Equity Premium in an Asset Pricing Model with Robust Control

Eric F. Y. Lam*

Gregory C. Chow†

(Working paper)

*Department of Economics and Finance, City University of Hong Kong, Kowloon, Hong Kong †Department of Economics, Princeton University, New Jersey, USA

1. Introduction

The return on aggregate stock in excess of the risk free rate in the United States in the 19th century has caught much attention in the finance literature [see Mehra and Prescott (1985), Mehra and Prescott (1988), Kurz and Beltratti (1996), Kocherlakota (1996), Mehra (2003), Mehra and Prescott (2003), among others]. In this paper, we analyze the equity premium by recognizing that a model is at best an approximation to the complex reality in the financial sector and such approximation is subject to misspecification. A consumer-investor is assumed to make consumption and asset allocation decisions that are robust to model misspecification. This is an application of the robust control theory of Hansen et al. (1999, 2000, 2002 and 2003). Our study is based on the robust asset pricing model of Chow and Zheng (2002). Using standard power utility function and aggregate consumption data, the model with reasonable values of its parameters can explain the observed equity premium. The main asset pricing implication when the investor-consumer prefers robustness is that she bears a burden of pessimism as she participates in the financial market. The equity premium implied by the model satisfies the Hansen-Jagannathan (1991) volatility bound.

The plan of the paper is as follows. Section 2 reviews the robust pricing model of Chow and Zheng (2002), derives the asset pricing equations when the consumer-investor carries out a robust control policy in view of model misspecification, and studies the pricing implications of preference for robustness. Section 3 estimates the long run mean incentive adjustment due to pessimism and the degree of robustness of US consumer-investor based on historical sample moments of returns and consumption growth, and decomposes the empirical equity premium into the fraction priced by the intertemporal marginal rate of substitution (compensation for the positive correlation with consumption) and the fraction due to preference for robustness (incentive adjustment for bearing the burden of pessimism). Section 4 calibrates the lower bounds of the long run mean incentive adjustment for pessimism, computes the distances between the point estimates from section 3 and the lower bounds, and sets an upper bound on the mean equity premium. Section 5 discusses some possible future studies on robust asset pricing. Section 6 concludes.

2. Robust Pricing Equations

The robust control psychology as studied by Hansen et al. (1999, 2000, 2002 and 2003) takes model uncertainty into consideration and seeks an appropriate adjustment. In the context of the intertemporal portfolio selection model of Samuelson (1969) as reformulated by Chow and Zheng (2002), a representative rational and robust consumer-investor chooses between consumption and investment in a risky asset and a less risky one over time by solving a max-min two-agent dynamic game. If the distribution of asset returns can be misspecified, the consumer-investor is assumed to use returns drawn from the distribution plus a correction vector which may be negative to allow for unfavorable circumstances. The consumer-investor assumes that nature as a fictitious second player sets the correction vector to lower her expected utility but subject to a cost constraint. This represents a conservative or pessimistic attitude. To the extent that nature can set the correction vector to reduce the return on a risky investment, the consumer-investor would require a higher premium to choose the venture.

¡ ¢ = T Let Rt+1 [R1,t+1 R2,t+1] be a vector of gross returns respectively to an aggregate

stock index and to a bond in an arbitrage-free financial market, meaning that one £

dollar invested in asset i at the beginning of period t will result in Ri,t+1 dollars at the end of period t or beginning of period t+1. The tilde marks are placed above random variables at time t for clarity purpose. Denote the covariance matrix of ¤ = σ Rt+1 by ( ij )2×2 . The consumer-investor is assumed to construct a self- financing portfolio with the two assets and consume Ct during period t. Let wt − and (1 wt ) be the proportion of investment allocated to the aggregate stock index and the risk-free bond. Following Samuelson (1969), the beginning-of- period value Zt+1 of such a portfolio is restricted by

= − [ − ] ¥ Zt+1 (Zt Ct ) wt 1 wt Rt+1 .

