Asset-pricing implications of biologically based non-expected utility

Review of Economic Dynamics 16 (2013) 497–510

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Review of Economic Dynamics www.elsevier.com/locate/red

Asset-pricing implications of biologically based non-expected utility Emil P. Iantchev 1 Department of Economics, Syracuse University, Syracuse, NY 13244, United States

a r t i c l e

i n f o

Article history: Received 22 October 2011 Revised 6 August 2012 Available online 17 August 2012 JEL classification: D53 D81 E44 G12 Keywords: Recursive utility Limited commitment Equity return predictability Cross-sectional distribution of equity returns

a b s t r a c t Results in population ecology suggest that evolutionary successful species should have an adaptive (reference-based) S-shaped utility function that is intrinsically more sensitive to aggregate than uninsured idiosyncratic shocks—the former cannot be diversified demographically. To test the asset-pricing relevance of these ideas, I embed the non-expected utility specification implied by evolutionary theory into an economy with partial risk sharing due to limited commitment. For the benchmark specification (CRRA = 6 over gains), Monte Carlo simulations of a Markov growth economy produce the following results: (i) matching the degree of consumption-smoothing in the cross section, the Sharpe ratio for a Lucas tree is 0.33, an increase of 44 percent relative to expected utility; (ii) the risk-free rate is low, stable and counter-cyclical, hence equity returns, unlike in the expected utility case, have the correct pattern of predictability; (iii) in the cross section, excess returns across equity classes exhibit both a value premium and a size discount with risk adjusted returns that are at least two times higher than their expected utility counterparts. © 2012 Elsevier Inc. All rights reserved.

1. Introduction Evolutionary biology suggests that successful species must behave as if they maximize an adaptive (reference-dependent) S-shaped hedonic utility function that features an arithmetic weighted average (conditional expectation) over idiosyncratic states of nature but a geometric weighted average over aggregate states of nature. This objective, which I will refer to as biologically based non-expected utility (BBNEU), leads to behavior that maximizes the probability of the species’ long run survival in the presence of both demographic (idiosyncratic) and environmental (aggregate) uncertainty. Since only aggregate shocks can lead to the extinction of the entire population of an agent’s descendents, investors behaving according to this objective will tend to be intrinsically more averse to aggregate than idiosyncratic shocks. Hence investors will require a premium to hold assets positively correlated with aggregate consumption growth but may not have as much incentive to completely diversify idiosyncratic consumption risk. The goal of this paper is to analyze the asset-pricing relevance of BBNEU by describing the pricing kernel implied by the specification and testing it with US data on consumption and equity returns. In order to distinguish between aversion to idiosyncratic and aggregate shocks, I embed the BBNEU preferences into a model of limited commitment that features partial sharing of idiosyncratic risk. The pricing kernel determining the values of contingent claims then depends on the maximum stochastic discount factor across agent types. The combination

E-mail address: [email protected]. I would like to thank the editor Urban Jermann and an anonymous referee for their helpful comments and suggestions which substantially improved the final version of this paper. Comments by participants at the UBC Winter Finance Conference, particularly the discussant Valery Polkovnichenko, contributed to a major revision of a previous version. I have also benefit from discussions with Francois Gourio, Jeffrey Kubik, Benjamin Mathew, and Arthur Robson. All remaining errors are mine. 1

1094-2025/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.red.2012.08.002

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of an adaptive S-shaped utility function, non-expected objective, and limited commitment generates rich variation in the equilibrium stochastic discount factor across states and over time. Individual marginal rates of substitution feature additional terms that reflect reference dependence and extra curvature with regards to aggregate shocks. These additional terms increase the volatility of the stochastic discount factor relative to the expected utility case. Thus for the same level of consumption smoothing and idiosyncratic risk aversion, the BBNEU model can generate a higher equity premium than its expected utility counterpart. I generate quantitative results for asset returns through Monte Carlo simulations of a two state stochastic growth economy with the Markov property. Values for the transition matrix, idiosyncratic income shares, and correlation between idiosyncratic income and consumption growth are set to match moments of the US consumption and income distributions as estimated by Heaton and Lucas (1996) and Blundell et al. (2008). Preference parameters are then calibrated to match moments of the return distributions for US T-bills and intermediate (5 yr) bonds. Limited commitment plays a crucial role here. Without it, idiosyncratic shocks are completely diversified and the BBNEU model cannot generate a risk-free rate that has the low volatility observed in the data. I then use Monte Carlo simulations to explicitly calculate the return distributions for equity. I consider broad market equity—a Lucas tree paying the aggregate endowment—and the six Fama/French size and book-to-market portfolios. For a coefficient of relative risk aversion towards static idiosyncratic bets equal to 6, the BBNEU model produces an excess return for the Lucas tree of 2.53 percent, more than doubling the excess return produced by the expected utility model with limited commitment and the same level of risk aversion. More importantly, the Sharpe ratio produced by the BBNEU model, 0.33, is close to the value of 0.37 for US equity. With appropriate leveraging, the BBNEU model is thus broadly consistent with the premium observed for US equity. Consumption data is no longer “excessively smooth” relative to the limited commitment framework—the BBNEU model is consistent with the moments of both equity returns and the consumption response to income. The non-linear way in which aggregate shocks enter the preference specification leads BBNEU investors to exhibit variation in risk aversion over time and across states. On average, risk aversion is pro-cyclical because of the unpleasant nature of unexpected negative shocks. The risk-free rate is thus counter-cyclical, falling in expansions and rising in contractions. In addition, the adaptive nature of hedonic utility generates history dependence in risk attitudes. Investors gradually become less (more) risk averse as the economy improves (worsens). As a result, not only are equity prices more volatile in the BBNEU than in the expected utility case, they also exhibit momentum. In both an expansion and a recession, current equity prices are higher if the economy is coming out of an expansion than if it is coming out of a recession. The BBNEU model can thus match the pattern of excess return predictability observed in the data. Specifically, when we regress the simulated excess returns for the Lucas tree on its dividend–price ratio, the value obtained for the annual horizon is 2.13. This estimate is not significantly different from the empirical value of 5.3 for post-WW2 equity. Unlike other models of predictability, however, the result here is not due to counter-cyclical risk aversion. Instead, the BBNEU model generates predictability in equity returns through a combination of pro-cyclical equity prices and a counter-cyclical risk-free rate. Finally, to investigate the implications of BBNEU for the cross-sectional distribution of equity returns, I calibrate and price Markov processes with 2 idiosyncratic states for dividend growth conditional on each aggregate state. The 4 state Markov process is sufficiently flexible to allow control over the correlation between dividend and aggregate consumption growth in addition to independently setting the mean and standard deviation of each dividend growth process. The mean excess returns produced by the BBNEU specification are consistent with the empirical evidence in the sense that (i) there is a value premium within each size class and (ii) a size discount within each book-to-market category. With Sharpe ratios around 0.15, risk adjusted returns for the six classes are understandably lower than for the Lucas tree—the dividend growth processes are much less than perfectly correlated with aggregate consumption growth. Nevertheless, the benchmark BBNEU specification produces Sharpe ratios that are at least twice as large as the risk adjusted returns produced by the expected utility model. 1.1. Related literature In terms of research on the evolutionary foundations of preferences, this paper is an application of the indirect evolutionary approach. Rather than reverse engineering preferences to fit certain empirical facts, this approach subjects preferences to the analysis of evolutionary selection in order to identify their fundamental characteristics. The obtained preference specification can then be used to derive implications about economic behavior that can be tested empirically. In designing the preference specification for this paper, I have used several studies on the properties of hedonic utility, mainly from evolutionary theory and neuroscience. Valuation of relative consumption is motivated by the seminal paper of Tremblay and Schultz (1999), and the more recent work by Elliott et al. (2008). Similarly, the S-shaped nature of hedonic utility over idiosyncratic states is based on the work by Rayo and Becker (2007), while the differential treatment of idiosyncratic and aggregate shocks is taken from Robson (1996). While the interested reader can refer to Robson and Samuelson (2010) for a comprehensive survey of the theoretical literature on the subject, it will be useful to explicitly consider a simple example that illustrates the ecological difference between idiosyncratic and aggregate shocks.

