Asset pricing in the frequency domain: theory and empirics

Asset pricing in the frequency domain: theory and empirics Ian Dew-Becker and Stefano Giglio Duke Fuqua and Chicago Booth 11/27/13 Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 1 / 55

Introduction What risks are people willing to pay to avoid? What determines risk premia? Early models: what s happening right now (CAPM, CCAPM) Later models: news about the future (EZ, ICAPM) Risk prices depend on dynamics of shocks In this paper we derive a new frequency-domain representation of asset prices Frequency domain is the natural place to study implications of dynamics Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 2 / 55

Introduction Spectral decomposition of risk prices in all affi ne models Consumption as an example today Separates preferences from dynamics Difference from Hansen et al. Allows non-parametric estimation Not possible in time domain Focus on economically important frequencies Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 3 / 55

What we do Suppose agents care about current and future consumption growth When a fundamental shock ε hits it moves consumption now and in the future The impulse-response function to ε tells us how it moves consumption Alternatively: ε induces many fluctuations in consumption These sum up to the total response of consumption G ε (ω) captures ω-frequency response: impulse transfer function It s the Fourier transform of the IRF Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 4 / 55

Which shock has the largest risk premium? Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 5 / 55

Why this is useful 1 See how dynamics are priced in different models Make statements about "long-run risk", low- vs. high-frequency rigorous 2 Non-parametric estimation Test economic intuition behind models instead of strict parametrizations Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 6 / 55

What we find: Theory Preferences have strong implications for Z (ω) Power utility: flat Z (ω) Internal habits: high weight on high frequencies Epstein Zin: power at lowest frequencies (230 years on avg.) Simple test: slope of Z Standard models surprisingly restrictive All functions are monotone Cannot isolate business-cycle frequencies Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 7 / 55

What we find: Empirics Structural Epstein Zin fails No significant coeffi cients "Long-run" is too long Implies consumption does not price equities Frequency domain generalization works Define "long-run" as cycles longer than 8 years Long-run shocks are significantly priced Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 8 / 55

Results apply to affi ne models generally Not just consumption-based models Related papers: Term structure of interest rates VIX futures curve Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 9 / 55

Outline 1 Theory 1 Basic frequency decomposition 2 Application to utility functions 2 Empirics; estimate the weighting function Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 10 / 55

Asset pricing With no arbitrage, there exists a stochastic discount factor (SDF), 1 = E t [R t+1 M t+1 ] for all returns R t+1 Excess returns depend on covariances with the SDF, E t (R t+1 R f,t+1 ) = ( ) M t+1 cov R t+1, E t M t+1 cov (r t+1, m t+1 ) r = log R, m = log M What moves M t+1, and by how much? Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 11 / 55

Affi ne models Assumption 1: Log SDF (m t+1 ) depends on the dynamics of a state variable x t m t+1 E t m t+1 = z k E t+1 x t+k+1 k=0 where E t+1 E t+1 E t E t is expectation operator; {z k } a set of known weights Common models: x t is consumption growth or equity returns Epstein Zin: innovation to SDF depends on future consumption growth (long-run risk) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 12 / 55

Assumption 2: x t driven by a VMA process x t = B 1 Γ (L) ε t for a vector of innovations ε t, selection vector B 1 = [1, 0, 0,...], Γ (L) lag polynomial, Γ (L) = Γ k L j k=0 We do not assume anything about higher moments: Normality Homoskedasticity Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 13 / 55

Standard risk prices Impulse response function: Innovations to the SDF are g j,k = de tx t+k dε j,t m t+1 E t m t+1 = j ( ) z k g j,k ε j,t+1 k=0 where {g j,k } is the IRF of x to ε j Risk price for ε j is k=0 z k g j,k Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 14 / 55

Result 1: Under assumptions 1 and 2, ( 1 π m t+1 E t m t+1 = j 2π π Risk price for ε j is ) Z (ω) G j (ω) dω ε j,t+1 1 π Z (ω) G j (ω) dω = 2π z k g j,k π k=0 G j (ω) is the impulse transfer function, frequency analog of IRF G j (ω) cos (ωk) g j,k k=0 Z (ω) is the unique price of risk at frequency ω Z (ω) z 0 + 2 k=1 Z and G j separate preferences and dynamics z k cos (ωk) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 15 / 55

Examples of IRFs and impulse transfer functions Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 16 / 55

Examples of IRFs and impulse transfer functions Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 17 / 55

Examples of IRFs and impulse transfer functions Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 18 / 55

