Question

Consider the following information: Rate of Return if State Occurs State of Economy Boom Good Probability of State of Economy 0.25 2.15 0.30 0.30 Stock A 0.23 0.12 -0.02 -0.18 Stock B Stock C 0.39 0.26 0.15 0.16 -0.12 -0.03 0.18 0.11 Poor Bust a. Your portfolio is invested 35 percent each in A and C and 30 percent in B. What is the expected return of the portfolio? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.) Expected retum b-1. What is the variance of this portfolio? (Do not round intermediate calculations. Round your answer to 5 decimal places.) Variance b-2. What is the standard deviation? (Do not round intermediate calculations. Enter your answer os o percent rounded to 2 decimal places.) Standard deviation

Answer 1

State	Probability	A	Probability Weighted return of A	B	Probability Weighted return of B	C	Probability Weighted return of C
Boom	0.25	23.00%	0.25 x 23 = 5.75%	39.00%	0.25 x 39 =9.75%	26.00%	0.25 x 26 = 6.50%
Good	0.15	12.00%	0.15 x 12= 1.80%	15.00%	0.15 x 15=2.25%	16.00%	0.15 x 16 = 2.40%
Poor	0.3	-2.00%	0.30 x -2= -0.60%	-12.00%	0.30 x -12= -3.60%	-3.00%	0.30 x -3 = -0.90%
Bust	0.3	-18.00%	0.30 x -18= -5.40%	-18.00%	0.30 x -18= -5.4%	-11.00%	0.30 x -11 = -3.30%
Expected return			5.75+1.8-0.60-5.40 = 1.55%		9.75+2.25-3.60-5.40 = 3.00%		6.50+2.40-0.90-3.3 = 4.70%

So if the weights of the portfolio are given we can calculate the portfolio expected return as follows:

Stock	Expected return	Portfolio weight	Weighted return
A	1.55%	35%	1.55 x 0.35 = 0.54250%
B	3.00%	30%	3 x 0.30= 0.09000%
C	4.70%	35%	0.35 x 4.70 = 0.22090%
			0.54250+0.09+0.02209 = 3.08750%

For portfolio variance and standard deviation, we need to find asset variance and standard deviation and the covariance and correlations of combinations AB, BC, AC

Variance and standard deviation of assets is calculated below:

State	Probability	A	P(X - Expected return of A)2
Boom	0.25	23.00%	0.25 x (23-1.55)2 =1.15%
Good	0.15	12.00%	0.15 x (23-1.55)2 =0.16%
Poor	0.3	-2.00%	0.30 x (-2-1.55)2 = 0.04%
Bust	0.3	-18.00%	0.30 x (-18-1.55)2 = 1.15%
Variance = sum of P(X - Expected return of A)^2			1.15+0.16+0.04+1.15= 2.50%
Standard Deviation = squareroot of variance			2.50.5 = 15.81%

State	Probability	B	P(X - Expected return of B)2
Boom	0.25	39.00%	0.25 x (39-3)2 =1.72%
Good	0.15	15.00%	0.15 x (15-3)2 =0.14%
Poor	0.3	-12.00%	0.30 x (-12-3)2 = 0.24%
Bust	0.3	-18.00%	0.30 x (-18-3)2 = 1.44%
Variance = sum of P(X - Expected return of A)^2			1.72+0.14+0.24+1.44= 5.45%
Standard Deviation = squareroot of variance			5.450.5 = 23.35%

State	Probability	C	P(X - Expected return of B)2
Boom	0.25	26.00%	0.25 x (26-4.7)2 =1.13%
Good	0.15	16.00%	0.15 x (16-4.7)2 =0.19%
Poor	0.3	-3.00%	0.30 x (-3-4.7)2 = 0.18%
Bust	0.3	-11.00%	0.30 x (-11-4.7)2 = 0.74%
Variance = sum of P(X - Expected return of A)^2			1.13+0.19+0.18+0.74= 2.24%
Standard Deviation = squareroot of variance			2.240.5 = 14.98%

Next we calculate the co variance and correlation of combinations:

AB:

State	Probability	P(X - Expected return of A) x (X - Expected return of B)
Boom	0.25	0.25 x (23-1.55) x (39-3) = 1.72%
Good	0.15	0.15 x(12-1.55) x (15-3) = 0.14%
Poor	0.3	0.30 x (-2-1.55) x(12-3) = 0.24%
Bust	0.3	0.30 x (-18-1.55) x (-18 -3) = 1.44%
Covariance		1.72+0.14+0.24+1.44 = 3.53%
Correlation=Covariance/standard deviation of A/ Standard deviation of B		3.53/15.81/23.35 = 0.96

AC:

State	Probability	P(X - Expected return of A) x (X - Expected return of C)
Boom	0.25	0.25 x (23-1.55) x (26-4.7) = 1.22%
Good	0.15	0.15 x(12-1.55) x (16-4.7) = 0.20%
Poor	0.3	0.30 x (-2-1.55) x(-3-4.7) = 0.07%
Bust	0.3	0.30 x (-18-1.55) x (-11 -4.7) = 0.81%
Covariance		1.22+0.22+0.07+0.81= 2.30%
Correlation=Covariance/standard deviation of A/ Standard deviation of C		2.30/15.81/14.98 = 0.97

BC:

State	Probability	P(X - Expected return of B) x (X - Expected return of C)
Boom	0.25	0.25 x (39-3)x (26-4.7) = 1.99%
Good	0.15	0.15 x(15-3) x (16-4.7) = 0.20%
Poor	0.3	0.30 x (-12-3) x(-3-4.7) = 0.27%
Bust	0.3	0.30 x (-18-3) x (-11 -4.7) = 0.93%
Covariance		1.99+0.2+0.27+0.93= 3.40%
Correlation=Covariance/standard deviation of A/ Standard deviation of C		3.40/23.35/14.98 = 0.97

$Standard \ Deviation_{Portfolio } = \sqrt_{(Weight_{A} \times \ Standard \ Deviation_{A })^2+(Weight_{B} \times \ Standard \ Deviation_{B})^2+(Weight_{C} \times \ Standard \ Deviation_{C })^2+2(Weight_{A} \times \ Standard \ Deviation_{A })(Weight_{B} \times \ Standard \ Deviation_{B }) correlation_{A,B}+2(Weight_{A} \times \ Standard \ Deviation_{A })(Weight_{C} \times \ Standard \ Deviation_{C }) correlation_{A,C}+2(Weight_{B} \times \ Standard \ Deviation_{B })(Weight_{C} \times \ Standard \ Deviation_{C }) correlation_{B,C}}$$Standard \ Deviation_{Portfolio } = \sqrt_{(0.35 \times \ 15.81)^2+(0.30 \times \ 23.35)^2+(0.35 \times \ 14.98)^2+2(0.35 \times 15.81)(0.30 \times 23.35) 0.96+2(0.35 \times 15.81)(0.35 \times \ 14.98) 0.97+2(0.30 \times \ 23.35)(0.35 \times 14.98) 0.97}$

Variance = square of standard deviaton so portfolio variance = 17.58² = 3.09%