The Dynamic Power Law Model Bryan Kelly University of Chicago Abstract I propose a new measure of common, time-varying tail risk for large cross sections of stock returns. Stock return tails are described by a power law in which the power law exponent is allowed to transition smoothly through time as a function of recent data. It is motivated by asset pricing theory and is estimable via quasi-maximum likelihood. Estimates indicate substantial time variation in stock return tails, and that the risk of extreme returns rises in weak economic conditions.
1 Introduction The mere potential for infrequent events of extreme magnitude can have important effects on asset prices. Tail risk, by nature, is an elusive quantity, which presents economists with the daunting task of explaining market behavior with rarely observed phenomena. This crux has led to notions such as peso problems (Krasker 1980) and the rare disaster hypothesis (Rietz 1988; Barro 2006), as well as skepticism about these theories due to the difficulty in testing them. The goal of this paper is to devise a panel approach to estimating conditional tail risk in stock returns. I assume that tail risks of all firms are driven by a common underlying process. I arrive at a conditional tail estimate by turning to the cross section of extreme events at each point in time. In large cross sections, a small number of stocks are likely to experience a tail event each period, and these are informative about the common conditional tail. Because extreme events occur infrequently, there is an inherent difficulty in estimating dynamic tail risk models from univariate time series. Exploiting the cross section bypasses data limitations faced in univariate approaches. Define the tail as the set of return events exceeding some high threshold u. I assume that the tail of asset return i behaves according to P (R i,t+1 > r R i,t+1 > u and F t ) = ( ) r ai ζ t. (1) u Equation (1) states that extreme return events obey a power law. The key parameter of the model, a i ζ t, determines the shape of the tail and is referred to as the tail exponent. Low values of a i ζ t correspond to fat tails and high probabilities of extreme returns. Past finance research has studied unconditional power law tails of returns. In contrast, Equation (1) is a statement about the conditional return tail. I allow ζ t to vary over time as a function of the conditioning information set F t. The time-varying component of the tail exponent, ζ t, is common to all assets and may therefore be viewed as economy-wide extreme event risk in returns. I refer to the tail structure in (1) as the dynamic power law model. I propose a specification for the dynamics of the power law exponent that captures autoregressive behavior of stock return extreme event probabilities. The model is straightforward to estimate and test with quasi-maximum likelihood estimation (QMLE) techniques. This accommodates dependence among returns as well as heterogeneity in tail exponents. I apply the estimator to a large cross section of US stock returns in the post-war sample. This data set demonstrates substantial variation in tail risk. Tail risk appears countercyclical, rising following recent negative stock market returns and in periods of high unemployment and low industrial production. I provide an example of an economic model that gives rise to a dynamic power law 1
in stock returns. I show that, in equilibrium, stock return tail risk is intimately linked to focal asset pricing quantities such as expected stock returns and dividend-price ratios. This theory is consistent with my empirical finding that tail risk and the aggregate dividend-price ratio comove strongly. 1.1 Literature Review Since at least Mandelbrot (1963) and Fama (1963), economists have argued that unconditional distributions of financial returns are heavy-tailed. In this paper I build on a large econometric literature that studies the extremal properties of asset returns in dynamic conditional models. Conditional return distributions are often modeled via GARCH processes with heavy-tailed i.i.d. innovations. Perhaps the earliest example of this approach is Bollerslev (1987) who incorporates Student t shocks in GARCH. Subsequent developments, such as the EVT-GARCH approach of McNeil and Frey (2000) have performed especially well in risk management applications (see, e.g., Kuester, Mittnik and Paolella (2006)). The current paper differs in explicitly allowing for timevariation in the power law exponent, above and beyond any volatility dynamics in returns. A second related line of research studies estimation of conditional tail indices that are a function of observables. Examples of this approach include Gardes and Girard (2008, 2010), Gardes and Stupfler (2013) and Goegebeur, Guillou and Schorgen (2013), among many others. In my approach, the power law exponent takes a GARCH-like autoregressive form that depends on the history of realized returns for all assets in the panel. The rest of the paper is organized as follows. In Section 2, I present the model and discuss estimation. I report empirical results for the dynamic tail behavior of US stock return in Section 3. In Section 4, I discuss an economic model in which the tail behavior of stock returns is fundamentally linked to key quantities studied in the financial economics literature such as the equity risk premium and price-dividend ratio, and Section 5 concludes. 2 Empirical Methodology 2.1 The Dynamic Power Law Model Assumption 1 parameterizes the evolution of ζ t in Equation (1). Assumption 1 (Dynamic Power Law Model). Let R t = (R 1,t,..., R n,t ) denote the cross section of returns in period t. 1 Let K t denote the number of R t elements exceeding threshold u in period 1 R denotes arithmetic return, which directly maps the tail distribution here with theoretical results in Section 2. 2
