Question

Provide all the details/steps in answering the questions.

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. (5 points) Which stock has the most systematic risk? Provide all the steps and equations.
2. (5 points) Which stock has the most unsystematic risk? Explain why. Provide all the steps and equations.
3. (10 points) What is the standard deviation of a portfolio which is comprised of $8,400 invested in stock A and $3,600 in stock B?

Answer 1

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

= √ 0.0016

= 0.04

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 × 0.04

= √ (0.49) (0.048) + (0.09) (0.0024) + 0.0168

= √ 0.02352 + 0.000216 + 0.0168

= √ 0.040536

=  20.13%

The standard deviation of the portfolio is 20.13%

note:- answer might differ a bit due to rounding off