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Example.2 Consider a population of N 0 individuals that reproduce asexually and live for only one period. Each period, an individual produces g H or g L descendents, say with g H > g L , each with probability 0.5. Suppose first that for every generation t, the realization of g is independent across individuals in the population. If the population N t is sufficiently large, a law of large numbers applies to the cross section and population growth during period t is given by E [ g ] = 0.5g H + 0.5g L . Thus over time, population numbers evolve according to

N t = E [ g ]t N 0 , and the evolutionary optimal (population-maximizing) strategy corresponds to maximizing the objective E [ g ], the expected value of direct offsprings. Now suppose that instead of independently, the realizations of g are completely governed by the environment so they are identical for all individuals in a generation. Population numbers now evolve according to

 Nt =

t 

 gs N0,

s =1

where g s is the realization of the growth factor for generation s. Clearly, population growth is no longer deterministic so it is not obvious that maximizing the objective E [ g ] continues to be the evolutionary optimal strategy. In fact it does not! To see this, transform the product into sum by applying natural logarithm:

1 t

1 t

ln N t =

t

s =1

ln g s +

1 t

ln N 0 .

By the strong law of large numbers, for t sufficiently large, the population evolves according to

ln N t = E [ln g ] · t + ln N 0 . Evolutionary optimal behavior thus now requires the maximization of E [ln g ] rather than expected growth. In reality, both idiosyncratic and environmental shocks, as well as individual actions, play a role in determining the realization of g for each individual. However, as shown by Robson (1996) and others, the point illustrated in this simple example generalizes to these more realistic situations. The curvature of the objective function will be less convex whenever aggregate, as opposed to uninsured idiosyncratic, risk is involved. As far as research on asset pricing is concerned, this paper is an example of the “exotic preferences” approach thoroughly surveyed by Backus et al. (2004). The preference specification used here deviates from expected utility and, as shown later, is a special case of the general structure described by Chew and Epstein (1990). The S-shaped hedonic utility function in this paper shares some common elements, namely reference dependence and diminishing sensitivity, with the specifications employed by Barberis et al. (2001), Yogo (2008), and Andries (2011), but abstracts from others, such as loss aversion and expected utility. These studies assume complete risk sharing and generate empirically plausible equity returns through the first-order risk aversion implied by loss averse preferences. In addition, Yogo (2008) incorporates a slow-moving habit as a reference point to match the predictability of excess returns, while Andries (2011) considers an expectations-based reference rule to improve the cross-sectional fit of equity returns. The BBNEU model accomplishes both with limited risk sharing but without loss aversion. In principal, BBNEU can accommodate loss aversion, but its presence is not required for any of the results. Nevertheless, in the context of limited commitment, it could be useful to compare the pricing implications of the BBNEU and loss aversion specifications and see whether the two models can mimic each other. In order to differentiate aversion to uninsured idiosyncratic and aggregate shocks, the model in this paper features partial risk sharing due to limited commitment. Asset markets are complete but are subject to endogenous solvency constraints motivated by the agents’ ability to default on their promises. If the penalty for default is not too harsh or the agents are sufficiently impatient, some of the idiosyncratic variability in income will remain undiversified in equilibrium. This literature originates from the works of Kehoe and Levine (1993, 2001), Kocherlakota (1996), and Alvarez and Jermann (2000, 2001) and recognizes that partial sharing of idiosyncratic risk will increase the volatility of the stochastic discount factor used for pricing contingent claims. Models with limited commitment can thus generate a low risk-free interest rate and a realistic equity premium for low levels of risk aversion. The problem with limited commitment models is that they require unrealistically low levels of idiosyncratic risk sharing in order to generate the necessary equity premium. This drawback was acknowledged in the quantitative work of Alvarez and Jermann (2001) and has more recently been strengthened by the work of Krueger et al. (2007). Using consumption data from the CEX, they show that consumption growth is too smooth for the limited commitment model to generate a reasonable equity premium for low values of risk aversion. Hence these models resolve the “equity premium puzzle” but

2

This is adapted from Section 10 in Houston and McNamara (1999).

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Table 1 Risk-neutral probabilities and stochastic discount factors implied by the BBNEU model with (SC) and without solvency constraints (FI) as well as the expected utility and GDA models. BBNEU_SC

BBNEU_FI

EU_SC

GDA

π H∗ H π H∗ L πL∗H ∗ πLL

0.336 0.664 0.587 0.413

0.376 0.624 0.523 0.477

0.318 0.682 0.585 0.415

0.382 0.618 0.098 0.902

zt −1 = H

zt −1 = L

zt −1 = H

zt −1 = L

mH H mH L mL H m LL

0.827 1.236 0.978 0.911

0.807 1.037 0.726 0.879

0.827 1.037 0.726 0.879

0.807 1.012 0.733 0.888

0.754 1.219 0.990 0.932

0.865 1.054 0.172 2.090

Notes: (i) CRRA = 3 for all specifications; (ii) other preference parameters calibrated to match moments of the risk-free rate for the stochastic process used in Routledge and Zin (2010).

