Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 13 ppt

Chapter 13 Asset Pricing 13.1 Introduction Chapter showed how an equilibrium price system for an economy with complete markets model could be used to determine the price of any redundant asset That approach allowed us to price any asset whose payoff could be synthesized as a measurable function of the economy’s state We could use either the Arrow-Debreu time- prices or the prices of one-period Arrow securities to price redundant assets We shall use this complete markets approach again later in this chapter However, we begin with another frequently used approach, one that does not require the assumption that there are complete markets This approach spells out fewer aspects of the economy and assumes fewer markets, but nevertheless derives testable intertemporal restrictions on prices and returns of different assets, and also across those prices and returns and consumption allocations This approach uses only the Euler equations for a maximizing consumer, and supplies stringent restrictions without specifying a complete general equilibrium model In fact, the approach imposes only a subset of the restrictions that would be imposed in a complete markets model As we shall see, even these restrictions have proved difficult to reconcile with the data, the equity premium being a widely discussed example Asset-pricing ideas have had diverse ramifications in macroeconomics In this chapter, we describe some of these ideas, including the important ModiglianiMiller theorem asserting the irrelevance of firms’ asset structures We describe a closely related kind of Ricardian equivalence theorem We describe various ways of representing the equity premium puzzle, and an idea of Mankiw (1986) that one day may help explain it 1 See Duffie (1996) for a comprehensive treatment of discrete and continuous time asset pricing theories See Campbell, Lo, and MacKinlay (1997) for a summary of recent work on empirical implementations – 386 – Asset Euler equations 387 13.2 Asset Euler equations We now describe the optimization problem of a single agent who has the opportunity to trade two assets Following Hansen and Singleton (1983), the household’s optimization by itself imposes ample restrictions on the co-movements of asset prices and the household’s consumption These restrictions remain true even if additional assets are made available to the agent, and so not depend on specifying the market structure completely Later we shall study a general equilibrium model with a large number of identical agents Completing a general equilibrium model may impose additional restrictions, but will leave intact individual-specific versions of the ones to be derived here The agent has wealth At > at time t and wants to use this wealth to maximize expected lifetime utility, ∞ β j u(ct+j ), Et < β < 1, (13.2.1) j=0 where Et denotes the mathematical expectation conditional on information known at time t, β is a subjective discount factor, and ct+j is the agent’s consumption in period t + j The utility function u(·) is concave, strictly increasing, and twice continuously differentiable To finance future consumption, the agent can transfer wealth over time through bond and equity holdings One-period bonds earn a risk-free real gross interest rate Rt , measured in units of time t + consumption good per timet consumption good Let Lt be gross payout on the agent’s bond holdings between periods t and t + , payable in period t + with a present value of −1 Rt Lt at time t The variable Lt is negative if the agent issues bonds and thereby borrows funds The agent’s holdings of equity shares between periods t and t + are denoted Nt , where a negative number indicates a short position in shares We impose the borrowing constraints Lt ≥ −bL and Nt ≥ −bN , where bL ≥ and bN ≥ A share of equity entitles the owner to its stochastic dividend stream yt Let pt be the share price in period t net of that period’s dividend The budget constraint becomes −1 ct + Rt Lt + pt Nt ≤ At , (13.2.2) See chapters and 17 for further discussions of natural and ad hoc borrowing constraints 388 Asset Pricing and next period’s wealth is At+1 = Lt + (pt+1 + yt+1 )Nt (13.2.3) The stochastic dividend is the only source of exogenous fundamental uncertainty, with properties to be specified as needed later The agent’s maximization problem is then a dynamic programming problem with the state at t being At and current and past y , and the controls being Lt and Nt At interior solutions, the Euler equations associated with controls Lt and Nt are −1 u (ct )Rt = Et βu (ct+1 ), u (ct )pt = Et β(yt+1 + pt+1 )u (ct+1 ) (13.2.4) (13.2.5) These Euler equations give a number of insights into asset prices and consumption Before turning to these, we first note that an optimal solution to the agent’s maximization problem must also satisfy the following transversality conditions: −1 lim Et β k u (ct+k )Rt+k Lt+k = 0, (13.2.6) lim Et β k u (ct+k )pt+k Nt+k = (13.2.7) k→∞ k→∞ Heuristically, if any of the expressions in equations (13.2.6 ) and (13.2.7 ) were strictly positive, the agent would be overaccumulating assets so that a higher expected life-time utility could be achieved by, for example, increasing consumption today The counterpart to such nonoptimality in a finite horizon model would be that the agent dies with positive asset holdings For reasons like those in a finite horizon model, the agent would be happy if the two conditions (13.2.6 ) and (13.2.7 ) could be violated on the negative side But the market would stop the agent from financing consumption by accumulating the debts that would be associated with