Question

State      Probability           A             B                           

Boom         .25                   12%       4%         

Bust            .75                   6%         18%      

               what is the standard deviation of the return on the portfolio?

Answer 1

Expected return on individual stock = Sum(Probability*Return)

Variance of individual stock = Sum(Probability*Squared Deviation) = [Probability*(X-R)2]

Std. Deviation of individual stock = Square root (Variance of Stock)

Covariance = Sum[Probability*(X-RA)*(X-RB)]

State	Probability	Return on A(%)	Probability*Return	Deviation(X-RA)	(X-RA)^2	Probability*(X-RA)^2
Boom	0.25	12	3	4.5	20.25	5.0625
Bust	0.75	6	4.5	-1.5	2.25	1.6875
		Expected Return RA=	7.5		Variance =	6.75
					Std. Deviation=	2.598076
State	Probability	Return on B(%)	Probability*Return	Deviation (X-R)	(X-R)^2	Probability*(X-R)^2
Boom	0.25	4	1	-10.5	110.25	27.5625
Bust	0.75	18	13.5	3.5	12.25	9.1875
		Expected Return R=	14.5		Variance =	36.75
					Std. Deviation=	6.062178
State	Probability	Deviation(X-R)	Deviation (X-R)	(X-R)*(X-R)	Probability*(X-R)*(X-R)
Boom	0.25	4.5	-10.5	-47.25	-11.8125
Bust	0.75	-1.5	3.5	-5.25	-3.9375
				Covariance=	-15.75

Variance of Portfolio = w12 * ơ12 + w22 * ơ22 + 2 * w1 * w2 * Covariance(1,2)

• wi = Portfolio weight of stock i
• ơi2 = Individual variance of stock i

Assuming equal amount is invested in both the stocks, wA=wB=0.5

So, Variance of portfolio = 0.25*6.75 + 0.25*36.75 + 2*0.5*0.5*(-15.75)

= 1.6875 + 9.1875 - 7.875

= 3 %2

Hence, Std. Deviation of portfolio = Sqrt(3)

= 1.732 %