Macroeconomic Tail Risks and Asset Prices

Transcription

1 Macroeconomic Tail Risks and Asset Prices David Schreindorfer June 17, 018 Abstract I document that dividend growth for the aggregate U.S. stock market is more correlated with consumption growth in bad economic times than in good times. In a consumption-based asset pricing model with a generalized disappointment averse investor and small, i.i.d. consumption shocks, this feature results in a realistic equity premium despite low risk aversion. The model is consistent with the main facts about stock market risk premia inferred from equity index options, remains tightly parameterized, and allows for analytical solutions for asset prices. W. P. Carey School of Business, Arizona State University, PO Box , PO Box david.schreindorfer@asu.edu. I thank Sreedhar Barath, Oliver Boguth, Itamar Drechsler, Lars-Alexander Kuehn, Rajnish Mehra, Seth Pruitt, and Bryan Routledge for helpful comments and suggestions. The paper has also benefited from the comments of seminar participants at Arizona State University Finance) and conference participants at the SFS Cavalcade 018. To be presented at the WFA 018 and the Summer Meetings of the Econometric Society

2 1 Introduction This paper proposes a consumption-based explanation for risk premia in aggregate stock markets with two distinguishing features. First, the theory predicts that the equity premium compensates investors predominantly for exposure to macroeconomic tail risks drops in consumption of the magnitude typically observed in recessions. Despite the fact that these risks are modelled as small, i.i.d. shocks, the theory rationalizes the equity premium based on low risk aversion for the representative investor. Second, the model remains quantitatively consistent with the main facts about stock market risk premia that can be inferred from equity index options. Options represent an invaluable diagnostic tool for equilibrium asset pricing models see, e.g. Backus et al. 011). These models give rise to a pricing kernel that maps the market return distribution f into a related risk-neutral distribution f. Risk premia are reflected in differences between the two distributions: the equity premium equals the difference is means, the variance premium equals the difference in variances, etc. Jointly, the two distributions provide a complete description of stock market risks and investors attitudes toward such risks. Empirically, prices of options with different strikes allow us to infer f Breeden and Litzenberger 1978). A successful theory of stock market risk premia should therefore be consistent with basic properties of f as well as differences between the two distributions. The best-known feature of the risk-neutral distribution is that it is significantly left-skewed Rubinstein 1994). Martin 017) compares six leading asset pricing models in terms of their ability to capture this non-normality. While it is well-known that the models of Campbell and Cochrane 1999) and Bansal and Yaron 004) are log-normal, Martin finds that even extensions of the long run risks model that were explicitly designed to capture the variance premium imply risk-neutral distributions that are essentially Gaussian. In contrast, the rare disaster model of Wachter 013) exceeds his measure of non-normality in the data by a factor of ten. The inability of mainstream asset pricing theories to capture the risk-neutral skewness is troubling, because

3 options data suggests that differences between f and f in the far left tail lie at the heart of the equity premium puzzle. In particular, Bollerslev and Todorov 011) estimate that 85% of the equity premium is associated with states that feature returns of -10% or less over a few weeks. The theory proposed here is consistent both with this feature and the shape of the risk-neutral distribution. The model relies on two key deviations from the canonical framework. First, the representative agent s risk preferences feature generalized disappointment aversion GDA), an axiomatic extension of expected utility EU) theory that resolves the Allais Paradox and allows for asymmetric risk attitudes over gains and losses Gul 1991, Routledge and Zin 010). I show that GDA preferences can generate a high price for macroeconomic tail risks, while simultaneously relying on low risk aversion. This property stands in sharp contrast to EU risk preferences with low risk aversion, which imply that consumption shocks of the same magnitude are close to irrelevant for the agent Mehra and Prescott 1985). 1 Second, consumption and dividends follow random walks with innovations from a mixture distribution, where both growth rates are subject to independent Gaussian shocks and negatively exposed to a common exponential shock. This assumption implies that, as in the data, consumption and dividend growth rates are slightly left-skewed and have a low unconditional correlation, while large drops in dividends are nevertheless likely to be accompanied by large drops in consumption. The strong tail dependence between consumption and dividends has not been documented before, and it results in a large equity premium because tail events in consumption are associated with high marginal utility for the representative investor. When fundamentals are calibrated to annual U.S. consumption and dividend data over without relying on asset pricing moments), the model replicates numerous empirical asset market facts. It matches the equity premium, the risk-free rate, the price-dividend ratio and Sharpe 1 Embedding EU risk preferences into a recursive utility framework does not alter this conclusion. For example, Bansal and Yaron 004, Table V) show that i.i.d. consumption growth shocks contribute almost nothing to the equity premium in their model. 3