t. 2 The tail of individual returns for stock i (i = 1,..., n), conditional upon exceeding u and given information F t, obeys the probability distribution 3 with corresponding density F u,i,t (r) = P (R i,t+1 > r R i,t+1 > u, F t ) = f u,i,t (r) = ζ t+1 u ( ) r (1+ζt+1 ). u ( ) r ζt+1 u The common element of exponent processes, ζ t+1, evolves according to and the observable update of ζ t+1 is 1 1 1 = π 0 + π 1 ζ t+1 ζ upd + π 2 (2) ζ t t 1 ζ upd t = 1 K t ln R k,t K t u. k=1 Equation (2) shows that ζ t is (t 1)-measurable. The timing convention used here is thus consistent with Equation (1). Also note that the a i coefficients do not enter Assumption 2. Below I discuss how fixing the a i parameters to equal one for all stocks results in a quasi-maximum likelihood estimator (QMLE) of the more general model that allows for heterogeneous a i. In particular, the QMLE, which ignores a i heterogeneity, remains a consistent estimator of the π 1 and π 2 parameters even when different a i s enter the data generating process. The benefit of this approach is that one can consistently estimate the ζ t dynamics without having to estimate the a i nuisance parameters (which may number in the thousands). The above specification exploits the comparatively rich information about tail risk in the cross section of returns, as opposed to relying, for example, on short samples of high frequency univariate data or options prices. In particular, ζ t is a deterministic function of past extreme returns and may be viewed as an extreme value analogue of GARCH (Engle (1982), Bollerslev (1986)). Process (2) is convenient for capturing autoregressive behavior in the power law exponent, while possessing a quasi-likelihood function that is simple to maximize (as discussed in the next section). Conditioning information enters the evolution of 1/ζ t+1 via the update term 1/ζ upd t, which is a This is without loss of generality as the model is equally applicable to log returns. In estimation, I work with daily returns. Because of the small scale of daily returns, the approximation ln(1 + x) x is accurate to a high order and the distinction between arithmetic and log returns is negligible. 2 I assume for notational simplicity that these are the first K t elements of R t. This is immaterial since elements of R t are exchangeable in my estimation. 3 This formulation applies similarly to the lower tail of returns. 3
summary statistic calculated from the cross section of tail observations on date t. The role of the update is to summarize information about prevailing tail risk from recent extreme return observations. I assume that the update is equal to Hill s (1975) estimator applied to the time t cross section of extreme events. Hill s estimator is a maximum likelihood estimator of the crosssectional tail distribution and serves as a summary of tail risk from the cross section each period. To see why this makes sense as an update, note that when u-exceedences (i.e., R i,t /u) obey a power law with exponent a i ζ t, the log exceedence is exponentially distributed with scale parameter a i ζ t. By the properties of an exponential random variable, E t 1 [ln(r i,t /u)] = 1/(a i ζ t ). As a consequence, the expected value of update 1/ζ upd t is the cross-sectional harmonic average tail exponent, E t 1 [ 1 K t K t k=1 ln R k,t u ] = 1 āζ t, where 1 ā 1 n n i=1 1 a i. (3) The left hand side is an average over the entire cross section due to the fact that the identities of the K t exceedences are unknown at time t 1. 4 This important property will be used to establish consistency and asymptotic normality of the dynamic power law estimation procedure that follows. Recursively substituting for ζ t shows that 1 ζ t+1 = π 0 1 π 2 + π 1 j=0 π j 2 1 ζ upd t j thus, 1/ζ t+1 is an exponentially-weighted moving average of daily updates. Estimating fully-specified versions of this model is difficult, and essentially infeasible without multistep estimation. It requires specifying a dependence structure among return tails and estimating stock-specific a i parameters. Incorporating both considerations adds an enormous number of parameters: Estimating the a i constants adds n parameters while imposing dependence structures like those implied by the models in Section 4 below adds another nk parameters, where K is the number of factors in a given model. 5 These are nuisance parameters when the goal is to estimate aggregate tail risk, as opposed to estimating univariate distributions., 2.2 Estimating the Dynamic Power Law Model My estimation strategy uses a quasi-likelihood approach, and is an example of a widely used econometric method with early examples dating at least back to Neyman and Scott (1948), Berk 4 While the identities of the exceedences are unknown, the number of exceedences is known since the tail is defined by a fixed fraction of the cross section size. 5 In Section 4, stock return tails obey a one-factor structure. 4
(1966), and the in-depth development of White (1996). The general idea is to use a partial or even mis-specified likelihood to consistently estimate an otherwise intractable model. The proofs that I present can also be thought of as a special case of Hansen s (1982) GMM theory. To avoid the nuisance parameter problem, I treat assets as though they are independent with identical tail distributions each period. The independence assumption avoids the need to estimate factor loadings for each stock, and the identical assumption avoids having to estimate each a i coefficient. These simplifications, however, alter the likelihood from the true likelihood associated with Assumption 1, to a quasi -likelihood, written below. I show that maximizing the quasi-likelihood produces consistent and asymptotically normal estimates for the parameters that govern tail dynamics, π 1 and π 2. Ultimately, the estimated ζ t series is shown to be the fitted cross-sectional harmonic average tail exponent. Since the average exponent series differs from ζ t only by the multiplicative factor ā, the two are perfectly correlated. Before stating the main proposition I discuss two important objects, the log quasi-likelihood and the score function (the derivative of the log quasi-likelihood with respect to model parameters). I refer to the tail model in Assumption 1 as the true model. Suppose, counterfactually, that all returns in the cross section share the same exponent, which is equal to the cross-sectional harmonic average exponent. Then the tail distribution of all assets becomes with corresponding density ( ) āζt+1 Ri,t+1 F u,i,t (R i,t+1 ) = u f u,i,t (R i,t+1 ) = āζ t+1 u ( Ri,t+1 u ) (1+āζt+1 ). Tildes signify that this distribution is different than the true marginal, F u,i,t. Under cross-sectional independence, the corresponding (scaled) log quasi-likelihood is L({R t } T t=1; π) = 1 T T 1 ln f(r t+1 ; π, F t ) = 1 T t=0 T 1 t=0 K t+1 k=1 ( ln āζ t+1 (1 + āζ t+1 ) ln R ) k,t+1, (4) u u where u-exceedences are included in the likelihood and non-exceedences are discarded. Define the 5
