Question

Consider the following information on a portfolio of three stocks:

                                                                                                    

		Rate of Return
State of Economy	Probability of State Occurring	Stock A (%)	Stock B (%)	Stock C (%)
Boom	.15	7	15	28
Normal	.70	9	12	17
Bust	.15	10	2	−35

The portfolio is invested 40 percent in each of Stock A and Stock B and the rest in Stock C.

1. Which of the three stocks has the lowest risk? Why?

2. What is the expected return on this portfolio? If the T-bill rate is 4.03 percent, what is the expected risk premium on the portfolio?

3. What is the standard deviation of returns on the portfolio?

Answer 1

a) First we need to calculate the expected return and standard deviation of the 3 stocks

States	Probability	A	Probability Weighted Return	P(X - Expected return of A)^2
Boom	0.15	7.00%	0.15x7%=1.05%	0.15(0.07-0.0885)^2=0.00513374999999999%
Normal	0.7	9.00%	0.7x9%=6.3%	0.7(0.09-0.0885)^2=0.0001575%
Bust	0.15	10.00%	0.15x10%=1.5%	0.15(0.1-0.0885)^2=0.00198375%
Expected Return = sum of Probability Weighted Return	8.850%
Variance= sum of P(X - Expected return of A)^2	0.007%
Standard deviation = Square root of variance	0.853%

States	Probability	B	Probability Weighted Return	P(X - Expected return of B)^2
Boom	0.15	15.00%	0.15x15%=2.25%	0.15(0.15-0.1095)^2=0.02460375%
Normal	0.7	12.00%	0.7x12%=8.4%	0.7(0.12-0.1095)^2=0.00771750000000001%
Bust	0.15	2.00%	0.15x2%=0.3%	0.15(0.02-0.1095)^2=0.12015375%
Expected Return = sum of Probability Weighted Return	10.950%
Variance= sum of P(X - Expected return of B)^2	0.152%
Standard deviation = Square root of variance	3.905%

States	Probability	C	Probability Weighted Return	P(X - Expected return of C)^2
Boom	0.15	28.00%	0.15x28%=4.2%	0.15(0.28-0.1085)^2=0.44118375%
Normal	0.7	17.00%	0.7x17%=11.9%	0.7(0.17-0.1085)^2=0.2647575%
Bust	0.15	-35.00%	0.15x-35%=-5.25%	0.15(-0.35-0.1085)^2=3.15333375%
Expected Return = sum of Probability Weighted Return	10.850%
Variance= sum of P(X - Expected return of C)^2	3.859%
Standard deviation = Square root of variance	19.645%

As A has the lowest standard deviation, it is the least risky

b)Portfolio Expected return is calcualted by solving the following equation:

If the T bill has the return of 4.03% then the excess return is 10.09-4.03 = 6.06%

b) First we need to calculate the covariance and correlation between the stocks to calculate portfolio standard deviation:

State	Probability	A	B	P(X - Expected return of A) x (X - Expected return of B)
Boom	0.15	7.00%	15.00%	0.15(0.07-0.0885)x(0.15-0.1095)=-0.01123875%
Normal	0.7	9.00%	12.00%	0.7(0.09-0.0885)x(0.12-0.1095)=0.0011025%
Bust	0.15	10.00%	2.00%	0.15(0.1-0.0885)x(0.02-0.1095)=-0.01543875%
Expected return	8.85%	10.95%
Standard Deviation	0.85%	3.90%
Covariance = sum of P(X - Expected return of A) x (X - Expected return of B)	-0.03%
Correlation= covariance/product of standard deviation	-0.77

State	Probability	A	C	P(X - Expected return of A) x (X - Expected return of C)
Boom	0.15	7.00%	28.00%	0.15(0.07-0.0885)x(0.28-0.1085)=-0.04759125%
Normal	0.7	9.00%	17.00%	0.7(0.09-0.0885)x(0.17-0.1085)=0.0064575%
Bust	0.15	10.00%	-35.00%	0.15(0.1-0.0885)x(-0.35-0.1085)=-0.0790912500000001%
Expected return	8.85%	10.85%
Standard Deviation	0.85%	19.65%
Covariance = sum of P(X - Expected return of A) x (X - Expected return of C)	-0.12%
Correlation= covariance/product of standard deviation	-0.72

State	Probability	B	C	P(X - Expected return of B) x (X - Expected return of C)
Boom	0.15	15.00%	28.00%	0.15(0.15-0.1095)x(0.28-0.1085)=0.10418625%
Normal	0.7	12.00%	17.00%	0.7(0.12-0.1095)x(0.17-0.1085)=0.0452025%
Bust	0.15	2.00%	-35.00%	0.15(0.02-0.1095)x(-0.35-0.1085)=0.61553625%
Expected return		10.95%	10.85%
Standard Deviation		3.90%	19.65%
Covariance = sum of P(X - Expected return of A) x (X - Expected return of B)	0.76%
Correlation= covariance/product of standard deviation	1

So we have the below correlation matrix:

Correlation Matrix	A	B	C
A	1.00	-0.77	-0.72
B	-0.77	1.00	1.00
C	-0.72	1.00	1.00

Using this we can calculate the portfolio standard deviation: