Diverse Beliefs and Time Variability of Asset Risk Premia

Diverse and Risk The Diverse and Time Variability of M. Kurz, Stanford University M. Motolese, Catholic University of Milan August 10, 2009 Individual State of SITE Summer 2009 Workshop, Stanford University 1/33

Defining A Risk on An Asset Diverse and Realized Risk The Assume: π t+1 = p t+1 + D t+1 R t p t p t Much past data available to compute empirical moments. No one knows true probability distributions. m = probability implied by the empirical distribution m is a unique and stationary probability. All agree on m The is the conditional expectations under m Individual State of E m t [π t+1 I t ] = 1 p t E m t [p t+1 + D t+1 R t p t I t ] 2/33

The Issue Diverse and Risk The The questions we aim to answer are: Individual State of What are the factors determining the Risk? 3/33

The Issue Diverse and Risk The The questions we aim to answer are: Individual State of What are the factors determining the Risk? Why does it fluctuate over time? 3/33

Some Answers Bond Market: macroeconomic Variables Fama and Bliss (1987) Internal dynamics of past yields Diverse and Risk The Individual State of 4/33

Some Answers Bond Market: macroeconomic Variables Fama and Bliss (1987) Internal dynamics of past yields Diverse and Risk The Campbell and Shiller (1991) Unexplained shocks to the bond market Individual State of 4/33

Some Answers Bond Market: macroeconomic Variables Fama and Bliss (1987) Internal dynamics of past yields Diverse and Risk The Campbell and Shiller (1991) Bernanke and Kuttner (2003) Cochrane and Piazzesi (2005) Unexplained shocks to the bond market Federal Reserve policy shocks Past yields and Business Cycles Individual State of 4/33

Some Answers Bond Market: macroeconomic Variables Fama and Bliss (1987) Internal dynamics of past yields Diverse and Risk The Campbell and Shiller (1991) Bernanke and Kuttner (2003) Cochrane and Piazzesi (2005) Piazzesi and Swanson (2004) Unexplained shocks to the bond market Federal Reserve policy shocks Past yields and Business Cycles Recessions forecasts, i.e. Non-Farm Payroll Individual State of 4/33

Some Answers Bond Market: macroeconomic Variables Fama and Bliss (1987) Internal dynamics of past yields Diverse and Risk The Campbell and Shiller (1991) Bernanke and Kuttner (2003) Cochrane and Piazzesi (2005) Piazzesi and Swanson (2004) Unexplained shocks to the bond market Federal Reserve policy shocks Past yields and Business Cycles Recessions forecasts, i.e. Non-Farm Payroll Individual State of Kurz and Motolese (2008) The importance of Market 4/33

Some More Answers Diverse and Risk Stock Market: framed as a problem of forecasting returns. Using Belief Variables some studies focus on earning forecasts dispersion: The Individual State of 5/33

Some More Answers Diverse and Risk Stock Market: framed as a problem of forecasting returns. Using Belief Variables some studies focus on earning forecasts dispersion: Miller (1977), Diether et al. (2002), Lee et al. (2002), Park (2005), Baker and Wurgler (2006), Campbell and Diebold (2007). The Individual State of 5/33

Diverse and Risk Effects of diverse beliefs on the risk premium Restrictions diverse beliefs rationality imposes Measuring market belief and testing hypotheses complementary to Kurz and Motolese (2008) on the Bond Market The Individual State of The Approach: 1. Outline basic model of belief 2. Formulate an Asset Pricing model 3. Deduce conclusion about risk premium 4. Test empirically using a reference model from literature 6/33

A Sketch of the Basic Model of Belief Risky asset payoff {D t, t = 1, 2,...} with true probability Π. Non stationary with structural breaks. Π is unknown. True process is Stable: Relative frequencies converge Has empirical distribution Substantial past data. All compute empirical probability m: Common knowledge. Assume: under m process is Markov with mean µ and transition F d t+1 = λ d d t + ρ d t+1, ρd t+1 N ( 0, σ 2 d) where dt = D t µ Diverse and Risk The Individual State of 7/33

A Sketch of the Basic Model of Belief (cont.) Note Π m. Theorem: m is stationary, unique and espressed by F. Diverse and Risk The Individual State of 8/33

A Sketch of the Basic Model of Belief (cont.) Note Π m. Theorem: m is stationary, unique and espressed by F. Q is a Rational Belief if data generated under it reproduces m We assume agents beliefs Q i are Markov Disagreement persists: Data shows diverse forecasts Must hold diverse transitions F i t Rationality implies (as a minimum): (A) A Rational agent cannot hold a constant F i F (B) Ft i fluctuate over time with the restriction (C) 1/N N t=1 (F t i F ) = 0 for all i: Rational Agents are right on average Rationality = Dynamics Diverse and Risk The Individual State of 8/33