gradient of ln f t (R t+1 ; π) with respect to π (the time-t element of the score function) as s t (R t+1 ; π) π ln f t (R t+1 ; π) = ln f t (R t+1 ; π) π ζ t+1 ζ t+1 ( K Kt+1 t+1 = ā ln R ) k,t+1 π ζ t+1. (5) u ζ t+1 With these expressions in place, I present the central econometric result. Proposition 1. Let the true data generating process of {R t } T t=1 satisfy the dynamic power law model of Assumption 1 with parameter vector π. Define the quasi-likelihood estimator ˆπ QL as k=1 ˆπ QL = arg max π Π L({R t} T t=1; π). If π is interior to the parameter space Π over which maximization occurs, E[s t (R t+1 ; π)] 0 for π π, and E[sup π Π s t (R t+1 ; π) ] <, then ˆπ QL p π. Furthermore, if E[sup π Π π s t (R t+1 ; π) ] <, 1 T T 1 t=0 s t(r t+1 ; π ) d N(0, G) and E[ π s t (R t+1 ; π )] is full column rank, then T (ˆπ QL π ) d N(0, Ψ), where Ψ = S 1 GS 1, S = E[ π s t (R t+1 ; π )], and G = E[s t (R t+1 ; π )s t (R t+1 ; π ) ]. Proof. The assumptions are standard regularity conditions needed to appeal to the QML development of Newey and McFadden (1994). Before proceeding, I establish a key lemma upon which the remainder of the proposition relies. It shows that s t (R t+1 ; π) (which is based on the mis-specified model Fu,t ) has expectation equal to zero given that the true data generating process satisfies Assumption 1. Lemma 1. Under Assumption 1, E[s t (R t+1 ; π)] = 0. 6
By the law of iterated expectations, E[s t (R t+1 ; π)] = E [ E t [s t (R t+1 ; π)] ] = E E t K K t+1 t+1 ā ln R k,t+1 π ζ t+1 ζ t+1 u k=1 [( Kt+1 = E K ) ] t+1 π ζ t+1 ζ t+1 ζ t+1 = 0. The second equality follows from expression (5) and the t-measurability of ζ t+1. The third equality follows from Equation (3), proving the lemma. The remaining arguments for consistency and asymptotic normality exactly follow the arguments of Newey and McFadden. 2.3 Volatility and Heterogeneous Exceedence Probabilities Implicit in the formulation above is that each element of the vector R t has an equal probability of exceeding threshold u. However, heterogeneity in individual stock volatilities affects the likelihood that a particular stock will experience an exceedence. Let X be a power law variable such that P (X > u) = bu ζ. The u-exceedence distribution of X is P (X > x X > u) = ( x ζ. u) Now consider a volatility rescaled version of this variable, Y = σx. The exceedence probability of Y equals b ( ) u ζ, σ different than that of X. When σ > 1, Y has a higher exceedence probability than X. However, the shape of Y s u-exceedence distribution is identical to that of X. A reformulation of the estimator to allow for heterogeneous volatilities is easily established. Let each stock have unique u-exceedence probability p i, and consider the effect of this heterogeneity on the expectation of the tail exponent update. In this case, the expectation is no longer the harmonic average tail exponent, but is instead the exceedence probability-weighted average exponent, E t 1 K t+1 K t+1 k=1 ln R k,t+1 u = 1 ζ t+1 where ω i = p i / j p j. The entire estimation approach and consistency argument outlined above proceeds identically after establishing this point. The ultimate result is that the fitted ζ t series is no longer an estimate of the equal-weighted average exponent, but takes on a volatility-weighted character due to the effect that volatility has on the probability of tail occurrences. Another potential concern is contamination of tail estimates due to time-variation in volatility. I address this in my main specification by allowing the threshold u to vary over time. My procedure n i=1 ω i a i, 7
selects u as a fixed q% quantile, û t (q) = inf { R (i),t R t : q 100 (i) } n where (i) denotes the i th order statistic of (n 1) vector R t. In this case, u expands and contracts with volatility so that a fixed fraction of the most extreme observations are used for estimation each period, nullifying the effect of volatility dynamics on tail estimates. My estimates are based on q = 5 (or 95 for the upper tail). 6 An alternative way to address time-variation in volatility is by pre-filtering each stock return time series to account for its volatility dynamics. While this has the potential to improve estimates of the dynamic power law exponent, substantial measurement-related problems can arise with this approach. Variance-standardization involves dividing returns by an estimate of volatility. The resulting quantity can be highly sensitive to estimation error in the divisor, which can in turn impair tail estimates (as demonstrated in a Monte Carlo experiment below). On the other hand, when the researcher is confident that variance dynamics are accurately estimated (for example, in a sample of stock indices or large, liquid individual stocks), pre-filtering returns may reduce estimation noise in the tail model. An example of this approach is to estimate a volatility time series, ˆσ it, for each asset i with a univariate GARCH model and construct volatility-adjusted GARCH residuals, ˆɛ it = R it /ˆσ it. The dynamic power law model may then be applied to the panel of ˆɛ it rather than R it. I investigate the effect that volatility pre-filtering has on dynamic power law estimates in both simulations and my empirical analysis. In both cases, pre-filtering has fairly minor effects on the estimated tail exponent time series. Therefore, there may be little cost to the simpler approach of estimating the dynamic power law using the raw return data. 2.4 Monte Carlo Evidence I conduct a series of Monte Carlo experiments designed to assess finite sample properties of the dynamic power law estimator. Table 1 shows results confirming that the asymptotic properties derived above serve as accurate approximations in finite samples. They also demonstrate the 6 Threshold choice can have important effects on results. An inappropriately low threshold will contaminate tail exponent estimates by using data from the center of the distribution, whose behavior can vary markedly from tail data. A very high threshold can result in noisy estimates resulting from too few data points. While sophisticated methods for threshold selection have been developed (Dupuis 1999; Matthys and Beirlant 2000; among others), these often require estimation of additional parameters. In light of this, Gabaix et al. (2006) advocate a simple rule fixing the u-exceedence probability at 5% for unconditional power law estimation. I follow these authors by applying a similar simple rule in the dynamic setting. Unreported simulations suggest that q = 1 to 5 (or 95 to 99 for the upper tail) is an effective quantile choice in my dynamic setting. 8