such violations of (13.2.6 ) and (13.2.7 ) No other agent would want to make those loans Current and past y ’s enter as information variables How many past y ’s appear in the Bellman equation depends on the stochastic process for y For a discussion of transversality conditions, see Benveniste and Scheinkman (1982) and Brock (1982) Martingale theories of consumption and stock prices 389 13.3 Martingale theories of consumption and stock prices In this section, we briefly recall some early theories of asset prices and consumption, each of which is derived by making special assumptions about either Rt or u (c) in equations (13.2.4 ) and (13.2.5 ) These assumptions are too strong to be consistent with much empirical evidence, but they are instructive benchmarks First, suppose that the risk-free interest rate is constant over time, Rt = R > , for all t Then equation (13.2.4 ) implies that Et u (ct+1 ) = (βR)−1 u (ct ), (13.3.1) which is Robert Hall’s (1978) result that the marginal utility of consumption follows a univariate linear first-order Markov process, so that no other variables in the information set help to predict (to Granger cause) u (ct+1 ), once lagged u (ct ) has been included As an example, with the constant relative risk aversion utility function u(ct ) = (1 − γ)−1 c1−γ , equation (13.3.1 ) becomes t (βR)−1 = Et ct+1 ct −γ Using aggregate data, Hall tested implication (13.3.1 ) for the special case of quadratic utility by testing for the absence of Granger causality from other variables to ct Efficient stock markets are sometimes construed to mean that the price of a stock ought to follow a martingale Euler equation (13.2.5 ) shows that a number of simplifications must be made to get a martingale property for the stock price We can transform the Euler equation Et β(yt+1 + pt+1 ) u (ct+1 ) = pt u (ct ) See Granger (1969) for his definition of causality A random process z t is said not to cause a random process xt if E(xt+1 |xt , xt−1 , , zt , zt−1 , ) = E(xt+1 |xt , xt−1 , ) The absence of Granger causality can be tested in several ways A direct way is to compute the two regressions mentioned in the preceding definition and test for their equality An alternative test was described by Sims (1972) 390 Asset Pricing by noting that for any two random variables x, z , we have the formula Et xz = Et xEt z + covt (x, z), where covt (x, z) ≡ Et (x − Et x)(z − Et z) This formula defines the conditional covariance covt (x, z) Applying this formula in the preceding equation gives βEt (yt+1 + pt+1 )Et u (ct+1 ) u (ct+1 ) + βcovt (yt+1 + pt+1 ) , u (ct ) u (ct ) = pt (13.3.2) To obtain a martingale theory of stock prices, it is necessary to assume, first, that Et u (ct+1 )/u (ct ) is a constant, and second, that covt (yt+1 + pt+1 ) , u (ct+1 ) = u (ct ) These conditions are obviously very restrictive and will only hold under very special circumstances For example, a sufficient assumption is that agents are risk neutral so that u(ct ) is linear in ct and u (ct ) becomes independent of ct In this case, equation (13.3.2 ) implies that Et β(yt+1 + pt+1 ) = pt (13.3.3) Equation (13.3.3 ) states that, adjusted for dividends and discounting, the share price follows a first-order univariate Markov process and that no other variables Granger cause the share price These implications have been tested extensively in the literature on efficient markets We also note that the stochastic difference equation (13.3.3 ) has the class of solutions ∞ t pt = Et β j yt+j + ξt , (13.3.4) β j=1 where ξt is any random process that obeys Et ξt+1 = ξt (that is, ξt is a “martingale”) Equation (13.3.4 ) expresses the share price pt as the sum of discounted expected future dividends and a “bubble term” unrelated to any fundamentals In the general equilibrium model that we will describe later, this bubble term always equals zero For a survey of this literature, see Fama (1976a) See Samuelson (1965) for the theory and Roll (1970) for an application to the term structure of interest rates Equivalent martingale measure 391 13.4 Equivalent martingale measure This section describes adjustments for risk and dividends that convert an asset price into a martingale We return to the setting of chapter and assume that the state st that evolves according to a Markov chain with transition probabilities π(st+1 |st ) Let an asset pay a stream of dividends {d(st )}t≥0 The cum-dividend time- t price of this asset, a(st ), can be expressed recursively as a(st ) = d(st ) + β st+1 u [ci (st+1 )] t+1 a(st+1 )π(st+1 |st ) u [ci (st )] t (13.4.1) Notice that this equation can be written −1 a(st ) = d(st ) + Rt a(st+1 )˜ (st+1 |st ) π (13.4.2) st+1 or −1 ˜ a(st ) = d(st ) + Rt Et a(st+1 ), where −1 −1 Rt = Rt (st ) ≡ β st+1 u [ci (st+1 )] t+1 π(st+1 |st ) u [ci (st )] t (13.4.3) ˜ and E is the mathematical expectation with respect to the distorted transition density u [ci (st+1 )] t+1 π(st+1 |st ) π (st+1 |st ) = Rt β ˜ (13.4.4a) u [ci (st )] t −1 Notice that Rt is the reciprocal of the gross one-period risk-free interest rate, as given by equation (13.2.4 ) The transformed transition probabilities are rendered probabilities—that is, made to sum to one—through the multiplication by βRt in equation (13.4.4a) The transformed or “twisted” transition measure π (st+1 |st ) can be used to define the twisted measure ˜ πt (st ) = π (st |st−1 ) π (s1 |s0 )˜ (s0 ) ˜ ˜ ˜ π (13.4.4b) For example, π (st+2 , st+1 |st ) = Rt (st )Rt+1 (st+1 )β ˜ u [ci (st+2 )] t+2 π(st+2 |st+1 )π(st+1 |st ) u [ci (st )] t Cum-dividend means that the person who owns the asset at the end of time t is entitled to the time-t dividend 392 Asset Pricing The twisted