generate a “consumption smoothness puzzle”, a feature of the data first recognized by Campbell and Deaton (1989) and investigated more recently by Attanasio and Pavoni (2009) and Heathcote et al. (2009). The BBNEU model reconciles equity returns with consumption variability by introducing extra terms in the stochastic discount factor. Since volatility in the stochastic discount factor no longer comes only from volatility in the ratio of marginal utilities, but also from changes in expectations and hedonic utility levels, we can allow for a greater degree of idiosyncratic risk sharing while still generating a reasonable equity premium. At first glance, the extra aversion to aggregate shocks implied by the BBNEU model is reminiscent of the generalized disappointment aversion (GDA) preferences analyzed by Routledge and Zin (2010). Unlike GDA, however, the extra curvature implied by aggregate risk is applied to all, not just the disappointing, outcomes. In addition, because of the S-shaped hedonic utility, there is diminishing sensitivity with respect to idiosyncratic disappointments. Depending on which effect dominates, it is possible for BBNEU investors to exhibit less risk aversion than even expected power utility investors. To illustrate these points, let ( zt , zt +1 ) denote a sequence of aggregate states, each of which can be a recession (L) or an expansion (H ). Table 1 summarizes the risk-neutral probabilities, πz∗t zt +1 , and stochastic discount factors, m zt zt +1 , generated by various models for the Markov growth economy (π H H = π LL = 0.43) used by Routledge and Zin (2010). The models include the BBNEU specification with (BBNEU_SC) and without (BBNEU_FI) solvency constraints, as well as the GDA specification and the expected utility case with solvency constraints (EU_SC). As evident from Table 1, the BBNEU model with complete risk sharing (BBNEU_FI) does overweigh the adverse aggregate state (L), regardless of the current state of nature. However, this effect is not as strong as the influence of solvency constraints (EU_SC) when the economy is currently in an expansion and certainly not as strong as the GDA overweighing when the economy is in a recession. The BBNEU specification can thus be better thought of as “nudging” the limited commitment model in the correct direction rather than mimicking the GDA specification. Relative to expected utility, the combination of an S-shaped hedonic utility function and a concave transformation over aggregate states generates preferences with higher curvature in expansions (H ) than in contractions (L). Hence contrary to the GDA case, the BBNEU model generates a counter-cyclical risk-free rate. In addition, the reference-dependent nature of hedonic utility leads to history dependence in the stochastic discount factor. In combination, these two effects increase the volatility of equity prices, which helps in generating a higher equity premium and predictable excess returns. 2. Theory Consider a world characterized by stochastic growth. Let et denote aggregate endowment in period t. Assume that et grows at the stochastic factor λt = et /et −1 , which can take on 2 possible values {λ L , λ H }. Suppose that the evolution of λ is determined by an ergodic Markov chain. The economy consists of 2 types of infinitely lived agents of equal measure.  Let {it } denote the stochastic process for i’s share of the aggregate endowment, which implies it ∈ [0, 1] and i it = 1 for all t. Assume that in each aggregate state, the share it can take on one of 2 values,  h (λ) > 1/2 and  l (λ) < 1/2. 2.1. Preferences Let zt = (λt , t ) denote the distribution of shocks in period t where 2t = t . Let {c i } = {c it ( zt ): ∀t  0, zt ∈ Z t } be a stochastic process for the consumption level of the agent of type i, where Z t is the set of possible histories as of time t. In adaptive evolutionary models, hedonic utility responds only to deviations from expected value. Furthermore, it is argued that the evolutionary optimal hedonic utility function is an S-shaped “squashing function” that translates relative consumption

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into hedonic experience.3 Hedonic experience in turn influences reproductive behavior, which ultimately determines evolutionary success. Consistent with this story, I assume that given history zt −1 and current aggregate endowment et −1 ( zt −1 ), each agent ranks consumption in period t relative to the consumption of a representative agent. Following Rayo and Becker (2007), I assume that the reference consumption level xt +1 is determined by the expectation, conditional on current information. Namely,

 

xt +1 zt = 0.5E t [λt +1 ]et .4 Hedonic utility for agent i in period t is then given by a continuous, non-decreasing, and log-concave function u : cˆ it → [h, h] with h > 0, where cˆ it is a realization of the relative consumption process {ˆc }i = {c it (zt )/xt (zt −1 ): ∀t  0, zt ∈ Z t }. For a history zt , agent i then ranks the stochastic consumption process {c } according to the following recursive objective motivated by Robson (1996),

 

U (ˆc ) zt = u (ˆct ) ·

 



Pr(t +1 | λt +1 , zt )U (ˆc ) zt +1

Pr(λt +1 |λt ) β



,

λt +1 t +1

where β ∈ (0, 1] is the pure discount factor. The motivation behind this objective is evolutionary. In the presence of environmental uncertainty, the expected utility criterion is not fitness maximizing. With purely idiosyncratic risks, shocks are diversified across the population of all descendents of an individual. Hence the correct objective is to maximize the expected, asymptotic number of descendents. On the other hand, aggregate shocks affect all descendents of an individual in exactly the same way. Hence an unusually long sequence of adverse aggregate shocks can put the entire population of descendents at the risk of extinction, which is not true in the case of idiosyncratic shocks. Note that the expression within the curly brackets is a certainty equivalent. In fact, the above preferences can equivalently be represented via the Koopmans equations







U t (ˆc ) = W cˆ t , μ U t +1 (ˆc ) where W (x, μ) and

for t  0,

μ are a time aggregator and a certainty equivalent with the following properties:

W (x, μ) = u (x) · μβ , and

μ(U t +1 ) =

 

Pr(t +1 | λt +1 , zt )U t +1

Pr(λt +1 |λt ) .

λt +1 t +1

Hence these preferences are a special case of the preferences considered by Chew and Epstein (1990). I assume the following functional form for the utility function u:

u (ˆc ) =

h exp(α cˆ ) for cˆ  c ∗ , ∗ ∗ ( 1 −γ ) ( 1 −γ ) h exp(α c ) − (c ) /(1 − γ ) + (ˆc ) /(1 − γ ) for cˆ > c ∗ .

There are three (ad hoc) reasons for choosing this particular functional form. First, it exhibits the S-shape typically associated with hedonic utility functions.5 Second, if c ∗ < 1, in a steady state, attitudes towards static idiosyncratic bets will be governed by the power utility function familiar to economists. Finally, because u is log-concave, applying logs to U t (ˆc ) gives

ln U t (ˆc ) = ln u (ˆct ) + β

 λt +1



Pr(λt +1 | λt ) · ln

 Pr(t +1 | λt +1 , zt )U t +1 (ˆc ) ,

t +1

with ln u (c ) concave in c (linear for c < c ∗ ), which ensures that dynamic programming methods can be applied in analyzing behavior generated by the maximization of U t .