The imagined problem of nature is

∞ ¦ ©

t ¨ 1 T −1

β + θ min+∞ Et u(Ct ) Vt+1 Vt+1 {V } t+1 t=0 t=0 2

subject to the above constraint on Zt+1 with the correction vector Vt+1 added to ¤ β ∈ Rt+1 . (0,1] is the consumer-investor’s subjective discount factor, u(Ct ) is the θ ∈ ++ period utility function and is a parameter inversely measuring the θ T −1 preference for robustness. The term ( / 2)Vt+1 Vt+1 is introduced as a cost to nature in order to confine its setting of Vt+1 . The Lagrange expression associated with nature’s minimization problem is

∞ ¦

θ −

t T 1 = β − βλ − − − + + L Et u(Ct ) t+1 Zt+1 (Zt Ct )[wt 1 wt ](Rt+1 Vt+1 ) Vt+1 Vt+1 t=0 2

λ = θ → ∞ where { t+1 : t 0,1, } are Lagrange multipliers. As , nature sets Vt+1 equals to zero, and the model reduces to the standard one. By construction, the

* robust pricing structure nests the standard case. Using nature’s solution Vt+1 to eliminate Vt+1 in L , the consumer-investor solves her robust optimization problem by maximizing

§ ∞ ¦

* ©

* t ¨

L = E β {u(C ) − βλ + Z + − (Z − C )[w 1− w ](R + ) − B(Z ,C , w )} t t t 1 t 1 t t t t t 1 t t t t=0

with respect to Ct , Zt and wt . The function B(Zt ,Ct , wt ) is the burden in the period utility function due to pessimism and is defined [see Chow and Zheng (2002)] as

1 B(Z ,C , w ) ≡ {β (Z − C )E u '(C )}2[σ w 2 +σ (1− w )2 + 2σ w (1− w )]. t t t θ t t t t+1 11 t 22 t 12 t t

The larger the θ , the lower the burden and the less pessimistic the consumer- investor. As θ → ∞ , the consumer-investor bears no burden and prefers no robustness against model misspecification, and the model reduces to the standard one. In this sense, θ also inversely measures the degree of pessimism. The burden B can be estimated from aggregate data on overall portfolio value, consumption, portfolio weight, and variances and covariance of returns to the stock and the bond once the values of θ and are given.

From the time t first order conditions or Euler equations of the robust optimization problem, we derive the fundamental pricing equations for stock return and bond return studied in this paper

B (Z − C ) + (1− w )B

Zt t t t wt

= − 1 E I R (1a) t t+1 1,t+1 −

u '(Ct )(Zt Ct )

B (Z − C ) − w B

Zt t t t wt

= − 1 E I R . (1b) t t+1 2,t+1 − u '(Ct )(Zt Ct )

" ! ≡ β > It+1 u '(Ct+1) / u '(Ct ) 0 is the intertemporal marginal rate of substitution (IMRS) > in consumption between time t and t+1. The function u '(Ct ) 0 is the < corresponding marginal utility and u ''(Ct ) 0 due to diminishing marginal utility. Subtracting (1b) from (1a), the pricing equation for excess return on the stock over the bond is

= − wt # $ 0 E I R . (2) t t+1 e,t+1 −

u '(Ct )(Zt Ct )

& ' % ≡ − Re,t+1 R1,t+1 R2,t+1 is the equity premium. In equations (1a) and (1b), the partial derivative of the burden B(Zt ,Ct , wt ) with respect to Zt is

( 2 2 2 2 B = {Z − C }{β E u '(C + )} [σ w +σ (1− w ) + 2σ w (1− w )] = −B , Zt θ t t t t 1 11 t 22 t 12 t t Ct which is the change in burden in shifting a dollar of total wealth between consumption and investment. B = −B because raising investment by a dollar is Zt Ct reducing consumption by a dollar. The terms in the square bracket yield a strictly positive real number. Since θ and the terms in the curly brackets are also strictly

‡ ∈ ++ ∈ −− ) positive , B ) and B . Therefore, the marginal burden in raising Z Zt Ct t or shifting a dollar in wealth from consumption to investment is strictly positive.

In equations (1a), (1b) and (2), the partial derivative of the burden B(Zt ,Ct , wt ) with respect to wt is

‡ Since the portfolio is self financing to support consumption into the infinite future, it must be the > case that Zt Ct . * 1 2 B = {β (Z − C )E u '(C + )} [2σ w − 2σ (1− w ) + 2σ (1− 2w )] = −B − . wt θ t t t t 1 11 t 22 t 12 t 1 wt

which is the change in burden in shifting a fraction wt of investment from the ∈ ++ bond to the stock. By similar argument, B ) . Therefore, the marginal burden wt in raising wt or raising an investment fraction from the bond to the stock is strictly positive.

