The single index model has been estimated for a particular security is estimated

Question

The single index model has been estimated for a particular security is estimated for a particular
security with the following results:
Ra = .01 + .50Rm + Ea
Rb = .02 + 1.30Rm + Eb

The standard deviation of the market return was .25

a.What is the R2 for firm A?
b.What is the for firm B?What is the R2 for firm A?
c.A portfolio which is 60% security A and 40% security B would have a beta of . . . ?
d. Provide an estimate of for security A assuming that the average monthly market
return was .015 (decimal), and the average T-Bill rate was .005.What is the R2 for
firm A?

Explanation / Answer

a)R2 is return for a or b. Rb = .02 + 1.30Rm + Eb ,take expected values on both sides => E(Rb) = E(.02 + 1.30Rm + Eb)=> E(Rb) = E(.02)+ 1.30E(Rm) + E(Eb)=> E(Rb) = .02+ 1.30E(Rm) + 0= .02+ 1.30E(Rm)...as E(Ea) expected value of error term is 0 as per model assumptions.

Given,  E(Rb) = .36 =>.36= .02+ 1.30E(Rm)=>.36-.02=1.30E(Rm)=>E(Rm)=.34/1.3=0.2615

Thus Return for firm A=Ra = .01 + .50*0.2615 =.01+ 0.13075 =0.14075

b)Rb = .02 + 1.30Rm + Eb take variance n both sides of equation=> Var(Rb)=Var(.02 + 1.30Rm + Eb)

=> Var(Rb)=Var(.02) + 1.32Var(Rm) + Var(Eb)

As per model assumptions,Var(Ea) variance of error terms is 0,variance of constant term .02 is 0. Thus

=> Var(Rb)=0 + 1.32Var(Rm) =>Var(Rm)=1.32Var(Rm) =1.33*.252

=>stdDev of B=sqrt(Var(Rb))=sqrt(1.33*.252)=1.3*.25=.325

c)beta of A=slope of eqn Ra = .01 + .50Rm + Ea=.5

beta of B=slope of eqn Rb =.02 + 1.30Rm + Eb=1.3

portfolio beta=weighted average of betas of A and B=.6*.5+.4*1.3=.3+.52=.82

d)Average Return per month of A=average T-Bill rate/month+ .5*average monthly market
return = .005+ .015*.5= .005+.0075=.0125=1.25%

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