Question

There are only two possible states of the economy. State 1 has a 67% chance of occurring. In State 1, Asset A returns 6.50% and Asset B returns 9.50%. In State 2, Asset A returns -3.60% and Asset B returns -6.60%. A portfolio of just these two assets is invested 43% in Asset A (with Asset B comprising the remainder w/o any negative weights). What is the standard deviation of the portfolio's returns?

Answer 1

Standard deviation of return of portfolio is 6.36%

Step-1:Calculation of expected return
Expected Return of :
Stock A	=	(67%*6.50%)+(33%*-3.60%)	=	3.17%
Stock B	=	(67%*9.50%)+(33%*-6.60%)	=	4.19%
Step-2:Calculation of variance
Stock A	Chance	Return	Expected Return
	a	b	c	d=((b-c)^2)*a
	67%	6.50%	3.17%	0.000744296
	33%	-3.60%	3.17%	0.001511146
	Variance			0.002255441
Stock B	Chance	Return	Expected Return
	a	b	c	d=((b-c)^2)*a
	67%	9.50%	4.19%	0.001891274
	33%	-6.60%	4.19%	0.003839859
	Variance			0.005731133
Step-3:Calculation of Standard deviation
Standard Deviation of :
Stock A	=	Variance^(1/2)	=	0.002255441	^(1/2)	=	4.75%
Stock B	=	Variance^(1/2)	=	0.005731133	^(1/2)	=	7.57%
Step-4:Calculation of covariance
	Chance	Return of Stock A	Expected return of Stock A	Return of Stock B	Expected return of Stock B
	x	a	b	c	d	e=((a-b)*(c-d))*x
	67%	6.50%	3.17%	9.50%	4.19%	0.001186451
	33%	-3.60%	3.17%	-6.60%	4.19%	0.002408856
	Covariance				0.003595307
Step-5:Calculation of correlation coefficient
Correlation Coefficient	=	Covariance between Stock A and Stock B	/	(Standard deviation of Stock A*Standard deviation of Stock B)
	=	0.003595307	/	(4.75%*7.57%)
	=	0.003595307	/	0.003595307
	=	1.0000
Step-6: Calculation of standard deviation of portfolio
Standard deviation of portfolio	=	((WA)^2*(SDA)^2+(WB)^2*(SDB)^2+2*WA*WB*SDA*SDB*CorrA,B)^(1/2)
	=	((43%)^2*(4.75%)^2+(57%)^2*(7.57%)^2+2*43%*57%*4.75%*7.57%*1.00)^(1/2)
	=	6.36%
Where,
WA	=	Weight of Stock A		=	43%
WB	=	Weight of Stock B		=	57%
SDA	=	Standard deviation of Stock A	=	4.75%
SDB	=	Standard deviation of Stock B	=	7.57%
CorrA,B	=	Correlation Coeeficient		=	1.0000