Consider the following scenarios for the return on Stock 1 and S&P 500 Index. St

Question

Consider the following scenarios for the return on Stock 1 and S&P 500 Index.

State of the World

Probability

Stock 1 Return

S&P 500 Index Return

Depression

0.25

-0.01

-0.05

Normal

0.50

0.04

0.09

Boom

0.25

0.14

0.16

(1 point) Suppose that you want to construct a 3-asset portfolio which has a beta of 1.0. You will invest 40% of your money in Stock 1, 10% in the risk-free asset, and the remaining 50% in another stock (“Stock 2”). What should be the beta of Stock 2?

State of the World

Probability

Stock 1 Return

S&P 500 Index Return

Depression

0.25

-0.01

-0.05

Normal

0.50

0.04

0.09

Boom

0.25

0.14

0.16

Explanation / Answer

Mean Stock1 return=0.25*-0.01+0.5*0.04+0.25*0.14=5.25%

Mean S&P500 return=0.25*-0.05+0.5*0.09+0.25*0.16=7.25%

covar(Stock1,S&P500)=0.25*(-0.01-0.0525)*(-0.05-0.0725)+0.5*(0.04-0.0525)*(0.09-0.0725)+0.25*(0.14-0.0525)*(0.16-0.0725)=0.003719

var(S&P500)=0.25*(-0.05-0.0725)^2+0.5*(0.09-0.0725)^2+0.25*(0.16-0.0725)^2=0.005819

beta=covar(stock1,S&p500)/var(s&P500)=0.003719/0.005819=0.639113

Portfolio beta=0.4*beta of stock 1+0.1*0+0.5*beta of stock 2

=>1=0.4*0.639113+0.1*0+0.5*beta of stock 2

=>beta of stock 2=(1-0.4*0.639113)/0.5=1.48871

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