Asset Pricing and the Equity Premium Puzzle: A Review Essay

Transcription

1 Asset Pricing and the Equity Premium Puzzle: A Review Essay Wei Pierre Wang Queen s School of Business Queen s University Kingston, Ontario, K7L 3N6 First Draft: April I benefit from discussions with Wulin Suo and other PhD candidates in class. I thank Panos Michael for useful suggestions. I am responsible for all the errors. Please contact author via wwang@business.queensu.ca and visit his website at for most recently updated draft. This paper is prepared as the term paper for MGMT923-Topics in Continuous-time Finance (course director: Prof. W. Suo)

2 Abstract In their seminal 1985 article, Rajnish Mehra and Edward Prescott find that their simulated equity premium can not match the real observed equity premium of 6.12% when a Lucas (1978) type representative agent model is considered and a time- and state-separable utility function is used. Weil (1989) also find the socalled risk-free rate puzzle when a non-expected utility is specified. Meanwhile, several papers (e.g. Hansen and Singleton (1983), Ferson (1983), Hansen and Jagannathan (1991) etc.) reject the Euler equation with time-additive utility under a representative agent framework. That is, the traditional asset pricing model meets its new challenge not only to solve the equity premium puzzle but also to provide a more generalized framework to characterize consumers consumption behavior. As known, the main problem with the CRRA utility specification by M&P (1985) is that the relative risk aversion coefficient and the intertemporal rate of substitution coefficient can not be disentangled. Then, the subsequent researches, including rare-event declines in aggregate consumption, the recursive utility model by Epstein and Zin (1989) and Weil (1990), habit formation model by Constatinides (1990), Abel (1990) and Campbell and Cochrane (1999), idiosyncratic risk and incomplete markets considered by Mankiw (1986), Lucas (1994) and Telmer (1993), generalized heterogeneous consumers specification by Constantinides and Duffie (1996) and Brav et al. (2002), and the borrowing-constrained OLG model by Constantinides et al.(1999) etc, all tend to separate the two coefficients and make certain theoretical enrichment to the traditional asset pricing model in order to simulate a reasonable high equity premium and low risk-free interest rate. Some succeed and some are still in question. This paper intends to review some most influential and recent methods and empirical work on asset pricing, which in some way offers respective perspectives on the equity premium puzzle. It will also discuss some most recent issues and further studies are recommended.

3 1 Problem Arising Robert Lucas (1978) and Douglas Breeden (1979) examined the stochastic behaviour of equilibrium asset prices in a one-good pure exchange economy with identical consumers. Each consumer maximizes the same utility function. The single good in their economy is costly produced in a number of different productive units. An asset is defined as a claim to all or part of the output of one of these units. Productivity in each unit fluctuates stochastically through time, and so do equilibrium asset prices. The above described economy is so-called representative agent models of asset returns in which per capita consumption is perfectly correlated with the consumption stream of the typical investor. In this model, a security s risk can be measured using the covariance of its return with per capita consumption, from which the CCAPM is implied. From the maximization problem of a representative agent, with time- and state-additive preferences, we can derive the Euler equation. 1 Even some recent work, such as Hansen and Singleton (1982,1983) and Ferson (1983)etc, rejected the Euler equation restriction on asset returns by time- and state-separable preferences. But Euler equation is still considered to be the foundation for theoretical analysis of asset pricing and agents investment behaviour. In their seminal 1985 article, Rajnish Mehra and Edward Prescott (1985) (M&P (1985)) describe a particular empirical problem for the representative agent paradigm. They employ a variation of Lucas (1978) pure exchange economy where each consumer has the following maximization problem: max E 0 [ β t U(C t )], where 0 < β < 1 (1) C t t=0 subject to a certain flow budget constraint. The utility function is restricted to be of constant relative risk aversion and takes the form U(C, α) = C1 α 1 1 α First they assume there is one productive unit producing perishable consumption good and one equity share that is competitively traded. Production process is described by y t+1 = x t+1 y t and transition probability φ ij is defined as {x t+1 = 1 please see appendix A for detailed derivation (2) 1

4 λ j, x t = λ i }. Secondly, they assume all securities are traded ex-dividend. The price of the security can be determined by 2 : P t = E t [ j=1 β j U (C t+j ) U (C t ) d t+j] (3) After further assumptions made, the economy is characterized as a two-state symmetric economy such that λ 1 = 1 + µ + δ and λ 2 = 1 + µ δ with transition probabilities satisfying φ 11 = φ 22 and φ 12 = φ 21. The parameters were selected such that the average growth rate of per capita consumption, the standard deviation of the growth rate of per capita consumption and the first order serial correlation of this growth rate all matched the sample values for the US economy in the period between 1889 and Conditional on the specific utility functional they choose, parameter 1/α measures people s willingness to substitute consumption between successive time periods. That is, relative risk aversion is the reciprocal of elasticity of intertemporal substitution. 3 In order to choose an appropriate value for α, they consider studies implemented by Arrow (1971), Friend and Blume (1975), Kydland and Prescott (1982), and Tobin and Dolde (1971). Finally, parameter α is restricted to have a maximum value of ten. From their empirical analysis, the average real return on relatively risk-less short-term securities over period was 0.80 percent, and the average real return on the Standard and Poor s 500 composite stock index over this period was 6.98 percent per annum. This leads to an average equity premium of 6.18 percent. 4 By varying preference parameter α between zero and ten, and β between zero and one, they are unable to obtain matched sizable equity premium. The largest premium from simulation is only 0.35%. This is known as the so-called equity premium puzzle. 2 please see appendix B for derivation 3 please see appendix C for proof 4 They used the Kuznets-Kendrick-USNIA measure on non-durables and services for real per capita consumption; the annual average Standard and Poor s Composite Stock Price Index, the annual dividends on this index and the consumption price deflator on non-durables and services to find the equity s return; and the same consumption deflator and the yields on ninety-day Treasure Bills, from 1931 to 1978, Treasure Certificates from 1920 to 1930 and sixty- to ninety-day prime commercial paper from 1889 to 1920 to find risk-free asset s return. (from Rietz (1988)) 2

5 The artificial premium being so small implies that the households don t mind holding risky equity relative to risk-less bonds. This means that variations in C t due to changes in d t do not reduce utility that much. The larger this consumption smoothing motive the larger the price of the risk. Therefore, consumption smoothing motive implied by time separable preferences is relatively small. Time separable preferences do not give large enough degree of consumption smoothing motive in order to generate the kind of price for risk that we observe in the real world. 2 Asset Pricing Models and Possible Resolutions to the Puzzle 2.1 A Possible Third Catastrophic State Rietz s 1988 paper proposed an alternative way to explain the equity premium puzzle by considering a three-state economy with a possible third catastrophic state, other than two or four symmetric states in M&P (1985). In his model, he still considers an Arrow-Debrew economy. The basic settings for the model, including representative agent s objective function, budget constraint and how to calculate the return on equity and risk-free rate, are very much like what were used in M&P (1985). The only distinguishable point he makes is a low-possibility, depressionlike third state added to the original M&P (1985) model. By matching the first two moments and correlation coefficient of the growth rate of consumption, Rietz is able to estimate the equity premium with risk aversion parameter being 10 (the maximum value M&P used in their model) and different disaster probabilities and time preference parameter. Three examples with different parameter of crash magnitude can show that equity premium estimated from simulation are between 5 and 7 percent, and riskfree return is between 0 and 3 percent. Also, the results show that the risk aversion parameter required to explain the equity premium decreases as the probability of a crash increases. Therefore, Rietz concludes that his third-state specification can solve the puzzle. After Rietz s proposition of a possible solution, Mehra and Prescott (1988) criticize Rietz s solution and point out two fatal problems. One problem they 3

6 consider is that equating the real return on a nominal Treasury bill with that on a real bill is not reasonable if the unanticipated inflation is not very small. The other problem in Rietz s solution is that he did not choose an appropriate size for the risk aversion coefficient. Rietz only uses risk aversion parameter of 10 to estimate the equity premium but, as known, using one value of the curvature parameter to account for one feature of the data and another value for some other feature is not the scientific practice. Also, they criticize that the disaster scenario proposed by Rietz is not realistic. Therefore, they conclude that the standard theory still faces an unsolved practical puzzle. However, in a heterogenous agent framework, as will be discussed later, it is still possible for some extreme state described by Rietz to occur at individual level but it is not be observed in aggregate data. 2.2 Non-Expected Utility Model In Epstein and Zin s (1989) and (1991), and Weil s (1990) articles, they describe a generalization of the standard expected utility class. This is known as the nonexpected utility preferences. Then Epstein and Zin (1990) and Weil (1989) take one step further and try to use this generalized expected utility preferences to solve the premium puzzle Theoretical Background Development In their 1989 paper, Epstein and Zin consider a so-called recursive utility function which takes the following form: V (C 0, C 1, ) = W (C 0, V (C 1, C 2, ) (4) for some function W. Certainty equivalent functionals µ is defined as µ(δ x ) = X, X R +, i.e., if a gamble yields outcome X with certainty then X is the certainty equivalent of the gamble. Let W have the CES form, W (C, Z) = [C ρ + βz ρ ] 1/ρ, where 0 ρ < 1, and 0 < β < 1 (5) thus V is an intertemporal CES utility function with elasticity of substitution σ = (1 ρ) 1. In their theorem3.1, they prove that under specific conditions and when W takes the CES form there exists a solution to the recursive utility function. 4