3 Rayo and Becker (2007) explore the evolutionary basis of the optimal hedonic utility function, while Roiser et al. (2009) and Zhong et al. (2009) present genetic and neurochemical evidence supporting the S-shape of hedonic utility. 4 Note that this reference point rule incorporates both dynamic and peer comparisons. 5 Note that as in prospect theory, the function can in principal exhibit a kink at the inflection point c ∗ . However, that need not necessarily be the case. In fact, for given c ∗ and h, α can always be chosen so that limx↑c ∗ u  (x) = limx↓c ∗ u  (x).

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2.2. Asset markets Agents can trade one period state-contingent securities. Each security delivers a unit of the aggregate endowment if the particular state of the world gets realized. Let qt ( zt , z ) denote the time t, state zt price of a unit of the security payable at t + 1 if state z takes place. Let at +1 ( zt , z ) denote the holdings of this security with a lower limit, B t +1 ( zt , z ), that will be described shortly. An agent’s behavior is described by the following recursive program,







ln J it at , zt = max

cˆ t ,{a z }z

s.t.

ln u (ˆct ) + β

 t  t

it z et z + at = ct + 

a z  B it +1 zt , z





 λ









Pr λ λt · ln t







Pr







  λ , zt J it +1 a z , zt , z

 



a z  qt z , z ,

z

for all z ,

where cˆ t = ct /xt . The first-order condition for a z∗ is

q t ( zt , z ∗ ) xt ( zt −1 )

   β Pr z∗ zt 

J it +1 (a z∗ , ( zt , z∗ ))



u (ˆct )

Pr(  | λ∗ , zt ) J it +1 (a z , ( zt , z )) u  (ˆct )

,

where the equality holds if a z∗ > B it +1 ( zt , z∗ ). The envelope condition is





J it at , zt =

u  (ˆct ) J it (at , zt ) u (ˆct ) xt ( zt −1 )

.

Combining these conditions gives us the Euler equation

      qt zt , z∗  Pr z∗ zt · mit +1 zt , z∗ ,

where





mit +1 zt , z∗ =

β E t −1 [λt ] u  (ˆc it +1 ( zt , z∗ )) u (ˆc it ( zt )) J it +1 (a z∗ , ( zt , z∗ )) · · · · ,  t t ∗ λt E t [λt +1 ] u (ˆc it ( z )) u (ˆc it +1 ( z , z )) E [ J it +1 (a z , ( zt , z ))|λ∗ , zt ]

is the stochastic discount factor implied by i’s program. Compared with the expected utility case, we have several additional terms influencing the stochastic discount factor. First, the ratio E t −1 [λt ]/ E t [λt +1 ] of expected growth factors influences the way reference consumption is determined and hence affects contingent prices. We also have the inverse of hedonic utility growth, which amplifies the marginal rate of substitution. This comes from the non-linear way aggregate shocks are treated in the preference specification. Finally, we have a counteracting term that takes into account the possibility that some of the variability in the continuation value may be due to idiosyncratic shocks, which are treated less adversely than aggregate shocks. Since the effects of these additional terms go in opposite directions, the volatility of this stochastic discount factor may in theory be either higher or lower than in the expected utility case. However, note that if either markets are close to complete or idiosyncratic shocks are sufficiently correlated with the aggregate shock, the value of the last term will be close to 1. Since studies such as Heaton and Lucas (1996) find that the cross-sectional dispersion of income shares is constant across aggregate states, the case where the last term is approximately 1 seems more empirically relevant than the alternative. Following the limited commitment literature, I consider equilibria with solvency constraints. Such equilibria satisfy











J it +1 B it +1 zt +1 , zt +1  U itR+1 zt +1



for all t , i , and zt +1 .

In the case where U itR+1 ( zt +1 ) = U (i eˆ )( zt +1 ), we have the economy analyzed in Alvarez and Jermann (2000) with solvency constraints that are not too tight. However, the reservation utilities can in principle be higher than in autarky if, as in Chien and Lustig (2010), agents are excluded from asset markets only for a finite period of time after a default. Alternatively, the reservation utilities can be lower if the punishment for default includes wage garnishments in addition to exclusion from capital markets. In equilibrium, asset markets have to clear, which implies that in every period, for at least one type, the Euler equation must be satisfied with equality. Otherwise, the price qt will adjust. Hence in a competitive equilibrium, we must have

       qt zt , z = max Pr z zt mit +1 zt , z . i =1,2

Given stochastic processes for the consumption of each type, we can use the above condition to compute the moments of the stochastic discount factor maxi {mit +1 ( zt , z )}. We can then check, using Hansen–Jagannathan bounds6 and other methods, whether the stochastic discount factor implied by the preference specification is consistent with the asset prices and returns observed in the data. 6 For descriptions of Hansen–Jagannathan bounds and their applications to asset pricing, see Hansen and Richard (1987) and Hansen and Jagannathan (1991) as well as the survey by Cochrane and Hansen (1992).

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3. Quantitative implications I now describe some of the empirical properties of the specification presented in Section 2. In order to calibrate the necessary model parameters and compare the asset-pricing predictions of the model to the actual numbers, I use several published studies and datasets. I use Heaton and Lucas (1996) and Blundell et al. (2008) for properties of the income and consumption processes. All data on government bonds and bills (risk-free asset) is obtained from the 2010 edition of the Morningstar SBBI Classic Yearbook. The data on aggregate consumption growth, equity, and inflation is obtained from Robert Schiller’s dataset, which is available on his website.7 Finally data on asset classes is obtained from the Fama/French files.8 3.1. Calibrating the model 3.1.1. Income and consumption processes Aggregate endowment expands (state H ) and contracts (state L) according to a Markov process with transition matrix Π . I set the transition probabilities so that the model: (i) matches the empirical autocorrelation of −0.085 for US aggregate consumption growth over 1890–2008, as calculated from Shiller’s dataset; (ii) Pr( H )/ Pr( L ) = 2.85, as implied by the NBER Business Cycles series. Specifically, the transition matrix determining the evolution of λ is given by



Π=



0.718 0.803 . 0.282 0.197

For the individual income processes, I use a calibration that is based on the estimates by Heaton and Lucas (1996) using PSID panel data. The idiosyncratic income shares are set so that the model matches the persistence (ρ i = 0.529) and crosssectional variation (σ i = 0.251) in income shares reported in Table A3 of Heaton and Lucas (1996). The calibrated values are







