Question

Ms. Smith wants to invest in two securities, ABC and PQR, and the relevant information is below: State of the Economy Probability Return on ABC (%) Return on PQR (%) Bear 0.25 -15% 4% Moderate 0.5 5% 4% Bull 0.25 20% 4%

a. Calculate expected returns and standard deviations of two securities.

b. Ms. Smith shorts $1,000 of PQR and invests all proceeds from this short sale as well as $3,000 of her own money into ABC. What is the expected return and the standard deviation of her portfolio?

Answer 1

Total Portfolio value = Value of ABC + Value of PQR

=4000+-1000

=3000

Weight of ABC = Value of ABC/Total Portfolio Value

= 4000/3000

=1.3333

Weight of PQR = Value of PQR/Total Portfolio Value

= -1000/3000

=-0.3333

ABC
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(A)^2* probability
Bear	0.25	-15	-3.75	-18.75	0.008789063
Moderate	0.5	5	2.5	1.25	0.000078125
Bull	0.25	20	5	16.25	0.006601563
	Expected return %=	sum of weighted return =	3.75	Sum=Variance ABC=	0.01547
			Standard deviation of ABC%	=(Variance)^(1/2)	12.44
PQR
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(B)^2* probability
Bear	0.25	4	1	0	0
Moderate	0.5	4	2	0	0
Bull	0.25	4	1	0	0
	Expected return %=	sum of weighted return =	4	Sum=Variance PQR=	0
			Standard deviation of PQR%	=(Variance)^(1/2)	0
Covariance ABC PQR:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% For B(B)	(A)*(B)*probability
Bear	0.25	-18.75	0	0
Moderate	0.5	1.25	0	0
Bull	0.25	16.25	0	0
			Covariance=sum=	0
		Correlation A&B=	Covariance/(std devA*std devB)=	0
	Expected return%=	Wt ABC*Return ABC+Wt PQR*Return PQR
	Expected return%=	1.3333*3.75+-0.3333*4
	Expected return%=	3.67
	Variance	=( w2A*σ2(RA) + w2B*σ2(RB) + 2*(wA)*(wB)*Cor(RA, RB)*σ(RA)*σ(RB))
	Variance	=1.3333^2*0.12437^2+-0.3333^2*0^2+2*1.3333*-0.3333*0.12437*0*0
	Variance	0.0275
	Standard deviation=	(variance)^0.5
	Standard deviation=	16.58%