7 Considering Kreps and Porteus s preferences 5, they define CE as µ t = (E t [U α t+1]) 1/α. After CES utility function form is utilized, the non-expected utility function takes the following form: U t = {C ρ t + β[e t (U α t+1) ρ/α ]} 1/ρ (6) where 1 α measures the relative risk aversion and (1 ρ) 1 measures the elasticity of intertemporal substitution. As a special case, if α is equal to ρ and there is no uncertainty in the model the above utility functional form can be reduced to the one used by M&P (1985). Also, they specify that, by assumption, mean value functionals µ exhibit risk aversion in the sense of second degree dominance, and then µ ( ) E( ). Therefore, the least risk averse intertemporal utility function is the one for which µ( ) = E( ). Also, they prove that when W = W, V is more risk averse than V if and only if µ ( ) µ( ), which is equivalent to α α. Epstein and Zin (1991) continue their work (1989) to implement empirical investigation into the performance of non-expected utility model. Their data set includes monthly data from the United States differing in the measurement of consumption, asset returns and time periods. Specifically, consumption goods that are taken for analysis are nondurable goods and services, and the sample period extends from April 1959 to December For the empirical estimation they implement Generalized Methods of Moments (GMM) estimator. Their results show that the GMM estimates for different subsets of data generally lead to rejections of the expected utility hypothesis. The performance of the non-expected utility model is sensitive to the choice of consumption measure and instrumental variables. They find that, in general, the elasticity of intertemporal substitution is less than 1, relative risk aversion is close to one, and consumers prefer the late resolution of uncertainty Resolution of the Puzzle Epstein and Zin (1990) assume that intertemporal utility is recursive as in Epstein and Zin (1989). Thus the notions of risk aversion and intertemporal substitutability are partially disentangled. They also assume that risk preferences exhibit 5 Interested readers please refer to D, Kreps and E, Porteus, 1979, Dynamic choice theory and dynamic programming, Econometrica 36, , and Temporal Von Neumann-Morgenstern and induced preferences, Journal of Economic Theory 20,

8 first-order risk aversion. 6 Since given the smoothness of consumption data the standard deviation of the consumption growth rate is larger than its variance, firstorder risk aversion can possibly generate a sizable equity premium. In their model, the aggregator function W takes CES form as equation (2). Certainty equivalent takes the following form: and µ(ũt+1) = {E t, [Ũ α t+1]} 1/α for 0 α < 1 (7) µ(ũt+1) = exp{e t [log(ũt+1)]} for α = 0 (8) If a further restriction on α and ρ such that α = ρ is imposed it leads to the standard interemporal expected utility function as U 0 = {E 0 β t C ρ t } 1/ρ (9) t=0 In their model, Epstein and Zin first specify a growth process with growth rates of endowment following a first-order Markov process. After adding a risk-free asset to the portfolio choice set of the individual, they show that the individual s portfolio choice is determined by solving max µ{( X t+1 / M t+1 ) ρ 1 ρ (a Mt+1 + (1 a)r f ) I t } (10) 0 a 1 where X t+1 denotes the future growth rate, and M t+1 is the return to equity, a denotes the proportion of savings invested in equity. They first consider a special case with i.i.d consumption growth rate to estimate the equity premium implied by the recursive intertemporal utility function. After Yaari risk preference function is adopted an individual s portfolio choice now is determined by solving a linear problem, and finally the risk-free rate is determined by r f = β 1 µ 1 ρ ( X) (11) Their model resolves the risk-free rate puzzle proposed by Weil (1989) with different value of ρ and first-order risk aversion coefficient assigned. In contrast to 6 In simple terms, if the risk premium is proportional to the standard deviation of a gamble then we think of this as first-order risk aversion. If the risk premium for small risks about certainty is proportional to the variance of the gamble, then we think of this case as second-order risk aversion. 6

9 the results developed in Weil (1989), their results show that a smaller elasticity of intertemporal substitution generates a smaller risk-free rate, however, this reduction in the level of the risk-free rate does not imply a reduction in the equity premium. Secondly, they consider a case of autocorrelated consumption growth. Their simulation results show that the largest average equity premium generated is 3.5% and the smallest is 0.0%. The typical premium generated is between 1% and 2%. This result confirms that i.i.d case is fairly representative. The pattern of their simulation tables indicates that the average premium on the equity gets larger (i) as the agent becomes more risk averse (ii) as the endowment growth process gets more negatively autocorrelated (iii) as the distribution for endowment growth becomes more skewed (iv) as the variance of the endowment growth becomes larger (v) as substitutability decreases(increases) when endowment growth is negatively(positively) autocorrelated. To sum up, in contrast to the historical average risk premium of 6.2% and the largest premium obtained by M&P (1985) of 0.35%, Epstein and Zin s results show a low risk-free rate and an average equity premium of roughly 2%. In their words, their recursive utility specification only partially resolves the puzzle. Weil (1989) s setup is very much like that in Epstein and Zin (1990). He intends to extend the non-expected utility framework to provide implications for solving the equity premium puzzle. Non-expected utility specification allows for an independent parameterization of attitudes toward risk and attitudes toward intertemporal substitution. That is, it allows for disentangling of elasticity of intertemporal substitution and the coefficient of relative risk aversion. In M&P (1985) they show that the Arrow-Debreu, representative agent framework with time-additive, CES, expected utility preferences can not account, except for unplausibly high value of the coefficient of relative risk aversion (of order 40 or 50), for both low average level of risk-free rate and the high equity premium. The model they consider has the property that elasticity of intertemporal elasticity of substitution is the inverse of the coefficient of relative risk aversion. However, the risk-free rate is mainly controlled by the magnitude of the elasticity of intertemporal substitution, while the risk premium is a reflection of the coefficient of relative risk aversion. Therefore, for generating a reasonable high equity premium by increasing relative risk aversion coefficient they can not avoid the problem of an extremely high risk-free rate. Thus, Weil proposed that, intuitively, an independence parameterization should be able to provide additional freedom to replicate both low level of the risk-free rate and relatively high equity premium. However, unfortunately, both cases of an 7