1 ( H ) 0.663 = 0.337 1 ( L )





and







2 ( H ) 0.337 , = 0.663 2 ( L )

so that the high growth state ( H ) is the idiosyncratically good state for agent type 1, while the low growth state ( L ) is the idiosyncratically good state for agent type 2. The idiosyncratic shocks in this specification are deterministic functions of the aggregate state. Consequently,





mit +1 zt , z =

β E t −1 [λt ] u  (ˆc it +1 ( zt , z )) u (ˆc it ( zt )) · · · ,  t λt E t [λt +1 ] u (ˆc it ( z )) u (ˆc it +1 ( zt , z ))

(1)

since conditional on the aggregate state, J it +1 (a z , ( zt , z )) = E [ J it +1 (a z , ( zt , z ))|λ , zt ] for any z . This fact simplifies the computations substantially as we no longer need to estimate the value functions J i . There are other ways to approximate the income distribution for which the idiosyncratic state is stochastic, conditional on the aggregate shock. However, as long as idiosyncratic shocks are sufficiently correlated with the aggregate state, the results should not change significantly. Given Π , the values of λ H = 1.042 and λ L = 0.961 are set so that M1. E [λ] = 2.11%, M2. Std[λ] = 3.57%, which correspond to the mean and standard deviation of the growth rate of real per capita aggregate consumption for the period 1889–2008 obtained from Schiller’s dataset. To calibrate the consumption response to income shocks, I use the following moment conditions for the cross-sectional relation between consumption and income growth rates, conditional on the current aggregate state, M3. Cov[d ln c it , d ln it et | H ]/Var[d ln it et | H ] = 18%, M4. Cov[d ln c it , d ln it et | L ]/Var[d ln it et | L ] = 13.3%. These numbers are derived from the estimates of Blundell et al. (2008) that use panel data from the PSID and CEX.9 The values suggest that idiosyncratic risk sharing is counter-cyclical, stronger in downturns than in expansions, a finding

7

http://www.econ.yale.edu/~shiller/data.htm. http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. 9 Blundell et al. (2008) estimate income processes consisting of permanent (random walk) and transitory (MA(1)) idiosyncratic components. They estimate that 36% of the permanent and 95% of the transitory idiosyncratic shocks are insured. Combining these estimates with the variances of the permanent and transitory components as well as the estimate for the coefficient in the MA(1) process gives, as sample averages, the numbers in M3 and M4. 8

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Table 2 Preference parameter values used in subsequent simulations for the BBNEU and EU specifications with γ = 6.

BBNEU EU_SC

β

c∗

h exp(α c ∗ )

0.746 0.784

0.956

0.556

−

−

which has been confirmed for the US and other developed countries by Krueger et al. (2010). Moreover, these numbers are consistent with the response, between 10% and 20%, estimated by Cochrane (1991) again with PSID data but different methodology. Calibrating the consumption processes is different from the normal approach of endogenously solving for consumption. In the absence of precise data on the nature of solvency constraints, however, it is not obvious which strategy is superior. In order to endogenously determine consumption, Alvarez and Jermann (2001) assume that agents are relegated to autarky forever in the case of default. It could be argued that such punishments are unrealistic in their severity. More recent work has thus explored less severe schemes. Chien and Lustig (2010), for example, assume that agents can declare bankruptcy, so upon default they loose only their non-labor assets. However, because such punishments are less severe, they lead to even less sharing of idiosyncratic risk than the benchmark model that already has too little of it. The point is, we can generate many different patterns of consumption smoothing by varying our assumptions about the nature of solvency constraints. If data on solvency constraints is imprecise, calibrating the consumption processes directly may be a preferable strategy. In the benchmark specification, I assume that the consumption shares of aggregate endowment depend only on the current state of nature (zt ). Hedonic utility in period t, however, depends on consumption relative to expected consumption as of t − 1. For the given transition matrix, expected future consumption growth is higher in a recession than in an expansion. With history independent consumption shares, the level of hedonic utility will thus be lower when the economy is coming out of a recession (zt −1 = L) than when it is coming out of an expansion (zt −1 = H ). For such consumption shares to be consistent with the limited commitment model, the punishment for default must be more severe when the economy is coming out of a recession (zt −1 = L) than when it is coming out of an expansion (zt −1 = H ). Although possible, it is not obvious why the reservation utilities should vary in a way that makes the consumption shares history independent. In Appendix A.1, I thus consider an alternative specification in which the consumption shares are history dependent and the only variation in reservation utilities comes from the current state (zt ) and the value of it λt /0.5E t −1 [λt ], the current relative consumption of type i in autarky. The results are largely unaffected. The only major difference from the benchmark case is that with history dependent consumption shares, excess returns for equity become “too predictable”. The regression coefficient of the excess return on the dividend yield increases from 2.17 in the benchmark case to 34, a value much higher than the empirical estimate of 5.3 for US equity. 3.1.2. Preference parameters I follow Alvarez and Jermann (2001) and do not explicitly calibrate the value of γ , the coefficient of relative risk aversion over static idiosyncratic bets. Instead, I provide results for γ = 5 and 6 (in Appendix A.2, Tables A.1–A.5), values of γ that are both empirically plausible and provide reasonable asset-pricing results. Given the utility function

u (ˆc ) =

h exp(α cˆ ) for cˆ  c ∗ , ∗ ∗ ( 1 −γ ) ( 1 −γ ) h exp(α c ) − (c ) /(1 − γ ) + (ˆc ) /(1 − γ ) for cˆ > c ∗ ,

the preference parameters that need to be calibrated are then: c ∗ —the value of cˆ at which the hedonic utility function changes from being concave to convex, h exp(α c ∗ )—the level of u at this particular point, and β —the pure discount factor. The particular values are picked so that the implied stochastic discount factor produces plausible moments for the annual real return on a US Treasury bill ( R f ) and nominal return on an intermediate (5 year) US Treasury bond ( I B ) with a coupon payment of 5.3% annually: M5. E [ R f − 1] = 0.75%, M6. Std[ R f ] = 4.0%, M7. E [ R I B − 1] = 5.57%.10 Table 2 summarizes the calibrated values that are used in the subsequent simulations for the BBNEU model and an expected power utility specification with solvency constraints (EU_SC). The EU_SC model has fewer degrees of freedom. We thus require it to approximate only the mean of the risk-free rate (M5).

10 The values for these three moments, as well as the coupon payment, are derived from the 1926–2008 time-series contained in the Morningstar SBBI Yearbook.

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Table 3 Simulated moments of the real return (%) distribution for unleveraged equity for the benchmark BBNEU model with (BBNEU_SC) and without (BBNEU_FI) solvency constraints and the expected utility model with solvency constraints (EU_SC).