A Sketch of the Basic Model of Belief (cont.) Diverse and Risk The Definition: A Belief State g i t pins down i s perceived transition of all state variables. In the case of dividend, it takes the form dt+1 i = λ d d t + λ g d gi t + ρ id t+1, ρid t+1 N ( 0, ˆσ d 2 ) Individual State of g i t is observable E i t [ d i It t+1, gt i ] E m t [d t+1 I t ] = λ g d gi t Persistent Diversity = g i t are different across i 9/33

A Sketch of the Basic Model of Belief (cont.) Diverse and Risk The In Sum: if rational, g i t must fluctuate and have a zero mean Individual State of 10/33

A Sketch of the Basic Model of Belief (cont.) Diverse and Risk The In Sum: if rational, g i t must fluctuate and have a zero mean We represent state of belief with: gt+1 i = λ Z gt i + ρ ig t+1, ρig t+1 N ( 0, σg 2 ) (1) Individual State of 10/33

A Sketch of the Basic Model of Belief (cont.) Diverse and Risk The In Sum: if rational, g i t must fluctuate and have a zero mean We represent state of belief with: Why: gt+1 i = λ Z gt i + ρ ig t+1, ρig t+1 N ( 0, σg 2 ) (I) It is Compatible with empirical evidence in survey data (II) Can give an analytic-bayesian justification. See Kurz (2006) and will be discussed later in the conference. (1) Individual State of 10/33

Dynamics of Market Belief Diverse and Definition: Market belief is the distribution (g 1 t, g2 t,..., gn t ) observed by sampling hence with known moments. Risk The Individual State of 11/33

Dynamics of Market Belief Definition: Market belief is the distribution (gt 1, g2 t,..., gn t ) observed by sampling hence with known moments. N Define mean market state of belief by Z t = 1 N Average (1) 1 N N i=1 g i t+1 = λ Z 1 N N gt i + 1 N i=1 Key condition: ρ ig t are correlated hence 1 N ρ ig t+1 N = ρz t+1 0 i=1 N i=1 i=1 ρ ig t+1 Z t is a state variable with empirical distribution Z t+1 = λ Z Z t + ρ Z t+1, ρ Z t+1 N ( 0, σz 2 ) g i t Diverse and Risk The Individual State of 11/33

The Structure of : s Diverse and We thus expand the empirical distribution to {(d t+1, Z t+1 ), t = 1, 2,...} Risk The d t+1 = λ d d t + ρ d t+1 ρ d t+1 N ( 0, Σ ), i.i.d. Individual State of Z t+1 = λ Z Z t + ρ Z t+1 ρ Z t+1 where Σ = σd 2, 0. 0, σ 2 Z 12/33

The Structure of : s (cont.) Diverse and Individual i s perception model (together with (1)) takes the form: dt+1 i = λ dd t + λ g d gi t + ρid t+1 ρ id t+1 N ( 0, Σ i) (2) where Z i t+1 = λ Z Z t + λ g Z gi t + ρiz t+1 ˆσ Σ i d 2, = ˆσ Zd, ˆσ Zd ˆσ 2 Z ρ iz t+1 Parameter sign λ g d 0 and λg Z 0 orient the model: When gt i > 0, agent i believes t+1 dividend and market belief will persist above normal.. Risk The Individual State of 13/33

Diverse and Assumptions: Large number of agents. A single commodity consumption. Riskless technology producing R > 1 at t + 1 with 1 unit of input at t. A single aggregate risky asset with supply S=1. The sequence of dividends {D t, t = 1, 2,...} with unknown distribution Π. Under m {D t, t = 1, 2,...} is Markov with transition d t+1 = λ d d t + ρ d t+1, ρd t+1 N ( 0, σd 2 ) Risk The Individual State of where d t = D t µ 14/33

(Cont.) Some : θ i t = date t stock purchases of agent i. B i t = amount invested in the riskless asset. p t = price of the stock. Think of it as the S&P500. Agent i optimization problem is: Subject to: max E (θ i,b i t i ) β k 1 e k=t 0 1 @ ci k A τ I t ct i = θt 1 i (p t + d t + µ) + Bt 1R i θtp i t Bt i Initial values ( θ0, i B0 i ) i s belief as specified in (2) Advantage: only mean market belief matters. Diverse and Risk The Individual State of 15/33

