On Market Risk Premia Material in Pages 13 - 37 of Private - - PowerPoint PPT Presentation

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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

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Definition of The Risk Premium

• Actual Premium

. πt%1 ' pt%1 % Dt%1 & Rtpt pt

• m = probability induced by the data’s empirical distribution
• Easy to show that m is unique and stationary probability.
• Long Run Premium = E m[ pt%1 % Dt%1

pt ] & ¯ R

• “The” Premium is a conditional expectations under m

E m

t [πt%1] ' 1

pt E m

t [pt%1 % Dt%1 & Rtpt]

Problem: What are the factors determining the Premium

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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

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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).

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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 {Dt , t = 1, 2, ...} is non-stationary with unknown probability Π.
• Under m {Dt , t = 1, 2, ...} is Markov with unconditional mean µ and transition

.

dt%1 ' λddt%ρd

t%1 , ρd t%1-N(0 , σ2 d) where dt ' Dt&µ.

• m induces a marginal probability measure m on

hence (D 4,ö)

.

E m

t [dt%1|dt] ' λddt

Notation

• = date t stock purchases of agent i. Aggregate supply = 1

θi

t

• Bt = amount invested in the riskless asset
• pt = price of the stock. Think of it as the S&P 500

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An Infinite Horizon Model (Cont.)

(1) Maximize Max

(θi , B i)

E i

t [ j 4 k ' t

&βk&1e

&( c i

k

τ )

] Subject to (i) Budget Constraint , c i

t ' θi t&1[pt%dt%µ] % B i t&1R&θi tpt&B i t

(ii) Initial values (θi

0 , B i 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

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An Infinite Horizon Model (Cont.)

Assume for a moment: Agents believe is conditionally normal. pt%1 % dt%1 Hence demand functions (2) . θi

t(pt) '

τ Rσ2

ε

[E i

t (pt%1 % dt%1 % µ & Rpt]

assumed constant, the same for all agents. σ2

ε 'Var i[pt%1%dt%1%µ&Rpt|Ht]

We later clarify the exact value of σ2

ε

We now must be explicit about the belief of agent i, which is the main issue.

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The Structure of Beliefs: Preliminaries

• Under true

, { } is non-stationary

Π Dt , t ' 1,2,...

• Observation:

– central to our approach.

m … Π

• All know

may not be the truth

dt%1'λddt%ρd

t%1 ,

ρd

t%1 - N(0 , σ2 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.

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Individual Belief As A State Variable

• Individual are “anonymous”
• i’s belief state

pins down perceived transition of state variables g i

t

• The distribution of

is observed. g i

t

• The Problem: the dynamics of

? g j

t

• 1. The information structure

(i) Quantitative { }

dt , t ' 1,2,...

(ii) Qualitative statements unique to date t, do not repeat.

(Ct1 , Ct2,...,CtKt)

• Subjective map: for each subset of indices

Ak

CtAk Y Ψi

tAk

If only are realized, i forecasts of

CtAk (dt%1 & λddt) ' Ψi

tAk

• Subjective interpretation of the public information (Ct1 , Ct2,...,CtK)

.

Ψi

t ' j Kt! k ' 1

πi

tAkΨi tAk

• ffer alternate subjective forecast

(Ct1,Ct2,...,CtKt) (dt%1 & λddt) ' Ψi

t

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Individual Belief As A State Variable

• 2. A Bayesian Motivation
• i believes

has true transition with unobserved mean bt dt dt%1 & λddt ' bt % ρd

t%1

, ρd

t%1 - N(0 , 1

β )

• At t = 1 a prior belief about bt

bt - N(n , 1 ˆ α ) After observing i updates to . Needs a belief on bt+1. dt%1&λddt E i

t%1(bt|dt%1)

How to go from to ? E i

t%1(bt|dt%1)

E i

t%1(bt%1|dt%1)

Without new information must use . E i

t%1(bt|dt%1)

imply a subjective estimate (C(t%1)1,C(t%1)2,...C(t%1)Kt%1) Ψi

t%1

Assumption (*): i uses a subjective probability to form date t+1 prior µ (3) . E i

t%1(bt%1|dt%1,Ψi t%1) ' µE i t%1(bt|dt%1) % (1&µ)Ψi t%1

0 < µ < 1

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Theorem 1: Suppose

and Assumption (*) holds. Then for large Ψi

t - N(0 , 1

γ ) t the posterior is a Markov state variable such that if we E i

t%1(bt%1|dt%1,Ψi t%1)

let and then (3) implies g i

t ' E i t (bt|dt,Ψi t)

