In: Finance
Consider the following situation:
State of Economy |
Probability of State of Economy |
Returns if State Occurs |
|
Stock A |
Stock B |
||
Boom |
40% |
30% |
20% |
Average |
40% |
10% |
10% |
Recession |
20% |
-30% |
10% |
The expected return on the market portfolio is 10% and the US Treasury bill yields 2%. The capital market is currently in equilibrium.
1. Which stock has the most systematic risk?
2. Which stock has the most unsystematic risk? Explain why.
3. What is the standard deviation of a portfolio which is comprised of $8,400 invested in stock A and $3,600 in stock B?
Calculating Mean and Standard deviation for both stocks.
Stock A:
Expected Return of a stock with probability distribution is a weighted average of individual returns. It is calculated as (p1*r1)+(p2*r2)+(p3*r3)
= (0.4*0.3)+(0.4*0.1)+(0.2*-0.3)= 0.1
Standard deviation of a stock with probability distribution is calculated as: sqrt((summation(x^2*p(x)))-Mean^2)
summation(x^2*p(x))= (0.3^2*0.4)+(0.1^2*0.4)+(-0.3^2*0.2)= 0.058
Mean= 0.1= 10%
Standard Deviation= sqrt(0.058-0.1^2)= 0.2191= 21.91%
Stock B:
Expected Return of a stock with probability distribution is a weighted average of individual returns. It is calculated as (p1*r1)+(p2*r2)+(p3*r3)
= (0.4*0.2)+(0.4*0.1)+(0.2*0.1)= 0.14
Standard deviation of a stock with probability distribution is calculated as: sqrt((summation(x^2*p(x)))-Mean^2)
summation(x^2*p(x))= (0.2^2*0.4)+(0.1^2*0.4)+(0.1^2*0.2)= 0.022
Mean= 0.14= 14%
Standard Deviation= sqrt(0.022-0.14^2)= 0.049= 4.9%
Systematic Risk is calculated using CAPM.
For Stock A:
10%= 2%+Beta1*(10%-2%)
Beta1= 1.
For Stock B:
14%= 2%+Beta2*(10%-2%)
Beta2= 1.5
Stock B has more systematic risk.
Unsystematic Risk is the standard deviations of the stock which is calculated above.
Standard deviations of Stock A is 21.91% and of Stock B is 4.9%.
Stock A has more unsystematic risk.
For the given Portfolio, weights of Stock A is 8400/(8400+3600)= 70% and of Stock B is 30%.
Standard deviation of a 2 stock portfolio is calculated as sqrt((w1^2*sd1^2)+(w2^2*sd2^2)(2*w1*w2*c*sd1*sd2)); where w is weight of the stock, sd is standard deviation of the stock and c is the correlation between stocks.
So, Standard deviation= sqrt(0.7^2*0.2191^2+0.3^2*0.049^2+2*0.7*0.3*0.2191*0.049)
= 16.24%
A) Expected return of stock A = Probability × return
= 0.4 × 30% + 0.4 × 10% + 0.2 × (-30%)
= 12% + 4% - 6%
= 10%
As per CAPM model,
Expected return = Risk free rate + beta (market return - risk free rate )
10% = 2% + beta ( 10% - 2%)
10% = 2% + 8% beta
Beta = 8% / 8% = 1
Expected return of B = Probability × return
= 0.4 × 20% 0.4 × 10% + 0.2 × 10%
= 8% + 4% + 2%
= 14%
Expected return = risk free rate + beta (market return - risk free rate)
14% = 2% + beta (10% - 2%)
14% - 2% = 8% beta
Beta = 12% / 8% = 1.5
As the Beta of stock B is more than that of Stock A , systematic risk of Stock B is higher.
B) standard deviation of stock A = √ probability × ( return - expected return)^2
= √ 0.4 (0.30 - 0.1)^2 + 0.4 (0.1 - 0.1)^2 + 0.2 (-0.3 - 0.1)^2
= √ 0.4 (0.2)^2 + 0 + 0.2 (+0.4)^2
= √ 0.4 (0.04) + 0.2 (0.16)
= √ 0.016 + 0.032
= √ 0.48
= 21.91%
Standard deviation of stock B = √ probability × (return - expected return)^2
= √ 0.4 (0.20 - 0.14)^2 + 0.4 (0.10 - 0.14)^2 + 0.2 (0.10 - 0.14)^2
= √ 0.4 (0.06)^2 + 0.4 (-0.04)^2 + 0.2 (-0.04)^2
= √ 0.4 (0.0036) + 0.4 (0.0016) + 0.2 (0.0016)
= √ 0.00144 + 0.00064 + 0.00032
= √ 0.0024
= 4.90%
As the standard deviation of Stock A higher than Stock B, the unsystematic risk of Stock A is higher.
C) Covariance = √ probability (return of A - expected return of A) (return of B - expected return of B)
= √ 0.4 (0.3 - 0.10) (0.2 -0.14) + 0.4 (0.10 - 0.10) (0.10 - 0.14) + 0.2 (-0.3 - 0.1) (0.1 -0.14)
= √ 0.4 (0.2) (0.06) + 0 + 0.2 (-0.4) ( -0.04)
= √ 0.4 (0.012) + 0.2 (0.016)
= √ 0.0048 + 0.0032
= √ 0.0080
= 0.0894
Weight of stock A = 8,400 / 3,600 + 8,400
= 8,400 / 12,000
= 0.70
Weight of stock B = 1 - 0.70 = 0.30
Standard deviation of the portfolio = √ (weight of stock A)^2 (std deviation of stock A)^2 + (weight of stock B)^2 (std deviation of stock B) ^2 + 2 × weight of stock A × weight of stock B × covariance
= √ (0.70)^2 (0.2191)^2 + (0.30)^2 (0.0490)^2 + 2 × 0.7 × 0.3 × 0.0894
= √ (0.49) (0.048) + (0.09) (0.0024) + 0.0376
= √ 0.02352 + 0.000216 + 0.0376
= √ 0.0613
= 24.76%
The standard deviation of the portfolio is 24.76%
note:- answer might differ a bit due to rounding off