Question

The following table presents the performance of stock and bond funds under various scenarios.

Scenario	Probability	Rate of return of stock fund (%)	Rate of return of bond fund (%)
Severe recession	0.10	-35	-8
Mild recession	0.20	-10	13
Normal growth	0.50	15	6
Boom	0.20	30	-5

Suppose an investor forms a portfolio with stocks and bonds. Find the investment opportunity set in differing proportions. (10 marks)

Weights in stock fund	Portfolio expected return	Portfolio standard deviation
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

1

e. Draw the investment opportunity set. (3 marks)

f. Calculate the weight in stock fund, expected return and standard deviation of the minimum-variance portfolio. (6 marks)

g. Given the T-bill rate is 2%. Calculate the weight in stock fund, expected return and standard deviation of the optimal risky portfolio. (6 marks)

h. Calculate the Sharpe ratio for the optimal risky portfolio. (2 marks)

i. Draw the capital allocation line (CAL) on the diagram in part (e). Show the position of the optimal risky portfolio on the CAL. (3 marks)

j. Suppose an investor chooses to invest 80% of his investment fund in the optimal risky portfolio and 20% in a risk-free asset. Calculate the expected return and standard deviation of his portfolio. (4 marks)

Answer 1

First we need to find the expected return and standard deviation of stock and bond fund:

States	Probabilaity	Stock	Probability Weighted Return	P(X - Expected return of Stock)^2
Severe Recession	0.1	-35.00%	0.1x-35%=-3.5%	0.1(-0.35-0.08)^2=1.849%
Mild recession	0.2	-10.00%	0.2x-10%=-2%	0.2(-0.1-0.08)^2=0.648%
Normal Growth	0.5	15.00%	0.5x15%=7.5%	0.5(0.15-0.08)^2=0.245%
Boom	0.2	30.00%	0.2x30%=6%	0.2(0.3-0.08)^2=0.968%
Expected Return=sum of probability weighted returns			8.000%
Variance= Sum of P(X - Expected return of Bond)^2				3.710%
Standard deviation=square root of variance				19.261%

States	Probabilaity	Bond	Probability Weighted Return	P(X - Expected return of Bond)^2
Severe Recession	0.1	-35%	0.1x-8%=-0.8%	0.1(-0.08-0.038)^2=0.13924%
Mild recession	0.2	-10%	0.2x13%=2.6%	0.2(0.13-0.038)^2=0.16928%
Normal Growth	0.5	15%	0.5x6%=3%	0.5(0.06-0.038)^2=0.0242%
Boom	0.2	30%	0.2x-5%=-1%	0.2(-0.05-0.038)^2=0.15488%
Expected Return=sum of probability weighted returns			3.800%
Variance= Sum of P(X - Expected return of Bond)^2				0.488%
Standard deviation=square root of variance				6.983%

We also need to find the covariance and the correlation of the two:

State	Probability	Stock	Bond	P(X - Expected return of Stock) x (X - Expected return of Bond)
Severe Recession	0.1	-35.00%	-8.00%	0.1(-0.35-0.08)x(-0.08-0.038)=0.5074%
Mild recession	0.2	-10.00%	13.00%	0.2(-0.1-0.08)x(0.13-0.038)=-0.3312%
Normal Growth	0.5	15.00%	6.00%	0.5(0.15-0.08)x(0.06-0.038)=0.077%
Boom	0.2	30.00%	-5.00%	0.2(0.3-0.08)x(-0.05-0.038)=-0.3872%
Expected return		8.00%	3.80%
Standard Deviation		19.26%	6.98%
Covariance = Sum of P(X - Expected return of Stock) x (X - Expected return of Bond)				-0.13%
Correlation = Covariance/standard deviation of stock/standard deviation of bond				-0.100

Portfolio Expected return is calculated by solving the following equation:

Weight of stock is given, weight of bond will be 1- weight of stock

Portfolio standard deviation is calculated by solving the following equation:

e) Substituting values we can calculate the investment opportunity set:

Weight of stock	Portfolio return	Portfolio standard deviation
10%	4.22%	6.38%
20%	4.64%	6.46%
30%	5.06%	7.18%
40%	5.48%	8.39%
50%	5.90%	9.91%
60%	6.32%	11.61%
70%	6.74%	13.44%
80%	7.16%	15.33%
90%	7.58%	17.28%
100%	8.00%	19.26%

f) Minimum variance portfolio requires the below equation:

g) Optimal risky portfolio:

h)