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We can substitute the estiniatecl A matrix into the formula (7.2.24) and use the estimated variancc-covariancc matrix of the error vector e,+i to calculate the sample variances and covariance of the expcclcd-return and dividend components of stock returns. The estimated cxpected-i clurn component has a sample variance equal lo 0.75 limes the variance of realized returns, while the estimated dividend component has a sample variance only 0.12 times the variance of realized returns. The remaining variance of realized returns (0.13 of the total) is attributed to covariance between the expectcd-return and dividend components.л The reason for this result is that the log dividend-price ralio forecasts slock returns, and it is itself a highly persistent process. Tljus revisions in the log dividend-price ratio are associated with persistent changes in expected future returns, and this can justify large changes in stock prices. The estimated VAR process is somewhat more complicated than the simple AR(1) example developed earlier in this chapter, for it includes two forecasting variables, each of which is close to a univariate AR(1). However die main effect of the interest-rate variable is to increase the forecastabilily of one-period stock returns; it has a rather modesl effect on the long-run behavior of the system, which is dominated by the persistent movements of the log dividend-price ratio. Thus the long-run properties of the VAR system are similar to those of the AR( 1) example. From ihis and our previous analysis, one would expect that VAR systems like (7.2.25) could account for the pattern of long-horizon regression results, and this indeed seems to be die case as shown by Campbell (Ю91), Hodrick (1992), and Kandel and Stambaugh (1989). Of course, VAR systems impose more structure on the data; but Hodrick (1992) presents some Monlc Carlo evidence that when die struc- Sirc is correct, the finite-sample behavior of VAR systems is correspondingly ettcr than that of long-horizon regressions with a large horizon relative to ic sample size. [ 7.3 Conclusion ihe research described in this chapter has helped to transform the way fi-ancial economists view asset markets. Ii used to be thought that expected a?sct returns were approximately constant and that movements in prices cpuld he attributed to news about future cash payments lo investors. Today the importance of time-variation in expected returns is widely recognized, ,-!1Ovcr a longer period 1926 to 1988, the two variances and the covariance have roughly e(jual shares of the overall variance of realized stock rclutus. Asymptotic standard errors lor the variance decomposition can be calculated using the delta method explained in Section A.4 of tire Appendix. As in the price-dividend VAR discussed above, the decomposition is conditional oh the information variables included in the VAK system. and this has broad implications for both academics and investment professionals. Al the academic level, there is an explosion of research on the determinants of time-varying expected returns. Economists are exploring a great variety of itleas, from macroeconomic models of real business cycles to more heterodox models of investor psychology. We discuss some of these ideas in Chapter 8. At. a more practical level, dynamic asset-allocation models are becoming increasingly popular. The techniques discussed here can provide quantitative inputs for these investment strategies. In this context long-horizon relurn regressions may be attractive not only for their potential statistical advantages, but also because investment strategies based on long-horizon return forecasts are likely to incur lower transactions costs. Problevts-Chapter 7 7.1 In the late 1980s corporations began to repurchase shares on a large scale. In this problem you are asked to analyze the effect of repurchases on the relation between stock prices and dividends. Consider a firm with fixed cash How per period, X. The total market value of the firm (including the current cash How ,Y) is V. This is die present value of current and future cash flow, discounted at a constant rate It: V = (1 + li)X/li. Each period, die firm uses a fraction X of its cash flow to repurchase shares al cum-dividend prices, and then uses a fraction (1-Х) of its cash How to pay dividends on the remaining shares. The firm has Л/ shares outstanding al die beginning of period / (before it repurchases shares). 7.1.1 What are the cum-dividend price per share and dividend per share at time t? 7.1.2 Derive a relation between the dividend-price ratio, the growth rate of dividends per share G, and the discount rate R. 7.1.3 Show that the price per share equals the expected present value of dividends per share, discounted at rate R. Explain intuitively why this lot inula is correct, even though the firm is devoting only a portion of its cash flow to dividends. 7.2 Consider a stock whose log dividend il, follows a random walk with drift: (/,+ t = ц + <lt + cH i, where e,+ ~ /V(0, a). Assume thai the required log rate of return on the stock is a constant r. 7.2.1 Wc use (lie nutation / ) (lot fundamental value ) lo denote die expected present value of dividends, discounted using the required rate of return. Show that / ; is a constant multiple of the dividend /),. Write the ratio EJl), as a function of the parameters of the model. 7.2.2 Show thai another formula lor the stock price which gives the.same expected rate of return is /, = 1, + rlf, where X > 0 is a function of the other parameters of the model. Solve for X. 7.2.3 Discuss the strengths and weaknesses of this model of a rational hulihle as compared with the ftlant hard-Watson huhhle, (7.1.Mi) in the text. Note: This problem is based on Front and Obslfeld (1991). 