In the standard model, the IMRS is a valid representation of the stochastic discount factor (SDF). When the consumer-investor prefers robustness against

model uncertainty, the IMRS may under-discount or over-discount asset returns ,

in the sense of Et I+ t+1 Ri,t+1 being larger or smaller than unity as indicated by (1a) . or (1b) and Et I- t+1 Re,t+1 being larger than zero as indicated by (2). The above conditional expectations are corrected to unity and zero respectively via three u '(Ct ) scaled marginal burdens of pessimism. As given in equations (1a) and (1b), B / u '(C ) is the marginal burden in shifting a dollar in wealth from Zt t consumption to investment as a fraction of the period marginal utility. This term is subtracted from both the IMRS-discounted returns on the stock and the bond. w B / u '(C )(Z − C ) is the change of burden in shifting a dollar from the bond to t wt t t t the stock as a portion of the period marginal utility since B is the change of wt − burden with respect to a fraction wt of a portfolio of (Zt Ct ) dollars. The intuition for (1− w )B − / u '(C )(Z − C ) is analogous. The two changes are added to the t 1 wt t t t IMRS-discounted returns on the stock and the bond respectively. As given in equation (2),

B ω θ ≡ wt > σ ⊆ ω < ∧ ω < ( ; ) 0, {Z ,C , w , ⋅⋅ } , 0 θ 0 t − t t t ( ) t Ct u '(Ct )(Zt Ct )

is the marginal burden in shifting a fraction wt from the bond to the stock as a . portion of the period marginal utility. This term causes Et I- t+1 Re,t+1 to be greater than zero. Thus the robust consumer-investor requires a higher excess return from investing in the stock than otherwise.

By the law of iterated expectations, taking unconditional expectation of (2) yields

= . − ω θ 0 EI- t+1 Re,t+1 E ( t ; ) .

Decomposing the first term on the right hand side by applying the identity = − = ρ σ σ ρ σ Cov(X ,Y) EXY (EX )(EY ) X ,Y X Y , where (⋅,⋅) and (⋅) denotes correlation coefficient and standard deviation, we obtain

0 ρ σ σ + (EI/ + )(ER + ) − Eω( ;θ) = 0 . (3) It+1 ,Re,t+1 It+1 Re,t+1 t 1 e,t 1 t

Assume that the IMRS and the asset returns are ergodic stationary for the first two moments, (3) implies

ω θ ρ σ 1

E ( t ; ) I ,Re I ERe 2 2 − = . σ EI EI σ Re Re

Applying the Triangular Inequality to the left hand side of the above equation,

ω θ ρ σ ω θ ρ σ

E ( t ; ) I ,Re I E ( t ; ) I ,Re I

2 2 2 2 + − ≥ − . σ EI EI σ EI EI Re Re

It follows that 3 ω θ | ρ |σ

E ( t ; ) I ,Re I | ERe | 2 2 + ≥ σ EI EI σ Re Re

≤ ρ ≤ Since 0 | X ,Y | 1, we obtain

σ 5 ω θ

I | ERe | E ( t ; ) 2 2 ≥ − . (4) EI σ σ EI Re Re

By essentially using an inequality to eliminate the correlation between the IMRS and the excess return, we establish the lower mean-variance bound of the IMRS as the Sharpe ratio of the excess return less the unconditional mean of incentive adjustment due to pessimism as a fraction of the product of the standard deviation of excess return and the mean of the IMRS. When the consumer- investor prefers no robustness against model misspecification, lim Eω(C ;θ ) = 0 , θ →∞ t (4) reduces to

σ 6 I | ERe | 2 ≥ . (5) EI σ Re

The above is a parametric approach of stating the Hansen-Jagannathan (1991) lower volatility bound. If the above inequality is violated, the extra term in (4) can reduce the gap.

3. Robustness of US Consumer-investor

= 1−γ −γ Suppose the consumer-investor has the power utility function u(Ct ) Ct /(1 ) , where the curvature parameter γ is the constant coefficient of relative aversion ≡ " in this case. Define the gross consumption growth as G7 t+1 Ct+1 / Ct . Then the

−γ −γ

β = β 9

IMRS is (C8 t+1 / Ct ) Gt+1 . Assume that the consumption growth and the net =

< T

> ? ; excess return follow a joint lognormal distribution, i.e. G: t+1 Re,t+1 ~ ( , ) , where

@ A

2 A

@ T σ σ E

D G G,Re C

≡ µ µ ^ ≡ B E

G Re D σ σ 2 C B G,Re Re

are the corresponding mean vector and the covariance matrix. Then

F G ≡ φ λ 2 gt+1 ln Gt+1 ~ N ( g , g ) , whose mean and variance are

1 φ = ln(µ 2 ) − ln(µ 2 +σ 2 ) g G 2 G G λ 2 = +σ 2 µ 2 g ln(1 G / G ).

