Consider the following situation: State of Economy Probability of State of Econo
ID: 2628486 • Letter: C
Question
Consider the following situation:
State of Economy
Probability of State of Economy
Returns if State Occurs
Stock A
Stock B
Stock C
Boom
30%
25%
10%
5%
Normal
50%
10%
5%
20%
Recession
20%
-30%
-5%
10%
The market risk premium is 8% and the US Treasury bill yields 3%. The capital market is currently in equilibrium.
(a) (4 points) Which stock has the most systematic risk? Why? Provide all the equations.
(b) (4 points) Which stock has the most unsystematic risk? Why? Provide all the equations.
(c) (7 points) What is the standard deviation of a portfolio which is comprised of $5,000 invested in stock A, $7,000 in stock B, and $8,000 in stock C?
Please show how you got the answer!
State of Economy
Probability of State of Economy
Returns if State Occurs
Stock A
Stock B
Stock C
Boom
30%
25%
10%
5%
Normal
50%
10%
5%
20%
Recession
20%
-30%
-5%
10%
Explanation / Answer
Expected return of Stock A = 30%*25%+ 50%*10%+ 20%*(-30%)=6.5%
Standard deviation of Stock A = sqrt(30%*(25%-6.5%)^2+ 50%*(10%-6.5%)^2+ 20%*(-30%-6.5%)^2)= 19.37%
Beta of Stock A = (6.5%-3%)/8%=0.4375
Similarly for B
Expected return of Stock B= 30%*10%+ 50%*5%+ 20%*(-5%)=4.5%
Standard deviation of Stock B = sqrt(30%*(10%-4.5%)^2+ 50%*(5%-4.5%)^2+ 20%*(-5%-4.5%)^2)= 5.22%
Beta of Stock B = (4.5%-3%)/8%=0.188
Similarly for C
Expected return of Stock C= 30%*5%+ 50%*20%+ 20%*(10%)=13.5%
Standard deviation of Stock C= sqrt(30%*(5%-13.5%)^2+ 50%*(20%-13.5%)^2+ 20%*(10%-13.5%)^2)= 6.73%
Beta of Stock C = (13.5%-3%)/8%=1.3125
C has maximum beta and hence has most systematic risk
Std Deviation
Beta
A
19.37%
0.4375
B
5.22%
0.188
C
6.73%
1.3125
The standard deviation is made up of both systematic and unsystematic risk, whereas beta measures just systematic risk. Stock A has a high std deviation and a relatively low beta, which means most of its risk is unsystematic
A has most unsystematic risk
(c) (7 points) What is the standard deviation of a portfolio which is comprised of $5,000 invested in stock A, $7,000 in stock B, and $8,000 in stock C?
State of Economy
Probability of State of Economy
Returns if State Occurs
Return of portfolio
Stock A
Stock B
Stock C
Boom
30%
25%
10%
5%
11.75%
Normal
50%
10%
5%
20%
12.25%
Recession
20%
-30%
-5%
10%
-5.25%
Mean = 11.75%*30%+12.25%*50%+ 20%*(-5.25%)= 8.6%
standard deviation of a portfolio= = sqrt(30%*(11.75%-8.6%)^2+ 50%*(12.25%-8.6%)^2+ 20%*(-5.25%-8.6%)^2)=6.93%
Std Deviation
Beta
A
19.37%
0.4375
B
5.22%
0.188
C
6.73%
1.3125