10 i.i.d dividend growth process and a non-i.i.d growth process he considers demonstrate that separating risk aversion from intertemporal substitution can not provide a solution to the equity premium puzzle and instead it highlights the existence of a risk-free rate puzzle (Why the risk-free rate is so low?) besides the equity premium puzzle. 2.3 Habit Formation Theoretical Foundation for Internal Habit Formation Constantinides (1990) relaxes the time separability of von Neumann-Morgenstern preferences and allows for adjacent complementarity in consumption. This property is known as habit persistence. With habit persistence Constantinides is able to show that the relative risk aversion coefficient approximately equals to the parameter α in equation (2) and the elasticity of consumption can be substantially lower than the inverse of the relative risk aversion coefficient. In his model, Constantinides considers a continuous-time economy in which the forcing process is a diffusion, and the economy also allows for production rather than a pure exchange economy. There are two technologies, one is risk-less and the other is risky. The rate of return of risk-less technology follows process rdt and that of risky technology follows a diffusion process µdt + σdw(t). A representative consumer invests a fraction α t, 0 α t 1, of the capital W t in the risky technology and the remaining fraction in the risk-less technology. The maximization problem of utility for a representative consumer becomes where max c t,α t E 0 0 x t e at x 0 + b e ρt γ 1 [c t x t ] γ dt (12) t 0 e a(s t) c s ds (13) x t is an exponentially weighted sum of past consumption. Thus, the above utility function is not time separable but exhibits habit persistence. As a special case, if x 0 = b = 0, then the function becomes time-separable with constant relative risk aversion 1 γ. After certain admissible policies on consumption and investment are restricted, the derived utility of capital can be written as V (W 0, x 0 ) = max {admissible α s,c s,s 0} E e ρs γ 1 [c s x s ] γ ds (14)

11 Then in his theorem 1 and 2 Constantinides solves an optimal and unique admissible consumption and investment policy, and shows that the process x t /[c t x t ] and x t /c t have stationary probability distribution with some specific density function. In the next step, he shows that for the time non-separable model the product of the relative risk aversion and intertemporal elasticity of substitution is substantially below one, but this does not necessarily requires the stronger assumption of persistence of habit. For the time-separable model this product is equal to one. In order to resolve the equity premium puzzle, Constantinides considers the case that the ratio of proportion of firm s investment into risky technology to the leverage ratio is equal to one. The mean and variance of the consumption growth rate are independent of this ratio. He demonstrates that habit persistence improves on time-separable preferences. Then he shows that the mean equity premium is driven by the relative risk aversion coefficient while the variance of the equity premium is driven by the elasticity of consumption. He proceeds to show that habit persistence can generate the sample mean and variance of consumption growth rate with a low relative risk aversion coefficient. Thus, from his simulation results, the equity premium puzzle is resolved in the sense that the model generates the mean and variance of the consumption growth rate with the mean relative risk aversion coefficient as low as Also, he found that the product of the elasticity of substitution and the relative risk aversion coefficient is about 0.25 when the puzzle is resolved with a low relative risk aversion coefficient Empirical Investigation In the mean time, Ferson and Constantinides (1991) employ an empirical investigation into the Euler equation when both habit persistence in preferences and the durability in consumption expenditures are taken into consideration. The discrete-time model they set up with habit persistence is a transform of Constantinides (1990) s habit formation model under continuous-time framework. A time-nonseparable von Neumann-Morgenstern utility function is modelled as (1 α) 1 β t (Ct F t=0 h a s c F t s) 1 α (15) s=1 where α > 0,a s 0, s=1 a s = 1, h 0, and c F t = τ=0 δ τ c t τ with δ τ 0 and τ=0 δ τ = 1 is the total flow of consumption services at time t. They show that parameter α approximately equals the relative risk aversion coefficient but 9

12 it differs the inverse of the elasticity of intertemporal consumption. The product of the relative risk aversion and elasticity of consumption is equal to one only in the absence of habit persistence, which has been proved in Constantinides (1990). Now, rewrite equation (15) as (1 α) 1 t=0 β t C 1 α t (16) where C t τ=0 δ τ c t τ h s=1 τ=0 a s δ τ c t τ s = δ 0 τ=0 b τ c t τ and, b 0 = 1, b τ = (δ τ h ı=0 a i δ τ i )/δ 0, τ 1. At last, they obtain the Euler equation for habit persistence as E t [ β τ (C t+τ /C t ) α (b τ 1 R t+1 b τ ) 1] = 0 (17) τ=0 where C t is defined as before. Then, as a special case, in the absence of habit persistence (h=0) and durability (δ τ = 0, τ 1), we obtain b τ = 0, τ 1, and the Euler equation becomes time- and state-separable as: E t [β(c t+1 /c t ) α R t+1 1] = 0 (18) In their empirical investigation, they study the returns on Treasury bills, bonds and value-weighted portfolio of common stocks traded on the NYSE. They consider not only monthly data (which was used mostly in previous studies by Hansen and Singleton (1982) etc.) covering but also quarterly data covering and annual data covering The goal of their study is to test the validity of the Euler equation (18) and estimate the model parameters using generalized method of moments. They demonstrate that the error term from Euler equation follows an MA(0) process for time-separable model and it follows an MA(1) process in the one-lag model. They examine null hypothesis H 0 : b s = 0, s 1, with an MA(0) error term against H a : b s = 0, s 2, with an MA(1) error term. Therefore, their estimated model follows equation (16) but C t = c t + b 1 c t 1 where b 1 is the parameter representing habit persistence (b 1 < 0) and durability (b 1 > 0). They pursue to test whether the null of a time-separable model, i.e. b 1 = 0, can be rejected. If the hypothesis is rejected, they proceed to estimate the sign of parameter b 1 to determine whether habit persistence dominating durability of consumption expenditure is suggested. They make use of eight financial instrumental variables to test and estimate the model. For consumption expenditures, they consider per capita consumption 10