4 ratio of the market, the prices and returns of options with different strikes, the difference between the VIX and SVIX indices Martin 017), and the contribution of stock market tail risk to the overall equity premium Bollerslev and Todorov 011). At the same time, the model remains tightly parameterized and it yields analytical solutions for asset prices. 1.1 Mechanism Figure I contrasts the implications of the model s two main assumptions to those of more common alternatives. The left panel illustrates the tail dependence between consumption and dividend growth rates in the data and for alternative shock distributions. I plot the correlation between consumption and dividend growth, conditional on consumption growth contemporaneously falling below a given threshold indicated on the horizontal axis). For annual U.S. data over , the downside correlation is increasing in the left tail, i.e. growth rates are more strongly correlated in times of low consumption growth. I show that this pattern is robust, in that it is also present in post-war data, at the industry level, for alternative dividend measures, and that it takes on a very similar magnitude at different data frequencies see Appendix B). As is clear from the figure, the mixture model provides a good fit for empirical downside correlations when it is calibrated to match the unconditional correlation of 0.57 in the data the rightmost point on the graph). In contrast, when shocks come from a bivariate normal distribution with the same unconditional correlation, consumption and dividend growth have a downside correlation of only 0. in the far left tail. Because the normal distribution is unable to capture the large amount of co-tail risk that cash flows display in the data, asset pricing models based on normal shocks imply that shocks to the level of dividends carry unrealistically low risk premia, and that tail risks contribute unrealistically little to the equity premium. The mixture distribution overcomes this issue. The right panel of Figure I illustrates properties of alternative utility functions. I plot the consumption growth density for the model s monthly benchmark calibration with mixture shocks, Downside correlations were introduced by Ang and Chen 00). 4

5 Downside correlation Probability Density Function Pricing Kernel c density GDA pricing kernel EU pricing kernel Data Mixture distribution Normal distribution c percentile c in percent 0 Figure I: Model mechanism. The left panel shows the correlation between log consumption and dividend growth, conditional on consumption growth contemporaneously falling below a given percentile of its unconditional distribution, i.e. corr d t, c t c t x]. The solid line corresponds to annual U.S. data over , whereas the dotted dashed) line corresponds to a normal-exponential mixture normal) distribution that matches the unconditional correlation in the data. The empirical downside correlations are only computed for percentiles above the 4-th to ensure a minimum of 0 observations. The right panel shows a density for monthly log consumption growth that is calibrated to annual U.S. data over , and the pricing kernels for GDA dotted line) and EU dashed line), both calibrated to match the equity premium. along with the pricing kernels for GDA and EU risk preferences. Given an identical calibration for consumption and dividends, both utility functions are calibrated to match the equity premium. Two differences are important. First, the GDA calibration implies high aversion against left tail events in consumption growth, but no aversion against small drops in consumption. 3 The equity premium thus exclusively reflects macroeconomic tail risks and, because tail events in consumption and dividends are likely to coincide, it is also primarily associated with states of low market returns as documented by Bollerslev and Todorov 011). Out-of-the-money put options have large payoffs 3 The step function for the GDA pricing kernel may appear simplistic. While GDA can accommodate curvature, low degrees of curvature have a negligible effect on asset prices, very similar to the effect of low risk aversion for EU risk preferences in Mehra and Prescott 1985). For parsimony and simpler intuition, I therefore assume no curvature. 5

6 when these returns occur and therefore provide a valuable hedge to the agent. I show that this allows the model to replicate their high prices and large negative returns. Relative to GDA, the EU pricing kernel takes on higher values for small drops in consumption and much lower values for tail events. For empirically realistic consumption growth distributions, EU preferences thus imply that the equity premium mostly reflects relatively likely) small shocks and that relatively unlikely) left tail events are of secondary importance for risk premia. I show that, as a result, calibrations based on EU preferences remain inconsistent with various option moments. The issue is simple: EU risk preferences make it impossible to increase the agent s aversion against tail risks without simultaneously increasing the aversion against smaller risks. It is therefore not possible to generate an equity premium that mostly reflects tail risks. While it is well-known that EU does not provide a plausible theory of risk aversion for both small-stakes and large-stakes gambles Rabin 000 and Rabin and Thaler 001), this observation has previously not been linked to the pricing options or to the composition of the equity premium. The second important difference between the two utility functions lies in the amount of risk aversion they require to produce realistic risk premia. For the small, i.i.d. consumption and dividend shocks considered in this paper, EU preferences require a relative risk aversion coefficient of twenty-four to match the equity premium. I propose a comparable measure of risk aversion for GDA preferences. The idea is to first compute the consumption risk premium for a given model calibration with GDA preferences and, holding the endowment calibration fixed, then search for an EU calibration a risk aversion coefficient) that implies the same premium. Because the risk premium only depends on the amount of endowment risk and the agent s aversion against such risk, two calibrations that rely on the same endowment calibration and result in the same premium also reflect the same degree of risk aversion. Based on this metric, I find that the benchmark GDA calibration implies the same level of risk aversion as an EU calibration with a risk aversion coefficient of about seven. Hence, the model resolves the equity premium puzzle. 6

7 Model I consider a representative-agent pure-exchange economy with a single consumption good. This section details the model and derives analytical solutions for asset prices..1 Assumptions Fundamentals. Aggregate consumption in period t equals C t, whereas the aggregate dividend is D t. Equity is a claim to the dividends in all future periods. Consumption and dividends follow i.i.d. random walks, c t+1 ln d t+1 ln Ct+1 C t Dt+1 D t ) =g + σε c t+1 ) =g + ϕσε d t+1. 1) Both log growth rates have a constant mean of g and constant volatility. Consumption volatility equals σ 0, whereas dividend volatility is higher and equal to ϕσ. The parameter ϕ 1 captures leverage, as in Abel 1999) and Campbell and Cochrane 1999). Innovations have mean zero and standard deviation one, and are described by the Gaussian-exponential mixture distribution 4 ε c t = 1 ω η c t + ωη e t 1), ε d t = 1 ω η d t + ωη e t 1). ) The shocks η c and η d are serially uncorrelated standard normals with contemporaneous correlation 1 ϱ 1, and η e is an i.i.d. exponentially-distributed random variable with unit rate parameter. The benchmark calibration sets ϱ = 0, so that Gaussian shocks are contemporaneously independent and the only source of dependence between consumption and dividends consists of common exponential shocks. The ϱ 0 case is used to illustrate the model mechanism in Section 3. The mixture 4 The same mixture-model has recently been applied in a portfolio choice context by Dahlquist et al. 016), who use it to model the returns of stocks and bonds. 7