estimator s robustness to dependence among tail observations and volatility heterogeneity across stocks, both of which are suggested by the structural model discussed in Section 4. Table 2 explores the estimator s performance when the true tail exponent is conditionally stochastic. Even though the estimator presented here relies on a conditionally deterministic exponent process, its estimates achieve over 80% correlation with the true tail series on average. Figure 1 explores the effects of variance standardization when returns are conditionally heteroskedastic. I discuss these tests in more detail below. 2.5 Correct Specification of Tail Parameter Evolution The first Monte Carlo experiment I run is designed to assess the finite sample properties of the dynamic power law quasi-maximum likelihood estimator under different dependence and heterogeneity conditions. In all cases, the evolution of the parameter governing tail risk follows Equation (2), and therefore the statistical model s specification of this process is correct. I allow for misspecification in terms of dependence and in the level of the tail exponent across stock. In particular, data is generated by the following process: R i,t = b i R m,t + e i,t where R m,t and e i,t, i = 1,..., n, are independent Student t variates with a i ζ t degrees of freedom. A well-known property of the Student t is that its tail distribution is asymptotically equivalent to a power law with tail exponent equal to (minus) the degrees of freedom. In generating data I therefore set the degrees of freedom equal to ζ t, whose transition is described by Equation (2). The b i coefficients control cross section dependence and heterogeneity in volatility. The a i coefficients control the tail risk heterogeneity across observations. I consider four cases: 1. Independent and identically distributed observations: b i = 0 and a i = 1 for all i, 2. Dependent and identically distributed observations: b i N(1,.5 2 ) and a i = 1 for all i, 7 3. Independent and heterogeneously distributed observations: b i = 0 and a i N(1,.2 2 ) for all i, 4. Dependent and heterogeneously distributed observations: b i N(1,.5 2 ) and a i N(1,.2 2 ) for all i. The cross section size is n=1000 or 2500, and the time series length is T =1000 or 5000. Parameters used to generate data are fixed at π 1 = 0.05 and π 2 = 0.93, with an intercept that ensures the 7 Observations in this case are identical only in terms of their tail exponent. Differences in b i across stocks introduces volatility and dependence heterogeneity. 9
mean value of ζ t is three. In each simulation, the quasi-maximum likelihood procedure described in Proposition 1 is used to estimate the model and its asymptotic standard errors. Summary statistics for parameter and asymptotic standard error estimates are reported in Table 1. Also reported is the time series correlation and mean absolute deviation between the fitted tail series and the true ζ t series, averaged across simulations. The general conclusion of the experiment is that the asymptotic theory of Proposition 1 is a good approximation for the finite sample behavior of the dynamic power law estimator. This is true not only when data are i.i.d., but also when observations are dependent and heterogenous. In all cases, the fitted tail series achieves a correlation of at least 96% with the true tail series. 2.6 Incorrect Specification of Tail Parameter Evolution Equation (2) is a stochastic process because ζ upd t is a function of time t returns. However, ζ t+1 is deterministic conditional upon time-t information. This discrepancy is largely innocuous when the time interval of observations is small. In the limit of small time intervals, tail risk processes in the structural models and the exponent process in the econometric model can be specified to line up exactly. A conditionally deterministic tail exponent process, then, can be thought of as a discrete time approximation to a continuous time stochastic process. The advantage of the approximation is that straight-forward likelihood maximization procedures can be used for estimation. 8 The next experiment proceeds as in the i.i.d. case above, but the true tail diverges from that assumed in the statistical model. In particular, the true degrees of freedom parameter ζ t is conditionally stochastic and follows a first order Gaussian autoregression, 9 ζ t+1 = ζ(1 ρ) + ρζ t + ση t+1, η t+1 N(0, 1). (6) I fix ρ = 0.99, n=1000 and T =1000, and let σ =0.005 or 0.010. The parameter σ governs the variability of the process, and thus the range of tail risk values that the data can experience. Summary statistics for the true process and the fitted process are reported in Table 2, as well as 8 This property is the tail analogue to the relation between GARCH models (in which volatility is conditionally deterministic) and stochastic volatility models. Nelson (1990) shows that a discrete GARCH(1,1) return process converges to a stochastic volatility process as the time interval shrinks to zero. An important result of Drost and Werker (1996) proves that estimates of a GARCH model at any discrete frequency completely characterize the parameters of its continuous time stochastic volatility equivalent. The same notion lies behind treating the process in (2) as a discrete time approximation to a continuous time stochastic process for the tail exponent. 9 Examining the performance of the (conditionally deterministic) dynamic power law model for estimating a stochastic tail process is analogous to evaluating the ability of GARCH to capture the dynamics of a stochastic volatility process. 10
