Covariance Question

CFA Level I Quant
#1

Can someone please help me answer the following covariance question? I am pretty sure that I understand how the answer is calculated, but it’s taking me far too long to get there. Is there quick way to find the solution? Thanks!

Steve Miller Advisers, Inc., has determined four possible economic scenarios and has projected the portfolio returns for two portfolios for their client under each scenario. Personal’s economist has estimated the probability of each scenario as shown in the table below. Given this information, what is the covariance of the returns on Portfolio A and Portfolio B?

Scenario Probability Return on Portfolio A Return on Portfolio B A 15% 18% 19% B 20% 17% 18% C 25% 11% 10% B 40% 7% 9%

A) .890332

B) .001898

C) .002019

#2

the answer is B.

these are the steps to calculate covariance of 2 portfolios:

  1. calculate the expected return of each portfolio

  2. calculate the variance from the expected return of each possible return for both portfolios (ie return-expected return)

  3. find the product of variances (var Port A * var Port B) calculated under each of the 4 scenarios

  4. apply the probabilities to each of the resulting 4 products

  5. add the results

hope this makes sense!

#3

Cov(A,B)=E[(A-E(A))*(B-E(B))]

Calculate E(A)=0.15*18%+0.2*17%+0.25*11%+0.4*7%=11.65%

E(B)=12.55%

Now use the formula above Cov(A,B)=0.15*(18%-11.65%)*(19%-12.55%)+…+0.4*(7%-11.65%)*(9%-12.55%)=0.001899 -> answer is B