8 parameter 1 ω 1 controls the skewness and unconditional correlation of the innovations, which equal skewε c t] = skewε d t ] =ω 3 corrε c, ε d ] =1 ω )ϱ + ω. 3) For ω < 0, skewness is negative due to the positive skewness of the exponential distribution. This is illustrated in the right panel of Figure I in the introduction), which shows the density of monthly log consumption growth in the benchmark calibration. Skewness is reflected in a slightly thicker left tail than in the normal case, but tail outcomes are far from disastrous. For ϱ < 1, i.e. when Gaussian shocks are imperfectly correlated, consumption and dividends are more strongly correlated conditional on low values for consumption growth. The reason is that large negative growth rates for both series are more likely to be observed when the common exponential shock takes on a large value. This effect is illustrated in the left panel of Figure I, which shows the downside correlation for the mixture distribution and for a normal distribution with an identical unconditional correlation. In Appendix B, I show that the empirical pattern in Figure I is robust along various dimensions. The Gaussian-exponential mixture distribution represents an attractive way to model the high downside correlation for at least four reasons. First, it nests the bivariate normal distribution for ω = 0. I am therefore able to clearly illustrate which model predictions result from the endowment specification, which result from preference assumptions, and which result from their interaction. Second, it is very parsimonious. With the restriction ϱ = 0, it has one free parameter, just like the bivariate standard normal distribution. Without the restriction, it has one additional parameter. Third, it remains highly tractable. Expectations taken with respect to cash flow innovations in the model s asset pricing formulae can be evaluated analytically, so that the model is no more difficult to solve than a log-normal model. Fourth, it results in a smooth, unimodal consumption growth density. 8

9 Preferences. Following Epstein and Zin 1989), the representative agent s time t utility, U t, is given by the constant elasticity of substitution recursion U ρ t = 1 β)cρ t + βµ tu t+1 ) ρ. 4) 0 < β < 1 characterizes impatience and ρ 1 measures the preference for intertemporal substitution the elasticity of intertemporal substitution for deterministic consumption paths is 1 1 ρ ). Risk preferences are captured by the function µ t U t+1 ), which equals the certainty equivalent of random future utility using the time t conditional probability distribution. In the remainder of the paper, I use the shorthand notation µ t µ t U t+1 ). The certainty equivalent features Generalized Disappointment Aversion GDA) as in Routledge and Zin 010), and it is defined by the implicit function ] ] uµ t ) = E t uu t+1 ) θe t uδµ t ) uu t+1 ) )D t+1, 5) where D t+1 1{U t+1 δµ t } is a disappointment indicator and where the period utility function is given by x α 1 α for α 1, α 0 ux) =. 6) lnx) for α = 0 Equation 5 nests two well-known preference specifications as special cases. First, for θ = 0 the second term drops out and risk preferences simplify to expected utility. In this case, the certainty equivalent is given by the explicit function µ t = E t U α t+1 ]) 1 α, the utility function equals the specification used in Epstein and Zin 1991) and Bansal and Yaron 004), and α 1 measures static risk aversion the coefficient of relative risk aversion for static gambles is 1 α). Second, for δ = 1 risk preferences simplify to Gul s 1991) model of disappointment aversion DA). 5 In this case, all outcomes that fall below the certainty equivalent are considered disappointing and receive a penalty, 5 DA preferences were developed to resolve the Allais 1979) paradox. The asset pricing implications of recursive utility with DA risk preferences have been analyzed by Epstein and Zin 001) in an endowment economy and by Campanale et al. 010) in a production economy. Neither paper analyzes option prices or considers the implications of non-normal shocks. 9

10 the magnitude of which is governed by the parameter θ > 0. GDA preferences, which represent the most general version of Equation 5 θ > 0 and δ 1), place the disappointment threshold further into the tail of the conditional distribution of U t+1. In particular, only realizations of U t+1 that fall below a fraction δ of the certainty equivalent µ t are considered disappointing.. Solving the model Because consumption and dividend levels contain a unit root, the solution of the agent s utility maximization problem is characterized by utility ratios rather than utility levels. Define the timeinvariant ratios v Vt C t and m µtvt+1) C t, where V t is the value of utility in equilibrium. The value-to-consumption ratio follows by dividing 4) by C t, v ρ = 1 β) + βm ρ. 7) Similarly, dividing 5) by C t and re-arranging terms results in m α = E v α e α ct θd t+1) ] 1 + θδ α. 8) E D t+1 ] The disappointment event V t+1 δµ t can be expressed in terms of consumption growth as c t+1 x v, where x v ln ) δm v is a constant that depends on the equilibrium value of v. 6 The model solution takes three steps. First, the expectations in 8) are evaluated analytically by integrating over the Gaussian and exponential innovations in consumption growth. Next, m is substituted out of the resulting expression based on 7), which yields a single nonlinear equation in v. Because the equation is lengthy, I show it in A.9) in the appendix. Lastly, the equation is solved numerically with a standard nonlinear equation solver. As discussed on p.1308 of Routledge and Zin 010), the implicit function theorem guarantees that the nonlinear equation has a unique solution. The steps for the limiting case α 0 are analogous and also shown in Appendix A. 6 While x v also depends on m, the latter can be substituted out based on 7), which yields x v = lnδ) + 1 ρ β 1 + v ρ 1 β 1 ) ]. 10

