Asset Prices and the Return to Normalcy Ole Wilms (University of Zurich) joint work with Walter Pohl and Karl Schmedders (University of Zurich) Economic Applications of Modern Numerical Methods Becker Friedman Institute, University of Chicago Rosenwald Hall, Room 301 November 22, 2013 Professor Kenneth L. Judd
Motivation Financial market characteristics: high stock returns (high volatility) low risk free rate (low volatility) stock return predictability... Common assumption: random walk component in consumption
Motivation (Continued) Empirical evidence: trend-stationarity vs random walk (see Nelson and Plosser (1982), Dejong and Whiteman (1991), Perron (1989), Andreou and Spanos (2003) or Christiano and Eichenbaum (1990)) Research focus: impact of trend-stationarity in time-series on asset prices and returns (as in DeJong and Ripoll (2007), Tallarini (2000) or Rodriguez (2006))
The Economy Discrete, infinite time, complete markets Representative investor with CRRA-utility Consumption process: ln c t = (1 ρ c )g t + ρ c ln c t 1 + ɛ c,t, ɛ c,t N(0, σ 2 c), g t = ḡ + g t 1 Consumption modelled as in Tallarini (2000) but the paper only concentrates on special case with IES = 1 (EZ-Utility)
The Economy (Continued) Asset pricing equations: Risk free rate: Consumption claim: p t c t = βe t p f t = βe t { (ct+1 { (ct+1 c t c t ) } γ ) ( )} 1 γ pt+1 + 1 c t+1
Theoretical Results We have have closed form solutions for the model with a permanent and a temporary shock when ρ c = 0: ln c t = g t + ν t g t = ḡ + g t 1 + ɛ t ɛ t N(0, σ 2 ɛ), ν t N(0, σ 2 ν) σ ɛ = 0 trend-stationarity σ ν = 0 random walk
Theoretical Results (Continued) Table: Analytical Solutions σ ν σ ɛ E(r s t ) E(r f t ) EP γ = 2 γ = 6 0.01 0 0.0936 0.0513 0.0423 0.005 0.005 0.0617 0.0461 0.0157 0 0.01 0.0513 0.0305 0.0208 0.01 0 0.5883 0.1389 0.4494 0.005 0.005 0.2027 0.0888 0.1139 0 0.01 0.0101-0.0486 0.0588
Detrending Remove linear trend: c t = c t /(1 + g) t Detrended consumption: log(c t ) = (1 ρ c )µ c + ρ c log(c t 1) + ɛ c,t, ɛ c,t N(0, σ 2 c) Pricing of the consumption claim: { p (c t = β(1 + g) 1 γ ) ( )} 1 γ E t+1 pt+1 t + 1 c t c t+1 c t
Projection Method Define x t = pt c t We solve for x which is a function of c : { (c x(c ) = β(1 + g) 1 γ ) 1 γ E + ( x(c c + ) + 1 ) } c Solution functions are approximated by n-degree chebychev polynomials: x(c ) = n α i Φ i (c ) i=1 Here c + denotes the next periods state of c, Φ i are the basis functions and α i the unknown solution coefficients of the chebychev polynomials.
