Assume that security returns are generated by the single-index model, R_i = alph

Question

Assume that security returns are generated by the single-index model, R_i = alpha_i + beta_iR_M + e_i where Ri is the excess return for security i and RM is the market's excess return. The risk-free rate is 2%. Suppose also that there are three securities A, B, and C, characterized by the following data: a. If sigmaM= 20%, calculate the variance of returns of securities A, B and C. b. Now assume that there are an infinite number of assets with return characteristics identical to those of A, B and C, respectively. If one forms a well-diversified portfolio of type A securities, what will be the mean and variance of the portfolio's excess returns? What about portfolios composed only of type B or C stocks? c. Is there an arbitrage opportunity in this market? What is it? Analyze the opportunity graphically. For c. according to the solutions manual: "There is no arbitrage opportunity because the well-diversified portfolios all plot on the SML. Because they are fairly priced there is no arbitrage". How is this possible since the E(R_i)/beta_i are not all equal? Should not E(R_i) = E(r_i) - r_f, and that over beta_i be equal for all the portfolios if there is no arbitrage?

Explanation / Answer

a) Var(Ri)=i^2*M^2+(ei)^2

Var(RA)=A^2*M^2+(eA)^2=.8^2*.20^2+.25^2=0.0881

Var(RB)=B^2*M^2+(eB)^2=1^2*.20^2+.10^2=0.050

Var(RC)=C^2*M^2+(eC)^2=1.2^2*.20^2+.20^2=0.0976

b)mean return for portfolio consisting of A=limit n->inf((1/n)*E(RA)+(1/n)*E(RA)+.....+n times)

=limit n->inf(n*(1/n)*E(RA))=limit n->inf(E(RA))=E(RA)=10%

Variance for portfolio consisting of A

=limit n->inf((1/n^2)*Var(RA)+(1/n^2)*Var(RA)+.....+n times) + limit n->inf((1/n^2)*Cov(RA,RA)+(1/n^2)*Cov(RA,RA)+.....+n(n-1) times)

=limit n->inf(n*(1/n^2)*Var(RA) + limit n->inf(n(n-1)*(1/n^2)*Cov(RA,RA))

=limit n->inf((1/n)*Var(RA) + limit n->inf((1-1/n)*Var(RA))

=0*Var(RA) + ((1-0)*Var(RA))

=Var(RA)=0.0881

Similarly the mean return for portfolio consisting of B=E(RB)=12%

Variance for portfolio consisting of B=Var(RB)=0.05

the mean return for portfolio consisting of C=E(RC)=14%

Variance for portfolio consisting of C=Var(RC)=0.0976

c) The SML plots the relationship of CAPM as: ERi=Rf+i*(ERm-Rf)

ERi-Rf=i*(ERm-Rf)

=> ERi-Rf/i=slope of SML.

for A,ERi-Rf/i=ERA-Rf/A=.10-.02/.8=.10

for B,ERi-Rf/i=ERB-Rf/B=.12-.02/1=.10

for A,ERi-Rf/i=ERC-Rf/C=.14-.02/1.2=.10

Thus ERA-Rf/A=ERB-Rf/B=ERC-Rf/C thus ERi-Rf/i are all equal thus these portfolios for A,B,C must plot on the same SML. Thus they are fairly priced and thus there is no arbitrage.

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