11 .3 Asset Pricing Determining asset prices requires the pricing kernel, which equals the representative agent s marginal rate of intertemporal substitution IMRS), M t+1 Vt/ Ct+1 V t/ C t C t and C t+1, respectively, and substituting back into M t+1 yields M t+1 = β v m. Differentiating 4) with respect to ) 1 ρ µ t V t+1. Using the implicit function theorem to differentiate µ t in 5) with respect to V t+1 gives v ) α 1 m e α 1) c t+1 1+θD t+1 1+θδ α ED t+1], so that the IMRS is given by µ t V t+1 = M t+1 = β v } e α 1) ct+1 {{} 1 + θ1{ c t+1 x v }), 9) }{{} Small risks Tail risks where β v β v/m)α ρ 1+θδ α ED t+1] is a constant that depends on the equilibrium value of v. The pricing kernel takes a very simple form, consisting of a constant, a small risks factor that is familiar from the EU case, and a tail risks factor that overweights left tail outcomes of consumption growth by a factor of 1 + θ). Note that the i.i.d. environment implies that the pricing kernel is a function of consumption growth only. The benchmark calibration sets α = 1, so that the small risks term drops out and the pricing kernel reduces further to a step function see Figure I). Given a solution for v, asset prices can be computed analytically based on the Euler equation EM t+1 R i t+1] = 1, 10) where R i denotes the gross return of asset i. Using this approach, Appendix A proves Proposition 1 Analytical Asset Prices) Denote the τ-period pricing kernel by M t:t+τ = τ h=1 M t+h, the risk-free rate by R f t:t+τ = 1/EM t:t+τ ], and the risk-neutral expectation of a t + τ)-measurable ] random variable X t+τ by E X t+τ ] = E Mt:t+τ EM t:t+τ ] X t+τ. Let S t denote the ex-dividend value of the dividend claim at time t. Equations A.17), A.0), A.33), A.39), and A.4)-A.7) in Appendix A provide analytical expressions for the scalar constants B, Ψ, Γ, Λ, and the univariate functions I 1 X) to I 4 X) in terms of the equilibrium value-to-consumption ratio v and the parameters of the model. Given these, 11

12 Bonds. A τ-period risk-free bond has a price of EM t:t+τ ] = B τ, where B EM t+1 ] is the price of a one-period bond. The yield of a τ-period bond is ln B, independent of τ. Dividend Strips. The price of a τ-period dividend strip equals EM t:t+τ D t+τ ] = Ψ τ D t, where Ψ EM t+1 e dt+1 ] is the price-dividend ratio of a one-period dividend strip. The one-period div strip) holding period return of a τ-period dividend strip equals R t+1 = e dt+1 /Ψ, independent of τ. Equity. The market has a price-dividend ratio of e dt+1, and a cum-dividend return of R t+1 St+1+Dt+1 S t of a dividend strip. S t D t = Ψ St+1 1 Ψ, an ex-dividend return of S t = = e dt+1 /Ψ, equivalent to the return Forwards. The price of a 1-period forward on the dividend claim equals F t E S t+1 ] = F S t, where F = Ψ/B is the forward-to-spot ratio. Options. The price of a 1-period European put option with a strike price of K equals P K) = EM t+1 max{0, K S t+1 }], and it can be computed as P K) = PX) F t, where PX) EM t+1 max{0, X e dt+1 /F}] ] =β v e α 1)g σω)+ 1 α 1) σ ) X {I 1 X) + θi 3 X)} eg ϕσω {I X) + θi 4 X)} F is the put-to-forward ratio and X K/F t. The corresponding call-to-forward ratio CX) follows from the normalized put-call parity, CX) = PX) + B1 X) VIX. The squared VIX index is defined as V IX t R f t+1 Ft 0 P t K) K dk + F t ) C t K) K dk. 1

13 Defining Γ EM t+1 d t+1 ], it can be computed as ] V IX = ln E St+1 E ln S t SVIX. The squared SVIX index is defined as SV IX t R f t:t+τ Ft 0 St+1 S t P t K) St dk + F t Defining Λ EM t+1 e dt+1 ], it can be computed as ] SV IX =B Var St+1 = BΛ Ψ. S t )]) = ln F Γ/B). ] C t K) St dk. Proposition 1 shows that analytical asset pricing formulae, including a formula for option prices, can be obtained despite the presence of exponentially-distributed shocks in the model. Additionally, it shows that both the term structure of real interest rates and the term structure of the equity premium are constant functions of maturity, in line with the empirical evidence in Ang et al. 008) and van Binsbergen et al. 01). 7 While flat term structures are a feature of any i.i.d. model, the challenge lies in simultaneously generating large risk premia. Models that generate large premia from dynamics features often have counterfactual implications for term structures, 8 whereas i.i.d. models typically generate counterfactually small risk premia. The GDA model breaks this tension because it can generate large risk premia based on small, i.i.d. shocks alone. 7 van Binsbergen et al. 01) document that the term structure of the equity premium has a significantly positive intercept, and a slope that is negative but insignificantly different from zero. Boguth et al. 01) and Schulz 016) question some of the findings of van Binsbergen et al. 01), including the negative slope, but seem to agree that risk premia and Sharpe ratios of near-term dividend strips are significantly positive and likely comparable to those of the market. 8 Both long run risk models and recursive utility models with a time-varying probability of rare disasters imply that the term structure of real interest rates is steeply downward sloping in typical calibrations. This is shown by Beeler and Campbell 01) and Backus et al. 014), who argue that these models are inconsistent with the empirical evidence. van Binsbergen et al. 01) show that the models of Campbell and Cochrane 1999) and Bansal and Yaron 004) imply risk premia of close to zero for near-term dividend strips, in contrast to a significantly positive premium in the data. 13