µ ' λZ (4) . g i

t%1 '

λZg i

t % ρig t%1

, ρig

t%1 - N(0 , σ2 g)

• Individual beliefs are correlated via

. ρig

t

• 3. Implied Individual Perception

(5) . d i

t%1 ' λddt % λg dg i t % ρid t

, ρid

t - N(0 , ˆ

σ2

d)

Hence as before we have data to measure . E i[d i

t%1|Ht,g i t ] & E m[dt%1 |Ht ]' λg dg i t

Definition:

– market state of belief. N is large. Zt ' 1 N j

N i ' 1

g i

t

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The Role of the Market Belief State Variable

• is observed.

Zt

• is non stationary with empirical distribution

Zt (6) . Zt%1 ' λZZt % ρZ

t%1

• We thus expand the empirical distribution to {

}. (dt%1,Zt%1) , t ' 1,2,... (7a) dt%1 ' λddt % ρd

t%1

ρd

t%1

ρZ

t%1

• N

0 , σ2

d, 0,

0, σ2

Z

' Σ , i.i.d. (7b) Zt%1 ' λZZt % ρZ

t%1

• Individual i’s perception model (together with (4))

(8a) d i

t%1 ' λddt % λg d g i t % ρid t%1

ρid

t%1

ρiZ

t%1

• N

0 , ˆ σ2

d,

ˆ σZd ˆ σZd, ˆ σ2

Z

' Σi

(8b) Z i

t%1 ' λZZt % λg Zg i t % ρiZ t%1

and

• rient model: When

, i believes t+1 dividend and λg

d $ 0

λg

Z $ 0

g i

t > 0

market belief will be above normal .

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The Role of the Market Belief State Variable

(8a)-(8b) means E i

t

dt%1 Zt%1 & E m

t

dt%1 Zt%1 ' λg

dg i t

λg

Zg i t

. Note: i’s belief is a probability on Q i ((D×Z×G i)4,ö) Define the market belief by ¯ Q' 1 N j

N i ' 1

Q i

t

The market expectations operator ¯ Et(Xt%1)' 1 N j

N i ' 1

E i

t (Xt%1)

Theorem 2: The market belief

is not a proper probability and the market ¯ Q expectations operator violates iterated expectations: . ¯ Et(dt%2) … ¯ Et ¯ Et%1(dt%2)

• Market Belief is neither a proper probability nor Rational.

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Equilibrium Asset Prices

Stability Conditions: R = 1 + r > 1 , .

0< λd <1 , 0< λZ%λg

Z <1

This is natural since .

¯ Et[Zt%1] ' (λZ % λg

Z)Zt

Theorem 3: For the model with HB and under the stability conditions, there is a unique equilibrium price function taking the form with parameters

pt ' adt % bZt & cS

.

a ' λd R & λd > 0 b ' λd

g

R&(λZ%λg

Z)

[1 % λd R & λd ] > 0

.

c ' σ2

gR

τr > 0 Note: Price map confirms earlier conjecture the price is conditionally Normal

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Equilibrium Risk Premia

1.

Analytic Results Realized Premium πt%1 ' pt%1 % dt%1 % µ & Rpt pt Agent i’s Perceived Premium 1 pt E i

t (pt%1 % dt%1 % µ & Rpt)

The Market Perceived Premium 1 pt ¯ Et(pt%1 % dt%1 % µ & Rpt) The standard Risk Premium 1 pt E m

t [pt%1 % dt%1 % µ & Rpt]

What is the relationship among these in equilibrium?

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Relationship Among Premia

(9a) ¯ Et(pt%1%dt%1%µ&Rpt) ' R σ2

ε

τ (9b) . E i

t [pt%1 % dt%1 % µ & Rpt] ' R σ2 ε

τ % [(a%1)λg

d % bλg Z](g i t & Zt)

• by model orientation

(a%1)λg

d % bλg Z > 0

(9c) E m

t [pt%1%dt%1%µ&Rpt] ' R σ2 ε

τ & (λg

d % λg Z)Zt

• ; premium declines with market belief

λg

d % λg Z > 0

• (9c) is key result: negative when

and positive when .