summary statistics for π parameter estimates and their asymptotic standard errors. The deterministic tail process provides accurate estimates even when the true tail parameter is stochastic. The mean absolute error between the fitted and true series ranges from 0.264 to 0.525, and their correlation ranges from 81.8% to 87.5%. 2.7 Adjusting for Individual Stock Volatility Dynamics The next Monte Carlo experiment is designed to evaluate the effects of volatility dynamics on the performance of dynamic power law estimates. I assume that each return series is described by the following GARCH process R i,t = σ i,t ɛ i,t, R m,t = σ m,t ɛ m,t, σ 2 i,t = 0.01 + 0.95(R 2 i,t 1/2 + R 2 m,t 1/2) σ 2 m,t = 0.01 + 0.95R 2 m,t 1 where ɛ innovations are unit-variance Student t variables with ν t degrees of freedom. The ν t process is specified as a sine-wave with range 2.5 to 3.5 and completes two full cycles between time 1 and T. This process is designed to capture persistence and cross section comovement in volatility of individual stocks in addition to persistent variation in the tail exponent of returns. 10 Returns are simulated with panel dimensions of T = 500 or 1,000, and N = 500 or 1,000. For each day t, I estimate the cross section Hill estimate based on the N returns that day and a lower tail threshold equal to the fifth percentile. Estimates use either raw returns (R i,t ), the true volatility-standardized returns (ɛ i,t ), or estimated volatility-standardized returns (ˆɛ i,t = R i,t /ˆσ i,t ). In the latter case, I allow for varying degrees of measurement error summarized as an imperfect correlation between the true and fitted volatility series, ρ = Corr(σ i,t, ˆσ i,t ). I allow ρ to vary between 0.95 and 0.995 in increments of 0.005. Each simulation consists of a set of parameter values for T, N, and ρ and delivers three different Hill estimate time series: Hill R t, Hill ɛ t, and Hillˆɛ,ρ t. From these, I calculate the time series correlation of different Hill estimates, which are then averaged over 2,000 simulations. The results are plotted in Figure 1. Panel A reports the average correlation between the tail estimates from fitted GARCH residuals (Hillˆɛ,ρ t ) and the true residuals (Hill ɛ t) as a function of estimation noise in the fitted GARCH model, ρ. For comparison, I also plot the average correlation between tail estimates from raw returns (Hill R t ) and Hill ɛ t (shown as horizontal lines in the figure). When ρ is lower, the estimated volatilities are noisier, and therefore the standardized returns ˆɛ i,t are also noisier. This estimation noise weakens the accuracy of the tail estimates. We see that when ρ drops below 0.98, one may in fact achieve more accurate tail estimates using un-standardized returns. Panel B reports average correlations 10 Cross section comovement in volatilities enters via the common R 2 m,t 1 term in each σ 2 i,t equation. 11
of the true degrees of freedom, ν t, with Hill R t and Hillˆɛ,ρ t, and the conclusion is similar. When volatilities are very accurately estimated, tail estimates benefit from preliminary return filters to adjust for volatility dynamics. However, amid even a modest amount of estimation noise (e.g. when estimated volatility is 95% correlated with the truth), volatility standardization can do more harm than good to tail estimates. 3 Empirical Results 3.1 Tail Risk Estimates Estimates for the dynamic power law model use daily CRSP data from 1963 to 2008 for NYSE/ AMEX/NASDAQ stocks with share codes 10 and 11. Accuracy of extreme value estimators typically requires very large data sets because only a small fraction of data is informative about the tail distribution. Since the dynamic power law estimator relies on the cross section of returns, I require a large panel of stocks in order to gather sufficient information about the tail at each point in time. The number of stocks in CRSP fluctuates drastically over time. To maintain a more homogeneous sample over time, I use the largest 1,000 stocks by market capitalization each day. 11 I focus my empirical analysis on the tails of raw returns. For robustness, I report tail estimates based on GARCH model residuals in order to adjust for scale differences across returns. I also explore how results change when residuals from the Fama-French (1993) three-factor model are used to estimate tail risk. Both GARCH and factor model adjustments are a means of mitigating the effects of heterogeneity and dependence on the estimator s efficiency. 12 For each set of analyses I compare tail risk estimates for the lower tail to those of the upper tail. Threshold u t is chosen to be the 5% cross-sectional quantile each period. Standard errors are estimated based on the form of the asymptotic covariance matrix Ψ derived in Proposition 1. In particular, Ŝ is the log quasi-likelihood Hessian evaluated at ˆπ parameters and Ĝ is the sample average log quasi-likelihood gradient outer product evaluated at ˆπ. Further details on standard error estimation via likelihood Hessians and gradient outer products may be found in Hayashi (2011). 11 The dynamic power law estimator in Section 2 accommodates changes in size of the cross section over time, highlighting another attractive feature of the estimator. Estimates based on the full cross section of several thousand stocks are very similar to those reported for the largest 1,000. I focus on the latter sample to emphasize that time variation in the tail exponent is not being driven by the rather dramatic variation in number and composition of firms in the entire public universe. 12 As my asymptotic theory results and Monte Carlo evidence show, abstracting from dependence does not affect the estimator s consistency. It may, however, affect the variance of estimates. The asymptotic covariance derived in Proposition 1 accounts for this decreased efficiency, hence test statistics maintain appropriate size. 12
