Question

Consider the following probability distribution for stocks A and B.

Scenario	Probability	Return on Stock A	Return on Stock B
1	.35	12%	-15%
2	.4	4%	5%
3	.25	-4%	25%

1. What are the expected returns and standard deviations for stocks A and B?

2. What is the correlation coefficient between the two stocks?

3. Suppose the risk-free rate is 2%. What is the optimal risky portfolio, its expected return and its standard deviation?

4. Suppose that stocks A and B had the expected return and standard deviations as you calculated in question 1, while being perfectly negatively correlated. Again, assume the risk-free rate is 2%. Describe the global minimum variance portfolio in this case (that is, the proportions (w_E, w_D), the expected return and standard deviation).

Answer 1

1

Stock A
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(A)^2* probability
1	0.35	12	4.2	7.2	0.0018144
2	0.4	4	1.6	-0.8	2.56E-05
3	0.25	-4	-1	-8.8	0.001936
	1. Expected return %=	sum of weighted return =	4.8	Sum=Variance Stock A=	0.00378
			1. Standard deviation of Stock A%	=(Variance)^(1/2)	6.14
Stock B
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(B)^2* probability
1	0.35	-15	-5.25	-18	0.01134
2	0.4	5	2	2	0.00016
3	0.25	25	6.25	22	0.0121
	1. Expected return %=	sum of weighted return =	3	Sum=Variance Stock B=	0.0236
			1. Standard deviation of Stock B%	=(Variance)^(1/2)	15.36
Covariance Stock A Stock B:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% For B(B)	(A)*(B)*probability
1	0.35	7.2	-18	-0.004536
2	0.4	-0.8	2	-6.4E-05
3	0.25	-8.8	22	-0.00484
			Covariance=sum=	-0.00944
		2. Correlation A&B=	Covariance/(std devA*std devB)=	-1

3

To find the fraction of wealth to invest in Stock A that will result in the risky portfolio with maximum Sharpe ratio
the following formula to determine the weight of Stock A in risky portfolio should be used
w(*d)= ((E[Rd]-Rf)*Var(Re)-(E[Re]-Rf)*Cov(Re,Rd))/((E[Rd]-Rf)*Var(Re)+(E[Re]-Rf)*Var(Rd)-(E[Rd]+E[Re]-2*Rf)*Cov(Re,Rd)
							Where
						Stock A	E[R(d)]=	4.80%
Stock B	E[R(e)]=	3.00%
Stock A	Stdev[R(d)]=	6.14%
Stock B	Stdev[R(e)]=	15.36%
	Var[R(d)]=	0.00377
	Var[R(e)]=	0.02359
T bill	Rf=	2.00%
Correl	Corr(Re,Rd)=	-1
Covar	Cov(Re,Rd)=	-0.0094
Stock A	Therefore W(*d)=	0.7144
Stock B	W(*e)=(1-W(*d))=	0.2856
	Expected return of risky portfolio=	4.29%
	Risky portfolio std dev (answer Risky portfolio std dev)=	0.00%
Where
Var = std dev^2
Covariance = Correlation* Std dev (r)*Std dev (d)
Expected return of the risky portfolio = E[R(d)]*W(*d)+E[R(e)]*W(*e)
Risky portfolio standard deviation =( w2A*σ2(RA)+w2B*σ2(RB)+2*(wA)*(wB)*Cor(RA,RB)*σ(RA)*σ(RB))^0.5

4

To find the fraction of wealth to invest in Stock A that will result in the risky portfolio with minimum variance
the following formula to determine the weight of Stock A in risky portfolio should be used
w(*d)= ((Stdev[R(e)])^2-Stdev[R(e)]*Stdev[R(d)]*Corr(Re,Rd))/((Stdev[R(e)])^2+(Stdev[R(d)])^2-Stdev[R(e)]*Stdev[R(d)]*Corr(Re,Rd))
							Where
						Stock A	E[R(d)]=	4.80%
Stock B	E[R(e)]=	3.00%
Stock A	Stdev[R(d)]=	6.14%
Stock B	Stdev[R(e)]=	15.36%
	Var[R(d)]=	0.00377
	Var[R(e)]=	0.02359
T bill	Rf=	2.00%
Correl	Corr(Re,Rd)=	-1
Covar	Cov(Re,Rd)=	-0.0094
Stock A	Therefore W(*d)=	0.7144
Stock B	W(*e)=(1-W(*d))=	0.2856
	Expected return of risky portfolio=	4.29%
	Risky portfolio std dev (answer Risky portfolio std dev)=	0.00%
Where
Var = std dev^2
Covariance = Correlation* Std dev (r)*Std dev (d)
Expected return of the risky portfolio = E[R(d)]*W(*d)+E[R(e)]*W(*e)
Risky portfolio standard deviation =( w2A*σ2(RA)+w2B*σ2(RB)+2*(wA)*(wB)*Cor(RA,RB)*σ(RA)*σ(RB))^0.5