13 expenditures on consumer non-durable goods. Their empirical result from test via monthly data shows that time-separable utility model is strongly rejected by goodness-of-fit test. They find that whether they use the first or the second lags of financial variables, they obtain negative b 1 coefficients, which suggests that habit persistence dominates durability. They also find that when they increase the order of the moving-average process assumed for the errors, that is, in the time-separable model they use an MA(1) assumption and in the one-lag model we use an MA(2) assumption, the results they obtain are similar to those before. In the next step, they switch their test and estimation from using monthly data to quarterly data and annual data. The hypothesis of a separable-utility is strongly rejected and their results provide further evidence that habit persistence dominates durability. In order to test the robustness of their empirical results they retest their hypothesis by using seasonal adjustment via dummy variables and using consumer durable goods expenditures. The goodness-of-fit tests indicate an improved fit with the negative b 1 coefficient. They conclude that their earlier finding that habit persistence dominates durability is also valid for durable goods. Therefore, Ferson and Constatinides (1991) find habit persistence from empirical data and this result confirms that habit formation can be a possible resolution of the equity premium puzzle Relative Consumption Mentioned in Abel (1990), Gali (1994) and Kocherlakota (1996) etc, an individual s utility can be a function not only of his own consumption but also of aggregate levels of consumption which can be measured by per capita consumption. Thus, an individual s investment decisions will be affected not only by his attitudes toward his own consumption risk but also by his attitudes toward variability in societal consumption. In this way, a representative agent s objective function can be adjusted from equation (1) and (2) to max c E t [β s c 1 α t+s Ct+sC γ t+s 1 λ 1]/(1 α) (19) s=0 where c t+s denotes the individual s level of consumption in period t + s, C t+s denotes the level of per capita consumption in period t + s, γ and λ are the coefficients that measure the sensitivity of marginal utility of own consumption to fluctuation to current and past level of per capita consumption. Therefore, an 11

14 individual derives utility from how well she is doing today relative to how well the average person is doing today and last period. In contrast to Constantinides (1990), per capita consumption can be thus regarded as the external habit formation for each individual. With different coefficients of γ and λ assigned, individuals can be generally divided into two groups. If γ and λ are both negative for some individuals, then we think they are jealous type. That is, they are unhappy when others are doing very well. If γ and λ are both positive, then these individuals are of patriotic type, which implies that they are happy when per capita consumption for the whole nation is high. Specifically, in steady state, individual s consumption is equal to per capita consumption. The representative agent regards per capita consumption as exogenous when choosing how much to invest in stocks and bonds. After writing down the first order conditions, for any specification of the discount rate and relative risk aversion coefficient we are able to find settings for γ and λ such that equity premium puzzle are resolved. However, in this type of setting, risk free rate puzzle still stays in question Recent Empirical Work Campbell and Cochrane (1999) empirically show that with habit formation they are able to calibrate their model to fit the unconditional equity premium and riskfree interest rate. Their model predicts many puzzles that face the standard power utility consumption-based model, including the equity premium and risk-free rate puzzle and the low unconditional correlation between consumption growth and stock returns. Their model generates long-horizon predictability of excess stock and bond returns from the dividend/price ratio and mean reversion in returns. It also generates high stock and return volatility despite smooth and unpredictable dividend streams, and persistent movements in return volatility. In their model, they specify that habit formation is external, that is, an individual s habit level depends on the history of aggregate consumption rather than on the individual s own past consumption. This specification is similar as in Abel (1990) but different from that in Constantinides (1990). Secondly, they specify that habit moves slowly in response to consumption. This is in contrast to empirical specification in Ferson and Constantinides (1991). Also they specify that habit adapts nonlinearity to the history of consumption. The nonlinearity specification is also in contrast to that in Ferson and Constantinides (1991) and it keeps habit 12