Projection Method Residual function: R(c ; α) x(c ) β(1 + g) 1 γ E Collocation projection: { (c ) + 1 γ( x(c c + ) + 1 ) } c Galerkin projection: R(c i ; α) = 0, i = 1,..., n c R(c ; α)φ i (c ) = 0, i = 1,..., n
Projection Method We solve for x(c ) on a range of ±6σ c mean of c around the unconditional We take chebychev nodes for the state-space
Integral in pricing equation: + ( ) c 1 γ + ( x(c c +) + 1 ) f (c+ c )dc+ Gauss-Hermite quadrature for expectation Integral in galerkin projection: +6σc 6σ c R(c i ; α)φ i (c ) = 0, i = 1,..., n Gauss-chebychev quadrature
Data We use three datasets to check robustness; First dataset (Mehra & Prescott 1985): 1889-1978 Second dataset (Mehra & Prescott 1985): 1889-2004 Third dataset (Robert Shiller): 1889-2009
Estimation Procedure 1. Estimate linear trend g and detrend consumption: c t = c t /(1 + g) t 2. Estimate AR(1) process for detrended consumption by OLS: log(c t ) = (1 ρ c )µ c + ρ c log(c t 1) + ɛ c,t, ɛ c,t N(0, σ 2 c) Table: Parameter Estimates µ c σ c g ρ c 95% conf. interv. (ρ c ) MP 1889 1978 1.12 0.035 0.018 0.92 (0.84, 1.00) MP 1889 2004 1.11 0.031 0.018 0.92 (0.84, 0.99) SH 1889 2009 2.64 0.034 0.021 0.90 (0.82, 0.98)
Moments of the Consumption Process Table: Comparison of Empirical and Model Moments 89 78 Model 89 04 Model 89 09 Model E(ln c t+1 c t ) 0.0175 0.0181 0.0173 0.0178 0.0200 0.0206 σ(ln c t+1 c t ) 0.0357 0.0357 0.0319 0.0319 0.0352 0.0352 ρ(ln c t+1 c t ) -0.1362-0.0501-0.1203-0.0426-0.0640-0.0498 ρ(ln c t+1 c t, lnct ) -0.1980-0.1946-0.2071-0.2059-0.2274-0.2237
Empirical Moments of Returns Table: Empirical Moments of Realized Returns for Different Periods E(rt s ) Vol(rt s ) E(rt f ) Vol(rt f ) EP MP 1889 1978 0.0698 0.1654 0.008 0.0567 0.0618 MP 1889 2004 0.0776 0.1660 0.0134 0.0520 0.0642 SH 1889 2009 0.0760 0.1873 0.0197 0.0580 0.0563
Structure of the Following Tables Data shows empirical moments found in the data (as given on the last slide) Bench refers to the benchmark model. We take the basic model by Mehra and Prescott (1985) where consumption growth follows an AR(1) process. CC is our trend-stationary model of the pricing of the consumption claim
Results Table: Results for the dataset from 1889-2009 ρ c E(r s t ) Vol(r s t ) E(r f t ) Vol(r f t ) EP Data 0.9 0.0760 0.1873 0.0197 0.0580 0.0563 γ = 2 Bench 0.0514 0.0395 0.0487 0.0055 0.0027 CC 0.95 0.0533 0.0423 0.0511 0.0082 0.0022 0.9 0.0545 0.0657 0.0497 0.0163 0.0047 0.85 0.0558 0.0841 0.0490 0.0246 0.0068 γ = 8 Bench 0.1557 0.0632 0.1395 0.0243 0.0162 CC 0.95 0.1837 0.0855 0.1687 0.0364 0.0150 0.9 0.1894 0.1681 0.1487 0.0717 0.0407 0.85 0.2026 0.2480 0.1324 0.1071 0.0701
Risk Free Rate Puzzle Kocherlakota (1990) shows that in growth economies well defined equilibria can exist even though the discount factor is larger than one and the agent might still prefer consumption today over future consumption. β > 1 is used e.g. in Piazzesi, Schneider and Tuzel (2007). We show that an equilibrium exists if β < 1 (1+g) 1 γ