Zt > 0 Zt < 0

• Result remains true for percentage premium since

E m

t [ pt%1%dt%1%µ&Rpt

pt ] ' R σ2

ε

τ & (λg

d % λg Z)Zt

adt % bZt & p0

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Endogenous Uncertainty and Risk Premium: Decomposition

(I) Volatility = Its effect on mean premium via

. R σ2

ε

τ

• σ2

ε ' Vart(pt%1 % dt%1)

' Var[(a%1)ρid

t%1 % bρiZ t%1] ' ((a%1) , b)TΣi((a%1) , b)

C It increases volatility of returns since b > 0 (II) Risk Perception reflected in

& (λg

d % λg Z)Zt

• Sign is of great interest
• When Zt > 0 the market views the long position as less risky and a lower

risk premium is awarded to it.

• Think of Z the same as Non Farm Payroll

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2. Empirical Test

• Results in Kurz and Motolese (2005)
• Estimate premia in futures, bond and the stock markets
• Dependent variable = realized excess holding returns of: Fed Funds futures,

3 month T Bills, 6 month T Bills and the S&P500.

• Data restrict to holding period from 1 - 6 months for Fed Funds futures,

from 1 - 12 months for T Bills and one quarter for the S&P500.

• Bond and Federal Funds Futures:1980:1 to 2003:10.
• forecast data of Blue Chip Financial Forecasts.
• In theory

and are for one asset and one period g i

t

Zt

• Use forecasts of interest rates to construct beliefs as in theory
• Use

for maturity k and holding period h Z (k,h)

t

• means mean market believes interest rate on maturity k will be

Z (k,h)

t

> 0 lower than normal at t + h.

• Orientation of

is as in the theory. Z (k,h)

t

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Variables on Distribution of Beliefs

– Cross sectional variances of individual beliefs σ(k,h)

t

• - beliefs about the slope of yield curve

SZ (6&F,h)

t

' Z (6,h)

t

& Z (F,h)

t

Traditional Variables: Recession, Monetary Policy and Past Yields

–year over year growth rate of Non Farm Payroll at t-1 NFPt&1 – rate of inflation at t-1 CPIt&1 – measured cumulative intensity of monetary policy F Cum

t

– three principal components of past interest rates R Fj

t

, j ' 1,2,3

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Results

Table 1A: Predictability of Excess Returns, Fed Funds Constant

NFPt&1 CPIt&1 F Cum

t

R F1

t

R F2

t

R F3

t

σ(F,h)

t

Z (F,h)

t

SZ (6&F,h)

t

R 2

h=4 0.503* (0.231)

• 0.177*

(0.040) 0.087* (0.036)

• 0.007

(0.033) 0.234* (0.107)

• 0.013

(0.046)

• 0.098

(0.057)

• 0.945*

(0.493)

• 0.573*

(0.126)

• 0.871*

(0.273) 0.289 h=6 0.633* (0.312)

• 0.232*

(0.052) 0.169* (0.047)

• 0.005

(0.042) 0.373* (0.141) 0.052 (0.068) 0.039 (0.108)

• 0.988*

(0.488)

• 0.930*

(0.188)

• 1.661*

(0.482) 0.436 Table 2: Predictability of Excess Returns, 3 Months Treasury Bills Constant

NFPt&1 CPIt&1 F Cum

t

R F1

t

R F2

t

R F3

t

σ(3,h)

t

Z (3,h)

t

SZ (6&F,h)

t

R 2

h=6 0.820* (0.174)

• 0.185*

(0.026) 0.078* (0.025) 0.006 (0.021) 0.360* (0.072) 0.032 (0.036)

• 0.044

(0.042)

• 0.820*

(0.189)

• 0.516*

(0.095)

• 0.636*

(0.153) 0.447 h=10 1.272* (0.133)

• 0.168*

(0.025) 0.067* (0.020) 0.027 (0.015) 0.561* (0.063)

• 0.016

(0.030)

• 0.013

(0.025)

• 0.887*

(0.120) 0.437* (0.063)

