Question

Hyaonth Macaw invests 48% of her funds in stock l and the balance instock J. The standard de iation of returns on i is 15%, and on J t is 2es use decimals, not percents, in your calculations.) a. Calculate the variance of portfolio returns, assuming the correlation between the retuns is 1. (Do not round intermediate calculations. Round your answer to 4 decimal places.) Portfolio variance b. Calculate the variance of portfolio returns, assuming the correlation is 7.(Do not round intermediate calculations. Round your answer to 4 decimal places.) Portfolio variance c. Calculate the variance of portfolio retuns, assuming the correlation is 0. (Do not round intermediate calculations. Round your answer to 4 decimal places Portfolio variance

Answer 1

In a two asset portfolio Var(RiRj) = wi^2σi^2 + wj^2σj^2 + 2wiwjCov(Ri,Rj)

wi = the portfolio weight of the asset i
wj = the portfolio weight of the asset j
σi = the standard deviation of returns on asset i
σj = the standard deviation of returns on asset j
Var(RiRj) = Variance of the two asset portfolio returns.

Cov(Ri,Rj) = the covariance between the returns on the two assets
This covariance can be further simplified as Cov(Ri,Rj)=σ(i)σ(j)*corr(Ri,Rj)

corr(Ri,Rj) = the correlation between the returns on asset i and j
σi = the standard deviation of returns on asset i
σj = the standard deviation of returns on asset j

On simplifying variance equations by substituting the value of Cov(Ri,Rj) with σ(i)σ(j)*corr(Ri,Rj), we get
Var(RiRj) = wi^2σi^2 + wj^2σj^2 + 2wiwjσ(i)σ(j)*corr(Ri,Rj)

Weight is given by how much an investor is invested into a particular asset (here stocks).
Given that, Macaw has invested 48% in stock i, so weight on asset i is given by wi=48%
Weight of asset j is wj=100%-48%=52%
Standard deviation of returns on asset i =15%
Standard deviation of returns on asset j =29%

Now, Var(RiRj) = wi^2σi^2 + wj^2σj^2 + 2wiwjσ(i)σ(j)*corr(Ri,Rj)
We will use the following values:
wi=.48
wj=.52
σi=.15
σj=.29
So, Var(RiRj)=.48^2*.15^2 + .52^2*.29^2 +2*.48*.52*.15*.29*corr(Ri,Rj)
=.2304*.0225 + .2704*.0841 + .0217152*corr(Ri,Rj)
=.005184 + .02274064 + .0217152*corr(Ri,Rj)
Var(RiRj)=.02792464 + .0217152*corr(Ri,Rj)

Part a:
When correlation is 1:
Var(RiRj)=.02792464 + .0217152*corr(Ri,Rj)
=.02792464 + .0217152
=.04963984 or .0496 (rounded upto 4 decimal places)

Part b:
When correlation is .7:
Var(RiRj)=.02792464 + .0217152*.7
=.02792464+.01520064=.04312528 or .0431 (rounded upto 4 decimal places)

Part c:
When correlation is 0:
Var(RiRj)=.02792464 + .0217152*corr(Ri,Rj)
=.02792464 or .0279 (rounded upto 4 decimal places)