covariance10 pts
A portfolio variance20 pts
Which rules are true15 pts
A correlation10 pts
Why diversification works15 pts
0/ 7 answered
1Numeric answer15 ptsNot answered
A return (in percent) is -2, 1 or 4 with probabilities 0.2, 0.5 and 0.3. Compute the expected value E[X].
E[X]=accepted: ± 0.001
2Numeric answer15 ptsNot answered
A return (in percent) is -2, 1 or 4 with probabilities 0.2, 0.5 and 0.3. Compute Var(X)=E[X2]−E[X]2.
Var(X)=accepted: ± 0.005
3Numeric answer10 ptsNot answered
For two returns, E[X]=1, E[Y]=2 and E[XY]=3. Find Cov(X,Y).
Cov=accepted: ± 0.001
4Numeric answer20 ptsNot answered
Two assets have σ1=0.2, σ2=0.3 and correlation ρ=0.5. Compute the variance of a 50/50 portfolio. Give it to four decimals.
σp2=accepted: ± 0.0002
5Choice15 ptsNot answered
Select every rule that always holds.
6Numeric answer10 ptsNot answered
Cov(X,Y)=0.9, σX=1.5 and σY=1.0. Find the correlation ρ.
ρ=accepted: ± 0.001
7Explain in words15 ptsNot answered
In two sentences, explain why a portfolio of two assets that are not perfectly correlated has less risk than the average of the two.
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