14 3 Quantitative implications This section shows numerical results for different model calibrations. The agent s decision interval is assumed to be monthly, and I target moments of annual time-aggregated) quantity and return data over , and moments of monthly option prices over To compute moments in the model, I simulate one million samples of equivalent length as the data, compute the moment of interest in each sample, and then report the median value. 3.1 Calibration Data. Consumption is measured as the sum of real nondurables and services consumption per capita from the Bureau of Economic Analysis. Dividends equal the sum of the cash dividends and repurchases of all common stocks in CRSP SHRCD 10 or 11), where repurchases are computed as in Bansal et al. 005). Including repurchases is important because they have become a significant component of firms total payout since the 1980 s. 9 The value-weighted average of the same firms returns serves as a measure of the market return. Real risk-free rates are latent, and the presence of an inflation risk premium in nominal rates implies that they cannot be measured without imposing additional assumptions. For simplicity, I follow the common approach of approximating the annual nominal risk-free rate by compounding four quarterly risk-free rates that are inferred from T-bills. 10 Dividends, market returns, and risk-free rates are converted to real terms using the CPI. The option dataset is discussed in Section Fama and French 001), Grullon and Michaely 00), and Brav et al. 005) present empirical evidence on the trend in firms payout policies and discuss potential explanations. Boudoukh et al. 007) illustrate that the total payout yield is a better predictor of returns in the time series than the dividend yield, and also more strongly related to cross-sectional differences in expected returns. 10 Data for monthly T-bills is not available for the whole sample. However, Ang et al. 008) estimate that the real term structure in the U.S. has been flat on average in post war data, so that the average 3-month rate should provide a good approximation of the average 1-month rate. 14

15 Endowment calibration. The model s implications for moments of annual consumption and dividend growth are shown in Table I. I set g = /1 to match the average consumption growth rate, and assume the same growth rate for dividends. The volatility of consumption and dividend growth are matched by setting σ = 0.063/ 1 and ϕ = I assume independent Gaussian shocks ϱ = 0) and match the unconditional correlation between consumption and dividend growth by setting the mixture parameter to ω = The four free endowment parameters are thus uniquely pinned down by four quantity moments. As is well-known, time-aggregation of an i.i.d. random walk results in positive first-order correlation Working 1960). The model calibration results in an autocorrelation of 0.4 for both consumption and dividend growth, close to the empirical autocorrelation of 0. for dividends. Consumption growth has a higher autocorrelation in the data, but Kroencke 017) pointed out out that this is a result of data filtering in the National Income and Product Accounts NIPA), and showed that unfiltered consumption data has about the same autocorrelation as dividends. Overall, the random walk model thus provides a very good and parsimonious description of the basic quantity moments in the data. In addition to the benchmark calibration with mixture shocks, I consider an endowment with Gaussian shocks to illustrate the model mechanism. In this case, exponential shocks are eliminated ω = 0) and the unconditional correlation between consumption and dividend growth is matched by assuming a correlation of ϱ = between Gaussian shocks. The remaining three endowment parameters are calibrated to the same values as in the benchmark. Table I shows that this calibration has almost identical implications for the basic moments of annual consumption and dividend growth. 11 Different from the mixture model, however, the Gaussian shock specification implies a very low downside correlation between consumption and dividend growth see Figure I in the introduction. 11 The two calibrations imply identical values for the mean, variance, and unconditional) correlation of monthly growth rates. Slight differences in annual moments arise from time aggregation. 15

16 Table I: Economic fundamentals E c] σ c] AC1 c] E d] σ d] AC1 d] corr c, d] A: Data B: Mixture shocks C: Gaussian shocks Annual growth rates are computed by aggregating levels to yearly, then computing the growth rate, then taking logs. Moments represent small sample medians based on 1 million samples of equivalent length as the sample. Parameters equal ω = and ϱ = 0 for the mixture calibration and ω = 0 and ϱ = for the Gaussian calibration. Common parameters are g =.0183/1, σ =.063/ 1, and ϕ = Data moments for dividends are computed separately for 1 overlapping annual subsamples that are each shifted by one month and then averaged. Preference calibration. For each endowment calibration, Panel A of Table II shows one model calibration with GDA risk preferences and one calibration with EU risk preferences. Time preferences are identical in all four cases: the elasticity of intertemporal substitution EIS) is assumed to equal 1/1 ρ) = 1.5 and the rate of time preference is 1 percent per year β = /1 ). Given endowment and time preference parameters, risk preferences are calibrated to match the historically observed equity premium. For EU calibrations and 4), this mapping is exact as risk preferences are characterized by the single parameter α. Of course, the calibrations based on EU risk preferences rely on a degree of risk aversion above twenty) that is typically considered unreasonable, but these calibrations merely serve as a means to illustrate the model mechanism. For GDA calibrations 1 and 3), I set α = 1 so that the utility function has no curvature, the only source of risk aversion consists of disappointments, and the pricing kernel reduces to a step function. This assumption eliminates one free parameter and it simplifies the intuition behind the utility function, but it is inconsequential for the main results. In particular, moderate degrees of curvature have a negligible effect on asset pricing moments, similar to their effect in the well-known EU case Mehra 16