15 always below consumption and marginal utility always finite and positive even in an endowment economy. Representative agent, in Campbell and Cochrane s specification, maximizes the utility function E t { β t (C t X t ) 1 α 1 } (20) 1 α t=0 where β is the subjective discount factor, X t is the level of habit. Surplus consumption ratio is defined as S t = (C t X t )/C t and S t = 0 corresponds to an extremely bad state in which consumption is equal to habit. The local curvature of the utility function is defined as η t Ctucc(Ct,Xt) u c(c t,x t) = α/s t. Then from the definition of relative risk aversion and the envelop condition u c = V W we have the following: RRA W V W W = ln V W ( ) V W ln W = ln u c( ) ln C t ln C t ln C t = η t (21) ln W t ln W t that is, relative risk aversion coefficient can be written as the product of utility curvature and the elasticity of consumption to individual wealth with aggregate held constant. If date t consumption moves proportionally to an individual wealth shock, risk aversion is the same as utility curvature. In their specification, consumption rises more than proportionally to an increase in idiosyncratic wealth, so risk aversion is greater than curvature. With external habit specification, St a = (Ct a X t )/Ct a, where C a denotes per capita consumption and in equilibrium C t = Ct a and S t = St a. They model both consumption growth and dividend growth as i.i.d. lognormal processed with some level of correlation. The log surplus consumption ratio evolves as a heteroskedastic AR(1) process, s a t+1 ln S a t+1 = (1 φ) s + φs a t + λ(s a t )(c a t+1 c a t g) (22) where s denote the steady state value, φ is parameter, g is the mean growth rate of consumption, and λ(s a t ) is called the sensitivity function. The above equation implies that habit itself adjusts slowly and geometrically to consumption c a t with coefficient φ. They choose the sensitivity function to satisfy three conditions: (1) the risk-free interest rate is constant; (2) habit is predetermined at the steady state; (3) habit moves nonnegatively with consumption everywhere. From their specification, the intertemporal marginal rate of substitution is M t+1 β u c(c t+1, X t+1 ) u c (C t, X t ) = β( S t+1 S t C t+1 C t ) α (23) 13

16 Thus, from Euler equation the real risk-free rate can be determined by R f t = 1/E t (M t+1 ). After some conditions are specified for the sensitivity function, log risk-free rate can be written as r f t = ln(β) + αg ᾱ s σ 2 2 = ln(β) + αg α (1 φ) (24) 2 which is a constant. But different functional forms for the sensitivity function can generate risk-less interest rate that vary significantly with the state variable s. Then a more general specification is to choose λ(s t ) so that the interest rate is linear function of the state s t rather a constant, for example, r f t = r f 0 B(s t s). In the next step, they compare their model with two data sets: (1) postwar ( ) value-weighted NYSE stock index returns, 3-month Treasury bill rate, and per capita nondurables and services consumption and (2) a century-long annual data set of S&P 500 stock, commercial paper returns ( ) and per capita consumption ( ). Parameters g, σ(standard deviation of consumption growth), r f, φ, α, σ w (standard deviation of dividend growth), ρ(correlation between consumption growth and dividend growth), β and s are chosen from estimates of historical data. From their simulation results, first they find a stationary distribution of the surplus consumption ratio, and the same value of price/dividend ratio of the dividend claim and the price/dividend ratio of the consumption claim. But the correlation between consumption growth and dividend growth is as low as 0.2. They find that conditional variance in stock returns are highly autocorrelated, which implies the ARCH effect. They also find there is little contemporaneous correlation between consumption growth and stock returns. Their simulated statistics show that the model can match the mean and the standard deviation of excess stock returns with a constant low interest rate and a discount factor β = 0.89 less than one by any choice of other parameters. Pointed by some previous papers, for a time-separable utility function, in order to simulate a high equity premium and low risk-free rate we need an implausible high relative risk aversion coefficient and sometimes we need discount factor to be greater than 1. But Campbell and Cochrane s model can first avoid the risk-free rate puzzle. They only need α to be as low as 2 and β = 0.89 to simulate observed real interest rate. Their model predicts a much less sensitive relationship across countries or over time between mean consumption growth and interest rate. Consumption is a pure random walk at any time horizon in their model, so the standard deviation of consumption growth grows roughly with the square root of the horizon. As defined before, the k-period stochastic discount factor is M t,t+k = β k ( S t+k S t C t+k C t ) α. At long horizon, the standard deviation of the discount 14

17 factor is dominated by the standard deviation of the consumption growth term since the state variable S t is stationary. Then we return to the equity premium puzzle. However, we should note that even S t is stationary St α is not stationary. In this way, we are able to simulate a high enough premium in the long run. As a conclusion, any model that wishes to explain the equity premium at long and short runs by means of an additional, stationary state variable must find some similar transformation so that the equity premium stays high at long horizons. Finally, they consider the issue proposed by Constantinides and Duffie (1996) who show how to disaggregate any representative agent marginal utility process to individual agents with power utility and low risk aversion in incomplete markets by allowing the cross-sectional variance of idiosyncratic income to vary with posited marginal utility process. Campbell and Cochrane allow many different reference groups with different levels of wealth by letting each agent s habit be determined by the average consumption of his reference group rather than by average consumption in the economy as a whole. Then each agent still has the identical power utility function. With identical surplus consumption ratios and consumption growth rates, marginal utility growth is unchanged despite the heterogeneity in group consumption levels. They also allow some individual heterogeneity and they can show that all individuals agree on asset prices and have no incentive to trade away from their endowments. They also consider the case with internal habit formation. With internal habits, consumption today raises tomorrow habits, lowering the overall marginal utility today. But asset prices are determined by ratios of marginal utilities. If internal habits simply lower marginal utilities at all dates by the same proportion then a switch from external habit to internal habit has no effect on allocations and asset prices. But we should note that that this internal-habit version model generates risk-free rates that are higher and vary with the state variable S, so the excess return is less sensitive and predictable. Daniel and Marshall (1997) consider both time-separable utility model and the habit model of Constantinides (1990). They assume the log-consumption expenditure and log-prices of equity follow random walk with drift processes, and their error terms are correlated. Theoretical risk-free rate is defined as the inverse of the expected nominal intertemporal marginal rate of substitution while the theoretical equity premium is defined as the negative covariance between the return on equity and the nominal intertemporal marginal rate of substitution. They consider that the error vector from the vector linear regression for the joint nominal equity return and intertemporal marginal rate of substitution process follow a vector ARCH model. Their goal is to test the hypothesis that theoretical equity premium and the 15

