/ * statistic
figure 6.1. Distributions for the CAPM Tenhlnlercepl Test Statistic for Four Hypotheses
when aa = 0.0007 and
ii ~ /клшФО.З) (0.0.29)
when ая = 0.001.
A plot of the four distributions from (6.6.26), (6.6.27), (6.6.28), and (16.29) is in Figure 6.1. The vertical bar on the plot represents the value
I 91 which Fama and French calculate for the test statistic. From this figure, notice that the distributions under the null hypothesis and the risk-based alternative hypothesis arc quite close together.10 This reflects the impact of ilie upper bound on the noncenlrality parameter. In contrast, the nonrisk-bltscd alternatives distributions arc far to the right of the other two distributions, consistent with the uuboundedness of the noncenlrality parameter (Or these alternatives.
I Given that Kama and French find a lest statistic of 1.91, these results suggest that the missing-risk-factors argument is not the whole story. From F gurc 6.1 one can see that 1.91 is still in the upper tail when the distribution of Ji in the presence of missing risk factors is tabulated. The -value using
II is distribution is 0.03 for the monthly data. Hence it seems unlikely that n issing factors completely explain the deviations.
The data offer some support for the nonrisk-bascd alternative views. The test statistic falls almost in the middle of the nonrisk-bascd allerna-
Sre MarKinlay (11)87) for detailed analysis ol ilie risk-based alternative
In this chapter we have developed the econometrics for estimating and testing multifactor pricing models. These models provide an allraclive alternative lo the single-factor CAPM, but users of such models should be aware of two serious dangers that arise when factors are chosen to lit existing dala without regard lo economic theory. First, the models may overfit the data because of data-snooping bias; in this case they will not be able lo predict ass,-t returns in the future. Second, the models may capture empirical regularities thai are due to market inefficiency or investor irrationality; in this case they may continue lo (it (he dala but they will imply Sharpe ratios for factor portfolios thai are too high lo be consistent with a reasonable underlying model of market equilibrium. Both these problems can be mitigated if one derives a factor structure from an equilibrium model, along the lines discussed in Chapter 8. In the end, however, the usefulness of multifactor models will not be fully known until sufficient new dala become available lo provide a true out-of-samplc check on their performance.
6.1 Consider a multiple regression of the return on any asscl or portfolio R on the returns of any set of portfolios from which the entire minimum-variance boundary can be generated. Show that die intercept of this regression will be zero and that the factor regression coefficients for any asscl will sum to uniiy.
6.2 Consider two economics, economy Л and economy H. The mean excess-return vector and the covariance matrix is specified below for each ol the economies. Assume there exist a riskfree asset, N risky assets with mean excess return д and nonsingular covariance matrix 12, and a risky laclor portfolio with mean excess return and variance aj. The factor portfolio is not a linear combination of the N assets. (This criterion can be met by eliminating one of the assets which is included in the laclor portfolio
live with the lower standard deviation of the elements of a. Several of the nonrisk-bascd alternatives could equally well explain the results. Different nonrisk-bascd views can give the same noncenlrality parameter and tcsi-siaiisiic distribution. The results are consistent with ihe data-snooping alternative ofl.o and MacKinlay (1990b), with the related sample selection biases discussed by Brccn and Korajczyk (1993) and Kolhari, Slianken, and Sloan (1995), and with the presence of markei inefficiencies.
6. Multifactor Iricing Models
if necessary.) For both economics Л and Ii:
/1 = a + Pni, ((3.7.1)
П = [iffaj + ЙЛа; + la;. (6.7.2)
(liven die above mean and covariance matrix and the assumption that the lactor porllolio ji is a traded asset, what is the maximum squared Sharpe ratio lor the given economies?
6.3 Returning lo the above problem, ihe economies are further specified. Assume the elements of a are < ross-seclionally independent and identically
и, ~ llDIO.rr;) i = 1.....N. (6.7.3)
The specification of the distribution of the elements of 6 conditional on a differentiates economics Л and /(. For economy A:
ii, I a ~ Ш)( 0) / = 1.....N, (6.7.4)
and for economy />:
Л, I a ~ lll)((U;) i = 1.....N. (6.7.3)
Unconditionally the cross-sectional distribution of the elements of 6 will
he the same for both economies, but for economy A conditional on a, 6 is fixed. What is the maximum squared Sharpe ratio for each economy? What is the maximum squared Sharpe ratio for each economy as the N increases lo infinity?
Тик first part of this book has examined the behavior of stock returns in some detail. The exclusive focus on returns is traditional in empirical research on asset pricing; yet it belies the name of the field to study only returns and not to say anything about asset prices themselves. Many of the most important applications of financial economics involve valuing assets, and for these applications it is essential to be able to calculate the prices that are implied by models of returns. In this chapter we discuss recent research that tries to bring attention back to price behavior. We deal with common stock prices throughout, but of course the concepts developed in this chapter are applicable to other assets as well.