measure πt (st ) is called an equivalent martingale measure We ˜ explain the meaning of the two adjectives “Equivalent” means that π assigns ˜ positive probability to any event that is assigned positive probability by π , and vice versa The equivalence of π and π is guaranteed by the assumption that ˜ u (c) > in (13.4.4a) We now turn to the adjective “martingale.” To understand why this term is applied to (13.4.4a), consider the particular case of an asset with dividend stream dT = d(sT ) and dt = for t < T Using the arguments in chapter or iterating on equation (13.4.1 ), the cum-dividend price of this asset can be expressed as aT (sT ) = d(sT ), at (st ) = u [ci (sT )] T Est β T −t aT (sT ), u [ci (st )] t (13.4.5a) (13.4.5b) where Est denotes the conditional expectation under the π probability measure Now fix t < T and define the “deflated” or “interest-adjusted” process at+j = ˜ at+j , Rt Rt+1 Rt+j−1 (13.4.6) for j = 1, , T − t It follows directly from equations (13.4.5 ) and (13.4.4 ) that ˜˜ Et at+j = at (st ) ˜ (13.4.7) where at (st ) = a(st ) − d(st ) Equation (13.4.7 ) asserts that relative to the ˜ twisted measure π , the interest-adjusted asset price is a martingale: using the ˜ twisted measure, the best prediction of the future interest-adjusted asset price is its current value Thus, when the equivalent martingale measure is used to price assets, we have so-called risk-neutral pricing Notice that in equation (13.4.2 ) the adjustment for risk is absorbed into the twisted transition measure We can write equation (13.4.7 ) as ˜ E[a(st+1 )|st ] = Rt [a(st ) − d(st )], (13.4.8) The existence of an equivalent martingale measure implies both the existence of a positive stochastic discount factor (see the discussion of Hansen and Jagannathan bounds later in this chapter), and the absence of arbitrage opportunities; see Kreps (1979) and Duffie (1996) Equilibrium asset pricing 393 ˜ where E is the expectation operator for the twisted transition measure Equation (13.4.8 ) is another way of stating that, after adjusting for risk-free interest and dividends, the price of the asset is a martingale relative to the equivalent martingale measure Under the equivalent martingale measure, asset pricing reduces to calculating the conditional expectation of the stream of dividends that defines the asset For example, consider a European call option written on the asset described earlier that is priced by equations (13.4.5 ) The owner of the call option has the right but not the obligation to the “asset” at time T at a price K The owner of the call will exercise this option only if aT ≥ K The value at T of the option is therefore YT = max(0, aT − K) ≡ (aT − K)+ The price of the option at t < T is then ˜ Yt = Et (aT − K)+ Rt Rt+1 · · · Rt+T −1 (13.4.9) Black and Scholes (1973) used a particular continuous time specification of π ˜ that made it possible to solve equation (13.4.9 ) analytically for a function Yt Their solution is known as the Black-Scholes formula for option pricing 13.5 Equilibrium asset pricing The preceding discussion of the Euler equations (13.2.4 ) and (13.2.5 ) leaves open how the economy, for example, generates the constant gross interest rate assumed in Hall’s work We now explore equilibrium asset pricing in a simple representative agent endowment economy, Lucas’s asset-pricing model We imagine an economy consisting of a large number of identical agents with preferences as specified in expression (13.2.1 ) The only durable good in the economy is a set of identical “trees,” one for each person in the economy At the beginning of period t, each tree yields fruit or dividends in the amount yt The fruit is not storable, but the tree is perfectly durable Each agent starts life at time zero with one tree See Lucas (1978) Also see the important early work by Stephen LeRoy (1971, 1973) Breeden (1979) was an early work on the consumption-based capital-asset-pricing model 394 Asset Pricing The dividend yt is assumed to be governed by a Markov process and the dividend is the sole state variable st of the economy, i.e., st = yt The timeinvariant transition probability distribution function is given by prob{st+1 ≤ s |st = s} = F (s , s) All agents maximize expression (13.2.1 ) subject to the budget constraint (13.2.2 )–(13.2.3 ) and transversality conditions (13.2.6 )–(13.2.7 ) In an equilibrium, asset prices clear the markets That is, the bond holdings of all agents sum to zero, and their total stock positions are equal to the aggregate number of shares As a normalization, let there be one share per tree Due to the assumption that all agents are identical with respect to both preferences and endowments, we can work with a representative agent 10 Lucas’s model shares features with a variety of representative agent asset-pricing models (See Brock, 1982, and Altug, 1989, for example.) These use versions of stochastic optimal growth models to generate allocations and price assets Such asset-pricing models can be constructed by the following steps: Describe the preferences, technology, and endowments of a dynamic economy, then solve for the equilibrium intertemporal consumption allocation Sometimes there is a particular planning problem whose solution equals the competitive allocation Set up a competitive market in some particular asset that represents a specific claim on future consumption goods Permit agents to buy and sell at equilibrium asset prices subject to particular borrowing and shortsales constraints Find an agent’s Euler equation, analogous to equations (13.2.4 ) and (13.2.5 ), for this asset Equate the consumption that appears in the Euler equation derived in step