− Other than the weights {wt ,(1 wt )} of the assets in the portfolio, upper case letter denotes the level of a variable and low case letter denotes the natural logarithm of a variable. As normality is preserved under linear transformation,

−γ

2 2

I −γ H = −γφ γ λ gt+1 ln Gt+1 ~ N( g , g ) .

−γ

The first and second (centered) moments of GJ t+1 are

K −γ −γφ +γ 2λ 2 = g g / 2 EGt+1 e

K −γ − γφ +γ 2λ 2 γ 2λ 2 = 2 g g g − VarGt+1 e (e 1).

It follows that the first and second (centered) moments of the IMRS are

M L −γ −γφ +γ 2λ 2

g g / 2 = β = β = N EIt+1 EGt+1 e EI (6a)

O −γ − γφ +γ 2λ 2 γ 2λ 2 2 2 g g g

σ = VarI + = β VarGI + = β e (e −1) = σ . (6b) It+1 t 1 t 1 I

0 Q Similarly, rP + ≡ ln R + ~ N (φ ,λ ) . Given the moments of r + , the moments e,t 1 e,t 1 re re e,t 1 of the net excess return are

S R φ +λ 2 / 2 = re re − = ERe,t+1 e 1 ERe

R 2φ +λ 2 λ 2 re re re σ = VarR + = e (e −1) = σ . Re,t+1 e,t 1 Re

M −γ

The covariance between ln Gt+1 and ln RT e,t+1 is

V W X U

−γ Cov(ln G + ,ln R + ) = −γCov(g + ,r + ) = −γλ . t 1 e,t 1 t 1 e,t 1 g,re

The covariance between the IMRS and net excess return is related to the above

Z [ Z Y

−γ [−γφ +φ +0.5(γ 2λ 2 +λ 2 )] −γλ = β = β g re g re g ,re − Cov(It+1, Re,t+1) Cov( Gt+1 , Re,t+1) e (e 1) .

Furthermore, the variances of the log normally distributed stock and bond returns, ln R ≡ r ~ N (φ ,λ 2 ), i =1,2, are given by i i ri ri

2φ +λ 2 λ 2 = ri ri ri − = σ VarRi,t+1 e (e 1) ii .

By ergodic stationarity for the first two moments,

] p b 1 T I ≡ I → EI T t=1 t

] p 1 T 2 Var(I) ≡ (I − I) →σ 2

T −1 t=1 t I _ ^ p 1 T R ≡ R → ER , i =1,2 i T t=1 i,t i

^ p 1 T 2 Var(R ) ≡ (R − R ) →σ 2 .

i T −1 t=1 i,t i Ri a ` p 1 T R ≡ R → ER e T t=1 e,t e

` p 1 T 2 Var(R ) ≡ (R − R ) →σ 2

e T −1 t=1 e,t e Re

a ] ` a b b p 1 T Cov(I, R ) ≡ (I − I)(R − R )→Cov(I, R ). e T −1 t=1 t e,t e e

The sample moments of IMRS are derived from the sample moments of gross consumption growth. The above sample statistics are consistent estimators of the corresponding population moments of the two time series. Therefore, the long run sample moments of gross consumption growth and returns on stock and bond documented by Mehra and Prescott (1985) and replicated on panel A of table 1 should be close to the population counterparts. Campbell et al (1997) extend the stock return data to include 19 more years of observations and use a different fixed income instrument is to proxy the risk-free rate. Their data are replicated on panel B of table 1.

~Table 1

3.1 Estimating preference for robustness

ω β γ Using equation (3), a point estimate of E ( t ) , given and , can be obtained via the historical covariance between consumption growth and excess return as

L L

ω = + T E ( t ) Cov(It+1, RT e,t+1) (EIt+1)(ERe,t+1) (7)

Treat the proportions of investment allocated to the stock and the bond as time

−γ

2 −2γ § −κ −κ ** ≈ = 8 ≈ ≈ invariant, wt w 0.9 , and assume that (Et Ct+1 ) Ct , ECt (ECt ) , −2γ ≈ σ ≈ †† θ ω Cov(Zt ,Ct ) 0 and 12 0 , the approximate for a particular E ( t ) is

2 β [2σ w − 2σ (1− w)] −γ θ ≈ 11 22 (EZ − EC )(EC ) . (8) ω t t t E ( t )

Since the amount of wealth can be measured in any monetary unit, we treat EZt

‡‡ as a numeraire and ECt as a fraction 0.064 of it.

