Question

The standard deviation of the market-index portfolio is 30%. Stock A has a beta of 1.50 and a residual standard deviation of 40%. a-1. Calculate the total variance for an increase of 0.10 in its beta? (Do not round intermediate calculations. Round your answer to 4 decimal places.) Total variance 0.3904 a-2 Calculate the total variance for an increase of 3.35% in its residual standard deviation? (Do not round intermediate calculations. Round your answer to 4 decimal places.) Total variance b. An investor who currently holds the market-index portfolio decides to reduce the portfolio allocation to the market index to 90% and to invest 10% in stock A. Which of the following changes will have a greater impact on the portfolio's standard deviation? Increase of 3.35% in residual standard deviation. Increase of 10 in beta

Answer 1

Part a-1:
Total variance is equal to systematic variance+residual variance
=beta^2 * variance of market index portfolio + variance of the stock
Given that, beta=1.5
Given that there is an increase of .1 in beta. So, beta value that we are going to use=1.5+.1=1.6

Here, standard deviation of market-index portfolio is 30%=.3
Residual standard deviation of stock A=40%=.4
Variance=(Standard deviation)^2
Variance of market index portfolio=(0.3)^2=0.09
Variance of stock A=(0.4)^2=0.16

Total variance=beta^2*variance of market index portfolio + variance of the stock

Substituting the values we get,
Total variance=1.6^2*0.09+0.16=2.56*0.09+0.16=0.3904

Part a-2:
Given that, the residual standard deviation (of stock A) increases by 3.35%=0.0335

New residual standard deviation=0.4+0.0335=0.4335
New variance of stock=0.4335^2=0.18792225
Variance of market index portfolio=(0.3)^2=0.09
Given that, beta=1.5
Total variance=beta^2*variance of market index portfolio + variance of the stock
Substituting the values we get,
1.5^2*0.09+0.18792225=0.2025+0.18792225=0.39042225

Part b:
Answer: Increase of .1 in beta.

First option:
Now the portfolio allocation is: 90% in market index and 10% in stock A.

Increase in 3.35% in residual standard deviation we calculated as:
Total variance=beta^2*variance of market index portfolio + variance of the stock
Substituting the values we get,
1.5^2*0.09+0.18792225=0.2025+0.18792225=0.39042225

Now, with the proportion of allocation, we will have:
Total variance=beta^2*variance of market index portfolio*Percentage allocation in market index portfolio + variance of the stock*Percentage allocation in stock
=1.5^2*0.09*90%+0.18792225*10%=0.18225+0.018792225=0.201042225
Standard deviation=Variance^1/2=0.201042225^1/2=0.448377324

Second option:
Increase of .1 in beta:
We have calculated this scenario as:
Total variance=beta^2*variance of market index portfolio + variance of the stock
Substituting the values we get,
Total variance=1.6^2*0.09+0.16=2.56*0.09+0.16=0.3904
Now the portfolio allocation is: 90% in market index and 10% in stock A.
Total variance=beta^2*variance of market index portfolio*Percentage allocation in market index portfolio + variance of the stock*Percentage allocation in stock
Total variance=1.6^2*0.09*90%+0.16*10%=0.20736+0.016=0.22336
Standard deviation=Variance^1/2=0.472609776

So, increase of .1 in beta will have greater impact.