Question

The following table lists possible rates of return on Company A and B.

State of the Economy	Probability	Company A	Company B
Deep recession	0.05	-20%	-30%
Mild recession	0.25	10	5
Average	0.35	15	20
Mild boom	0.20	20	25
Strong boom	0.15	25	30

(a)Based on the above data calculate by using the appropriate formulae

the standard deviations of returns for Company A and B   

the covariance of returns between Company A and B

the correlation between Company A and B

(b)If you wish to diversify risk would it be advisable to form a portfolio of both securities A and B? State your reasons. (No computations are required to answer this part of the question.)

(c)Find the minimum variance one can get by forming a portfolio of A and B. Short-selling either stock is allowed – i.e., weights need not be all positive.

Answer 1

(a)

For Company A:

State of the Economy	pi	RA	piRA	RA - E(R)	(RA - E(R))2	pi(RA - E(R))2
Deep recession	0.05	- 20	- 1.00	- 15.5	240.25	12.01
Mild recession	0.25	10	2.50	- 12.0	144.00	36.00
Average	0.35	15	5.25	- 9.25	85.56	29.95
Mild boom	0.20	20	4.00	- 10.5	110.25	22.05
Strong boom	0.15	25	3.75	- 10.75	115.56	2.33
Total			14.5			102.34

Standard Deviation of Company A =  

Standard Deviation of Company A =

Standard Deviation of Company A = 10.12 %

For Company B:

State of the Economy	pi	RB	piRB	RB - E(R)	(RB - E(R))2	pi(RB - E(R))2
Deep recession	0.05	- 30	- 1.50	- 17.75	315.06	15.75
Mild recession	0.25	5	1.25	- 15.00	225.00	56.25
Average	0.35	20	7.00	- 9.25	85.56	29.95
Mild boom	0.20	25	5.00	-11.25	126.56	25.31
Strong boom	0.15	30	4.5	-11.75	138.06	20.71
Total			16.25			147.97

Standard Deviation of Company B =  

Standard Deviation of Company B =

Standard Deviation of Company B = 12.16 %

Computation of the Co-variance of Company A & B.We have,

State of the Economy	pi	RA - E(R)	RB - E(R)	pi(RA - E(R))(RB - E(R))
Deep recession	0.05	- 15.5	- 17.75	13.76
Mild recession	0.25	- 12.0	- 15.00	45.00
Average	0.35	- 9.25	- 9.25	29.95
Mild boom	0.20	- 10.5	- 11.25	23.62
Strong boom	0.15	- 10.75	- 11.75	18.94
Total				131.27

Co-Variances of Company A & B = p_i(R_A - E(R))(R_B - E(R))

Co-Variances of Company A & B =131.27

Computation of the Correlation Coefficient of Company A & B.We have,

Correlation Coefficient = Co-variances / SD(A) x SD(B)

Correlation Coefficient = 131.27 / 12.16 x 10.12

Correlation Coefficient = 1.0

(b) Since, the correlation coefficient of company A & B are 1.0. It is perfectly positive correlated,diversification does not reduce risk. It means both are moving in the same direction. Both are making profit or loss together at any particular time. If we wish to diversify risk, we should not form a portfolio from Company A & B.

(c) Computation of the minimum variances portfolio when correlation coefficient is perfectly positive correlated( +1.0).We have,

Weight of Company A = SD(B) / ( SD(A) + SD(B))

Weight of Company A = 12.16 / ( 12.16 +10.12)

Weight of Company A = 12.16 / 22.28 = 0.55

Weight of company B = 1 - 0.55 = 0.45

Hence,the minimum variance of Company A is 0.55 and minimum variance of Company B is 0.45 to forming a portfolio of A & B.