Table 3 reports estimates for the dynamic power law evolution (2). The lower tail exponent of raw returns ζ t varies around a mean of 1.5, with ˆπ 1 = 0.066 and ˆπ 2 = 0.931. Tests of the null hypothesis of constant tail risk are strongly rejected with p-values below 0.001. The fact that ˆπ 1 + ˆπ 2 > 0.99 implies that the tail exponent is highly persistent. The upper tail is slightly fatter (consistent with the results of Jansen and de Vries 1991) and possesses similar persistence and variability. When stock returns are converted to factor model residuals, or pre-filtered for GARCH(1,1) effects, estimation results are qualitatively unchanged. 13 I plot the fitted lower tail series ( ζ t ) based on raw returns in Figure 2 along with point-wise two-standard-error confidence bands. Figure 3 shows the fitted lower tails based on raw returns versus those based on GARCH-standardized residuals. These two sets of tail estimates possess very similar time series behavior and are 82.4% correlated. They are plotted alongside the log dividendprice ratio for the aggregate market. The estimated lower tail risk series appears moderately countercyclical, with the tail of raw returns sharing a monthly correlation of 43% with the log dividend-price ratio. The beginning of the sample sees high tail risk, immediately following a drop in the CRSP value-weighted index of 28% in the first half of 1962 (the first major market decline during the post-war period). Tail risk declines steadily until December of 1968, when it reaches its lowest levels in the sample. This corresponds to a late 1960 s bull market peak, the level of which is not reached again until the mid-1970 s. Tail risk rises throughout the 1970 s, accelerating its ascent during the oil crisis. It remains around its mean value of -2 to -2.5 for most of the remaining sample. Tail risk recedes in the four bull market years leading up to the 1987 crash, rising quickly in the following months. During the technology boom tail risk retreats sharply but briefly, spiking to its highest post-2000 level amid the early 2003 market trough. At this time the value-weighted index was down 49% from its 2000 high and NASDAQ was 78% of its peak. Finally, during the late 2008 financial crisis, tail risk sees a modest climb after falling in the first half of 2008. Figure 4 shows the threshold series u t for raw returns alongside monthly realized volatility of the CRSP value-weighted index. The threshold for the lower tail has a 60% correlation with volatility. As the figure and correlation show, the threshold appears to successfully absorb volatility changes. In Table 4, I report monthly correlations between tail risk estimates and macroeconomic variables. Lower (upper) tail estimates have correlation with unemployment of 60% (50%) indicating countercyclicality in tail risk. As a brief exploration into the determinants of tail risk, I estimate a regression of the tail process on its own lag and lags of a collection of macroeconomic variables, including the aggregate dividendprice ratios, unemployment, inflation, growth in industrial production, the aggregate stock market 13 Mittnik, Paolella and Rachev (2000) investigate the constancy of asset return tail indices after removing GARCH effects and find little evidence for time variation. My findings in favor of time-variation are likely benefitting from increased power based on the inclusion of a large panel of individual stocks. 13
return and realized equity volatility. I show the estimated impacts of these variables on future tail risk in Table 5. Coefficients have been scaled to be interpreted as the response of tail risk ( ζ t, in number of standard deviations) to a one standard deviation increase in the dependent variable. The most important variables for determining tail risk are past unemployment and past market returns. A return that is one standard deviation above its mean predicts that the lower tail becomes thinner by 0.045 standard deviations (t= 3.2), while a one standard deviation increase in past unemployment increases the lower tail exponent by 0.055 standard deviations (t=3.1). Other potentially important determinants of tail risk are the dividend-price ratio and industrial production. 4 Tail Risk in Economic Models of Asset Prices Tail risk plays a central role in our economic understanding of asset price behavior (see, for example, Barro (2006), Drechsler and Yaron (2011), Gabaix (2012), and Wachter (2013)). In this section I develop a stylized asset pricing model based on economic fundamentals (cash flows and investor preferences) in which stock returns obey a dynamic power law in equilibrium. It possesses the interesting feature that the heaviness of stock return tails is intimately connected to central asset pricing phenomena such as a large equity premium. The model underscores the usefulness of dynamic power law models for understanding pricing behavior. Empirical predictions of this model are studied in depth in Kelly and Jiang (2013). Investor preferences over consumption are recursive (Epstein and Zin (1989)). These are summarized by the economy-wide intertemporal marginal rate of substitution, which is also the stochastic discount factor that prices assets in the economy. Written in its log form, this is m t+1 = θ ln β θ ψ c t+1 + (θ 1)r c,t+1 where θ = 1 γ, γ is the risk aversion coefficient, ψ is the intertemporal elasticity of substitution 1 1 ψ (IES), c t+1 is log consumption growth, and r c,t+1 is the log return on an asset paying aggregate consumption as its dividend each period. I assume γ > 1 and ψ > 1, which implies θ < 0 (Bansal and Yaron (2004)). These parameter restrictions ensure that investors have a preference for early uncertainty resolution. 14