BBNEU_SC BBNEU_FI EU_SC US data, 1926–2008

E [ R ex eq ]

Std[ R eq ]

Sharpe

Std[ R f ]

2.53 5.35 1.15 7.28

7.77 36.8 5.04 19.7

0.33 0.15 0.23 0.37

4.49 30.1 6.30 4.02

Notes: (i) ex = excess return, eq = Lucas tree paying the aggregate endowment as a dividend; (ii) data for equity returns from Schiller’s data set; (iii) CPI used as deflator.

3.2. Equity returns I now present results on the nature of the equity returns implied by the BBNEU model, both in the cross section and over time. In addition to the benchmark specification (BBNEU_SC), I also present results for the BBNEU model with complete risk sharing (BBNEU_FI),11 and the expected power utility (EU_SC) model. Appendix A.2 contains results for two other specifications, the BBNEU_SC model with γ = 5 and with history-dependent consumption shares (BBNEU_HDCS). All results are based on explicit calculations for moments of the return distributions for (i) a claim to the aggregate endowment (unleveraged equity) and (ii) the six Fama/French size and book-to-market portfolios.12 For each of the initial states, the sample moments are computed using Monte Carlo simulations of the aggregate endowment process with n = 500,000. In addition, for the six Fama/French portfolios, the dividend growth process for each asset class is calibrated to match moments of its empirical distribution as well as its correlation with the growth of aggregate consumption. Table 3 summarizes the unconditional moments of the equity return distribution. The mean and volatility of the excess returns implied by the benchmark model are smaller than their empirical counterparts. This is partially due to the fact that in reality equity has leverage, while in the model it does not. Since leverage increases volatility, the empirical counterparts to the simulated moments will be smaller than the values estimated from the data. The Sharpe ratio is one way to control for this difference and as evident from column 3, in terms of the Sharpe ratio, for γ > 5 the BBNEU specification produces reasonable approximations to the empirical ratio of 0.37. For a given value of γ , the expected utility model with solvency constraints generates a lower Sharpe ratio because the volatility in consumption implied by M3 and M4 alone is not sufficient to generate the required variation in marginal rates of substitution.13 The extra terms introduced by the BBNEU specification help in this case. First, because of the referencedependent nature of hedonic utility, changes in expectations have a direct influence on the marginal rate of substitution. In addition, agents with BBNEU preferences are relatively more averse towards unexpected aggregate shocks and hence require a high return to accept bets correlated with the aggregate state. On the other hand, they are not as averse towards idiosyncratic shocks and do not have as much incentive to completely diversify idiosyncratic risk. In combination with the solvency constraints implied by limited commitment, the BBNEU specification increases the volatility of the stochastic discount factor and is thus able to generate predictions consistent with both equity prices and the response of consumption to idiosyncratic income shocks. It should be noted that limited commitment plays a crucial role in generating both the equity premium and a stable riskfree rate. For example, even with full risk sharing, the BBNEU model can generate a reasonably volatile stochastic discount factor, especially if we further increase aversion to static idiosyncratic bets. However, for γ = 6 the implied risk-free rate is almost 7 times more volatile than the empirical standard deviation of 4%. There appears to be a synergy between the BBNEU and limited commitment models. It is the combination of reference dependence, extra sensitivity to aggregate risk, and partial risk sharing induced by limited commitment that creates implications consistent with the moments of the equity and risk-free return distributions. 3.2.1. Predictability Table 4 shows that the BBNEU_SC model produces the correct pattern of predictability for equity returns, in the sense that high/low current dividend yields predict high/low subsequent excess returns. Given that the model is based on a two state Markov process, the precision of these estimates is not high. However, for the BBNEU model, the same pattern emerges for γ = 5 and the effect is even stronger for the specification with history-dependent consumption shares (results in Appendix A.2). This is in sharp contrast with the expected utility specification (EU_SC) which produces a large coefficient with the wrong sign. The expected utility specification cannot generate the correct pattern of predictability because equity prices are not sufficiently volatile to generate a counter-cyclical dividend yield. The BBNEU_SC model, on the other hand, produces a stochastic discount factor that is more volatile and exhibits history dependence. As evident from Table 5, the value of the

11

The preference parameters in this specification are as in Table 1 but we have a representative agent that consumes the aggregate endowment stream. Developed and analyzed in Fama and French (1993). 13 This is the well-known “excess smoothness” result first described by Campbell and Deaton (1989). For the case of expected utility and limited commitment, the “excess smoothness” result is confirmed by Krueger et al. (2007). 12

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Table 4 Correlations for excess return (equity—risk free) and dividend growth with the dividend–price ratio of equity for the benchmark BBNEU model with (BBNEU_SC) and without (BBNEU_FI) solvency constraints and the expected utility model with solvency constraints (EU_SC).

BBNEU_SC BBNEU_FI EU_SC US data, 1947–1996

R tex+1 = a + b( D t / P t ) Implied b

D t +1 / D t = a + b( D t / P t ) Implied b

2.17 0.03 −122 5.3 (2.0)

−0.01 0.00 0.55 2.0 (1.1)

Note: US data taken from Table 20.1 in Cochrane (2001), standard errors in parentheses.

Table 5 Values for the stochastic discount factor, mt ,t +1 , generated by the benchmark (BBNEU_SC) and expected utility (EU_SC) models with solvency constraints. BBNEU_SC

mH H mH L mL H m LL

EU_SC

zt −1 = H

zt −1 = L

0.716 1.821 0.966 0.813

0.687 1.748 0.923 0.776

0.611 1.825 1.131 0.995

BBNEU_SC discount factor is higher when the economy is coming out of an expansion (zt −1 = H ) than when it is coming out of a recession (zt −1 = L). As a result, equity prices in the BBNEU_SC model exhibit momentum. Specifically, in both an expansion (zt = H ) and a recession (zt = L), current equity prices are higher if the economy is coming out of an expansion (zt −1 = H ) than if it is coming out of a recession (zt −1 = L). To understand why equity prices exhibit momentum, consider the discount factor implied by the BBNEU_SC model for the states ( zt −1 , zt , zt +1 ) = ( H H H ) and ( L H H ) respectively. For ( H H H ), expression (1) simplifies to

m H H ( zt −1 = H ) =

β = 0.716, λH

which is equivalent to the discount factor of a CRRA investor with log utility (CRRA = 1). On the other hand, when the economy is coming out of a recession ( zt −1 = L ), we have

m H H ( zt −1 = L ) =

β · λH



E [λ| H ] E [λ| L ]

γ

·

u (ˆc ( L H )) u (ˆc ( H H ))

= 0.687.