Example 1: power utility Log pricing kernel is exactly m t+1 E t m t+1 = α( c t+1 E t c t+1 ) Weights are z 0 = α, z k = 0 for k > 0 Z power (ω) = α All shocks have equal weight no matter how long they last Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 20 / 55

Example 2: Internal habits Period utility: U (C t ) = (C t bc t 1 ) 1 α 1 α b determines size of habit (and risk aversion) We log-linearize the SDF to get weighting function Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 21 / 55

Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 22 / 55

Example 3: External habits Period utility: C is aggregate consumption U (C t ) = (C t b C t 1 ) 1 α 1 α ( Ct+1 b C t M t+1 = C t b C t 1 ) α Future dynamics do not matter marginal utility depends only on today s consumption Weighting function is flat like power utility Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 23 / 55

Example 4: Epstein Zin preferences Assume homoskedastic, log-normal consumption growth SDF can be written as ( E t+1 m t+1 = ρ E t+1 c t+1 + (α ρ) E t+1 ) θ j c t+1+j j=0 [α risk aversion; ρ inverse EIS; θ linearization parameter near 1] (α ρ) is long-run risk term Total mass of Z is α Z (ω) = α + 2 (α ρ) j=1 θ j cos (ωj) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 24 / 55

For θ 1, Z approaches Z (ω) = (α ρ) δ (ω) + ρ δ (ω) is a point mass at zero (periodic extention of Dirac δ) Standard calibrations say primarily frequency zero matters (α ρ) /α 1 Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 25 / 55

Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 26 / 55

Models have strong, surprising implications for Z (ω) What do we mean by long run? EZ preferences have more than half the weight on cycles longer than 230 years! Interpretation: take a permanent consumption shock. Half of its price comes from cycles >230 years. 3 4 of its price from cycles >75 years. Clear, sharp differences between E Z, power, habit formation utility Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 27 / 55

Implications for calibration Models are usually calibrated to match unconditional moments With a non-trivial weighting function, consumption dynamics matter For internal habits, autocorrelation matters For Epstein Zin, need to calibrate long-run standard deviation Std. dev. of innovations to Beveridge Nelson trend LRR models sometimes as high as 4% per quarter Empirically, no more than 2% per quarter Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 28 / 55

A limitation All the weighting functions are monotone Maybe consumers dislike mainly business-cycle frequency shocks? Huge policy literature suggests BC is relevant Standard models do not allow that Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 29 / 55

Multiple priced variables What about higher moments? E.g. priced disaster risk or volatility shocks Price of risk for a shock ε j is m m indexes priced state variables 1 π Z m (ω) G m,j (ω) dω 2π π Each priced variable gets a weighting function Z m (ω) For Epstein Zin, Z m (ω) is almost always isolated near zero (only long-run volatility shocks should be priced) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 30 / 55

Estimation We don t necessarily need FD for estimation Anything done in the FD, in principle, works in time domain Why is the frequency decomposition useful? Generalize models Parameterize estimation in terms of frequencies directly (not possible in time domain) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 32 / 55

Two-step estimation: Estimate reduced-form risk prices Rotate into frequency domain (using IRF) State variables follow a VAR X t = ΦX t 1 + ε t Consumption growth is first element of X Remainder of X should forecast consumption growth Rotate risk prices on ε t Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 33 / 55

X t = ΦX t 1 + ε t Estimate reduced-form prices for ε t ( p) GMM on cross-section of equity returns (1 = E [MR]) Each ε t has an ITF G k (ω; Φ); risk prices are p k = 1 π Z (ω) G k (ω; Φ) dω 2π π With K shocks, can estimate Z up to K degrees of freedom Need basis functions for Z (ω) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 34 / 55

Parametrizing Z: the utility basis Models we have explored before: Z U (ω) = q 1 If q 3 = 0, we have EZ If q 1 = 0, we have internal habit If q 1 = q 3 = 0, we have power utility Note: we have an extra parameter θ. θ j cos (ωj) + q 2 + q 3 cos (ω) j=1 Poorly identified, so use standard calibration (0.975) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 35 / 55

Parametrizing Z: the bandpass basis Groups together frequencies Write Z (ω) directly as a step function. Z BP (ω) = q 1 Z (0,2π/32) (ω) + q 2 Z (2π/32,2π/6) (ω) + q 3 Z (2π/6,π) (ω) Three components lower-than-bc: > 32 quarters BC: 6-32 quarters higher-than-bc: <6 quarters Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 36 / 55

Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 37 / 55

Rotating the risk prices SDF is E t+1 m t+1 = ( ) z k B 1 Φ j ε t j=0 = Frequency-domain risk prices {}}{ q Rotation into { Frequency domain }} 1 π π Z 1 (ω) cos (ωk) dω { π 1 π π π Z 2 (ω) cos (ωk) dω B 1 Φ j j=0 1 π π π Z 3 (ω) cos (ωk) dω }{{} Reduced-form risk prices ε t Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 38 / 55

1 Estimate dynamic effects of shocks on consumption growth (Φ, IRF) 2 Estimate reduced-form risk prices on ε t ( p) 3 Choose basis functions 4 Rotate p into frequency domain Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 39 / 55

Empirics: Data 25 size- and B/M-sorted portfolios; 49 industry portfolios, post-war data Priced variables: Consumption, GDP, investment components Cochrane (1996) argues investment priced Predictive variables (X ): Two principal components of 13 standard predictors: P/E; P/D; term spread; default spread; unemployment; value spread; short-term interest rate; equity issuance; I/K; cay Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 40 / 55

Empirics: Estimation 1 Estimate VAR, X t = ΦX t 1 + ε t 2 Estimate reduced-form prices of ε t ( p) Use moment condition 0 = E [(R i,t+1 R f,t+1 ) exp ( pε t+1 )] 3 Rotate p to get parameters in spectral weighting function Parameters are estimated in two stages We construct standard errors from combined GMM moment Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 41 / 55

Predictive variables Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 42 / 55

Estimated impulse transfer functions Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 43 / 55

Immediately clear it will be diffi cult/impossible to estimate Epstein Zin Need more power at low frequencies Could impose theoretical restrictions Cointegration Similar to Blanchard and Quah (1989) Still must estimate long-run behavior of something Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 44 / 55

Main estimates Weak results with utility basis no frequencies seem to matter Highly significant results for bandpass basis Long-run shocks significantly priced Coeffi cients map to risk aversion Results robust to using one-step GMM (though weaker) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 45 / 55

Estimated weighting functions Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 46 / 55

Time-domain weights Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 47 / 55

Equity Pricing: results Notes on the estimation We have used the effi cient matrix for the asset pricing conditions (GMM) Results robust to including Industry portfolios, and using identity matrix for GMM Focusing on frequency domain allowed us to: Estimate a more economically intuitive version of long-run Focus on important frequencies for which we have power (<230 years!) Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 48 / 55

Alternative state variables Utility: weak, erratic results Bandpass: low frequencies priced for all variables Consistent with PIH Doesn t distinguish utility models Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 49 / 55

Estimating structural models is treacherous Epstein Zin says investors only care about frequency zero This is not testable; makes the model look useless Allowing more frequencies to be priced helps Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 50 / 55

Returns-based models Pricing kernel also stated in terms of returns CAPM: E t+1 m t+1 E [r m ] Var [r m ] E t+1r m t+1 Epstein Zin, power utility (Campbell, 1993): E t+1 m t+1 = α E t+1 r w,t+1 + (1 α) E t+1 Two factor model: current returns, discount-rate news r w,t+j+1 j=1 So use returns as priced variable instead of macro aggregates Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 51 / 55

Estimates with returns as priced variable Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 52 / 55

Related literature Hansen and Sheinkman (2009), Hansen and Borovicka (2012): How m t+1 evolves following a shock (IRF of m) We study how the 1-period innovation in m depends on the evolution of consumption after a shock Hansen, Heaton and Li (2008), Lettau and Wachter (2007): Price zero-coupon dividend claims Combines cash-flow and SDF dynamics, (link E t m t+1 to c t+1 ) Alvarez and Jermann (2005): Permanent vs. transitory component of the pricing kernel Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 53 / 55

Related literature We focus on what drives the volatility of SDF today Disadvantages: we don t price zero-coupon claims Advantages: It s all you need to price returns: 0 = E t [R e t+1 (M t+1 E t M t+1 )] Study separately preferences and dynamics To study evolution of the whole m you need to specify how the risk-free rate evolves, which depends on the dynamics of consumption. Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 54 / 55

Conclusion We derive a frequency-domain representation of affi ne asset pricing models Assumptions required (existence of MA representation, SDF depends on dynamics) are standard and minimal 1) Obtain sharp implications in many models 2) Allows us to easily generalize models and focus on economically relevant frequencies Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing 11/27/13 55 / 55