Question

The standard deviation of the market-index portfolio is 35%. Stock A has a beta of 1.40 and a residual standard deviation of 45%.

a-1. Calculate the total variance of stock A if its beta is increased by 0.20?
a-2. Calculate the total variance of stock A if its residual standard deviation is increased by 7.54%?
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 (an increase of 0.20 in beta or increase of 7.54% in residual standard deviation) will have a greater impact on the portfolio's standard deviation?
i. Both will have the same impact.
ii. Increase of .20 in beta will have a greater impact.
iii. Increase of 7.54% in the residual standard deviation will have a greater impact.

Answer 1

a-1. Variance = Beta² * Standard deviation of market² + Residual standard deviation²

Beta increases by 0.2

= 1.4 + 0.2

= 1.6

Total variance

= 1.6² * 35%² + 45%²

= 51.61%

a-2. If residual value increases by 7.54% it becomes:

= 45 + 7.54

= 52.54%

Total variance

= 1.4² * 35%² + 52.54%²

= 51.61%

b. ii. Increase of .20 in beta will have a greater impact.

Portfolio standard deviation helps ascertain the effects of systematic risk on the portfolio. Beta is used to represent that systematic risk. A change in Beta will therefore affect standard deviation more because standard deviation shows the impact of beta as a representation of systematic risk.