For the transition matrix Π , expected future growth is higher if the economy is currently in a recession than in an expansion, namely E [λ| H ] < E [λ| L ] and hence cˆ ( L H ) < cˆ ( H H ). Consequently, m H H ( zt −1 = L ) < β/λ H , which is equivalent to an increase in the risk aversion of a CRRA investor.14 Given this variation in risk aversion, in order to compensate investors, current equity prices will be lower (expected excess return higher) when the economy is coming out of a recession than when it is coming out of an expansion. In addition, unlike the expected utility case, the BBNEU stochastic discount factor is on average higher when the economy is currently in an expansion (zt = H ) than in a recession (zt = L). The risk-free rate in the BBNEU model is thus countercyclical. Table 6 shows that this pattern is consistent with the empirical evidence. Ceteris paribus, a counter-cyclical risk-free rate will make excess returns more pro-cyclical, further contributing to the predictability result for equity. To summarize, the BBNEU model produces return predictability because risk sensitivity varies over time and across states. This variation, however, is not ad hoc. Rather, it is generated endogenously through the reference-dependent nature of hedonic utility and the non-linear way in which aggregate shocks enter the preference specification. In addition, unlike models such as Campbell and Cochrane (1999) and Routledge and Zin (2010), the predictability result here is not due to counter-cyclical risk aversion. On the contrary, risk aversion is pro-cyclical because of the unpleasant nature of unexpected negative shocks. Instead, the BBNEU model generates predictability in equity excess returns through a combination of procyclical equity prices and a counter-cyclical risk-free rate. 3.2.2. Cross-sectional distribution To investigate the implications of the model for the cross-sectional distribution of equity returns, this section presents Mote Carlo simulations for the return distributions of the 6 size and book-to-market portfolios developed by Fama and French (1993). In order to match the dividend growth process for each equity class, I use a Markov process with 2 idio-

14

For the benchmark BBNEU_SC model, the value of m H H ( zt −1 = L ) = 0.687 corresponds to a power utility investor with CRRA of 1.98.

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Table 6 f Correlations between the risk-free rate (%), rt , and aggregate endowment growth (%), et /et −1 , for the benchmark BBNEU model with (BBNEU_SC) and without (BBNEU_FI) solvency constraints and the expected utility model with solvency constraints (EU_SC). f

rt = a + b(et /et −1 ) Implied b

−1.14 −8.29 1.77 −0.53 (0.22)

BBNEU_SC BBNEU_FI EU_SC US data, 1926–2008

Notes: Data on US T-bills from Morningstar SBBI, data on aggregate consumption growth from Shiller’s dataset, robust standard error in parentheses.

Table 7 Calibrated values for the dividend growth (D) processes of the six Fama/French size and book-to-market portfolios.

Hh

D D Hl D Lh D Ll Pr (hh| H H ) Pr (ll| L H ) E [ ln D t ] (%) Std[ ln D t ] (%) ρ ( ln D t ,  ln C t )

SmGr

SmNe

SmVa

LaGr

LaNe

LaVa

1.30 0.77 1.34 0.68 0.58 0.25 3.04 28.4 0.05

1.32 0.88 1.27 0.66 0.75 0.21 6.28 25.1 0.23

1.62 0.95 1.56 0.01 0.85 0.13 15.6 53.7 0.41

1.12 0.91 1.28 0.77 0.75 0.05 1.79 16.0 −0.03

1.14 0.95 1.22 0.68 0.76 0.14 2.07 16.8 0.27

1.37 0.93 1.56 0.28 0.89 0.15 9.03 39.1 0.25

Notes: (i) D Z x —dividend growth factor in aggregate state Z and idiosyncratic state x; (ii) ρ (·)—correlation with aggregate consumption growth; (ii) dividend growth calibrations based on annual data, 1927–2008, from the Fama/French files; (iii) CPI used as the deflator.

syncratic states for dividend growth—high (h) and low (l)—conditional on each aggregate state. This results in 4 states— Hh, Hl, Lh, Ll—and 12 degrees of freedom for each portfolio. Regrettably, the time series available for dividend growth is not sufficiently long to allow calibration of the unrestricted 4 state dividend growth process. In order to achieve identification, I thus impose additional restrictions on the transition matrix, namely on Pr(xt xt +1 | zt zt +1 ), the transition probabilities between the idiosyncratic states xt and xt +1 conditional on the aggregate transition from zt to zt +1 15 :

Pr(hh| H H ) = Pr(ll| H H ) = Pr(hh| LL ) = Pr(ll| LL ) and

Pr(hh| H L ) = Pr(ll| H L ) = Pr(hh| L H ) = Pr(ll| L H ). These conditions assume that (i) idiosyncratic states are symmetric conditional on the aggregate state and (ii) the persistence of the idiosyncratic process does not depend on the aggregate state. With these restrictions in place, we can calibrate the dividend growth process for each portfolio using six empirical moments—the mean and standard deviation of dividend growth, the correlation with aggregate consumption growth, the mean conditional on expansion (Hh) and contraction (Ll), and the empirical value of Pr( Hh | Hh). Table 7 presents, for each equity class, the calibrated values for the dividend growth, transition probabilities, and the implied mean and standard deviation of the dividend growth, as well as its correlation with aggregate consumption growth. The advantage of the 4 state Markov process is that we can control the correlation between dividend and aggregate consumption growth in addition to independently setting the mean and standard deviation of each dividend growth process. In this sense, the dividend processes for these portfolios are closer approximation to reality than the dividend process of a fictional asset such as the Lucas tree. This benefit comes at the expense of greater uncertainty regarding the values of the calibrated parameters. Nevertheless, given that the same dividend growth processes are used in all simulations, differences in return distributions across models should be indicative of each model’s empirical applicability. Table 8 summarizes the implications of the BBNEU model for the cross-sectional return distribution of the six equity classes. The mean excess returns are consistent with the empirical evidence in the sense that (i) there is a value premium within each size class and (ii) a size discount within each book-to-market category. Relative to other classes, value stocks

15 I also experimented with imposing restrictions, such as symmetry, on the values of dividend growth in each state. Such restrictions, however, significantly reduce the model’s ability to match the correlations between dividend and aggregate consumption growth, a problem which is avoided using the above restrictions on the transition probabilities.