17 and Prescott 1985). In the GDA calibration with mixture shocks calibration 1), the disappointment threshold δ and disappointment magnitude θ are chosen to match the equity premium exactly, and to match various option pricing moments as well as possible. This mapping is overidentified as preferences provide a single degree of freedom after matching the equity premium, while Sections 3.4 and 3.5 consider a continuum of option moments. The implied disappointment probability of 0.63% per month implies that these events occur about once every 13 years on average. For the GDA calibration with Gaussian shocks calibration 3), δ and θ are chosen to match the equity premium, and to imply the same disappointment probability ED] as the benchmark calibration. The latter property is useful for contrasting the implications of both shock distributions because it ensures a minimal degree of comparability between the associated pricing kernels. 3. Basic asset pricing moments Panel B of Table II shows that all four calibrations are similarly successful at replicating basic asset pricing moments. In addition to matching the equity premium, the model resolves the risk-free rate puzzle due to the assumed high EIS the well-known channel of Weil 1989). Different from models with long run risks, however, the value of the EIS is inconsequential for risk premia due to the i.i.d. environment. 1 Additionally, all calibrations are reasonably successful at replicating the Sharpe ratio and log price-dividend ratio of the market. Return volatility is somewhat lower than in the data, but that is appropriate because the i.i.d. assumption does not generate any excess volatility Note that, for any value of ρ, the pricing kernel in 9) is observationally equivalent to one with a different value of ρ and a compensating adjustment of β and δ. 13 Differences in the volatility of annual dividend growth in Table I) and annual returns in Table II) result from i) differences in the time-aggregation methods for dividends and returns ii) the fact that returns are cum-dividend and reported in levels whereas dividend growth is in log units. The choice to report returns in levels was made to allow for an easy comparison with the equity premium decomposition considered in Section

18 Table II: Asset prices Data 1) ) 3) 4) GDA EU GDA EU Mixture Mixture Gaussian Gaussian A: Risk preferences θ δ α Implied ED] %) B: Basic asset prices ER R f ] %) σr R f ] %) SRR] Epd] ER f ] %) C: Martin 017) diagnostic V IX SV IX D: Bollerslev and Todorov 011) decomposition ERP 10%) ER R f ] %) Panel A: Risk preference calibrations and the monthly disappointment probability they imply. Time preferences parameters are β = /1 and ρ = 1 1/1.5 in all four calibrations. Endowment calibrations are shown in Table I. Panel B: Time-aggregated annual asset pricing moments. Model moments equal small sample medians based on 1 million samples of equivalent length as the sample. Data moments are computed separately for twelve overlapping annual subsamples that are each shifted by one month and then averaged. Panel C: Difference between the monthly VIX and SVIX indices, both expressed in annualized percent. The data moment represents the average daily value over Panel D: Fraction of the equity premium due to states with short-horizon returns of -10% or less. The empirical value over is taken from Bollerslev and Todorov 011). See the main text for details. 18

19 Despite implying similar moments, the four calibrations rely on quite different mechanisms. A comparison between GDA calibrations 1 and 3 shows that a much higher disappointment magnitude of θ = 13. versus θ = 5.99) is required to match the equity premium when shocks come from a normal rather than a mixture distribution, despite an identical disappointment probability of 0.63% per month. The reason is that the bivariate normal distribution results in a much lower downside correlation than the mixture distribution see the left panel of Figure I. When bad news for equity holders low dividend growth) is less likely to occur in times of high marginal utility low consumption growth), these times have to be more painful higher θ) to yield a large equity premium. Relative to GDA, EU implies that the pricing kernel takes on elevated values for a larger part of the consumption growth distribution see the right panel of Figure I. The equity premium therefore relies less on a high correlation between consumption and dividend growth in the extreme tail, and more on their unconditional comovement. Because the two endowments are calibrated to the same unconditional correlation, the EU model relies on similar degrees of risk aversion α = 6.46 versus α = 3.37) to match the equity premium in both cases. These comparisons show that the importance of tail dependence in innovations for asset prices crucially depends on the investor s risk preferences. I return to this observation in Section 3.6, where I show that the equity premium reflects compensation for very different risks under different risk preference specifications. 3.3 Risk aversion To calibrate the model, I set preference parameters to match asset pricing moments. This section assesses whether the resulting calibrations imply a reasonable degree of risk aversion. In intertemporal models such as the one considered here, the representative investor faces a lottery over random continuation utilities. Because continuation utilities depend on both preference parameters and the endowment process, so does risk aversion. A natural approach for assessing the level of risk aversion in dynamic settings is therefore to compute the risk premium on the 19

