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Páginas: 9 (2215 palabras) Publicado: 16 de noviembre de 2012
CHAPTER 10: SINGLE-INDEX AND MULTIFACTOR MODELS
l. a. To optimize this portfolio one would need: n n n2 – n 2 ________ n2 + 3n 2 b. In a single index model: or equivalently, using exess returns = = = 60 60 1770 estimates of means estimates of variances estimates of covariances

_______________________ = 1890 estimates ri – rf = α i + β i(rM – rf) + ei Ri = α i + β i RM + ei

the variance ofthe rate of return on each stock can be decomposed into the components: (l) β i σM (2) σ2(ei)
2 2

The variance due to the common market factor The variance due to firm specific unanticipated events
2

In this model Cov(ri,rj) = β iβ jσM . The number of parameter estimates would be: n = 60 estimates of the mean E(ri), n = 60 estimates of the sensitivity coefficient β i, n = 60 estimates ofthe firm-specific variance σ2(ei), and 1 estimate of the market mean E(rM) 1 estimate for the market variance σM –––––––––––––––––––––––––––––––––––––––––––– 182 estimates Thus, the single index model reduces the total number of required parameter estimates from 1,890 to 182, and in general from (n2 + 3n)/2 to 3n + 2.
2

2. a.

The standard deviation of each individual stock is given by: σi =[β i σM + σ2(ei) ]1/2 10-1
2 2

Since β A = .8, β B = 1.2, σ(eA ) = 30%, σ(eB) = 40%, and σM = 22% we get: σA = (.82 × 222 + 302)1/2 = 34.78% σB = (1.22 × 222 + 402)1/2 = 47.93% b. The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities: E(rp) = wA E(rA ) + wBE(rB) + wfrf where wA , wB, and wf are the portfolio weights of stock A,stock B, and T-bills, respectively. Substituting in the formula we get: E(rp) = .30 × 13 + .45 × 18 + .25 × 8 = 14% The beta of a portfolio is similarly a weighted average of the betas of the individual securities: β P = wA β A + wBβ B + wfβ f The beta of T-bills (β f ) is zero. The beta of the portfolio is therefore: β P = .30 × .8 + .45 × 1.2 + 0 = .78 The variance of this portfolio is : σP = βP σM + σ2(eP) where β P σM is the systematic component and σ2(eP) is the nonsystematic component. Since the residuals, ei are uncorrelated, the non-systematic variance is: σ2(eP) = wA σ2 (eA ) +wB σ2(eB) + wf σ2(ef) = .302 × 302 + .452 × 402 + .252 × 0 = 405 where σ2(eA ) and σ2(eB) are the firm-specific (nonsystematic) variances of stocks A and B, and σ2(ef), the nonsystematic variance ofT-bills, is zero. The residual standard deviation of the portfolio is thus: 10-2
2 2 2
2 2

2

2

2

σ(eP) = (405)1/2 = 20.12% The total variance of the portfolio is then: σP = .782 × 222 + 405 = 699.47 and the standard deviation is 26.45%. 3. a. The two figures depict the stocks' security characteristic lines (SCL). Stock A has a higher firmspecific risk because the deviations of theobservations from the SCL are larger for A than for B. Deviations are measured by the vertical distance of each observation from the SCL. Beta is the slope of the SCL, which is the measure of systematic risk . Stock B's SCL is steeper, hence stock B's systematic risk is greater. The R2 (or squared correlation coefficient) of the SCL is the ratio of the explained variance of the stock's return to totalvariance, and the total variance is the sum of the explained variance plus the unexplained variance (the stock's residual variance). β i σM β i σM + σ2(ei)
2 2 2 2 2 2 2

b.

c.

R2 =

Since stock B's explained variance is higher (its explained variance is β B σM , which is greater since its beta is higher), and its residual variance σ2(eB) is smaller, its R2 is higher than stock A's. d.Alpha is the intercept of the SCL with the expected return axis. Stock A has a small positive alpha whereas stock B has a negative alpha; hence stock A's alpha is larger. The correlation coefficient is simply the square root of R2, so stock B’s correlation with the market is higher.

e.

4. a.

Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more...
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