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Table 8 Simulated and empirical real excess returns (%) for the six Fama/French equity classes for the benchmark BBNEU model with (BBNEU_SC) and without (BBNEU_FI) solvency constraints and the expected utility model with solvency constraints (EU_SC). SmGr

SmNe

SmVa

LaGr

LaNe

BBNEU_SC

E [ R ex ] Sharpe Mkt Beta

2.18 0.09 0.81

3.27 0.14 1.26

8.53 0.16 3.31

1.66 0.12 0.59

2.76 0.16 1.06

LaVa 4.91 0.13 1.83

BBNEU_FI

E [ R ex ] Sharpe Mkt Beta

4.90 0.10 0.93

6.18 0.12 1.17

10.4 0.14 1.99

4.34 0.11 0.82

5.65 0.14 1.07

7.25 0.12 1.38

EU_SC

E [ R ex ] Sharpe Mkt Beta

0.74 0.03 0.52

1.45 0.06 0.71

3.48 0.07 1.25

0.42 0.03 0.42

1.18 0.08 0.63

1.91 0.05 0.83

US data

E [ R ex ] Sharpe Mkt Beta

8.90 0.27 1.38

12.6 0.44 1.27

14.8 0.47 1.34

7.06 0.34 0.97

7.93 0.37 1.01

10.9 0.41 1.21

Notes: (i) Mkt Beta is the linear projection coefficient of the excess return for each portfolio on the excess return of a value-weighted basket of all portfolios, with weights set at the mean value for the period 1927–2008; (ii) US data is from the Fama/French files (equity) and Morningstar SBBI Classic Yearbook (risk-free rate).

have more volatile and more idiosyncratically persistent dividend growth processes. Their dividends thus grow persistently by a lot in expansions but also shrink persistently by a lot in contractions. In combination, these two features make value stocks unattractive to BBNEU investors, hence the higher equilibrium return. However, judging by the betas on the valueweighted market return, the BBNEU model appears to produce a value premium that is larger than what is observed empirically. In risk adjusted terms, returns for the six classes are understandably lower than for the Lucas tree since the dividend growth processes are much less than perfectly correlated with aggregate consumption growth. Nevertheless, the benchmark BBNEU specification produces Sharpe ratios that are at least twice as large as the risk adjusted returns produced by the expected utility model. Recall from Table 5 that the BBNEU_SC stochastic discount factor takes on twice as many values as its expected utility counterpart. The results in Table 8 suggest that because of this feature, the BBNEU_SC model can approximate the empirical covariance between marginal utility and asset returns better than the expected utility specification, especially when idiosyncratic variation is added to asset payoffs. Note that such a goodness-of-fit improvement need not be automatic, which suggests that the changes introduced by the BBNEU_SC specification are empirically beneficial. 4. Conclusion Consumption-based asset-pricing models have struggled to generate predictions that match the empirical distributions of consumption and asset returns. Yet such models are the only economic models of asset valuation we have. This paper has argued that introducing evolutionary considerations into investor preferences can improve the goodness-of-fit of the consumption-based asset-pricing model with limited commitment. Changing the preference specification from expected utility to evolutionary based non-expected utility improves the performance of the limited commitment model in several dimensions—the risk adjusted return to equity, the cyclicality of the risk-free rate, the predictability of equity returns, and the cross-sectional distribution of equity returns. The evolutionary based specification can achieve this because it generates marginal rates of substitutions that approximate the empirical rates better than expected utility can. At the same time, the modifications to preferences are not ad hoc but rather follow from a logical analysis of the type of preferences that are likely to survive the process of natural selection. Since evolutionary theory provides a systematic way of generating novel predictions about preferences, to the extent that asset valuations are ultimately determined by human investors, this approach is likely to produce further contributions to the study of asset pricing. Appendix A. Robustness checks A.1. History-dependent consumption shares (HDCS) Suppose zt = H . Let α H H and α L H denote the consumption shares, as a function of the history ( zt −1 , zt ), for the type whose participation constraint is binding in state H . For this type, consumption shares will be given by (1 − α H L ) and (1 − α LL ) for zt = L. The values of α H H and α L H are chosen so that

 ln u and

α H H λH 0.5E H [λ]



  + β E J t +1 (·) zt = H = ln u



 H λH 0.5E H [λ]



  + β E U tR+1 (·) zt = H

E.P. Iantchev / Review of Economic Dynamics 16 (2013) 497–510

 ln u

α L H λH 0.5E L [λ]



  + β E J t +1 (·) zt = H = ln u



 H λH



0.5E L [λ]

509

  + β E U tR+1 (·) zt = H .

α depend on the current and past state, the above two conditions imply  HH H   H H   LH H   H λH α λ  λ α λ − ln u = ln u − ln u ln u 0.5E H [λ] 0.5E H [λ] 0.5E L [λ] 0.5E L [λ]

Given that the values of





and a similar condition for the other type. The values for α zt −1 zt are then chosen so that a condition like the above is satisfied for each current state, as well as conditions M3 and M4. A.2. Results

Table A.1 Preference parameter values used in subsequent simulations for the BBNEU_SC with specifications.

BBNEU_SC BBNEU_HDCS

γ = 5 and HDCS with γ = 6

β

c∗

h exp(α c ∗ )

0.777 0.726

0.946 0.953

0.707 0.497

Table A.2 Simulated moments of the real return (%) distribution for unleveraged equity for the BBNEU_SC with with γ = 6 specifications.

BBNEU_SC BBNEU_HDCS

γ = 5 and HDCS

E [ R ex eq ]

Std[ R eq ]

Sharpe

Std[ R f ]

2.13 2.29

7.60 6.82

0.281 0.336

4.00 4.27

Notes: (i) ex = excess return, eq = Lucas tree paying the aggregate endowment as a dividend; (ii) data for equity returns from Schiller’s data set; (iii) CPI used as deflator.

Table A.3 Correlations for excess return (equity—risk free) and dividend growth with the dividend–price ratio of equity for the BBNEU_SC with γ = 5 and HDCS with γ = 6 specifications.

BBNEU_SC BBNEU_HDCS

R tex+1 = a + b( D t / P t ) Implied b

D t +1 / D t = a + b( D t / P t ) Implied b

12.4 34.1

−0.03 −0.07

Table A.4 f Correlations between the risk-free rate (%), rt , and aggregate endowment growth (%), et /et −1 , for the BBNEU_SC model with γ = 5 and HDCS specification with γ = 6. f

rt = a + b(et /et −1 ) Implied b

−1.04 −1.19

BBNEU_SC BBNEU_HDCS

Table A.5 Simulated and empirical real excess returns (%) for the six Fama/French asset classes for the BBNEU_SC with SmGr

SmNe

SmVa

γ = 5 and HDCS with γ = 6 specifications. LaGr

LaNe

LaVa

BBNEU_SC

E [ R ex ] Sharpe Mkt Beta

1.76 0.07 0.80

2.74 0.12 1.31

7.02 0.14 3.32

1.19 0.09 0.54

2.36 0.14 1.12

4.39 0.11 1.97

BBNEU_HDCS

E [ R ex ] Sharpe Mkt Beta

1.91 0.08 0.74

3.08 0.14 1.30

9.64 0.18 4.03

1.38 0.10 0.46

2.60 0.16 1.09

4.25 0.10 1.78

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