20 Table III: Risk aversion 1) ) 3) 4) 5) 6) GDA EU GDA EU EU EU Mixture Mixture Gaussian Gaussian Mixture Gaussian A: Risk preferences θ δ α B: Risk premium %) The table shows the risk premium on the consumption claim for different model calibrations. Time preferences parameters are β = /1 and ρ = 1 1/1.5 in all four calibrations. Endowment calibrations are shown in Table I. consumption claim. Formally, the risk premium is defined as the fraction of consumption that the agent would give up today and at every future date and state) in order to exchange his endowment for an alternative endowment with the same mean consumption growth rate but no risk: v risk premium = 1, 11) vno risk where v no risk is the value-to-consumption ratio for the risk-free endowment. 14 The risk-premium is akin to Lucas s 1987) measure for the welfare costs of business cycles. Table III shows the risk premium for the four model calibrations in Table II, and for two additional calibrations discussed below. With EU preferences, the premium exceeds 80% for both endowment specifications calibrations and 4). While the literature has not established a bound for reasonable levels of the consumption risk premium, the magnitude appears high both on introspective grounds and relative to other models. Specifically, Epstein et al. 014) compute a 14 v no risk can be computed analytically: Iterating on 4) yields v no risk 1 β = 1 βee c ]. Using Lemma ρ.b.i in the appendix, the mean consumption growth rate can be derived as Ee c ] = exp{g σω+0.5σ 1 ω )} 1 σω. 0

21 premium of 57% in the model of Bansal and Yaron 004) case II, γ = 10), 9% in Barro 009), and 65% in Wachter 013). 15 Static risk aversion is high as well in the EU calibrations I consider. The coefficient of relative risk aversion in static gambles, 1 α, exceeds 0 for both EU calibrations, twice the upper bound considered reasonable by Mehra and Prescott 1985). The EU model therefore does not resolve the equity premium puzzle, even when cash flows feature a high degree of tail dependence with consumption. In contrast, both GDA calibrations result in a risk premium of around 40%, which is at the lower end of the values in leading theories. To provide a direct comparison with the static measure of risk aversion in the EU model, Table III considers two additional EU calibrations. First, an EU model with a static risk aversion of 1 α = 7.11 and mixture shocks calibration 5) implies the same risk premium as the GDA model with mixture shocks calibration 1). Because both models rely on the same endowment calibration, an identical risk premium implies an identical degree of risk aversion. The benchmark GDA calibration thus implies the same level of risk aversion as an EU model with a risk aversion of about seven. Second, an EU model with 1 α = and Gaussian shocks calibration 6) implies the same risk aversion as the GDA model with Gaussian shocks calibration 3). For both endowments, the GDA model thus implies a level risk aversion that is generally considered low. In particular, it implies less risk aversion than the calibrated model of Bansal and Yaron 004). The GDA model therefore resolves the equity premium puzzle of Mehra and Prescott 1985) For the BY and Wachter models, Epstein et al. 014) compute the risk premium for the unconditional mean of the state variables. Concavity of the utility function implies that this number represents a lower bound for the mean of the risk premium across states. 16 Cochrane 017) surveys the macro-finance literature and argues that, in order to achieve a full resolution of the equity premium puzzle, a model must replicate the observed equity premium based on low risk aversion and low consumption volatility over both short and long horizons. He concludes that no model to date has achieved this goal. The resolution suggested here satisfies Cochrane s criteria. 1

22 3.4 Option prices In this section, I use i) implied volatility curves and ii) the VIX-SVIX diagnostic suggested by Martin 017) to examine option prices in the model. Data. A panel of option price quotes over January, 1990 to December 31, 015 is obtained from Market Data Express, the official data vendor of the Chicago Board Options Exchange CBOE). The dataset contains end-of-day information of all European exercise style options written on the S&P 500 underlying SPX). I apply standard filters to remove observations with low liquidity and obvious data errors. Because the model is calibrated at a monthly frequency, I interpolate option prices to a constant maturity of 30 calendar days. Details on the data filters and interpolation method are discussed in the online appendix. I compute data moments separately for each day of the sample and report the average. Dealing with heterogeneous sample periods. For two reasons, it would be inappropriate to use the model s ability to match the level of option prices as a metric of it s empirical success. First, the model was calibrated to fundamentals over , a period over which the volatility of monthly market returns was about three percentage points higher than over the option sample. To the extend that the model captures return volatility over the long sample, it should therefore overshoot the level of option prices that was observed over Second, the model s i.i.d. structure does not generate any excess volatility. Given a reasonable level of cash flow volatility, this channel implies that the model should undershoot the level return volatility and thus the level of option prices. To avoid ad hoc choices in dealing with this issue, I decide to focus on the model s ability to match relative option prices, and adjust the level of option prices in the model as follows. 17 First, I compute the average Black and Scholes 1973) implied volatility IV) of 17 An alternative approach, followed by Bekaert and Engstrom 017), is to find a model-implied mapping between option moments and other moments that can be observed over the longer sample. Option moments in the earlier sample can then be computed based on the fitted value from this mapping. I do not follow this approach here because time-series variation in the implied volatility curve is not known to be well-described

