Question

You are expecting either a recession or steady growth next year. Recession has a 19% probability of happening. In steady growth, stock ABC returns 12.75% and stock XYZ returns 9.75%. In a recession, stock ABC returns -6.50% and stock XYZ returns -4.30%. You are going to put together a portfolio of these two stocks with positive portfolio weight in each and allocate 61% of the portfolio to ABC with the remainder to XYZ. What is the variance of the portfolio?

Answer 1

Variance of portfolio is   0.00456

Step-1:Calculation of expected return
Expected Return of :
Stock ABC	=	(19%*-6.50%)+(81%*12.75%)	=	9.09%
Stock XYZ	=	(19%*-4.30%)+(81%*9.75%)	=	7.08%
Step-2:Calculation of variance
Stock ABC	Chance	Return	Expected Return
	a	b	c	d=((b-c)^2)*a
	19%	-6.50%	9.09%	0.004619395
	81%	12.75%	9.09%	0.001083562
	Variance			0.005702957
Stock XYZ	Chance	Return	Expected Return
	a	b	c	d=((b-c)^2)*a
	19%	-4.30%	7.08%	0.0024608
	81%	9.75%	7.08%	0.000577225
	Variance			0.003038024
Step-3:Calculation of Standard deviation
Standard Deviation of :
Stock ABC	=	Variance^(1/2)	=	0.005702957	^(1/2)	=	7.55%
Stock XYZ	=	Variance^(1/2)	=	0.003038024	^(1/2)	=	5.51%
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
	19%	-6.50%	9.09%	-4.30%	7.08%	0.003371558
	81%	12.75%	9.09%	9.75%	7.08%	0.000790859
	Covariance				0.004162418
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)
	=	0.004162418	/	(7.55%*5.51%)
	=	0.004162418	/	0.004162418
	=	1.0000
Step-6: Calculation of Variance of portfolio
Variance of portfolio	=	(WA)^2*(SDA)^2+(WB)^2*(SDB)^2+2*WA*WB*SDA*SDB*CorrA,B
	=	(61%)^2*(7.55%)^2+(39%)^2*(5.51%)^2+2*61%*39%*7.55%*5.51%*1.00
	=	0.00456
Where,
WA	=	Weight of Stock ABC		=	61%
WB	=	Weight of Stock XYZ		=	39%
SDA	=	Standard deviation of Stock ABC	=	7.55%
SDB	=	Standard deviation of Stock XYZ	=	5.51%
CorrA,B	=	Correlation Coefficient		=	1.0000