Question

Your portfolio is invested 20 percent each in A and C, and 60 percent in B. What is the standard deviation of the portfolio?

		Rate of Return if State Occurs
State of Economy	Probability of State of Economy	Stock A	Stock B	Stock C
Boom	.15	.37	.47	.27
Good	.45	.22	.18	.11
Poor	.35	-.18	-.22	-.08
Bust	.05	-.04	-.07	-.05

• 29.78%

• 21.36%

• 25.63%

• 10.96%

• 17.26%

Answer 1

Step 1: Calculation of Expected Return & Variance of all the three stocks

1. Calculation of  Expected Return & Variance of Stock A

State	Probability	Return	Return * prob	dA	dA^2	dA^2 * prob
Boom	0.15	0.37	0.0555	0.2805	0.07868025	0.011802038
Good	0.45	0.22	0.0990	0.1305	0.01703025	0.007663613
Poor	0.35	(0.18)	-0.0630	-0.2695	0.07263025	0.025420588
Bust	0.05	(0.04)	-0.0020	-0.1295	0.01677025	0.000838513
		Total (ER)	0.0895		Total ()	0.04572475

(I) Expected Return (ER(A)) = 0.0895  or 8.95%

where dA = Return - ER

(II) 0.04572475

2. Calculation of  Expected Return & Variance of Stock B

State	Probability	Return	Return * prob	dB	dB^2	dB^2 * prob
Boom	0.15	0.47	0.0705	0.3990	0.15920100	0.02388015
Good	0.45	0.18	0.0810	0.1090	0.01188100	0.00534645
Poor	0.35	(0.22)	-0.0770	-0.2910	0.08468100	0.02963835
Bust	0.05	(0.07)	-0.0035	-0.1410	0.01988100	0.00099405
		Total (ER)	0.0710		Total ()	0.059859

(I) Expected Return (ER(B)) = 0.0710   or 7.10%

where dB = Return - ER

(II) 0.059859

3. Calculation of  Expected Return & Variance of Stock C

State	Probability	Return	Return * prob	dC	dC^2	dC^2 * prob
Boom	0.15	0.27	0.0405	0.2105	0.04431025	0.006646538
Good	0.45	0.11	0.0495	0.0505	0.00255025	0.001147613
Poor	0.35	(0.08)	-0.0280	-0.1395	0.01946025	0.006811088
Bust	0.05	(0.05)	-0.0025	-0.1095	0.01199025	0.000599513
		Total (ER)	0.0595		Total ()	0.01520475

(I) Expected Return (ER(C)) = 0.0595   or 5.95%

where dC = Return - ER

(II) 0.01520475

Step 2 : Calculation of Covariance between all the stocks

1. Calculation of Covariance between Stock A & Stock B

State	Probability	dA	dB	dA*dB	dA*dB* Prob
Boom	0.15	0.2805	0.399	0.1119195	0.016787925
Good	0.45	0.1305	0.109	0.0142245	0.006401025
Poor	0.35	-0.2695	-0.291	0.0784245	0.027448575
Bust	0.05	-0.1295	-0.141	0.0182595	0.000912975
				Total	0.0515505

Cov (A,B) = 0.0515505

2. Calculation of Covariance between Stock B & Stock C

State	Probability	dB	dC	dB*dC	dB*dC* Prob
Boom	0.15	0.399	0.2105	0.0839895	0.012598425
Good	0.45	0.109	0.0505	0.0055045	0.002477025
Poor	0.35	-0.291	-0.1395	0.0405945	0.014208075
Bust	0.05	-0.141	-0.1095	0.0154395	0.000771975
				Total	0.0300555

Cov (B,C) = 0.0300555

3. Calculation of Covariance between Stock A & Stock C

State	Probability	dA	dC	dA*dC	dA*dC* Prob
Boom	0.15	0.2805	0.2105	0.05904525	0.008856788
Good	0.45	0.1305	0.0505	0.00659025	0.002965613
Poor	0.35	-0.2695	-0.1395	0.03759525	0.013158338
Bust	0.05	-0.1295	-0.1095	0.01418025	0.000709013
				Total	0.02568975

Cov (A,C) = 0.02568975

Step 3 : Calculation of Standard Deviation of Portfolio

Weight of A (wA) = 0.2
Weight of B (wB)= 0.6
Weight of C (wC)= 0.2
0.04572475
0.059859
0.01520475
Cov (A,B) = 0.0515505
Cov (B,C) = 0.0300555
Cov (A,C) = 0.02568975

0.21360 approx or 21.36%

21.36%