Equation (3) also divides the mean equity premium into two parts

d d c Eω( ;θ) Cov(I + , R + ) = t − t 1 e,t 1 ERe,t+1 . (9) EIt+1 EIt+1

§ Campbell and Cochrane (1999) assume log consumption follows a random walk with drift

= + ε ε σ 2 ct+1 ct t+1, t+1 ~ IID(0, ε ) .

Multiply both sides by −γ , take conditional expectation, apply Jensen’s inequality to the LHS and −γ > −γ take anti-log, EtCt+1 Ct . ** −κ < −κ By Jensen’s inequality, E(Ct ) [E(Ct )] . †† This is the case if the bond is risk-free. ‡‡ If the consumer-investor allocates 90% of the investment to stock and the rest to bond, the summary statistics in panel A of table 1 suggests that the long run mean annual return of the portfolio is about 6.40%. The portfolio can self-finance consumption into the indefinite future if the consumer-investor retains about 93.60% of end of the period portfolio value annually. ω θ The first is attributable to E ( t ; ) and the second is priced by the IMRS given certain values of β and γ . The covariance in the second term equals

e f

h f e > <

−γ i

β 0 if Cov(Ct+1, Re,t+1) 0

e f j Cov(C , R ) h −γ t+1 e,t+1 < > Ct 0 if Cov(Ct+1, Re,t+1) 0.

If the excess return is more positively correlated with consumption, the mean required by the consumer-investor is higher as indicated by a larger positive term added to the first term in equation (9). The intuition is that the excess return raises the fluctuation of the consumption path of a risk-averse consumer-investor. Expressed as fractions of the risk premium these two components are

ω θ

E ( t ; ) l ≡ k PR1 (10a)

ERe,t+1EIt+1

m n

Cov(It+1, Re,t+1) m ≡ − n PR2 (10b) ERe,t+1EIt+1

Based on panel B of table 1, we compute and report the point estimates of the N σ ω θ θ γ β EI , I , E ( t ; ) , 1/ , PR1 and PR2 under plausible values for and in table 2.

~table 2

ω θ θ γ = E ( t ; ) and 1/ are estimated to be 0.05090 and 0.00387 when 2 and β = ω θ ω θ 0.99 . The estimate of E ( t ; ) is close to the value 0.0485 for ( t ; ) suggested by Chow and Zheng (2002) to solve the equity premium puzzle without resort to estimation. At γ = 5 and β = 0.99 , 74.25% of the empirical equity premium can be attributed to the unconditional expectation of the marginal burden of pessimism in shifting investment from the bond to the stock while 25.75% is due to compensation for the excess return’s positive correlation with consumption.

3.2 Bounding equity premium with robustness

In the standard case without robust control, the IMRS lower volatility bound or the inequality (5) is substantially violated empirically using past US data, plausible values of γ in the CRRA power utility function. Restating (5), the anomaly§§ is

− γφ +γ 2λ 2 γ 2λ 2 o e 2 g g (e g −1) | ER | < e −γφ +γ 2λ 2 . g g / 2 σ e Re

Attempts have been made to solve this puzzle by using different forms of preference [e.g. Epstein and Zin (1989, 1990 and 1991), Constantinides (1990), Abel (1990), Ferson and Constantinides (1991), and Campbell and Cochrane (1999)] or by substituting consumption out of the model [e.g. Campbell (1993) and Li (1997)]. In our robust control framework, the unconditional expectation ω θ E ( t ; ) in (4) is used. We calibrate the model such that the IMRS lower volatility bound (4) is just satisfied using historical US data, plausible values of γ β ω θ and . Define the value of E ( t ; ) just satisfying the lower volatility bound by

r q

Eω( ;θ ) ≡ inf {Eω(⋅) : Eω(⋅) ≥| ERp | EI −σ σ } . (11) t t min s ++ e θ∈ I Re

Any smaller value violates the lower volatility bound. Due to the fact that ω θ ω θ E ( t ; ) lies on or above E ( t ; )min , the interest lies on the distance between the point estimate and the lower bound