to the equilibrium consumption derived in step This procedure will give the asset price at t as a function of the state of the economy at t In our endowment economy, a planner that treats all agents the same would like to maximize E0 ∞ β t u(ct ) subject to ct ≤ yt Evidently the solution is to t=0 set ct equal to yt After substituting this consumption allocation into equations (13.2.4 ) and (13.2.5 ), we arrive at expressions for the risk-free interest rate and 10 In chapter 8, we showed that some heterogeneity is also consistent with the notion of a representative agent Stock prices without bubbles 395 the share price: −1 u (yt )Rt = Et βu (yt+1 ), u (yt )pt = Et β(yt+1 + pt+1 )u (yt+1 ) (13.5.1) (13.5.2) 13.6 Stock prices without bubbles Using recursions on equation (13.5.2 ) and the law of iterated expectations, which states that Et Et+1 (·) = Et (·), we arrive at the following expression for the equilibrium share price: ∞ β j u (yt+j )yt+j + Et lim β k u (yt+k )pt+k u (yt )pt = Et k→∞ j=1 (13.6.1) Moreover, equilibrium share prices have to be consistent with market clearing; that is, agents must be willing to hold their endowments of trees forever It follows immediately that the last term in equation (13.6.1 ) must be zero Suppose to the contrary that the term is strictly positive That is, the marginal utility gain of selling shares, u (yt )pt , exceeds the marginal utility loss of holding the asset forever and consuming the future stream of dividends, ∞ Et j=1 β j u (yt+j )yt+j Thus, all agents would like to sell some of their shares and the price would be driven down Analogously, if the last term in equation (13.6.1 ) were strictly negative, we would find that all agents would like to purchase more shares and the price would necessarily be driven up We can therefore conclude that the equilibrium price must satisfy ∞ βj pt = Et j=1 u (yt+j ) yt+j , u (yt ) (13.6.2) which is a generalization of equation (13.3.4 ) in which the share price is an expected discounted stream of dividends but with time-varying and stochastic discount rates Note that asset bubbles could also have been ruled out by directly referring to transversality condition (13.2.7 ) and market clearing In an equilibrium, the representative agent holds the per-capita outstanding number of shares 428 Asset Pricing Hansen and Jagannathan want to approach the data with a class of stochastic discount factors To begin, Hansen and Jagannathan note that one candidate for a stochastic discount factor is −1 y ∗ = x (Exx ) q (13.14.4) This can be verified directly, by substituting into equation (13.14.2 ) and verifying that q = E(y ∗ x) Besides equation (13.14.4 ), many other stochastic discount factors work, in the sense of pricing the random returns x correctly, that is, recovering q as their price It can be verified directly that any other y that satisfies y = y∗ + e is also a stochastic discount factor, where e is orthogonal to x Let Y be the space of all stochastic discount factors 13.14.3 A Hansen-Jagannathan bound Given data on q and the distribution of returns x, Hansen and Jagannathan wanted to infer properties of y while imposing no more structure than linearity of the pricing functional (the law of one price) Imposing only this, they constructed bounds on the first and second moments of stochastic discount factors y that are consistent with a given distribution of payoffs on a set of primitive securities For y ∈ Y , here is how they constructed one of their bounds: Let y be an unobserved stochastic discount factor Though y is unobservable, we can represent it in terms of the population linear regression 26 y =a+xb+e (13.14.5) where e is orthogonal to x and −1 b = [cov (x, x)] cov (x, y) a = Ey − Ex b 26 See chapter for the definition and construction of a population linear regression Hansen-Jagannathan bounds 429 Here cov(x, x) = E(xx) − E(x)E(x) We have data that allow us to estimate the second-moment matrix of x, but no data on y and therefore on cov(x, y) But we have data on q , the vector of security prices So Hansen and Jagannathan proceeded indirectly to use the data on q, x to infer something about y Notice that q = E(yx) implies cov(x, y) = q − E(y)E(x) Therefore −1 b = [cov (x, x)] [q − E (y) E (x)] (13.14.6) Thus, given a guess about E(y), asset returns and prices can be used to estimate b Because the residuals in equation (13.14.5 ) are orthogonal to x, var (y) = var (x b) + var (e) Therefore [var (x b)] ≤ σ (y) , (13.14.7) where σ(y) denotes the standard deviation of the random variable y This is the lower bound on the standard deviation of all 27 stochastic discount factors with prespecified mean E(y) For various specifications, Hansen and Jagannathan used expressions (13.14.6 ) and (13.14.7 ) to compute the bound on σ(y) as a function of E(y), tracing out a frontier of admissible stochastic discount factors in terms of their means and standard deviations Here are two such specifications First, recall that a (gross) return for an asset with price q and payoff x is defined as z = x/q A return is risk free if z is constant (not random) Then note that if there is an asset with risk-free return z RF ∈ x, it follows that E(yz RF ) = z RF Ey = , and therefore Ey is a known constant Then there is only one point on the frontier that is of interest, the one with the known E(y) If there is no risk-free asset, we can calculate a different bound for every specified value of E(y) Second, take a case where E(y) is not known because there is no risk-free payout in the set of returns Suppose, for example, that the data set consists of “excess returns.” Let xs be a return on a stock portfolio and xb be