Risk Premia and the Conditional Tails of Stock Returns
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1 Risk Premia and the Conditional Tails of Stock Returns Bryan Kelly NYU Stern and Chicago Booth
2 Outline Introduction An Economic Framework Econometric Methodology Empirical Findings Conclusions
3 Tail Risk in the Big Picture Value of assets depends on the potential for infrequent, extreme payoff events 1 Peso problems (Krasker, 1980) 2 Potential for rare disasters can explain equity puzzles (Rietz, 1988; Barro, 2006; Weitzman 2007; Gabaix 2009; Wachter 2009) Plausible mechanism OR convenient (though unrealistic) explanation?
4 The Trouble with Tail Risk Tail risk is difficult to measure, even unconditionally Few risks are static: Feasibility of conditional tail measures? My solution: An economically-motivated conditional tail risk measure extracted from the cross section of asset returns
5 Objectives 1 Structural understanding of how tail risk is priced I derive tractable expressions for expected returns as a function of a tail risk state variable I derive the distribution of return tail events implied by the model 2 I econometrically identify the conditional tail distribution of returns Directly estimable from the cross section of asset prices by exploiting restriction implied by economic theory 3 I evaluate theories relating tail risk to risk premia using my estimated series
6 Preview of Empirical Results Tail risk varies substantially over time and is highly persistent Tail measure predicts market returns over horizons of one month to five years, outperforms commonly studied predictors A one standard deviation increase in tail risk increases expected returns by 4.4% per year Large explanatory power for cross section of returns Stocks that covary highly with tail risk earn annual expected returns 2% to 6% lower than stocks that with low tail risk covariation
7 Outline Introduction An Economic Framework Econometric Methodology Empirical Findings Conclusions
8 Structural Models and Tail Risk Emergence in varied theoretical settings, for example 1 Long run risks + heavy-tailed shocks (similar to Eraker and Shaliastovich 2008, Drechsler and Yaron 2009) 2 Time-varying rare disasters (similar to Gabaix 2009, Wachter 2009) 3 Long run risks + large swings in confidence (similar to Bansal and Shaliastovich 2009)
9 A Tail Risk State Variable in the Long Run Risks Framework Epstein-Zin preferences: m t+1 = θ ln β θ ψ c t+1 + (θ 1)r c,t+1 Dynamics of the real economy: c t+1 = µ + x t + σ c σ t z c,t+1 + Λ t W c,t+1 x t+1 = ρ x x t + σ x σ t z x,t+1 σt+1 2 = σ 2 (1 ρ σ ) + ρ σ σt 2 + σ σ z σ,t+1 Λ t+1 = Λ(1 ρ Λ ) + ρ Λ Λ 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 Λt W i,t+1 f W (w) = 1 2 exp( w ), w R
10 What Is Tail Risk? Gaussian baseline: is variance sufficient to characterize risk of extreme events? Illustrative example: Normal-Laplace distribution Since at least Mandelbrot (1963) and Fama (1963), economists have argued for power law return tails ( x ) ζ P(R > x R > u) =, u some high threshold u ζ tail risk measure Does ζ change through time? What kind of world?
11 What Is Tail Risk? Var(X N L )=σ 2 +2Λ 2 P(X N L >x X N L >u)=exp( [x u]/λ) Normal (µ,σ 2 ) Laplace (Λ) Normal Laplace (µ,σ 2,Λ) u
12 What Is Tail Risk? Return Gaussian baseline: is variance sufficient to characterize risk of extreme events? Illustrative example: Normal-Laplace distribution Since at least Mandelbrot (1963) and Fama (1963), economists have argued for power law return tails ( x ) ζ P(R > x R > u) =, u some high threshold u ζ tail risk measure Does ζ change through time? What kind of world?