Fundamental dynamics of the real economy are described by the following endowment processes: c t+1 = µ + σ t z c,t+1 + Λ t W c,t+1 σt+1 2 = σ 2 (1 ρ σ ) + ρ σ σt 2 + σ σ z σ,t+1 Λ 2 t+1 = Λ 2 (1 ρ Λ ) + ρ Λ Λ 2 t + σ Λ z Λ,t+1 d i,t+1 = µ i + φ i c t+1 + σ i σ t z i,t+1 + q i Λ t W i,t+1. (7) Log consumption growth ( c t+1 ) is subject to two shocks which I assume are independent. The first, σ t z c,t+1, is composed of a standard normal i.i.d. draw, z c,t+1, and is subject to conditional volatility σ t. Conditional variance is itself a stochastic process and evolves as a Gaussian autoregression. 14 This shock captures thin-tailed consumption growth risk. The second shock, Λ t W c,t+1, captures heavy-tailed consumption risk. W c,t+1 is an i.i.d. draw from a unit Laplace distribution f W (w) = 1 exp( w ), w R. 2 W c,t+1 is symmetric and has a higher probability of extreme realizations than the normal distribution. W c,t+1 is scaled by Λ t and, like conditional variance, the square of Λ t evolves as a Gaussian stochastic process. As it evolves, the risk of extreme consumption events fluctuates. High values of the tail risk process Λ t fatten the tails of consumption shocks while low values shrink the tails. The structure of the dividend growth process for equity of firm i mimics that of consumption. It possesses exposure to aggregate fundamental shocks through the term φ i c t+1. It is also subject to an idiosyncratic standard Gaussian shock, z i,t+1, which is scaled by the conditional volatility of consumption. Finally, individual firm cash flows include an idiosyncratic heavy-tailed shock, W i,t+1, which is an independent unit Laplace variable and is interacted with the economy-wide tail risk process, Λ t. The model decouples the behavior of moderately sized cash flow shocks from extreme shocks. The distribution of smaller shocks will depend on both σ and Λ, since both the Gaussian and Laplace distributions can generate small shocks. However, the Laplace shocks are tail-dominant so that the probably law governing extreme events depends principally on the tail risk process. In the appendix, I show that the tail distribution of arithmetic returns for each stock, R i,t+1 = exp(r i,t+1 ), satisfies the dynamic power law structure in Equation (1). In particular, the tail 14 The Gaussian form of the variance evolution is commonly used in the literature (see Bansal and Yaron (2004) and extensions). While in theory it admits negative conditional variance, it s unconditional mean σ 2 is assumed to be sufficiently large relative to the its innovations variance σ 2 σ to ensure that variance becomes negative with a probability arbitrarily close to zero. The same implicit assumption is made for the Λ 2 t process. It is straightforward to modify these processes to strictly preclude negative conditional variances, though these modifications complicate the model solution without providing additional economic insights. 15
distribution of arithmetic returns is asymptotically equivalent to a power law, P t (R i,t+1 > r R i,t+1 > u) P t (R i,t+1 < r R i,t+1 < u) ( ) r ai ζ t u ( ) r ai ζ t u where ( ) denotes asymptotic tail equivalence, a i is a function of deep model parameters, and ζ t = 1/Λ t. I also show that the tail risk process, which governs the conditional power law exponent for returns, also determines other key asset pricing quantities such as the price-dividend ratio and the expected excess stock return: and pd i,t+1 = A i,0 + A i,σ σ 2 t+1 + A i,λ Λ 2 t+1 E t [r i,t+1 r f,t ] + 1 2 V ar t(r i,t+1 ) = β i,c λ c (σ 2 t + 2Λ 2 t ) + β i,σ λ σ σ 2 σ + β i,λ λ Λ σ 2 Λ where A, β, and λ terms are endogenous functions of deep model parameters. To summarize, the structural model predicts a close link between the risk of extreme events in stock returns, extreme events in the real economy, and risk premia across assets and over time. Direct estimation of conditional tail risk from consumption and dividend data is infeasible due to their infrequent observation and poor measurement. The model presented here highlights the path to an alternative estimation strategy since the power law tail exponent of stock returns, ζ t, is also driven by Λ t. Because returns are frequently and precisely observed, estimates of their tail distribution identify the Λ t process. Most importantly, the structure that the model places on the return tail distribution implies that the cross section can be exploited to extract conditional tail risk estimates at high frequencies. 5 Conclusion A measure of extreme event risk is essential to understanding the behavior of asset prices. If this risk changes through time, extreme value techniques based on aggregate data will be incapable of providing conditional tail measures. I present a new dynamic tail risk model that overcomes this difficulty. The model uses the cross section of individual stock returns to inform estimates of conditional tail risk at each point in time. Tests show that extremal risk is significantly time-varying and persistent. This conclusion holds for raw returns as well as factor model residuals. Furthermore, evidence shows that tail risk is countercyclical, rising following drops in the stock market level and 16
amid high volatility and unemployment. perspective of a structural economic model. I show how these results can be understood from the 17
References Bansal, R., and A. Yaron (2004): Risks for the long run: A potential resolution of asset pricing puzzles, Journal of Finance, pp. 1481 1509. Barro, R. (2006): Rare disasters and asset markets in the twentieth century, The Quarterly Journal of Economics, 121(3), 823 866. Berk, R. H. (1966): Limiting behavior of posterior distributions when the model is incorrect, The Annals of Mathematical Statistics, 37(1), 51 58. Bollerslev, T. (1986): Generalized autoregressive conditional heteroskedasticity, Journal of econometrics, 31(3), 307 327. Bollerslev, T. (1987): A conditionally heteroskedastic time series model for speculative prices and rates of return, The review of economics and statistics, pp. 542 547. Bollerslev, T., G. Tauchen, and H. Zhou (2009): Expected stock returns and variance risk premia, Review of Financial Studies, 22(11), 4463 4492. Campbell, J., and R. Shiller (1988): The dividend-price ratio and expectations of future dividends and discount factors, Review of Financial Studies, 1(3), 195 228. Drechsler, I., and A. Yaron (2011): What s vol got to do with it?, Review of Financial Studies, 24(1), 1 45. Drost, F. C., and B. J. Werker (1996): Closing the GARCH gap: Continuous time GARCH modeling, Journal of Econometrics, 74(1), 31 57. Dupuis, D. (1999): Exceedances over high thresholds: A guide to threshold selection, Extremes, 1(3), 251 261. Embrechts, P., C. Kluppelberg, and T. Mikosch (1997): Modelling extremal events for insurance and finance. Springer Verlag. Engle, R. (1982): Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica: Journal of the Econometric Society, pp. 987 1007. Epstein, L., and S. Zin (1989): Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica, 57(4), 937 969. Eraker, B., and I. Shaliastovich (2008): An equilibrium guide to designing affine pricing models, Mathematical Finance, 18(4), 519 543. 18
