Question

You are thinking about investing your money in the stock market. You have the following two stocks in mind: stock A and stock B. You know that the economy can either go in recession or it will boom. Being an optimistic investor, you believe the likelihood of observing an economic boom is two times as high as observing an economic depression. You also know the following about your two stocks:

State of the Economy	RA	RB
Boom	–2%	10%
Recession	40%	6%

REQUIRED:

1. Calculate the expected return for stock A and stock B.   
2. Calculate the total risk (variance and standard deviation) for stock A and for stock B.
3. Calculate the expected return on a portfolio consisting of equal proportions in both stocks.   
4. Calculate the expected return on a portfolio consisting of 10% invested in stock A and the remainder in stock B.   
5. Calculate the covariance between stock A and stock B.                     
6. Calculate the correlation coefficient between stock A and stock B.    
7. Calculate the variance of the portfolio with equal proportions in both stocks using the covariance from answer e.                                                 

H.Calculate the variance of the portfolio with equal proportions in both stocks using the portfolio returns and expected portfolio returns from answer c.          

Answer 1

ANSWER

a) Propability of boom is twice than recession. P ( boom) is 2/3 and P ( receission) is 1/3.

	A	B	C	D = A*B	E = A*C
	Probability	RA	RB	E(RA)	E(RB)
Boom	0.67	-0.02	0.10	-0.0133	0.0667
Recession	0.33	0.40	0.06	0.1333	0.02
Expected Return	12.00	0.0867

E(R_A) = 12%

E(R_B) = 8.67%

b)

A	B	C	D = B-C	E = D2	F = A * E
Probability	RA	E(RA)	RA - E(RA)
0.67	-0.02	0.12	-0.14	0.0196	0.013067
0.33	0.40	0.12	0.28	0.0784	0.026133
				Total	0.0392

Variance of R_{A =} 0.0395

Standard deviation is square root of variance.

SD (R_A) = 0.19799 (19.799%)

A	B	C	D = B-C	E = D2	F = A * E
Probability	RB	E(RB)	RB - E(RB)
0.67	0.10	0.0867	0.01	0.000178	0.000119
0.33	0.06	0.0867	-0.03	0.000711	0.000237
				Total	0.000356

Variance of R_{B =} 0. 000356

Standard deviation is square root of variance.

SD (R_B) = 0.018856 (1.886%)

c) Portfolio weights: W_A=0.5 and W_B=0.5:

Expected portfolio return

	A	B	C = A*B
	Weights	Expected Return
RA	0.5	0.12	0.06
RB	0.5	0.0867	0.0433
		Total	0.1033

Expected portfolio return is 0.10335 (10.335%)

d) Portfolio weights: W_A=0.1 and W_B=0.9:

Expected portfolio return

	A	B	C = A*B
	Weights	Expected Return
RA	0.1	0.12	0.012
RB	0.9	0.0867	0.0780
		Total	0.0900

Expected portfolio return is 0.0900 (9.00 %)

e)

COV (R_A,R_B) =

A	B	C	D = B-C	E	F	G = E-F	H = A * D*G
Probability	RA	E(RA)		RB	E(RB)
0.67	-0.02	0.12	-0.14	0.10	0.0867	0.0133	-0.00124
0.33	0.4	0.12	0.28	0.06	0.0867	-0.0267	-0.00249
Total	-0.00373

COV (R_A,R_B) = -0.00373

f) CORR(R_A,R_B) =

Formula = COV (R_A,R_B) / SD(R_A) * SD(R_B)

–0.0037333 / (0.19799* 0.018856) = –1 (Rounding! Remember the correlation coefficient cannot be less than –1)

g) VAR(R_P) = 0.5² × 0.018856² + 0.5² × 0.19799² + 2 × 0.5 × 0.5 × –0.0037333 =

        –0.008022

SD(R_P) = 8.957%

h) Expected portfolio return

	A	B	C = A*B
	Weights	Expected Return
RA	0.5	0.12	0.06
RB	0.5	0.0867	0.0433
		Total	0.1033

Expected portfolio return is 0.10335 (10.335%)

Expected return of boom = 0.5 * -0.05 +0.5 *0.1 = 0.04 ( 4%)

Expected return of recession = 0.5 *0.4 +0.5 *0.06 = 0.23 ( 23%)

Standard Deviation of portfolio

	A	B	C	D = B-C	E = D2	F = A * E
	Probability	Expected Return	E(RP)	E - E(RP)
Boom	0.67	0.04	0.10335	-0.06	0.004013	0.002675
Recession	0.33	0.23	0.10335	0.13	0.01604	0.005347
						0.008022

Variance is 0.008022

Standard deviation is square root of variance

SD is 0.089567