a return on a risk-free bond Let z = xs − xb be the excess return Then E [yz] = 27 The stochastic discount factors are not necessarily positive Hansen and Jagannathan (1991) derive another bound that imposes positivity 430 Asset Pricing Thus, for an excess return, q = , so formula (13.14.6 ) becomes 28 −1 b = − [cov (z, z)] E (y) E (z) Then −1 var (z b) = E (y) E (z) cov (z, z) E (z) Therefore, the Hansen-Jagannathan bound becomes −1 σ (y) ≥ E (z) cov (z, z) E (z) E (y) (13.14.8) In the special case of a scalar excess return, (13.14.8 ) becomes σ (y) E (z) ≥ E (y) σ (z) (13.14.9) The left side, the ratio of the standard deviation of the discount factor to its mean, is called the market price of risk Thus, the bound (13.14.9 ) says that the market price of risk is at least E(z) The ratio E(z) thus determines a straightσ(z) σ(z) line frontier in the [E(y), σ(y)] plane above which the stochastic discount factor must reside For a set of returns, q = and equation (13.14.6 ) becomes b = [cov (x, x)]−1 [1 − E (y) E (x)] (13.14.10) The bound is computed by solving equation (13.14.10 ) and b cov (x, x) b ≤ σ (y) (13.14.11) In more detail, we compute the bound for various values of E(y) by using equation (13.14.10 ) to compute b , then using that b in expression (13.14.11 ) to compute the lower bound on σ(y) Cochrane and Hansen (1992) used data on two returns, the real return on a value-weighted NYSE stock return and the real return on U.S Treasury bills They used the excess return of stocks over Treasury bills to compute bound (13.14.8 ) and both returns to compute equation (13.14.10 ) The bound (13.14.10 ) is a parabola, while formula (13.14.8 ) is a straight line in the [E(y), σ(y)] plane 28 This formula follows from var(b z) = b cov(z, z)b Hansen-Jagannathan bounds 431 13.14.4 The Mehra-Prescott data In exercise 13.1, we ask you to calculate the Hansen-Jagannathan bounds for the annual U.S time series studied by Mehra and Prescott Figures 13.14.1 and 13.14.2 describe the basic data and the bounds that you should find 29 Figure 13.14.1 plots annual gross real returns on stocks and bills in the United States for 1889 to 1979, and Figure 13.14.2 plots the annual gross rate of consumption growth Notice the extensive variability around the mean returns of (1.01, 1.069) apparent in Figure 13.14.1 1.6 1.5 1.4 1.3 1.2 1.1 0.9 0.8 0.7 0.6 0.8 0.85 0.9 0.95 1.05 1.1 1.15 1.2 1.25 Figure 13.14.1: Scatter plot of gross real stock returns (y axis) against real Treasury bill return (x-axis), annual data 1889–1979 The circle denotes the means, (1.010, 1.069) Figure 13.14.3 plots the Hansen-Jagannathan bounds for these data, obtained by treating the sample second moments as population moments in the preceding formulas For β = 99 , we have also plotted the mean and standard deviation of the candidate stochastic discount factor βλ−γ , where λt is t the gross rate of consumption growth and γ is the coefficient of relative risk aversion Figure 13.14.3 plots the mean and standard deviation of candidate 29 These bounds were computed using the Matlab programs hjbnd1.m and hjbnd2.m 432 Asset Pricing 1.15 1.1 1.05 0.95 0.9 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 Figure 13.14.2: U.S annual consumption growth, 1889– 1979 discount factors for γ = (the square), γ = 7.5 (the circle), γ = 15 (the diamond), and γ = 22.5 (the triangle) Notice that it takes a high value of γ to bring the stochastic discount factor within the bounds for these data This is Hansen and Jagannathan’s statement of the equity premium puzzle 13.15 Factor models In the two previous sections we have seen the equity premium puzzle that follows upon imposing that the stochastic discount factor is taken as βλ−γ , where t λt is the gross growth rate of consumption between t and t + and γ is the coefficient of relative risk aversion In response to this puzzle, or empirical failure, researchers have resorted to “factor models.” These preserve the law of one price and often the no-arbitrage principle, but they abandon the link between the stochastic discount factor and the consumption process They posit a model-free process for the stochastic discount factor, and use the overidentifying restrictions from the household’s Euler equations from a set of N returns Rit+1 , i = 1, , N , to let the data tell what the factors are Factor models 433 1.8 1.6 1.4 1.2 σ(y) 0.8 0.6 0.4 0.2 0.86 0.88 0.9 0.92 0.94 E(y) 0.96 0.98 1.02 Figure 13.14.3: Hansen-Jagannathan bounds for excess return of stock over bills (dotted line) and the stock and bill returns (solid line), U.S annual data, 1889–1979 Thus, suppose that we have a time series of data on returns Ri,t+1 The Euler equations are Et Mt+1 Rit+1 = 1, (13.15.1) for some stochastic discount factor Mt+1 that is unobserved by the econometrician Posit the model k log (Mt+1 ) = α0 + αj fjt+1 (13.15.2) j=1 where the k factors fjt are governed by the stochastic processes m fjt+1 = βj0 + βjh fj,t+1−h + ajt+1 , (13.15.3) h=1 where ajt+1 is a Gaussian error process with specified covariance matrix This model keeps Mt+1 positive The factors fjt+1 may or may not be observed Whether they are observed can influence the econometric procedures that are feasible If we substitute equations (13.15.2 ) and (13.15.3 ) into equation (13.15.1 ) we obtain the N sets of moment restrictions k Et exp α0 + m βjh fj,t+1−h + ajt+1 Rit+1 = αj βj0 + j=1 h=1 (13.15.4) 434 Asset Pricing If current and lagged values of the factors fjt are observed, these conditions can be used to estimate the coefficients αj , βjh by the generalized method of moments If the factors are not observed, by making the further assumption that the