On Market Risk Premia Material in Pages 13 - 37 of Private Information and Diverse Beliefs: How Different? by Mordecai Kurz, Stanford University May 12, 2006 1
Definition of The Risk Premium • Actual Premium π t % 1 ' p t % 1 % D t % 1 & R t p t . p t • m = probability induced by the data’s empirical distribution • Easy to show that m is unique and stationary probability. Long Run Premium = E m [ p t % 1 % D t % 1 ] & ¯ • R p t • “The” Premium is a conditional expectations under m t [ π t % 1 ] ' 1 E m E m t [p t % 1 % D t % 1 & R t p t ] p t Problem: What are the factors determining the Premium 2
Some Answers: Macroeconomic Variables Fama and Bliss (1987) – past yields Cambpell and Shiller (1991) – Shocks to the bond market hence past yields Bernanke and Kuttner (2003) – Federal Reserve Policy shocks Cocharane Piazzesi (2005) – Past yields Piazzesi and Swanson (2004) – past yields and recessions forecasters such as Non Farm Payroll . We study problem in context of heterogenous beliefs • • Endogenous Uncertainty - Kurz (1974) = component of volatility and risk induced by market beliefs Our Interest: Effects of market beliefs on the risk premium 3
Literature on Heterogenous Beliefs Harrison and Kreps (1978) Varian (1985), (1989) Harris and Raviv (1993) Detemple and Murthy (1994) Kurz (1994), (1997a) Kurz and Beltratti (1997) Kurz and Motolese (2001) Kurz Jin and Motolese (2005a) (2005b) Motolese (2001) Nielsen (1996) Wu and Guo (2003), (2004). 4
An Infinite Horizon Model Assumptions • Large number of agents • A single commodity -- “consumption” • Riskless technology producing R > 1 at t+1 with 1 unit of input at t Dividend process {D t , t = 1, 2, ...} is non-stationary with unknown probability Π . • • Under m {D t , t = 1, 2, ...} is Markov with unconditional mean µ and transition d t % 1 ' λ d d t % ρ d t % 1 , ρ d t % 1 - N(0 , σ 2 d ) where d t ' D t & µ. . • m induces a marginal probability measure m on (D 4 , ö ) hence E m t [d t % 1 |d t ] ' λ d d t . Notation θ i • = date t stock purchases of agent i. Aggregate supply = 1 t B t = amount invested in the riskless asset • • p t = price of the stock. Think of it as the S&P 500 5
An Infinite Horizon Model (Cont.) & ( c i k ) 4 t [ j E i & β k & 1 e Maximize Max ] (1) τ ( θ i , B i ) k ' t c i t ' θ i t & 1 [p t % d t % µ] % B i t & 1 R & θ i t p t & B i Subject to (i) Budget Constraint , t ( θ i 0 , B i (ii) Initial values 0 ) Exponential utility is common in study of asset pricing. Examples 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 6
An Infinite Horizon Model (Cont.) Assume for a moment: Agents believe is conditionally normal . p t % 1 % d t % 1 Hence demand functions τ θ i [E i (2) t (p t ) ' t (p t % 1 % d t % 1 % µ & Rp t ] . R σ 2 ε σ 2 ε ' Var i [p t % 1 % d t % 1 % µ & Rp t |H t ] assumed constant, the same for all agents. We later clarify the exact value of σ 2 ε We now must be explicit about the belief of agent i, which is the main issue. 7
The Structure of Beliefs: Preliminaries • Under true , { } is non-stationary D t , t ' 1,2,... Π • Observation: m … Π – central to our approach. d t % 1 ' λ d d t % ρ d ρ d t % 1 - N(0 , σ 2 • All know may not be the truth t % 1 , d ) FACTS (i) Subjective modeling contribute more than 50% to forecasts (ii) Agents use m only as a reference from which to deviate (iii) Vast data on market forecast distribution of most variables. Two Central Points On PI vs. HB (i) With PI, data on market forecasts of exogenous variables lead an agent to update his own belief. Such forecasts contain new information! (ii) With HB all know they have the same information. Others’ forecast data do not lead an agent to update belief about exogenous variables as these reflect others’ “opinions” or subjective models. Use forecasts of “others” only to forecast endogenous variables. 8
Individual Belief As A State Variable • Individual are “anonymous” g i • i’s belief state pins down perceived transition of state variables t g i • The distribution of is observed. t g j • The Problem: the dynamics of ? t 1. The information structure (i) Quantitative { } d t , t ' 1,2,... (ii) Qualitative statements unique to date t, do not repeat. (C t1 , C t2 ,...,C tK t ) • Subjective map: for each subset of indices A k C tA k Y Ψ i tA k (d t % 1 & λ d d t ) ' Ψ i If only are realized, i forecasts of C tA k tA k • Subjective interpretation of the public information (C t1 , C t2 ,...,C tK ) K t ! t ' j Ψ i π i tA k Ψ i . tA k k ' 1 (d t % 1 & λ d d t ) ' Ψ i • offer alternate subjective forecast (C t1 ,C t2 ,...,C tK t ) t 9