23 an at-the-money ATM) option in the model and in the data. Next, I adjust IVs in the model by multiplying the IV at every strike price by the same ratio σ AT M data /σat model M. This adjustment implies that the ATM volatility level in the model perfectly matches that in the data by construction. However, because the same adjustment factor is applied at all moneyness levels, the adjustment does not help the model s ability to match the empirically-observed shape of the IV curve, i.e. it does not help in matching relative option prices. For example, a log-normal model would imply a horizontal implied volatility curve before and after the adjustment. Option returns considered in the next section) require no adjustment, as the proportional adjustment would apply equally to the price and payoff of the option. Implied volatilities. Figure II shows average implied volatilities in the data and for the four model calibrations in Table II. The benchmark calibration top-left panel) provides a near-perfect match for the implied volatility curve, both in terms of its slope and its lack of) curvature. A comparison with the alternative calibrations shows that this result is mostly a consequence of downside risk in cash flows. In particular, the calibration with Gaussian shocks generates a nearhorizontal curve for both preference specifications right panels), while the model with mixture shocks and EU preferences bottom-left panel) generates a volatility smirk that is almost as steep as in the data. These findings suggests that a significant part of the negative skewness in the risk-neutral distribution can indeed be explained by physical tail risk. A high price of tail risk, as implied by GDA preferences, is not sufficient for capturing the shape of the implied volatility curve. The benchmark) model s ability to capture this challenging data moment is noteworthy because downside risk in cash flows is controlled by a single parameter, the mixture parameter ω. In turn, ω was chosen to match the unconditional correlation between consumption and dividend growth, rather than to match asset pricing moments. Does the mixture distribution imply a realistic amount of tail risk? Calibration 1 implies a skewness of for monthly log market returns, very close to the data value of over the option sample. A second, more detailed assessment by any variables. 3

24 EU preferenecs GDA preferences 30 5 Mixture shocks Data Model 30 5 Gaussian shocks Moneyness Moneyness Moneyness Moneyness Figure II: Implied volatilities. Average IVs in annualized percent) for the 1-month maturity are plotted against moneyness. Each panel corresponds to one of the four calibrations in Table II. Moneyness is measured in volatility units as lnk/f ) SV IX. Data curves are computed separately for each day in the sample and then averaged. follows in Section 3.5, where I look at option returns. Martin 017) proposes the difference between the VIX index and a related simple VIX index, or SVIX, as a means to quantify deviations from log-normality. The squared VIX index captures the risk-neutral entropy of ex-dividend market returns, a measure of dispersion that can be expressed as a weighted average of variance and higher moments. 18 Similarly, the squared SVIX index measures the risk-neutral variance of ex-dividend market returns. Intuitively, the difference between the VIX 18 See Backus et al. 011) for a great discussion of entropy and its properties in the asset pricing context. 4

25 EU preferences GDA preferences 0-0 Data Model Mixture shocks 0-0 Gaussian shocks Moneyness Moneyness Moneyness Moneyness Figure III: Option returns. Average put option returns in %/month) for the 1-month maturity are plotted against moneyness. Each panel corresponds to one of the four calibrations in Table II. Put returns are given by max{0, K S t+1 }/P t K), and moneyness is measured in volatility units as lnk/f ) SV IX. Data curves are computed separately for each day in the sample an overlapping sample) and then averaged. and SVIX indices therefore captures the magnitude of higher risk-neutral moments. Both VIX and SVIX equal weighted averages of option prices along the strike dimension see the definition in Proposition 1. Different from option prices, however, both indices are straightforward to compute analytically in any model by relying on their respective interpretation as entropy and variance. VIX minus SVIX thus allows for an easy comparison of the importance of higher moments across models. Martin proves that log-normal models such as Campbell and Cochrane 1999) and Bansal and Yaron 004) imply that the difference between VIX and SVIX is always negative see 5

26 his Result In Martin s daily sample, the difference is positive on all days, typically far from zero, and has an average of My daily sample spans , and I find an average of 0.60 with no negative values either. Martin further illustrates that the variable rare disaster model of Wachter 013) implies a value that is an order of magnitude larger than in the data. A typical calibration of the rare disaster model thus implies considerably more tail risk than implied by options, while log-normal models imply considerably less. 0 Panel D of Table II shows the difference between VIX and SVIX indices in the four model calibrations. The benchmark calibration 1) yields a difference of 0.64, very close to the empirical value of Results for the remaining calibrations mirror the findings for implied volatilities. The model with EU preferences and mixture shocks calibration ) implies a VIX-SVIX of 0.41, reasonably close to the empirical value, whereas the calibrations with Gaussian shocks calibrations 3 and 4), which resulted in near-horizontal implied volatility curves, also result in a VIX-SVIX value of close to zero. Like average option prices, Martin s diagnostic thus shows that a large part of the risk-neutral skewness can be explained by physical tail risk. 3.5 Option returns To assess the relative importance of the quantity and price of tail risk, it is informative to consider option returns. In particular, if the high price of out-of-the-money put options solely reflected a large quantity of tail risk high expected cash flows), while the price of tail risk was low as implied by EU risk preferences, these options would have relatively low returns in absolute value). In contrast, the combination of a small quantity of tail risk and a high price of tail risk would result 19 Martin 017) shows that even extensions of the long run risks model that were explicitly designed to capture the variance premium Bollerslev et al. 009 and Drechsler and Yaron 010) imply a value for VIX-SVIX of essentially zero. The richer endowment specifications in these models therefore do not suffice for generating an empirically realistic degree of non-normality under the risk-neutral measure. 0 This conclusion confirms the findings of Backus et al. 011), who show that the Barro 006) model generates an implied volatility curve that is significantly steeper than in the data. 6