18 risk-free interest rate implied by the particular model converge to the estimated values at both short and long horizon. Their evidence suggests that the time-separable utility fails at all horizons with measures of consumption expenditures as either nondurables good plus services or nondurables alone. Constantinides model does not perform well at the quarterly horizon but displays a substantial improvement in fit when the horizon is lengthened to two years. In particular, when consumption is measured as nondurables plus services, the Constantinides model with curvature parameter set to 11 comes quite close to matching the moments of the two-year equity premium. Furthermore, these models perform surprisingly well to match the moments of the risk-free rate at long horizons between two and three years. Also, unlike the time-separable utility model, the mean of the theoretical risk-free rate in the Constantinides model is decreasing in the level of risk aversion. Constantinides model can better match the observed risk-free rate at longer time horizons. Therefore, his results show that habit formation model of Constantinides can be regarded as a rather robust explanation for the equity premium puzzle. Chapman (2001) appraised the importance of applying habit formation by Constantinides (1990) to solve the equity premium puzzle. But he also weakens the strength of habit formation for explaining the puzzle over a long horizon. He found that consumption growth actually behaves very different before 1948 and after 1948 such that the explanatory power of habit formation model is driven by the pre-1948 data. He demonstrates that intrinsic habit can not rationalize the unconditional moments of the discrete time consumption and real asset returns for value function curvature levels below when he uses data from 1949 to 2000 constructed in the same manner as in M&P (1985). 2.4 Idiosyncratic Risk, Incomplete Markets and Heterogeneous Consumers When asset markets are complete, individuals can write contract or trade against any contingency. That is, they can diversify any idiosyncratic risks to their consumption and income. As a result, any individual s consumption stream is perfectly correlated with per capita consumption stream since everyone will behave in the same way. However, when the number of assets can not span the uncertainty space, i.e., the issuance market is incomplete, individual consumption growth will feature risk not present in per capita consumption growth and so in- 16

19 dividual consumption growth will be more viable than per capita consumption. Many economists hope that this higher variable individual consumption growth might help to explain the equity premium puzzle, including (1) Mankiw (1986) who assumes heterogenous idiosyncratic risk to individual consumption ex post to explain the high equity premium; (2) Lucas (1994) and Telmer (1993), both of who show that dynamic incomplete market models tend to produce similar results proposed by M& P (1985) and equity premium puzzle is rather robust; (3) Constantinides and Duffie (1996) who show that with permanent instead of transitory shocks to individual labor income and absence of labor income insurance markets they are able to generate a risk-free rate lower than that in complete markets, and who also suggest negative correlation between risk premium and variance of cross-sectional distribution of individual and aggregate consumption growth as a potential source of the equity premium. This section will make a detailed examination to the above studies Idiosyncratic Risk to Aggregate Consumption Instead of assuming all the consumers are homogeneous, Mankiw (1986) assumes that there are an infinite number of individuals that are identical ex ante, however, that their consumption is not the same ex post. He also assumes that aggregate shocks to consumption are not dispersed equally across all consumers and these shocks only affect some consumers ex post. By assuming the absence of certain contingent-claim markets, i.e., markets are incomplete, he shows that representative agent models are ineffective as approximations to a complex economy with ex post heterogeneous consumers. Under the situation of the absence of complete markets individual consumption is more variable than per capita consumption. Therefore, using aggregate consumption data to estimate the equity premium becomes inappropriate. The relative risk aversion coefficient implied by the real equity premium data should be more than ten. In his model, Mankiw assumes a discrete state space with good state and bad state. When good state takes place per capita consumption in the economy takes value of µ. Otherwise, it takes value (1 φ)µ. Each state occurs with probability 1/2. A portfolio in the economy pays -1 in the bad state and pays 1+π in the good state. Each representative agent ex ante maximizes EU(C). From the first order condition equity premium can be approximated as π = [µu (µ)/u (µ)]φ = Aφ (25) where A is the coefficient of relative risk aversion and A = π/φ. 17

20 Then Mankiw assumes that in the bad state the fall in aggregate consumption φµ concentrated among a fraction λ of the population. In fact, λ measures the concentration of the aggregate shock. Under this assumption, the equity premium becomes: π = λ{ U ((1 φ/λ)µ) U (µ) } (26) U (µ) The main results follow from proposition 1 to 3 in his article. Proposition 1 states that if the utility function is quadratic, then the premium is independent of the concentration of the aggregate shock. Proposition 2 states that if the third derivative of the utility function is positive, then an increase in the concentration of the aggregate shock increases the premium. That is, one can not determine the size of equity premium from aggregate data alone. Proposition 3 shows no upper bound condition on the equity premium from the degree of risk aversion and the aggregate shock only. Therefore, Mankiw concludes that unless aggregate shocks to income affect all investors equally ex post, relative asset returns in general depend on the distribution of aggregate shocks among the population. It is not possible to infer investors risk aversion from the aggregate data alone. This opinion may possibly explain the puzzle M&P have when the equity premium is estimated when per capita consumption data alone was used ex post Idiosyncratic Risk to Income, Short Sales constraints and Incomplete Markets Lucas (1994) proposes a more general case than Mankiw (1986) by constructing an asset pricing model with undiversiable income shocks to income, and borrowing and short sale constraints at an infinite time horizon. The idiosyncratic shocks are constructed such that the security payoffs do not span the uncertainty space, i.e., agents can not ex ante insure against future idiosyncratic shocks to their income and thus the market is incomplete. In her model, there are I 7 types of consumers and an infinite number of identical agents for each type. Aggregate output Y t in each period consists of a stochastic dividend and the sum of individuals stochastic income. Consumers are heterogeneous in the sense that they receive different shocks to their income. She considers not only the same CRRA utility 7 I is set to two and both agents have the same preferences for the simulation analysis. 18