13 A Tail Risk State Variable in the Long Run Risks Framework Epstein-Zin preferences: m t+1 = θ ln β θ ψ c t+1 + (θ 1)r c,t+1 Dynamics of the real economy: c t+1 = µ + x t + σ c σ t z c,t+1 + Λ t W c,t+1 x t+1 = ρ x x t + σ x σ t z x,t+1 σt+1 2 = σ 2 (1 ρ σ ) + ρ σ σt 2 + σ σ z σ,t+1 Λ t+1 = Λ(1 ρ Λ ) + ρ Λ Λ 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 Λt W i,t+1 f W (w) = 1 2 exp( w ), w R
14 Prices and Excess Returns Proposition The log wealth-consumption ratio and log price-dividend ratio for asset (i) are linear in state variables, Proposition wc t+1 = A 0 + A x x t+1 + A σ σ 2 t+1 + A Λ Λ t+1 pd i,t+1 = A i,0 + A i,x x t+1 + A i,σ σ 2 t+1 + A i,λ Λ t+1. The expected return on asset (i) in excess of the risk free rate is E t[r i,t+1 r f,t ] = β i,c λ c(σ 2 c σ 2 t + 2Λ t) + β i,x λ xσ 2 xσ 2 t + β i,σ λ σσ 2 σ + β i,λ λ Λ σ 2 Λ 1 2 Var(r i,t+1). Proof
15 Key Implications 1 Tail risk forecasts excess stock returns High tail risk high future returns 2 Covariance with tail risk impacts cross section of expected returns High return tail risk beta low expected returns
16 Implied Distribution of Returns Proposition The lower and upper tail distributions of arithmetic returns are 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) where a i = max(φ i, q i ) 1 and ζ t = 1/ Λ t. ( ) r ai ζ t u ( ) r ai ζ t u
17 Key Implications Tail risk state variable drives risk premia and tail exponent 1 Tail risk (and thus tail exponent) forecasts excess stock returns High tail risk high future returns 2 Covariance with tail risk (and thus tail exponent) impacts cross section of expected returns High return tail risk beta low expected returns Other Structural Models
18 Outline Introduction An Economic Framework Econometric Methodology Empirical Findings Conclusions
19 Econometric Intuition P t(r i,t+1 < r «r ai ζ t R i,t+1 < u) u Single process drives tail dynamics of entire panel of returns
20 Definition: Dynamic Power Law Model Individual returns on asset (i), conditional upon exceeding threshold u and given F t, obey with exponent F u,i,t (r) = P(R i,t+1 > r R i,t+1 > u, F t ) = 1 1 = π 0 + π 1 ζ t+1 ζt upd + π 2 1 ζ t ( ) r ai ζ t u and observable update 1 ζ upd t = 1 K t ln R k,t K t u k=1
21 Definition: Dynamic Power Law Model Individual returns on asset (i), conditional upon exceeding threshold u and given F t, obey with exponent F u,i,t (r) = P(R i,t+1 > r R i,t+1 > u, F t ) = 1 1 = π 0 + π 1 ζ t+1 ζt upd and observable update 1 ζ upd t = 1 K t K t k=1 ln R k,t u + π 2 1 ζ t = π 0 1 π 2 + π 1 ( ) r ai ζ t u j=0 { Hill (1975) Estimator π j 2 applied to cross section 1 ζ upd t j
22 Quasi-Likelihood Estimator for Dynamic Power Law Model Proposal: Assume tail observations in time t cross section are identical and independent But theory (and years of empirical work) suggests... 1 Dependent observations (factor structure) 2 Heterogeneous volatility 3 Heterogeneous tail exponent Result: Despite mis-specification, estimator consistent and asymptotically normal
23 Quasi-Likelihood Estimator for Dynamic Power Law Model 1 Assume (provisionally) tail observations are cross-sectionally independent and each obey F u,i,t (X i,t+1 ; π) = f u,i,t (X i,t+1 ; π) = ζ t u ( ) x ζt u ( ) x (1+ ζ t) u 2 Construct log quasi-likelihood using only u-exceedences L(X ; π) = 1 T 3 Maximize T ln f u,t (X t+1 ; π) = 1 T t=0 T t=0 K t+1 k=1 ( 1 ζ t QML Estimator: ˆπ QL arg max L(X ; π) π Π ln X ) k,t+1 u
24 Asymptotic Properties of QML Estimator Proposition Let the true DGP of {R t } T t=1 be given by the Dynamic Power Law model with parameter values π. Under standard GMM regularity conditions, ˆπ QL p π and T (ˆπQL π ) d N(0, Ψ) where Ψ = S 1 GS 1, S = E[ π s(x t ; π )], and G = E[s(X t ; π )s(x t ; π ) ].
25 Lemma Proof Sketch First order condition of quasi-likelihood maximization s(x t+1 ; π) π ln f t (X t+1 ; π) = K t+1 ζ t K t+1 k=1 ln X k,t+1 u = 0 MLE identification condition: expected value of FOC equals zero Mis-specified MLE is GMM FOC moment condition holds E[s(X t+1 ; π)] = 0 when the true model is the Dynamic Power Law. Proof: E[s(X t+1 ; π)] = EˆE t[ K t+1 ζ t = E[ K t+1 K Xt+1 K 1 t+1 n ζ t = 0 when 1 ζt = ln X k,t+1 ] u k=1 P i a i ] ζ t 1 n P i a i ζ t.
