Question

Donald Rock is a fund-of-fund portfolio manager of the asset management company Capital Value Ltd. His company has a proprietary model to forecast the probability of different states of the economy and the corresponding returns from different funds. Consider the following information:

State of economy	Probability	Fund A	Fund M	Fund Z
Good	20%	15%	20%	35%
Normal	70%	10%	15%	20%
Bad	10%	-3%	-8%	-15%

a. What are the expected return and standard deviation of his portfolio if he chooses to invest 60% of his assets in Fund A, 30% in Fund M and 10% in Fund Z?

b. What are the expected return and standard deviation of his portfolio if he chooses to invest equally in Fund A and Z?

c. If the economy turns out to be in a normal state, what will be the expected return of his portfolio in part (a)?

Answer 1

a

Fund A
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(A)^2* probability
Good	0.2	15	3	5.3	0.0005618
Normal	0.7	10	7	0.3	6.3E-06
Bad	0.1	-3	-0.3	-12.7	0.0016129
	Expected return %=	sum of weighted return =	9.7	Sum=Variance Fund A=	0.00218
			Standard deviation of Fund A%	=(Variance)^(1/2)	4.67
Fund M
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(B)^2* probability
Good	0.2	20	4	6.3	0.0007938
Normal	0.7	15	10.5	1.3	0.0001183
Bad	0.1	-8	-0.8	-21.7	0.0047089
	Expected return %=	sum of weighted return =	13.7	Sum=Variance Fund M=	0.00562
			Standard deviation of Fund M%	=(Variance)^(1/2)	7.5
Fund Z
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(C)^2* probability
Good	0.2	35	7	15.5	0.004805
Normal	0.7	20	14	0.5	0.0000175
Bad	0.1	-15	-1.5	-34.5	0.0119025
	Expected return %=	sum of weighted return =	19.5	Sum=Variance Fund Z=	0.01673
			Standard deviation of Fund Z%	=(Variance)^(1/2)	12.93
Covariance Fund A Fund M:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% For B(B)	(A)*(B)*probability
Good	0.2	5.3000	6.3	0.0006678
Normal	0.7	0.3	1.3	2.73E-05
Bad	0.1	-12.70	-21.7	0.0027559
			Covariance=sum=	0.003451
		Correlation A&B=	Covariance/(std devA*std devB)=	0.985621875
Covariance Fund A Fund Z:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% for C(C)	(A)*(C)*probability
Good	0.2	5.3	15.5	0.001643
Normal	0.7	0.3	0.5	0.0000105
Bad	0.1	-1270.00%	-34.5	0.0043815
			Covariance=sum=	0.006035
		Correlation A&C=	Covariance/(std devA*std devC)=	0.999232103
Covariance Fund M Fund Z:
Scenario	Probability	Actual return% -expected return% For B(B)	Actual return% -expected return% for C(C)	(B)*(C)*probability
Good	0.2	6.3	15.5	0.001953
Normal	0.7	1.3	0.5	0.0000455
Bad	0.1	-21.7	-34.5	0.0074865
			Covariance=sum=	0.009485
		Correlation B&C=	Covariance/(std devB*std devC)=	0.978244645
	Expected return%=	Wt Fund A*Return Fund A+Wt Fund M*Return Fund M+Wt Fund Z*Return Fund Z
	Expected return%=	0.6*9.7+0.3*13.7+0.1*19.5
	Expected return%=	11.88
	Variance	=w2A*σ2(RA) + w2B*σ2(RB) + w2C*σ2(RC)+ 2*(wA)*(wB)*Cor(RA, RB)*σ(RA)*σ(RB) + 2*(wA)*(wC)*Cor(RA, RC)*σ(RA)*σ(RC) + 2*(wC)*(wB)*Cor(RC, RB)*σ(RC)*σ(RB)
	Variance	=0.6^2*0.0467^2+0.3^2*0.07497^2+0.1^2*0.12933^2+2*(0.6*0.3*0.0467*0.07497*0.98562+0.3*0.1*0.07497*0.12933*0.97824+0.6*0.1*0.99923*0.0467*0.12933)
	Variance	0.003994
	Standard deviation=	(variance)^0.5
	Standard deviation=	6.32%

b

Expected return%=	Wt Fund A*Return Fund A+Wt Fund M*Return Fund M+Wt Fund Z*Return Fund Z
Expected return%=	0.5*9.7+0*13.7+0.5*19.5
Expected return%=	14.6
Variance	=w2A*σ2(RA) + w2B*σ2(RB) + w2C*σ2(RC)+ 2*(wA)*(wB)*Cor(RA, RB)*σ(RA)*σ(RB) + 2*(wA)*(wC)*Cor(RA, RC)*σ(RA)*σ(RC) + 2*(wC)*(wB)*Cor(RC, RB)*σ(RC)*σ(RB)
Variance	=0.5^2*0.0467^2+0^2*0.07497^2+0.5^2*0.12933^2+2*(0.5*0*0.0467*0.07497*0.98562+0*0.5*0.07497*0.12933*0.97824+0.5*0.5*0.99923*0.0467*0.12933)
Variance	0.007744
Standard deviation=	(variance)^0.5
Standard deviation=	8.80%

c

Fund A
Scenario	Probability	Return%	=rate of return% * probability
Good	0	15	0
Normal	1	10	10
Bad	0	-3	0
	Expected return %=	sum of weighted return =	10
Fund M
Scenario	Probability	Return%	=rate of return% * probability
Good	0	20	0
Normal	1	15	15
Bad	0	-8	0
	Expected return %=	sum of weighted return =	15
Fund Z
Scenario	Probability	Return%	=rate of return% * probability
Good	0	35	0
Normal	1	20	20
Bad	0	-15	0
	Expected return %=	sum of weighted return =	20

Expected return%=	Wt Fund A*Return Fund A+Wt Fund M*Return Fund M+Wt Fund Z*Return Fund Z
Expected return%=	0.6*10+0.3*15+0.1*20
Expected return%=	12.5