Individual Belief As A State Variable 2. A Bayesian Motivation • i believes has true transition with unobserved mean b t d t t % 1 - N(0 , 1 d t % 1 & λ d d t ' b t % ρ d ρ d , β ) t % 1 b t - N( n , 1 • At t = 1 a prior belief about b t α ) ˆ E i d t % 1 & λ d d t t % 1 (b t |d t % 1 ) . Needs a belief on b t+1 . After observing i updates to E i E i How to go from t % 1 (b t |d t % 1 ) to t % 1 (b t % 1 |d t % 1 ) ? E i Without new information must use t % 1 (b t |d t % 1 ) . Ψ i (C (t % 1)1 ,C (t % 1)2 ,...C (t % 1)K t % 1 ) imply a subjective estimate t % 1 Assumption (*): i uses a subjective probability to form date t+1 prior µ E i t % 1 (b t % 1 |d t % 1 , Ψ i t % 1 ) ' µE i t % 1 (b t |d t % 1 ) % (1 & µ) Ψ i (3) 0 < µ < 1 . t % 1 10
t - N(0 , 1 Ψ i Theorem 1: Suppose γ ) and Assumption (*) holds. Then for large E i t % 1 (b t % 1 |d t % 1 , Ψ i t the posterior t % 1 ) is a Markov state variable such that if we g i t ' E i t (b t |d t , Ψ i let t ) and µ ' λ Z then (3) implies g i λ Z g i t % ρ ig ρ ig t % 1 - N(0 , σ 2 (4) , g ) . t % 1 ' t % 1 ρ ig • Individual beliefs are correlated via . t 3. Implied Individual Perception d i t % 1 ' λ d d t % λ g d g i t % ρ id ρ id σ 2 (5) , t - N(0 , ˆ d ) . t Hence as before we have data to measure E i [d i t % 1 |H t ,g i t ] & E m [d t % 1 |H t ] ' λ g d g i . t N Z t ' 1 g i N j Definition: – market state of belief. N is large. t i ' 1 11
The Role of the Market Belief State Variable • is observed. Z t • is non stationary with empirical distribution Z t λ Z Z t % ρ Z (6) Z t % 1 ' . t % 1 • We thus expand the empirical distribution to { }. (d t % 1 ,Z t % 1 ) , t ' 1,2,... d t % 1 ' λ d d t % ρ d (7a) ρ d σ 2 d , 0, 0 t % 1 t % 1 - N 0 , ' Σ , i.i.d. ρ Z σ 2 λ Z Z t % ρ Z 0, (7b) Z t % 1 ' t % 1 Z t % 1 • Individual i’s perception model (together with (4)) d i t % 1 ' λ d d t % λ g d g i t % ρ id ρ id σ 2 (8a) ˆ d , ˆ σ Zd 0 t % 1 t % 1 ' Σ i - N 0 , Z i t % 1 ' λ Z Z t % λ g Z g i t % ρ iZ ρ iZ σ 2 (8b) σ Zd , ˆ ˆ t % 1 t % 1 Z λ g λ g g i d $ 0 and Z $ 0 orient model: When t > 0 , i believes t+1 dividend and market belief will be above normal . 12
The Role of the Market Belief State Variable (8a)-(8b) means λ g d g i d t % 1 d t % 1 t E i & E m . ' t t Z t % 1 Z t % 1 λ g Z g i t Q i ((D×Z×G i ) 4 , ö ) Note: i’s belief is a probability on N Q ' 1 ¯ Q i N j Define the market belief by t i ' 1 N E t (X t % 1 ) ' 1 E i ¯ N j The market expectations operator t (X t % 1 ) i ' 1 ¯ Theorem 2 : The market belief Q is not a proper probability and the market E t (d t % 2 ) … ¯ ¯ E t ¯ expectations operator violates iterated expectations: E t % 1 (d t % 2 ) . Market Belief is neither a proper probability nor Rational. • 13
Equilibrium Asset Prices 0< λ Z % λ g 0< λ d <1 , Z <1 Stability Conditions: R = 1 + r > 1 , . E t [Z t % 1 ] ' ( λ Z % λ g ¯ Z )Z t This is natural since . Theorem 3: For the model with HB and under the stability conditions, there is a unique p t ' ad t % bZ t & cS equilibrium price function taking the form with parameters λ d a ' > 0 . R & λ d λ d λ d g b ' [1 % ] > 0 R & λ d R & ( λ Z % λ g Z ) c ' σ 2 g R τ r > 0 . Note: Price map confirms earlier conjecture the price is conditionally Normal 14
Equilibrium Risk Premia Analytic Results 1. π t % 1 ' p t % 1 % d t % 1 % µ & Rp t Realized Premium p t 1 E i Agent i’s Perceived Premium t (p t % 1 % d t % 1 % µ & Rp t ) p t 1 ¯ The Market Perceived Premium E t (p t % 1 % d t % 1 % µ & Rp t ) p t 1 E m The standard Risk Premium t [p t % 1 % d t % 1 % µ & Rp t ] p t What is the relationship among these in equilibrium? 15
Relationship Among Premia E t (p t % 1 % d t % 1 % µ & Rp t ) ' R σ 2 ¯ (9a) ε τ t [p t % 1 % d t % 1 % µ & Rp t ] ' R σ 2 E i τ % [(a % 1) λ g d % b λ g Z ](g i (9b) ε . t & Z t ) (a % 1) λ g d % b λ g • Z > 0 by model orientation t [p t % 1 % d t % 1 % µ & Rp t ] ' R σ 2 E m τ & ( λ g d % λ g (9c) ε Z )Z t λ g d % λ g • Z > 0 ; premium declines with market belief • (9c) is key result: negative when and positive when Z t > 0 Z t < 0 . Result remains true for percentage premium since • R σ 2 τ & ( λ g d % λ g ε Z )Z t t [ p t % 1 % d t % 1 % µ & Rp t E m ] ' p t ad t % bZ t & p 0 16
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