Question

Consider the following information on Stocks A, B, C and their returns (in decimals) in each state: State Prob. of State A B C Boom 20% 0.32 0.19 0.18 Good 45% 0.16 0.1 0.09 Poor 25% 0.02 0.01 0.03 Bust 10% -0.09 -0.06 -0.02 If your portfolio is invested 25% in A, 40% in B, and 35% in C, what is the standard deviation of the portfolio in percent? Answer to two decimals, carry intermediate calcs. to at least four decimals.

Answer 1

Solution:
	The standard deviation of the portfolio in percent	8.26%
Working Notes:
	First of all we calculate Return of portfolio at each state of Economy.
Return at Boom (rb)	Return of portfolio at Boom (rb)= Weighted average return of individual asset
	=Sum of ( return x weight of % invested)
	= 25% x 0.32 + 40% x 0.19   + 35% x 0.18
	=0.2190
	0.219
Return at Good   (r Good )	Return at Good   (r Good) = Weighted average return of individual asset
	=Sum of ( return x weight of % invested)
	= 25% x 0.16 + 40% x 0.1   + 35% x 0.09
	=0.1115
	0.1115
Return at Poor   (r Poor)	Return at Poor   (r Poor)= Weighted average return of individual asset
	=Sum of ( return x weight of % invested)
	= 25% x 0.02 + 40% x 0.01   + 35% x 0.03
	=0.0195
	0.0195
Return at Bust   (r Bust)	Return at Bust   (r Bust)= Weighted average return of individual asset
	=Sum of ( return x weight of % invested)
	= 25% x -0.09 + 40% x -0.06   + 35% x -0.02
	= -0.05350
	-0.0535
Expected return of portfolio(Er) = Sum of ((prob of each state) x (Return of portfolio at each state))
	=(20% x 21.90%) + (45% x 11.15%) + (25% x 1.95%) + (10% x -5.35%)
	=0.0935000
	0.0935
The variance of this portfolio = Sum of [(Prob. Of each state) x ( (Return of the portfolio at each state - Expected return of the portfolio))^2 ]
				=20% x (21.90% - 9.35%)^2 + 45% x (11.15% - 9.35%)^2 + 25% x (1.95% - 9.35%)^2 + 10% x (-5.35% - 9.35%)^2
	=0.0068257500
	0.00682575
	The standard deviation of Portfolio = Square root of the variance of portfolio
	The standard deviation of Portfolio = (0.0068257500)^(1/2)
	The standard deviation of Portfolio = 0.082618097
	The standard deviation of Portfolio = 0.082618
	The standard deviation of Portfolio = 0.0826
	The standard deviation of Portfolio	8.26%
Please feel free to ask if anything about above solution in comment section of the question.