26 Volatility and Other Considerations By varying threshold each period, accommodate time-varying volatility Cross sectional differences in volatility? Explicitly modeling dependence?
27 Outline Introduction An Economic Framework Econometric Methodology Empirical Findings Conclusions
28 Data Count Primary sample: Daily NYSE/AMEX/NASDAQ stock returns from CRSP Fama-French factors; Ken French s data library Federal Reserve macro data Goyal and Welch (2008) data OptionMetrics Other (VIX, Hao Zhou s variance risk premium)
29 Dynamic Power Law Estimates = π ζ 0 + π 1 + π t+1 ζt upd 2 ζ t Table: Both Tails Lower Tail Upper Tail ζ (0.021) (0.044) (0.018) π (0.014) (0.010) (0.058) π (0.015) (0.011) (0.092)
30 Dynamic Power Law Estimates: Exponent Series Exponent (both) Aggregate Price Dividend Ratio (Log) Tail Risk ( ζ t ) ρ(exponent, log P/D) = 14%
31 Dynamic Power Law Estimates: Exponent Series 6 Lower Upper 1.5 Aggregate Price Dividend Ratio (Log) Tail Risk ( ζ t ) ρ(lower Exponent, log P/D) = 15%, ρ(upper Exponent, log P/D) = 14%
32 Dynamic Power Law Estimates: Threshold Volatility Threshold (both) ρ(volatility, Threshold) = 60%
33 Testing Model Implications: Predicting Stock Returns Theory suggests increases in tail risk forecast increases in excess returns Predictive regressions of excess returns on aggregate market over short (one month) and long (up to five year) horizons Compare against common alternatives (dividend-price ratio, term spread, etc.) Robustness
34 Testing Model Implications: Predicting Stock Returns Univariate Prediction One month horizon One year horizon Five year horizon Coef. t-stat R 2 Coef. t-stat R 2 Coef. t-stat R 2 Tail Risk (-ζ lower) Book-to-market Cross section premium Default return spread Default yield spread Dividend payout ratio Dividend price ratio Dividend yield Earnings price ratio Inflation Long term return Long term yield Net equity expansion Stock volatility Term Spread Treasury bill rate Variance risk premium
35 Testing Model Implications: Predicting Stock Returns Bivariate Prediction One month horizon One year horizon Five year horizon Tail Tail Tail Tail Tail Tail Coef. t Coef. t R 2 Coef. t Coef. t R 2 Coef. t Coef. t R 2 Book-to-market Cross section premium Default return spread Default yield spread Dividend payout ratio Dividend price ratio Dividend yield Earnings price ratio Inflation Long term return Long term yield Net equity expansion Stock volatility Term Spread Treasury bill rate Variance risk premium
36 Testing Model Implications: Predicting Stock Returns Out-of-Sample Prediction (Lower Tail) Monthly Out-of-Sample R 2 = 1.3%
37 Testing Model Implications: The Cross Section of Returns Theory predicts 1 Differential exposure to tail risk state variable implies cross-sectional difference in expected returns 2 Negative price of tail risk: assets with high beta on tail risk have hedge value Test for cross-sectional relation between individual asset/portfolio return tails and returns 1 Returns on tail risk beta-sorted portfolios 2 Fama-MacBeth tests 3 Robustness to alternative characteristics
38 Testing Model Implications: The Cross Section of Returns Tail Beta-Sorted Portfolios: NYSE/AMEX/NASDAQ Stocks Tail Risk Beta Low High Diff. Diff (5-1) t-stat Panel A: Tail Risk Beta Only All Panel B: Market Beta / Tail Risk Beta Low β MKT High β MKT Panel C: Market Equity / Tail Risk Beta Small Big Panel D: Book-to-Market / Tail Risk Beta Growth Value
39 Testing Model Implications: The Cross Section of Returns Stage 2 Fama-MacBeth Results: NYSE/AMEX/NASDAQ Stocks -ζ (both) ζ (lower) ζ (upper) R. Vol R e MKT SMB HML Intercept R
40 Hedging Tail Risk Tail Risk Betas: 25 Size/BM Ptfs and VIX
41 Outline Introduction An Economic Framework Econometric Methodology Empirical Findings Conclusions
42 Conclusions Derive link between return tails and risk premia in an affine pricing framework with tail risk Present new methodology for capturing dynamic extreme risk in the economy Identify substantial time variation in tails Empirics consistent with predictions of structural model 1 Large variation in tail risk over time 2 Tail exponent forecasts excess market returns 3 Associated with large cross-sectional differences in average returns What next? 1 Unified pricing with other asset classes (options and credit)
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