Fama, E. (1963): Mandelbrot and the stable Paretian hypothesis, Journal of Business, 36(4), 420 429. Fama, E., and K. French (1993): Common risk factors in the returns on stocks and bonds, Journal of Financial Economics, 33(1), 3 56. Gabaix, X. (2012): Variable rare disasters: An exactly solved framework for ten puzzles in macro-finance, The Quarterly Journal of Economics, 127(2), 645 700. Gabaix, X., P. Gopikrishnan, V. Plerou, and H. Stanley (2006): Institutional investors and stock market volatility, Quarterly Journal of Economics, 121(2), 461 504. Gardes, L., and S. Girard (2008): A moving window approach for nonparametric estimation of the conditional tail index, Journal of Multivariate Analysis, 99(10), 2368 2388. (2010): Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels, Extremes, 13(2), 177 204. Gardes, L., and G. Stupfler (2013): Estimation of the conditional tail index using a smoothed local Hill estimator, Extremes, pp. 1 31. Gnedenko, B., and A. Kolmogorov (1968): Limit distributions for sums of independent random variables. Addison-Wesley. Goegebeur, Y., A. Guillou, and A. Schorgen (2013): Nonparametric regression estimation of conditional tails: the random covariate case, Statistics, pp. 1 24. Hansen, L. P. (1982): Large sample properties of generalized method of moments estimators, Econometrica: Journal of the Econometric Society, pp. 1029 1054. Hayashi, F. (2011): Econometrics. Princeton University Press. Hill, B. (1975): A simple general approach to inference about the tail of a distribution, The Annals of Statistics, 3, 1163 1174. Jansen, D. W., and C. G. De Vries (1991): On the frequency of large stock returns: Putting booms and busts into perspective, The review of economics and statistics, pp. 18 24. Kelly, B., and H. Jiang (2013): Tail Risk and Asset Prices, Chicago Booth Working Paper. Krasker, W. (1980): The peso problem in testing the efficiency of forward exchange markets, Journal of Monetary Economics, 6(2), 269 276. Kuester, K., S. Mittnik, and M. S. Paolella (2006): Value-at-risk prediction: A comparison of alternative strategies, Journal of Financial Econometrics, 4(1), 53 89. 19
Mandelbrot, B. (1963): The variation of certain speculative prices, Journal of business, 36(4), 394. Matthys, G., and J. Beirlant (2000): Adaptive threshold selection in tail index estimation, Extremes and Integrated Risk Management, pp. 37 49. McNeil, A. J., and R. Frey (2000): Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach, Journal of empirical finance, 7(3), 271 300. Mittnik, S., M. S. Paolella, and S. T. Rachev (2000): Diagnosing and treating the fat tails in financial returns data, Journal of Empirical Finance, 7(3), 389 416. Nelson, D. B. (1990): Stationarity and persistence in the GARCH (1, 1) model, Econometric theory, 6(03), 318 334. Newey, W., and K. West (1987): A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, 703 708. Newey, W. K., and D. McFadden (1994): Large sample estimation and hypothesis testing, Handbook of econometrics, 4, 2111 2245. Neyman, J., and E. L. Scott (1948): Consistent estimates based on partially consistent observations, Econometrica: Journal of the Econometric Society, pp. 1 32. Rietz, T. (1988): The equity risk premium: A solution, Journal of Monetary Economics, 22(1), 117 131. Wachter, J. A. (2013): Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?, The Journal of Finance, 68(3), 987 1035. White, H. (1996): Estimation, Inference and Specification Analysis, Econometric Society Monographs. Cambridge University Press. 20
Appendix A Economic Model Derivations I solve for prices in the model with procedures commonly employed in consumption-based affine pricing models, following Bansal and Yaron (2004), Eraker and Shaliastovich (2008), and Bollerslev, Tauchen and Zhou (2009), among others. 15 The first result proves that log valuation ratios in the economy are linear in the tail risk process. The log wealth-consumption ratio and log price-dividend ratio for asset i are linear in state variables, wc t+1 = A 0 + A σ σ 2 t+1 + A Λ Λ 2 t+1 pd i,t+1 = A i,0 + A i,σ σ 2 t+1 + A i,λ Λ 2 t+1. (8) These equilibrium equations are solved by positing valuation ratios that are log-linear and proceeding by the method of undetermined coefficients to obtain expressions for each A term in the log wealth-consumption ratio. The derivation begins from the Euler condition E t [exp(m t+1 + r c,t+1 )] = 1, and substitutes for r c,t+1 using the Campbell-Shiller identity, r c,t+1 = κ 0 +κ 1 wc t+1 wc t + c t+1. Evaluating the expectation involves computing the cumulant generating function of a Laplace variable. The following property of the Laplace distribution is informative for this end. Lemma 2. The cumulant generating function of a unit Laplace variable W t+1 evaluated at s is [ ln E t exp(swt+1 ) ] ( ) 1 = ln 1 s 2. The cumulant generating function of the cash flow growth shock is therefore ( 1 ln E t [exp (sλ t W t+1 )] = ln 1 s 2 Λ 2 t ). (9) The lemma shows that an additional assumption is needed to insure that the cumulant generating function is well defined for all parameter configurations. In particular, it requires that Λ 2 t < min [ φ 2 i, q 2 ] i. To obtain log prices that are linear in the state variable Λ 2 t, I use a first order Taylor expansion of (9) around zero: ln E t [exp (sλ t W t+1 )] s 2 Λ 2 t. Returning to the Euler equation, [ ( 1 = E t exp θ ln β + θ(1 1 ) ] ψ ) c t+1 + θ[κ 0 + κ 1 wc t+1 wc t ] [ { ( ) = E t exp θ ln β + (1 γ)µ + θ κ 0 + A 0 [κ 1 1] + κ 1 [A σ σ 2 (1 ρ σ ) + A Λ2 Λ (1 ρ Λ )] + Λ 2 t θa Λ [κ 1 ρ Λ 1] 15 Analytical results are stated subject to linear approximations such as the log return identity of Campbell and Shiller (1988), as used in the aforementioned articles. 21