Question

Assume you have created a 2-stock portfolio by investing $30,000 in stock X with a beta of 0.8, and $70,000 in stock Y with a beta of 1.2. Market risk premium is 8% and risk-free rate is 6%.

The followings are the probability distributions of Stocks X and Y’s future returns:

State of Economy          Probability rx                      rY

Recession 0.1                               -10%                -35%

Below average             0.2                               2% 0%

Average                        0.4                               12%                 20%

Above average 0.2                               20%                 25%

Boom                           0.1                               38%                 45%

1. Calculate the portfolio’s expected rate of return and the standard deviation of its future returns
2. Calculate the required rate of return of your portfolio.
3. Which stock in your portfolio is currently under-valued? Explain.

Answer 1

	(in %)
State of economy	probability(p)	R(x)	R(y)	p*x	p*y	p*((x-∑px)^2)	p*((y-∑py)^2)	p*(x-∑px)(y-∑py)
Recession	0.1	-10	-35	-1	-3.5	48.40	240.10	107.80
Below average	0.2	2	0	0.4	0	20.00	39.20	28.00
average	0.4	12	20	4.8	8	0.00	14.40	-0.00
above average	0.2	20	25	4	5	12.80	24.20	17.60
Boom	0.1	38	45	3.8	4.5	67.60	96.10	80.60
Expected Return				12.00	14.00	148.80	414.00	234.00
						Rx	Ry
Expected Return =		∑px for stock A and ∑py for stock B and ∑pz for stock C				12.00	14.00
Standard deviation=		∑p*((x-∑px)^2)^(1/2)				12.20	20.35
		working				148.8^(1/2)	414^(1/2)
Expected Return of portfolio =			(Return of stock A* weight of stock A) + (Return of stock B* weight of stock B)
		=	(12*0.3)+(14*0.7)
		=	13.4
Weights						0.3	0.7
						(30000/100000)	(70000/100000)
covariance (A,B)=	∑p*(x-∑px)(y-∑py)
=	234
Standard Deviation of Portfolio=		[{(weight of A)^2 * (sigma of A)^2} + {(weight of B)^2 + (sigma of B)^2} + {2*(weight of A)*(weight of B)*Covariance of A&B}]^(1/2)
	=	(((0.3)^2*(12.20)^2) + ((0.7)^2*(20.35)^2) + (2*0.3*0.7*234))^(1/2)
	=	17.74
Calculation of Beta Portfolio
Particulars	X	Y	Total
weights	0.3	0.7
Beta	0.8	1.2
Weighted beta	0.24	0.84	1.08
Reuired return of portfolio=	Rf+(Rm-Rf)*portfolio beta
	6+(8*1.08)
	14.64
Reuired return of stock=	Rf+(Rm-Rf)*stock beta
X	6+(8*0.8)=	12.40
Y	6+(8*1.2)=	15.60
Stock	Expected Return	Required Return	Alpha=Er-Re	Over/under valued
X	12.00	12.4	-0.40	Overvalued
Y	14.00	15.6	-1.60	Overvalued