21 function as in M & P (1985) and budget constraint as those in appendix B for each type of consumer but also short sale and borrowing constraints as: s it K s t, and, b it K b t (27) where the short sale constraint is assumed to be independent of time and current state, and the borrowing constraint is assumed to be linear in aggregate dividend. She makes the similar specification for growth of output that takes on two possible values and a 2 2 symmetric transition probabilities matrix as that in M & P (1985) in order to provide a comparable result for showing that equity premium puzzle is a rather robust phenomenon even when borrowing and short sale constraints are considered. She assumes that the fraction of aggregate output distributed as dividends, δ, is assumed to be constant. Thus labor income becomes (1 δ)y t. At any time t agent one receives labor income as x 1t = (1 δ)y t η 1t and similarly agent two receives x 2t = (1 δ)y t η 2t, where η 1t + η 2t = 1. Then she considers two distinct cases for joint distribution of idiosyncratic and aggregate shocks. Case 1 corresponds to a dynamic version of the income process considered by Mankiw (1986). In this case, η 1 = η 2 = 0.5 when aggregate output is high, and η 1 is drawn from i.i.d. binomial distribution when aggregate output is low. That is, in a recession like state, idiosyncratic shocks are positively correlated with aggregate shocks. Case 2 follows Weil (1992) which shows that the impact of idiosyncratic shocks on the equity premium depends on the third derivative of the utility function, and in a two-period model with CRRA utility idiosyncratic risks tend to increase the proportion of the equity premium (ratio) but has ambiguous effect on the level of the premium (difference). In this case, the idiosyncratic shocks are assumed to be independent of aggregate shocks. That is, the realization of the labor income η is assumed to be i.i.d. binomial and independent of aggregate output Y t. For the simulation analysis, the annual discount factor is fixed at The parameters that are varied to simulate different statistics are focused on the magnitude and distribution of idiosyncratic shocks η, the short sale constraint and borrowing constraint, and the relative risk aversion coefficient. First she considers the case that trading in both stock and bond markets is prohibited. Then in case 1 as mentioned above when η = 0.45 and α = 1.5 the equity premium and the risk-free rate simulated are respectively 5.6% and 2%. Case 2 even creates as large as 11% equity premium and negative risk-free rate. Intuitively we can expect that lower values of η can even produce lower risk-free rate and a higher equity premium. 19

22 The results obtained from four cases, which are correlated and uncorrelated idiosyncratic and aggregate shocks with trading restricted in stock market only, and correlated and uncorrelated idiosyncratic and aggregate shocks with trading restricted in bond market only, are amazingly similar. This is because consumers can effectively smooth their income shocks by trading. The highest equity premium simulated in all four case is about 0.8% which is much smaller than the real premium from historical data. Even when more severe borrowing constraints are imposed, for similar other parameters, the equity premium increases to only about 1.5%. With trading, agents with a bad idiosyncratic shock can effectively self-insure by selling financial assets to agents with a good luck. Therefore, idiosyncratic shocks to income are largely irrelevant to asset prices with trading even when the borrowing constraints are severe. Then, solving the equity premium puzzle requires more than closing forward market for labor income. she concludes that equity premium puzzle is still rather robust. Telmer (1993) considers a very similar incomplete market model as proposed by Lucas (1994). In his model, there are two heterogenous consumers with different consumption stream in the economy. The stochastic growth rate of aggregate income follows a first-order Markov process and is restricted to take on two values. Both agents receive half of the aggregate income when aggregate income growth is high and different proportions of it otherwise. The only tradable asset he considers is the risk-less bond. Both static and dynamic borrowing constraints are also considered in his model. For estimating the first two moments and first order autocorrelation coefficient for the consumption growth, he makes use of monthly data of the US economy over time period The discount factor is set to and aggregate income growth takes two fixed values in good and bad states for his numerical analysis. The effects of incomplete markets are characterized by varying the heterogeneity parameter from 0.5 (representative agent) to 0.65 (extreme heterogeneity). The effects of increased risk aversion are explored by varying the curvature parameter from 2.0 to 4.0. Numerical simulation is implemented in a consequence of economies including complete markets, incomplete markets without intertemporal trading, incomplete markets with trade in bonds only and unconstrained borrowing, and incomplete markets with trade in bonds only and borrowing constraints. His results show that in contrast to the representative agent model, the incomplete markets model tends to generate increased pricing kernel (or intertemporal marginal rate of substitution) variability while decreasing the risk-free rate. This effect will be even more observed as borrowing constraints are introduced 20