Question

You decide to invest in a portfolio consisting of 11 percent Stock X, 53 percent Stock Y, and the remainder in Stock Z. Based on the following information, what is the standard deviation of your portfolio? State of Economy Probability of State Return if State Occurs of Economy Stock X Stock Y Stock Z Normal .76 10.70% 4.10% 13.10% Boom .24 18.00% 26.00% 17.50%

Answer 1

Stock X
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(A)^2* probability
Normal	0.76	10.7	8.132	-1.752	0.000233282
Boom	0.24	18	4.32	5.548	0.0007387273
	Expected return %=	sum of weighted return =	12.45	Sum=Variance Stock X=	0.00097
			Standard deviation of Stock X%	=(Variance)^(1/2)	3.12
Stock Y
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(B)^2* probability
Normal	0.76	4.1	3.116	-5.256	0.0020995407
Boom	0.24	26	6.24	16.644	0.0066485457
	Expected return %=	sum of weighted return =	9.36	Sum=Variance Stock Y=	0.00875
			Standard deviation of Stock Y%	=(Variance)^(1/2)	9.35
Stock Z
Scenario	Probability	Return%	=rate of return% * probability	Actual return -expected return(A)%	(C)^2* probability
Normal	0.76	13.1	9.956	-1.056	8.47503E-05
Boom	0.24	17.5	4.2	3.344	0.000268376
	Expected return %=	sum of weighted return =	14.16	Sum=Variance Stock Z=	0.00035
			Standard deviation of Stock Z%	=(Variance)^(1/2)	1.88
Covariance Stock X Stock Y:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% For B(B)	(A)*(B)*probability
Normal	0.76	-1.7520	-5.256	0.000699847
Boom	0.24	5.548	16.644	0.002216182
			Covariance=sum=	0.002916029
		Correlation A&B=	Covariance/(std devA*std devB)=	1
Covariance Stock X Stock Z:
Scenario	Probability	Actual return% -expected return% for A(A)	Actual return% -expected return% for C(C)	(A)*(C)*probability
Normal	0.76	-1.752	-1.056	0.000140609
Boom	0.24	5.548	3.344	0.00044526
			Covariance=sum=	0.000585869
		Correlation A&C=	Covariance/(std devA*std devC)=	1
Covariance Stock Y Stock Z:
Scenario	Probability	Actual return% -expected return% For B(B)	Actual return% -expected return% for C(C)	(B)*(C)*probability
Normal	0.76	-5.256	-1.056	0.000421826
Boom	0.24	16.644	3.344	0.001335781
			Covariance=sum=	0.001757606
		Correlation B&C=	Covariance/(std devB*std devC)=	1
	Expected return%=	Wt Stock X*Return Stock X+Wt Stock Y*Return Stock Y+Wt Stock Z*Return Stock Z
	Expected return%=	0.11*12.45+0.53*9.36+0.36*14.16
	Expected return%=	11.42
	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	=0.11^2*0.03118^2+0.53^2*0.09353^2+0.36^2*0.01879^2+2*(0.11*0.53*0.03118*0.09353*1+0.53*0.36*0.09353*0.01879*1+0.11*0.36*1*0.03118*0.01879)
	Variance	0.003572
	Standard deviation=	(variance)^0.5
	Standard deviation=	5.98%