7.3 Consider a slock whose expected return obeys /il-Mil = r + x,. (7.1.27) Assume thai ,v, follows an AR(I) process, v,ii = fU?,n. 0 < ф < 1. (7.1.28) 7.3.1 Assume that £M is uncorrelated with news about future dividend payments on the slock. Using the loglinear approximate framework developed in Set lion 7.1.3, derive the autocovariance function of realized stock returns. Assume that ф < p, where p is the parameter of linearization in the loglinear framework. Show that the aulocovariances of stock returns are all negative and die off at rate ф. Give an economic interpretation of your formula for return aulocovariances. 7.3.2 Now allow , to be correlated with news about future dividend payments. Show that die aulocovariances of stock returns can be positive if i and dividend news have a suflicienlly huge positive covariance. 7.4 Suppose that the log fundamental value of a stock, v obeys the process where p is a constant, /> is die parameter of linearization defined in Section 7.1.3, dt is the log dividend on the assel, and , is a white noise error term. 7.4.1 Show thai if the price of die stock equals its fundamental value, then the approximate log sunk return defined in Section 7.1.3 is unforecastable. 7.4.2 Now suppose that the managers of the company pay dividends according to the rule d, =z c+ kv,-\ 4- (1 - X)dt \ + !) where с and X are constants (0 < Л < 1), and /, is a while noise error uncorrelaicd with e,. Managers partially adjust dividends towards fundamental value, where the speed of adjustment is given by X. Marsh and Merlon (198b) have argued for the empirical relevance of this dividend policy. Show that if the price of the slock equals its fundamental value, the log dividend-price ratio follows an AR( 1) process. What is the persistence of this process as a function of X and p? 7.4.3 Now suppose that the stock price does not equal fundamental value, but rathersatisfies/i, = v, - y(di - vt), where у > 0. That is, price exceeds fundamental value whenever fundamental value is high relative to dividends. Show that the approximate log stock return and the log dividend-price ratio satisfy the AR(1) model (7.1.27) and (7.1.28), where the optimal forecaster of the log stock return, x is a positive multiple of the log dividend-price ratio. 7.4.4 Show that in this example innovations in stock returns are negatively correlated with innovations in x,. 7.5 Recall the definition of the perfect-foresight stock price: !> = E£o P7 [d - P)rf,+ i+> + к - г]. (7.2.8) The hypothesis that expected returns are constant implies that the actual stock price p, is a rational expectation of/),*, given investors information. Now consider forecasting dividends using a smaller information set /,. Define }>, = E[p; I J,]. 7.5.1 Show that Var(/>,) > Var(/>,). Give some economic intuition for this result. 7.5.2 Show that Var(# - p,) > Var(/j* - p,) and that Var(/ * - p,) > Var( - pi)- Give some economic intuition for these results. Discuss circumstances where these variance inequalities can be more useful than the inequality in part 7.5.1. 7.5.3 Now define h+\ = k+P/>/-t-i+(l-pV/+i-/V П+i is the return that would prevail under the constant-expected-return model if dividends were forecast using the information set J,. Show that Var(r,+i) < Var(r,+). Give some economic intuition for this result and discuss circumstances where it can be more useful than the inequality in part 7.5.1. Note: This problem is based on Mankiw, Romer, and Shapiro (1985) and West (1988b). Intertemporal Equilibrium Models THIS CHAPTER RKIATES asset prices to the consumption and savings decisions of investors. The static asset pricing models discussed in Chapters 5 and 6 ignore consumption decisions. They treat asset prices as being determined by the portfolio choices of investors who have preferences defined over wealth one period in the future. Implicitly these models assume that investors consume all their wealth alter one period, or al least that wealth uniquely determines consumption so that preferences defined over consumption arc equivalent to preferences defined over wealth. This simplification is ultimately unsatisfactory. In the real world investors consider many periods in making their portfolio decisions, and in this intertemporal setting one must model consumption and portfolio choices .simultaneously. Intertemporal equilibrium models of asset pricing have the potential to answer two questions that have been left unresolved in earlier chapters. First, what forces determine the riskless intcresL rale (or more generally the rale of return on a zero-beta asset) and the rewards that investors demand for bearing risk? In the CAPM the riskless interest rate or zero-beta return and the reward for bearing market risk are exogenous parameters; ihe model gives no account of where they come from. In the APT the single price of market risk is replaced by a vector of factor risk prices, but again the risk prices are determined outside the model. We shall sec in this chapter that intertemporal models can yield insights into the determinants of these parameters. A second, related question has to do with predictable variations in asset returns. The riskless real interest rale moves over time, and in Chapter 7 we presented evidence that forecasts of stock returns also move over time. Importantly, excess stock returns appear to be just as forecastable as real slock returns, suggesting that the reward for bearing stock market risk changes over time. Are these phenomena consistent with market efficiency? Is it possible to construct a model with rational, utility-maximizing investors in which the equilibrium return on risky assets varies over time in the way dc- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 [ 49 ] 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 |