• 0.413*

(0.149) 0.663 Table 4C: Predictability and Time Variability of Excess Returns, 6 Months Treasury Bills Constant

NFPt&1 CPIt&1 F Cum

t

R F1

t

R F2

t

R F3

t

σ(6,h)

t

Z (6,h)

t

SZ (6&F,h)

t

R 2

h=6 1.964* (0.370)

• 0.388*

(0.063) 0.175* (0.050) 0.012 (0.047) 0.828* (0.158) 0.045 (0.081)

• 0.052

(0.087)

• 2.508*

(0.441)

• 1.434*

(0.195)

• 1.384*

(0.348) 0.519 h=10 2.717* (0.269)

• 0.387*

(0.056) 0.092* (0.038) 0.072* (0.032) 1.141* (0.133)

• 0.100

(0.063)

• 0.014

(0.052)

• 1.867*

(0.258)

• 0.906*

(0.129)

• 0.733*

(0.279) 0.677

Sum: Risk premium declines with market belief and with diversity of beliefs.

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GRAPHS

Figure 1 Excess Returns on Fed Funds Futures contract 6 months ahead. The gray line

(green in color) represents the fitted values from regression.

Figure 2 Excess Returns on 3 Months T-Bills 10 months ahead. The gray line (green in

color) represents the fitted values from regression.

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Figure 3 Excess Returns on 6 Months T-Bills 10 months ahead. The gray line (green in

color) represents the fitted values from regression).

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Rationality of Beliefs: Can It All Be Rationalized

• Not REE
• Reject Arrow-Debreu’s equating individual states with Maket States

Without Rationality restrictions theory has internal contradictions:

Example:

E i

t (pt%1 % dt%1 % µ & Rpt) ' (a%1)(λddt%λg dg i t )%b(λZZt%λg Zg i t )%µ&c&Rpt

while under m

E m

t [pt%1%dt%1%µ&Rpt] ' (a%1)(λddt)%b(λZ)Zt)%µ&c&Rpt

Rationality would insist the time average of is zero. True under the Bayesian

g i

t

procedure outlined.

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Rational Beliefs (RB), Kurz (1994), (1997a)

Definition: A belief is said to be an RB if it is a probability model which, if simulated, reproduces the empirical distribution known from the data.

• An RB cannot be rejected by the data
• In the perception models (4) and (14a) -(14b) the covariance matrix and

two parameters are defined by the agent’s belief (λg

Q , λg Z)

• For (4), (14a) -(14b) Rationality of Belief requires
• The empirical distribution of

λg

dg i t %ρid t

λg

Zg i t %ρiZ t%1

Equals the distribution of

ρd

t

ρZ

t%1

• N 0

0, σ2

d, 0,

0, σ2

Z

, i.i.d.

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What Does These Amount To?

Five implied rationality conditions: (1) (2) (3) (λg

d)2σ2 g

1 & λ2

Z

% ˆ σ2

d ' σ2 d

(λg

Z)2σ2 g

1 & λ2

Z

% ˆ σ2

Z ' σ2 Z

λg

dλg Zσ2 g

1 & λ2

Z

% ˆ σZd ' 0 (4) (5) . (λg

d)2λZσ2 g

1 & λ2

Z

% Cov(ˆ ρid

t , ˆ

ρid

t&1) ' 0

(λg

Z)2λZσ2 g

1 & λ2

Z

% Cov(ˆ ρiZ

t , ˆ

ρiZ

t&1) ' 0

(1)-(3) pin down the covariance matrix in (14a)-(14b). (4)-(5) pin down the serial correlation of the two terms . ( ˆ ρid

t , ˆ

ρiZ

t )

Further Restrictions on the "Free" (λg

d , λg Z)

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(I) restrict : ˆ σ2

d > 0 , ˆ

σ2

Z > 0

(λg

d , λg Z)

. |λg

d| < σd

σg 1 & λ2

Z

|λg

Z| < σZ

σg 1 & λ2

Z

(II) The covariance matrix in (14a)-(14b) is positive definite. This implies . 1 & λ2

Z

σ2

g

> (λg

Z) 2

σ2

Z

% (λg

d) 2

σ2

d

The "free" are restricted to a narrow range but sufficient to generate (λg

d , λg Z)

volatility in order of magnitude seen in the data.

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