The basic framework for our analysis is the discounled-cash-flow or present-value model. This model relates the price of a stock to its expected future j cash flows-its dividends-discounted to the present using a constant or j time-varying discount rate. Since dividends in all future periods enter the j present-value formula, the dividend in any one period is only a small component of the price. Therefore long-lasting or persistent movements in div- i idends have much larger effects on prices than temporary movements do. A similar insight applies to variation in discount rates. The discount rate between any one period and the next is only a small component of the long-horizon discount rate applied to a distant future cash flow; therefore persistent movements in discount rates have much larger effects on prices than temporary movements do. For this reason the study of asset prices is intimately related to the study of long-horizon asset returns. Section 7.1 uses the present-value model to discuss these links between movements in prices, dividends, and returns.
We mentioned at the end of Chapter 2 that there is some evidence for predictability of stock returns at long horizons. This evidence is statistically weak when only past returns are used to forecast future returns, as in Chap- ter 2, hut it becomes considerably stronger when other variables, such as the dividend-price ratio or the level of interest rates, are brought into the
analysis. In Section 7.2, wc use (lie formulas of Section 7.1 to help interpret these findings. We show how various test statistics will behave, both under the null hypothesis and under the simple alternative hypothesis that the expected stock return is time-varying and follows a persistent first-order autoregressive (AR(1)) process. A major theme of the section is that recent empirical findings using longer-horizon dala arc roughly consistent with this persistent AR(1) alternative model. We also develop the implications of the AR(1) model for price behavior. Persistent movements in expected returns have dramatic effects on stock prices, making them much more volatile than they would be if expected returns were constant.
The source of this persistent variation in expected slock returns is an important unresolved issue. One view is that the time-variation in expected returns and the associated volatility of stock prices arc evidence against the Efficient Markets Hypothesis (EMU), hut as we argued-in Chapter 1, die EMH can only be tested in conjunction with a model of equilibrium returns. This chapter describes evidence against the joint hypothesis that the EMH holds and that equilibrium stock returns are constant, but it leaves open the J>ssibility that a model with time-varying equilibrium stock returns can be constructed to fit the data. We explore this possibility further in Chapter 8. .
j 7.1 The Relation between Prices, Dividends, and Returns
In this section we discuss the present-value model of stock prices. Using с identity that relates slock prices, dividends, and returns, Section 7.1.1 esents the expected-prcsent-value formula for a stock with constant expected returns. Section 7.1.1 assumes away the possibility that there are -called rational bubbles in stock prices, but this possibility is considered in Section 7.1.2. Section 7.1.3 studies the general case where expected stock rt turns vary through time. The exact present-value formula is nonlinear in tl is case, but a loglinear approximation yields some useful insights. Section 7.1.4 develops a simple example in which the expected slock return is li nc-varying and follows an AR( 1) process.
We. first recall the definition of the return on a stock given in Chapter l. The net simple return is
1 Я,+, - - 1. (7.1.1)
This definition is straightforward, but it does use two notalional conventions that deserve emphasis. First, R,+ \ denotes the return on the stock held from time t to time I + 1. The subscript / + 1 is used because the return only becomes known at time t + \. Second, denotes the price of a share of stock measured at the end of period /, or eqnivalcnlly an ex-dividend price:
Purchase of the slock al price R, today gives one a claim lo next periods dividend per share D,+ ] but not to this periods dividend I),.
An iiltcrivativeiiieasurc()frclurnisthe)og(Mconiinii,iiislyc<)inp<)unde(l return, defined in Chapter 1 as
;+i = log(l + ЛУИ). (7.1.2)
Here, as throughout this chapter, we use lowercase letters to denote log variables.
7.1.1 The Linear lresenl-Value Relation with Constant Expected Returns
In this section wc explore the consequences of the assumption that the expecied stock return is equal to a constant R:
E, [ R,.
Taking expectations of the identity (7.1.1), imposing (7.1.3), and rearranging, wc obtain an equation relating the current slock price to the next periods expected stock price and dividend:
l\ = К/
1 + R
This cxpectational difference equation can be solved forward by repeatedly substituting out future prices and using the Law of Iterated Expectations- the result that E, [li,+1[X]] = E,[X], discussed in Chapter 1-lo eliminate future-dated expectations. After solving forward A periods we have
R, = Е/
1 + R
The second term on the right-hand side of (7.1.5) is the expected discounted value of the slock price A periods from the present. For now, wc assume that this term shrinks lo zero as the horizon A increases:
i + r) p,+k
I lu-se tiiniii> assumptions arc standard in tin- linam с literature. I lowcver some of the literature on volatility lests. lor example Shiller (IIIH1) and Campbell and Shiller (HIH7, l<J88a.l>). uses the alternative limine, convention thai the stork price is measined at the Ьсцппипк of the period or trailed ciuinlividencl. Dilleienccs between the formulas Kiven in tins chapter and those in the ungual volatility papers are due to ibis dillrrcmc in limine, contentions.
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