(a remark) Diverse and Risk The The Exponential utility model is common in studies of asset pricing. See for example Singleton (1987) Brown and Jennings (1989) Grundy and McNichols (1989) Wang (1994) He and Wang (1995) Duffie (2002) Dai and Singleton (2002) Allen, Morris and Shin (2005) and many others Individual State of 16/33

Asset Assume for a moment: agents believe p t+1 + d t+1 is conditionally normal. Define: Excess return per share V t+1 = p t+1 + (d t+1 + µ) Rp t. Define the state variables in the optimization: ψt i = ( 1, d t, Z t, gt i ) u = (u 0, u 1, u 2, u 3 ) We show that the demand function of agent i is: where θt(p i t, ψt) i = Rτ [ ( E i r ˆσ V 2 t Vt+1 I t, gt) i ] + uψ i t ˆσ 2 V = adjusted variance of V t assumed constant for all i. Diverse and Risk The Individual State of Stability Conditions: R = 1 + r > 1, 0 < λ d < 1, λ Z < 1, 0 < λ Z + λ g Z < 1. 17/33

Asset Prices and Risk Theorem: There is a unique equilibrium price function which takes the form p t = a d d t + a z Z t + P 0 with parameters a d = λ d + u 1 > 0 a z = (a d + 1) λ g d + (u 2 + u 3 ) R λ d R ( λ Z + λ g ) > 0 Z P 0 = (µ + u 0) r ˆσ2 V Rτ. Diverse and Risk The Individual State of The Theorem above confirms equilibrium price p t is conditionally normal Price exhibits excess volatility due to beliefs 18/33

Structure of the Risk : Main Result Diverse and We compute: E m t [π t+1 I t ] = 1 p t E m t [( ) ] r ˆσ 2 V Rτ u 1d t u 0 a z (R λ Z ) Z t Risk The Individual State of 19/33

Structure of the Risk : Main Result Diverse and We compute: E m t [π t+1 I t ] = 1 p t E m t [( ) ] r ˆσ 2 V Rτ u 1d t u 0 a z (R λ Z ) Z t Risk The Main Theorem: Since a z (R λ Z ) > 0, the Risk increases with ˆσ 2 V and declines with market belief Z t Also, ˆσ 2 V (a d + 1) 2 ˆσ 2 d + a2 Z ˆσ2 Z (I) Mean increases with variance of d and Z: belief state volatility increases mean premium (II) Time Changes in Market Belief reflected in a z (R λ Z ) > 0. Individual State of The risk premium on a long position lower when market belief about the future is favorable higher when market belief about the future is unfavorable 19/33

Diverse and What It Does Not Say: Agents are on their demand functions Not optimal to choose to be a contrarian; may be short when wants to be long and long when wants to be short Analogous to why we do not adopt a log utility Note: you don t change forecast of d t+1 when you find Z t < 0! What It Does Say: A formal theorem about market overshooting: today s price adjusts to Z t more then expected tomorrow s price Rational investing agents must form an opinion about future beliefs of others - the market. Diverse beliefs are central to this results (topic of conference) Risk The Individual State of 20/33

Some Comments Definition: An equilibrium exhibits Endogenous Uncertainty if its price map depends upon market belief. With Non-Exponential Utility: entire distribution (gt 1,..., gn t ) matters but Main Result holds if slope of Z t satisfies a Z > 0. Mean market belief Z t = 1 N g N t. i i=1 Diversity σt Z = 1 N (gt i Z t ) 2 N i=1 Empirical work considers first two moments of (Z t, σ Z t ) distribution Hypothesis: effect of cross sectional diversity σt Z on risk premium is negative (the Narrow Door Hypothesis ) Diverse and Risk The Individual State of 21/33

and Variables Description Diverse and Excess Stock Returns R t,t+6 six-month return on the CRSP value weighted index net of the return on a 90 day T-bill Risk The R t,t+12 1 year return on the CRSP value weighted index net of the return on a 90 day T-bill Individual State of Livingston Six-Month Growth Forecasts F g t+6,t+12 Mean forecast of six-month real GDP growth rate from 6 months to 12 month after t computed from individual Livingston survey responses about the level of nominal GDP and the CPI 6 and 12 months after date t 22/33

and Variables Description (cont.) Diverse and Risk Financial Predictors DP t dividend yield on the CRSP value-weighted portfolio The DEF t TERM t the yield spread between a broad corporate bond portfolio and the AAA yield spread the yield spread between a 10 year U.S. Treasury bond and a one-month Treasury bill Individual State of Macroeconomic Predictors CAY t Lettau and Ludvigson (2001) log consumptionwealth ratio 23/33

and Variables Description (cont.) Diverse and Risk The Variables Z g t+6,t+12 Belief Index of Real GDP Growth Rate between the end of period t+6 and t+12 Individual State of σ Z g Cross-sectional standard deviation of individual Livingston forecasts of 6-month growth rate in real GDP between the end of period t+6 and t+12 24/33