logs of returns are jointly normally distributed and by exploiting the assumption that the errors ajt are Gaussian, analytic solutions for Ri,t+1 as a function of current and lagged values of the k factors can be attained, and these can be used to form a likelihood function 30 This structure is known as an affine factor model The term “affine” describes the function (13.15.3 ) (linear plus a constant) This kind of model has been used extensively to study the term structure of interest rates There the returns are taken to be a vector of one-period holding-period yields on bonds of different maturities 31 13.16 Heterogeneity and incomplete markets As Hansen and Jagannathan (1991) and the preceding analysis of the log-linear model both indicate, the equity premium reflects restrictions across returns and consumption imposed by Euler equations These restrictions not assume complete markets A complete markets assumption might enter indirectly, to justify using aggregate consumption growth to measure the intertemporal rate of substitution The equity premium puzzle is that data on asset returns and aggregate consumption say that the equity premium is much larger than is predicted by Euler equations for asset holdings with a plausible coefficient of relative risk aversion 30 Sometimes even if the factors are unobserved, it is possible to deduce good enough estimates of them to proceed as though they are observed Thus, in their empirical term structure model, Chen and Scott (1993) and Dai and Singleton (forthcoming) set the number of factors k equal to the number of yields studied Letting Rt be the k × vector of yields and ft the k × vector of factors, they can solve equation (13.15.1 ) for an expression of the form R = g0 + g1 ft from −1 which Chen and Scott could deduce ft = g1 (Rt − g0 ) to get observable factors See Gong and Remolona (1997) for a discrete-time affine term-structure model 31 See Piazzesi (2000) for an ambitious factor model of the term structure where some of the factors are interpreted in terms of a monetary policy authority’s rule for setting a short rate Heterogeneity and incomplete markets 435 γ Gregory Mankiw (1986) posited a pattern of systematically varying spreads across individual’s intertemporal rates of substitution that could magnify the theoretical equity premium Mankiw’s mechanism requires (a) incomplete markets, (b) a precautionary savings motive, in the sense of convex marginal utilities of consumption, and (c) a negative covariance between the cross-sectional variance of consumption and the aggregate level of consumption To magnify the quantitative importance of Mankiw’s mechanism, it helps if there are (d) highly persistent endowment processes We shall study incomplete markets and precautionary savings models in chapters 16 and 17 But it is pertinent to sketch Mankiw’s idea here Consider a heterogeneous consumer economy Let M (git ) be the stochastic discount fac−γ tor generated by consumer i ’s Euler equations, say, M (git ) = βgit , where β is a constant discount factor, git is consumer i ’s gross growth rate of consumption, and γ is the coefficient of relative risk aversion in a CRRA utility function Here M (git ) is consumer i ’s intertemporal rate of substitution between consumption at t − and consumption at t evaluated at the random growth rate ci,t /ci,t−1 = git With complete markets, M (git ) = M (gjt ) for all i, j This equality follows from the household’s first-order conditions with complete markets (see Rubinstein, 1974) However, with incomplete markets, the M (git )’s need not be equal across consumers Mankiw used this fact to magnify the theoretical value of the equity premium Mankiw assumed that consumers share the same function M , but that the gross rate of consumption growth varies across households and that the crosssection distribution of g across households varies across time 32 Thus, assume Prob(git ≤ G) = Ft (G) and define the first moment of the cross-sectional distribution at time t as µ1t = gFt (d g) Also define higher moments µjt of git about the mean by µjt = (g − µ1t )j Ft (d g) Mankiw considered the consequences of time variation in the cross-section distribution of personal stochastic discount factors M (git ) Mankiw assumed an incomplete market setting in which for each household i , the following Euler equations hold for a risk-free gross return Rf t from t − to t and an excess 32 For a setting in which the cross section of M ’s varies over time, see the it model of Krusell and Smith (1998) described in chapter 17 In these incomplete markets models, the cross section distribution of wealth at a given date is among the state variables for the economy 436 Asset Pricing return of stocks over a bond, Rxt : Ei,t−1 [Rf t M (git )] = Ei,t−1 [Rxt M (git )] = 0, where Ei,t−1 is an expectation operator conditioned on person i ’s information at date t−1 Taking unconditional expectations and applying the law of iterated expectations gives the following unconditional versions of these Euler equations: E [Rf t M (git )] = (13.16.1a) E [Rxt M (git )] = (13.16.1b) Equations (13.16.1 ) express the idea that in an incomplete markets setting, any individual household’s marginal rate of substitution M (git ) is a legitimate stochastic discount factor It follows that any linear combination of households’ M (git )’s, in particular, the cross section mean M (git )d F (git ), is also a legitimate stochastic discount factor Therefore, (13.16.1 ) implies that E Rf t M (git ) dFt (git ) = (13.16.2a) E Rxt M (git ) dFt (git ) = (13.16.2b) Following Mankiw (1986) and Cogley (1999), use the second-order Taylor series approximation M (git ) ≈ M (µ1t ) + M (µ1t ) (git − µ1t ) + M (µ1t ) (git − µ1t ) (13.16.3) This implies the approximation M (git ) dFt (git ) = M (µ1t ) + M (µ1t ) µ2t , which leads to the following approximation of (13.16.2 ): ERt M (µ1t ) + M (µ1t ) µ2t = ERxt M (µ1t ) + M (µ1t ) µ2t = (13.16.4a) (13.16.4b) Heterogeneity and incomplete markets 437 For the risk-free return Rf t , equation (13.16.4a) implies ERf t = E M (µ1t ) + M (µ1t ) µ2t For an excess return, (13.16.4b ) and the definition of a covariance imply E (Rxt ) = −cov Rxt , M (µ1t ) + M (µ1t ) µ2t E M (µ1t ) + M (µ1t ) µ2t (13.16.5) When M (µ1t )µ2t = , equation (13.16.5 ) collapses to a version of the standard formula for the equity premium in a representative agent model When M (µ1t ) > [that is, when marginal utility is convex and when the variance µ2t of the cross section of distribution of git ’s covaries inversely with the excess return], the expected excess return is higher Thus, variations in the crosssection heterogeneity of stochastic discount factors can potentially boost the equity premium under three conditions: (a) convexity of the marginal utility of consumption, which implies that M > ; (b) an inverse correlation between excess returns and the cross-section second moment of the cross-section distribution of git ; and (c) sufficient dispersion in the cross-section distribution of git to make the covariance large in absolute magnitude The third aspect is relevant because in many incomplete markets settings, households can achieve much risk sharing and intertemporal consumption smoothing by frequently trading a small number of assets (sometimes only one asset) See the Bewley models of chapter 17 In Bewley models, households each have an idiosyncratic endowment process that follows an identically distributed but household-specific Markov process Households use purchases of an asset to smooth endowment fluctuations Their ability to so depends on the rate of return of the asset and the persistence of their endowment shocks Broadly speaking, the more persistent are the endowment shocks, the more difficult it is to self-insure, and therefore the larger is the cross-section variation in M (git ) that emerges Thus, higher persistence in the endowment shock process enhances the mechanism described by Mankiw Constantinides and Duffie (1996) 33 reverse engineer a general equilibrium with incomplete markets that features Mankiw’s mechanism Their economy is 33 Also see Attanasio and Weber (1993) for important elements of the argument of this section 438 Asset Pricing arranged so that no trade occurs in equilibrium, and it generates the volatility of the cross-section distribution of consumption growth as well as the negative covariation between excess returns and the cross-section dispersion of consumption growth required to activate Mankiw’s mechanism An important feature of Constantinides and Duffie’s example is that each household’s endowment process is very persistent (it is a random walk) Storesletten, Telmer, and Yaron (1998) are pursuing ideas from Mankiw and Constantinides and Duffie by using evidence from the PSID to estimate the persistence of endowment shocks They use a different econometric specification than that of Heaton and Lucas (1996), who found limited persistence in endowments from the PSID data, limited enough to shut down Mankiw’s mechanism Cogley (1999) checked the contribution of the covariance term in equation (13.16.5 ) using data from the Consumer Expenditure Survey, and found what he interpreted as weak support for the idea The cross-section covariance found by Cogley has the correct sign but is not very large 13.17 Concluding remarks Chapter studied asset pricing within a complete markets setting and introduced some arbitrage pricing arguments This chapter has given more applications of arbitrage pricing arguments, for example, in deriving Modigliani-Miller and Ricardian irrelevance theorems We have gone beyond chapter in studying how, in the spirit of Hansen and Singleton (1983), consumer optimization alone puts restrictions on asset returns and consumption, without requiring complete markets or a fully spelled out general equilibrium model At various points in this chapter, we have alluded to incomplete markets models In chapters 17 and 19, we describe other ingredients of such models Exercises 439 Exercises Exercise 13.1 Hansen-Jagannathan bounds Consider the following annual data for annual gross returns on U.S stocks and U.S Treasury bills from 1890 to 1979 These are the data used by Mehra and Prescott The mean returns are µ = [ 1.07 1.02 ] and the covariance matrix of 0274 00104 returns is 00104 00308 a For data on the excess return of stocks over bonds, compute Hansen and Jagannathan’s bound on the stochastic discount factor y Plot the bound for E(y) on the interval [.9, 1.02] b Using data on both returns, compute and plot the bound for E(y) on the interval [.9, 1.02] Plot this bound on the same figure as you used in part a c On the textbook’s web page (ftp://zia.stanford.edu/pub/sargent/webdocs/matlab), there is a Matlab file epdata.m with Kydland and Prescott’s time series The series epdata(:,4) is the annual growth rate of aggregate consumption ct /ct−1 Assume that β = 99 and that mt = βu (ct )/u (ct−1 ), where u(·) is the CRRA utility function For the three values of γ = 0, 5, 10 , compute the standard deviation and mean of mt and plot them on the same figure as in part b What you infer from where the points lie? Exercise 13.2 The term structure and regime switching, donated by Rodolfo Manuelli Consider a pure exchange economy where the stochastic process for consumption is given by, ct+1 = ct exp [α0 − α1 st + εt+1 ] , where (i) α0 > , α1 > , and α0 − α1 > (ii) εt is a sequence of i.i.d random variables distributed N (µ, τ ) Note: Given this specification, it follows that E[eε ] = exp[µ + τ /2] (iii) st is a Markov process independent from εt that can take only two values, {0, 1} The transition probability matrix is completely summarized by 440 Asset Pricing Prob [st+1 = 1|st = 1] = π (1) , Prob [st+1 = 0|st = 0] = π (0) (iv) The information set at time t,Ωt , contains {ct−j , st−j , εt−j ; j ≥ 0} There is a large number of individuals with the following utility function ∞ β t u (ct ), U = E0 t=0 where u(c) = c(1−σ) /(1 − σ) Assume that σ > and < β < As usual, σ = corresponds to the log utility function a Compute the “short-term” (one-period) interest rate b Compute the “long-term” (two-period) interest rate measured in the same time units as the rate you computed in a (That is, take the appropriate square root.) c Note that the log of the rate of growth of consumption is given by log (ct+1 ) − log (ct ) = α0 − α1 st + εt+1 Thus, the conditional expectation of this growth rate is just α0 − α1 st + µ Note that when st = , growth is high and, when st = , growth is low Thus, loosely speaking, we can identify st = with the peak of the cycle (or good times) and st = with the trough of the cycle (or bad times) Assume µ > Go as far as you can describing the implications of this model for the cyclical behavior of the term structure of interest rates d Are short term rates pro- or countercyclical? e Are long rates pro- or countercyclical? If you cannot give a definite answer to this question, find conditions under which they are either pro- or countercyclical, and interpret your conditions in terms of the “permanence” (you get to define this) of the cycle Exercises 441 Exercise 13.3 Growth slowdowns and stock market crashes, donated by Rodolfo Manuelli 34 Consider a simple one-tree pure exchange economy The only source of consumption is the fruit that grows on the tree This fruit is called dividends by the tribe inhabiting this island The stochastic process for dividend dt is described as follows: If dt is not equal to dt−1 , then dt+1 = γdt with probability π , and dt+1 = dt with probability (1 − π) If in any pair of periods j and j + , dj = dj+1 , then for all t > j , dt = dj In words, the process – if not stopped – grows at a rate γ in every period However, once it stops growing for one period, it remains constant forever on Let d0 equal one Preferences over stochastic processes for consumption are given by ∞ β t u (ct ), U = E0 t=0 where u(c) = c βγ (1−σ) < (1−σ) /(1 − σ) Assume that σ > , < β < , γ > , and a Define a competitive equilibrium in which shares to this tree are traded b Display the equilibrium process for the price of shares in this tree pt as a function of the history of dividends Is the price process a Markov process in the sense that it depends just on the last period’s dividends? c Let T be the first time in which dT −1 = dT = γ (T −1) Is pT −1 > pT ? Show conditions under which this is true What is the economic intuition for this result? What does it say about stock market declines or crashes? d If this model is correct, what does it say about the behavior of the aggregate value of the stock market in economies that switched from high to low growth (e.g., Japan)? Exercise 13.4 Manuelli The term structure and consumption, donated by Rodolfo Consider an economy populated by a large number of identical households The (common) utility function is ∞ β t u (ct ) , t=0 34 See also Joseph Zeira (1999) 442 Asset Pricing where < β < , and u(x) = x1−θ) /(1 − θ), for some θ > (If θ = , the utility is logarithmic.) Each household owns one tree Thus, the number of households and trees coincide The amount of consumption that grows in a tree satisfies ct+1 = c∗ cϕ εt+1 , t where < ϕ < , and εt is a sequence of i.i.d log normal random variables with mean one, and variance σ Assume that, in addition to shares in trees, in this economy bonds of all maturities are traded a Define a competitive equilibrium ˜ b Go as far as you can calculating the term structure of interest rates, Rjt , for j = 1, 2, c Economist A argues that economic theory predicts that the variance of the log of short-term interest rates (say one-period) is always lower than the variance of long-term interest rates, because short rates are “riskier.” Do you agree? Justify your answer d Economist B claims that short-term interest rates, i.e., j = , are “more responsive” to the state of the economy, i.e., ct , than are long-term interest rates, i.e., j large Do you agree? Justify your answer e Economist C claims that the Fed should lower interest rates because whenever interest rates are low, consumption is high Do you agree? Justify your answer f Economist D claims that in economies in which output (consumption in our case) is very persistent (ϕ ≈ ), changes in output (consumption) not affect interest rates Do you agree? Justify your answer and, if possible, provide economic intuition for your argument ... (13. 4.7 ) asserts that relative to the ˜ twisted measure π , the interest-adjusted asset price is a martingale: using the ˜ twisted measure, the best prediction of the future interest-adjusted... used to price assets, we have so-called risk-neutral pricing Notice that in equation (13. 4.2 ) the adjustment for risk is absorbed into the twisted transition measure We can write equation (13. 4.7... pricing than we took in chapter Recall that in chapter we described two alternative complete markets models, one with once-and-for-all trading at time of date- and history-contingent claims, the