Results Table: Results with adjusted β for the dataset from 1889-2009 β E(r s t ) Vol(r s t ) E(r f t ) Vol(r f t ) EP Data 0.0760 0.1873 0.0197 0.0580 0.0563 γ = 2 Bench 1.019 0.0224 0.0385 0.0197 0.0054 0.0027 CC 1.02 0.0244 0.0717 0.0197 0.0160 0.0048 γ = 8 Bench 1.106 0.0347 0.0580 0.0197 0.0217 0.0150 CC 1.116 0.0809 0.2428 0.0197 0.0640 0.0612
Robustness Across Datasets 0.1 MP Data 1889 1978 0.1 MP Data 1889 2004 0.1 Shiller Data 1889 2009 ρ c e + 0.05 0.09 ρ c e 0.09 0.09 0.08 e ρ c 0.05 Bench 0.08 0.08 0.07 0.07 0.07 Equity Premium 0.06 0.05 0.04 Equity Premium 0.06 0.05 0.04 Equity Premium 0.06 0.05 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0 2 4 6 8 10 γ 0 2 4 6 8 10 γ 0 2 4 6 8 10 γ
Model Two: Pricing of the Dividend Claim Data on dividends Consumption = Divided Income + Labor Income Vector autoregressive process for consumption and labor income to dividend ratio We assume a common linear trend g in consumption, prices, dividends and labor income
Pricing of Dividend Claim We have now two states: detrended consumption c and the ratio of labor income to dividend income δ: with x t = c t = d t + e t ( c t = d t + e t ) δ t = e t = e t d t dt x t = (I Φ) µ + Φx t 1 + ɛ t [ ] [ ] [ ] [ ] log c t ρc ρ, Φ = cδ µc ɛc,t, µ =, ɛ log δ t ρ δc ρ δ µ t = N(0, Σ) δ ɛ δ,t
Pricing of Dividend Claim Pricing equation for the dividend claim: p t d t = β(1 + ḡ) 1 γ E t { (c t+1 Define x t = pt d t. Now x t is a function of c t and δ t. c t ) 1 γ ( pt+1 1 )} +. d t+1 1 + δ t+1 Apply solution method as described before but with the two-dimensional state space (c, δ).
Robustness - Pricing of Dividend Claim Table: Results for the dataset from 1889-2009 E(r s t ) Vol(r s t ) E(r f t ) Vol(r f t ) EP Data 0.0760 0.1873 0.0197 0.0580 0.0563 γ = 2 β = 0.99 0.0561 0.0990 0.0502 0.0187 0.0059 β = 1.02 0.0244 0.0731 0.0197 0.0182 0.0048 γ = 7 β = 0.99 0.1784 0.2313 0.1361 0.0700 0.0423 β = 1.10 0.0760 0.2618 0.0197 0.0628 0.0563
Return Predictability Regression: Cumulative Returns on log Price Dividend Ratio Table: Predictability of Stock Returns h = 1 h = 3 h = 5 R 2 β R 2 β R 2 β Data 0.0317-0.0880 0.0644-0.1962 0.1048-0.3268 Bench 0.1207-1.2735 0.0617-1.2576 0.0431-1.3274 CC 0.0673-0.1493 0.1769-0.4634 0.2587-0.7927 DC 0.0204-0.0732 0.0565-0.2387 0.0888-0.4328
Conclusion Large impact of underlying process Model is able to match financial market characteristics like High equity premium Low risk free rate High stock and low bond volatilities Return predictability with standard preferences and risk aversion below 10
Appendix Empirical evidence: trend-stationarity vs. random walk Table: Test Statistics and Critical Values of Unit Root Tests ADF-Test PP-Test KPSS-Test ct dt pt ct dt pt ct dt pt MP 1889 1978-2.0325-3.6731-2.6208-2.3314-3.2025-2.4199 0.1016 0.0772 0.1219 MP 1889 2004-2.1832-4.2298-2.3758-2.5224-3.6364-2.2484 0.1576 0.0535 0.1327 SH 1889 2009-2.3218-4.1468-2.7335-2.5763-3.5601-2.5933 0.2088 0.0909 0.1496 Critical Values 1% 5% 10% 1% 5% 10% 1% 5% 10% -3.99-3.43-3.13-4.04-3.45-3.15 0.216 0.146 0.119
Appendix Hansen-Jagannathan Bounds 1 0.9 0.8 0.7 0.6 σ(m) 0.5 0.4 0.3 0.2 0.1 γ = 1 0 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 E(m)