Descriptive of Premia (1971:S1-2007:S2) Diverse and Risk The µ σ R t,t+6 R t,t+12 Individual State of R t,t+6 6.646 23.376 1.000 R t,t+12 5.098 16.668 0.692 1.000 25/33

Descriptive of Predictors (1971:S1-2007:S2) Diverse and Risk The µ σ F g t+1,t+2 DP t DEF t TERM t CAY t Individual State of F g t+6,t+12 3.090 1.125 1.000 DP t 3.104 1.258 0.143 1.000 DEF t 1.028 0.376 0.451 0.609 1.000 TERM t 1.011 1.182 0.297-0.236 0.056 1.000 CAY t 0.000 0.016-0.327 0.119-0.172 0.158 1.000 26/33

Descriptive of Variables (1971:S1-2007:S2) Diverse and Risk The µ σ Z g t+6,t+12 σ Z g Individual State of Z g t+6,t+12-0.603 1.890 1.000 σ Z g 2.319 1.012 0.576 1.000 27/33

Belief Index Z g t+6,t+12 Diverse and Risk The Individual State of 28/33

The Cross-sectional standard deviation σ Z g Diverse and Risk The Individual State of 29/33

The Reference Model (1971:S1-2007:S2) R t,t+6 R t,t+12 (A) (B) (A) (B) F g t+6,t+12-0.254-0.170 (0.032) (0.270) DP t 0.086 0.100 0.197 0.206 (0.563) (0.474) (0.244) (0.185) DEF t 0.138 0.225 0.000 0.058 (0.417) (0.149) (1.000) (0.719) TERM t 0.084 0.171 0.172 0.230 (0.489) (0.145) (0.246) (0.126) CAY t 0.244 0.161 0.264 0.209 (0.026) (0.188) (0.074) (0.240) Diverse and Risk The Individual State of R 2 0.067 0.097 0.112 0.119 S.E. 0.966 0.951 0.942 0.939 P-values in parenthesis. All R 2 are adjusted 30/33

Test of Theorem for 6-month horizon (1971:S1-2007:S2) R t,t+6 (A) (B) (1) (2) F g t+6,t+12-0.254 (0.032) Z g t+6,t+12-0.587-0.555 (0.000) (0.000) σ Z g -0.082 (0.506) DP t 0.086 0.100 0.469 0.499 (0.563) (0.474) (0.007) (0.009) DEF t 0.138 0.225 0.143 0.142 (0.417) (0.149) (0.213) (0.203) TERM t 0.084 0.171-0.078-0.085 (0.489) (0.145) (0.397) (0.378) CAY t 0.244 0.161 0.000 0.010 (0.026) (0.188) (0.998) (0.926) Diverse and Risk The Individual State of R 2 0.067 0.097 0.215 0.208 S.E. 0.966 0.951 0.886 0.890 P-values in parenthesis. All R 2 are adjusted 31/33

Test of Theorem for 12-month horizon (1971:S1-2007:S2) R t,t+12 (A) (B) (1) (2) F g t+6,t+12-0.170 (0.270) Z g t+6,t+12-0.422-0.363 (0.006) (0.016) σ Z g -0.150 (0.279) DP t 0.197 0.206 0.445 0.498 (0.244) (0.185) (0.006) (0.004) DEF t 0.000 0.058 0.016 0.016 (1.000) (0.719) (0.902) (0.903) TERM t 0.172 0.230 0.053 0.041 (0.246) (0.126) (0.642) (0.721) CAY t 0.264 0.209 0.101 0.119 (0.074) (0.240) (0.517) (0.446) Diverse and Risk The Individual State of R 2 0.112 0.119 0.176 0.177 S.E. 0.942 0.939 0.908 0.907 P-values in parenthesis. All R 2 are adjusted 32/33

Diverse and Risk The Data supports theoretical conclusion Standard variables DEF t, TERM t and CAY t predominantly reflect market beliefs not exogenous conditions More empirical proof that markets overshoot Individual State of More evidence the risk of future market belief (i.e. belief of others ) is a dominant market risk 33/33