Corrections to Theory of Asset Pricing (2008), Pearson, Boston, MA

Transcription

1 Theory of Asset Pricing George Pennacchi Corrections to Theory of Asset Pricing (8), Pearson, Boston, MA. Page 7. Revise the Independence Axiom to read: For any two lotteries P and P, P P if and only if for all (,] and all P : P + ( )P P + ( )P Moreover, for any two lotteries P and P y, P P y if and only if for all (,] and all P : P + ( )P P y + ( )P. Page 9. Revise the last paragraph to read: Next, suppose that utility is not quadratic but any general increasing, concave form. Are there particular probability distributions for portfolio returns that make expected utility, again, depend only on the portfolio return s mean and variance? Such distributions would need to be fully determined by their means and variances, that is, they must be two-parameter distributions whereby higher-order moments could be expressed in terms of the rst two moments (mean and variance). Many distributions, such as the gamma, normal, and lognormal, satisfy this criterion. But in the context of an investor s portfolio selection problem, such distributions need to satisfy other reasonable conditions. Since an individual is able to choose which assets to combine into a portfolio, all portfolios created from a combination of individual assets or other portfolios must have distributions that continue to be determined by their means and variances. In other words, we need a distribution such that if the individual assets return distributions depend on just mean and variance, then the return on a linear combination (portfolio) of these assets has a distribution that depends on just the portfolio s mean and variance. Furthermore, the distribution should allow for a portfolio that possibly includes a risk-free (zero variance) asset, as well as assets that may be independently distributed. The only distributions that satisfy these additivity, possible

2 risk-free asset, and possible independent assets restrictions is the stable family of distributions. However, the only distribution within the stable family that has nite variance is the normal (Gaussian) distribution. Thus, since the multivariate normal distribution satis es these portfolio conditions and has nite variance, it can be used to justify mean-variance analysis.. Page. Revise equation (.9) to add subscripts. 4. Page 6 equation (.). Change N to n: U ( R p + x i p ) < U ( R p x i p ) m = m = m V! m = m V i! m = m! m j ij (.) 5. Page 7. To indicate that it is a column vector, add a prime after b z = [b z b z ::: b nz ], z = ; :::; k. 6. Page 9. It is clearer to write equation (4.4) as e s = 6 4 W. W s W s W s+. W k = (4.4) 7. Page 7. In Exercise 4 part a., it should read Show whether or not p t = f t + b t, subject to the speci cations in (6.9) and (6.4), is a valid solution for the price of the risky asset. 8. Page 65. In footnote, revise the sentence We examine how to model an asset price process that is a mixture of a di usion process and a jump process in Chapter. See Chamberlain (98) and Liu (4).

3 9. Page 67 to 68. It should be noted that there may be other probability limits, in addition to the normally-distributed Brownian motion, for processes that have independent and identically distributed increments. For example, a compensated Poisson process can have zero mean and variance proportional to time but would not have a limiting distribution that is Brownian motion. To obtain normality, we also need to assume a continuous sample path. See Merton (99, Chapter ).. Page 85. Equation (9.) has an extraneous. It should read " # dh (t) =! i (t) ( i r) H (t) + rh (t) F (t) dt +! i (t) H (t) i dz i. Page 9. Equation (9.4) has an extraneous. It should read (9.) r P rr + [ (r r) + q r ] P r rp P = (9.4). Page. Exercise should state where, r, and are positive constants.. Page 7. In equation (.), s should be changed to u : Z Z = exp (u) dz (u) du (.) 4. Page. Under equation (.), the de nition should be r n r k + n=. 5. Page 57. The last term after the rst equality in equation (.45) should have a minus sign and, therefore, the last term after the second equality in equation (.45) will have a plus sign: = U (Ct ; t) + J t + J W [rw t Ct ] + a (x; t) J x + b (x; J t) W J xx ij j r ( i r) J W W = e t (t) ln + e t d +a (x; t) F x + t b (x; d (t) e t) F xx + d (t) ln [W t ] + F t + e t d (t) r ij j r ( i r) e t (.45)

4 Similarly, this makes the last terms in equations (.46) and (.49) to have plus (t) = ln [d (t)] + + d (t) ln [W t ] + e t F t + d (t) +a (x; t) e t F x + b (x; t) e t F xx + d (t) ij j r ( i r) (.46) = ln [d (t)] + e t F t + d (t) r + a (x; t) e t F x (.49) + b (x; t) e t F xx + d (t) ij j r ( i r) 6. Page 86 The rst line of equation (.) should iw np P ij! jw + ijw 7. Page. The exponent of the rst term in (4.) should have a minus sign, so that (4.) is corrected to read [J W J x ] J W J W W ( r) + (rw bx)j W + (b a)xj x J = 8. Page. In the second paragraph, the reference to (4.58) should be (4.57):...the utility function given in (4.56) and (4.57) is (ordinally) equivalent Page. In the rst sentence, change partial di erential equation to ordinary di erential equation.. Page The rst three sentences on this page should read Similar to the time-separable case, for an in nite horizon solution to exist, we need consumption to be positive in (4.7), which requires > r + [ r] = ( ) : 4

5 This will be the case when the elasticity of intertemporal substitution,, is su ciently small. For example, assuming >, this inequality is always satis ed when <.. Page 4. Correct the de nition of w t to be w t denotes the number of shares of the risky asset held by the individual at date t.. Page 5. In the rst paragraph below equation (5.5), the fourth sentence should be corrected to read Conversely, when = but R t < R, z t is larger than z t.. Page 8. Insert expected in the sentence This portfolio policy is referred to as the growth-optimum portfolio, because it maximizes the expected (continuously compounded) return on wealth. 4. Page 48. In the fourth paragraph, correct the variance so that the sentence reads The equilibrium would be the same if all traders received the same signal, y N(m; + n ) or if they all decided to share information on their private signals among each other before trading commenced. 5. Page 5. In equations (6.9), (6.), and in the text below i (6.), h remove the extraneous right parenthesis to change Varh i ~P j I i ) to Var ~P j I i. References [] Chamberlain, G. (98): A Characterization of the Distributions That Imply Mean-Variance Utility Functions, Journal of Economic Theory, 9, 85-. [] Liu, L. (4): A New Foundation for the Mean-Variance Analysis, European Journal of Operational Research, 58, 9-4. [] Merton, R.C. (99): Continuous-Time Finance. Blackwell, Cambridge, MA. 5