§§ This particular inequality can be trivially saved by assuming an unreasonably large value for γ . Eω( ;θ ) − Eω( ;θ) d ≡ t t min (12) Eω ω θ E ( t ; )min

Using the point estimates based on (7), we establish an upper bound on the mean equity premium, which is defined as

w x

y y

u u v

u σ σ + ω θ { z E ( t ; )

≡ ≤ I Re y

| ER | sup y | ER |:| ER | (13) } e max | e e EI

ω θ Based on panel A of table 1, we compute and report E ( t ; )min under plausible values for γ and β in table 3. With the results from the previous section, we also

compute and report dEω and | ER e |max .

~table 3

γ = β = ω θ For 2 and 0.99 , the lower bound of E ( t ; ) is 0.04801 and the ω θ γ = estimate of E ( t ; ) lies above its lower bound by 6.02%. When 5 , the γ ω θ discrepancy increases to 35.80%. As increases, E ( t ; ) lies further above its lower bound. Thus these lower bounds become worse point estimates as the consumer-investor become more risk averse. The upper bound on the mean equity premium is 6.48% for γ = 2 and β = 0.99 . When the consumer-investor becomes more risk averse, a higher absolute mean equity premium is admissible.

4. Future Studies

Estimates and statistical tests using raw data series will improve on the figures provided in this paper. We will investigate whether adding subsistence consumption to the empirical analysis would hinder the prominent effect of the robust pricing model. These investigations will shed further light on the general applicability of the robust control framework in asset pricing research. We will allow the degree of pessimism in the robust pricing model to be time varying and estimate the amount of pessimistic premium a financial market has offered over time and investigate whether pessimism has been self-dependent and contagious across regions. Time variation in equity premium is to be decomposed into variations due to investment or consumption risk and modeling risk.

5. Conclusive Remark

Based on the sample moments of Campbell et al (1997), the expectation of the discounted extra expected marginal utility of a dollar of investment allocated to stock rather than to bond as a fraction of the period marginal utility is estimated to be 0.05090 when the relative risk aversion is 2 and a subjective discount factor equals 0.99. This is close to the value suggested by Chow and Zheng (2002) to solve the equity premium puzzle without resort to estimation. This point estimate is above its lower bound and becomes further away from the bound as the consumer-investor become more risk averse but the gap is invariant to the taste on patience. The degree of robustness of a typical US consumer-investor is estimated to be 0.00387. For the same subjective discount factor and a plausible relative risk aversion of 4, 74.25% of the empirical equity premium can be attributed to the above mean while 25.75% is due to compensation for the excess return’s positive correlation with consumption. Calibration using summary statistics of Mehra and Prescott (1985) suggests that when the above expectation is higher than or equals to 0.04801, the mean-variance of the IMRS of the robust pricing model with CRRA power preference meets the Hansen- Jagannathan (1991) lower volatility bound with a coefficient of relative risk aversion of 2 and a subjective discount factor of 0.99. Essentially, the IMRS mean-variance lower bound or the Sharpe ratio of the equity premium is corrected by a standardized incentive adjustment due to pessimism. In the presence of the preference for robustness against model uncertainty, the absolute mean equity premium should be less than or equals to 6.48%. When the consumer-investor becomes more risk averse, a higher absolute figure is admissible.

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Table 1. Summary Statistics of US Annual Consumption Growth and Asset Returns

Panel A. Sample moments from 1889 to1978

− − G R1 1 R2 1 Re µ (⋅) 1.0183 0.0698 0.0080 0.0618

σ (⋅) 0.0357 0.1654 0.0567 0.1667

Panel B. Sample moments from 1889 to 1997

g r1 r2 re φ (⋅) 0.0172 0.0601 0.0183 0.0418 λ (⋅) 0.0328 0.1674 0.0544 0.1774 λ g ,(⋅) 0.0011 0.0027 -0.0002 0.0029

Note: Data in the 2nd to 5th columns in panels A and B are sourced from table 1 of Mehra and Prescott (1985) and table 8.1 of Campbell et al (1997) respectively. Real consumption growths are based on per capita real consumption on non-durables and services. Aggregate real stock returns are real returns on the stock index constructed by Grossman and Shiller (1981) prior to 1920s and Standard & Poor’s Composite Stock Price Index thereafter. Mehra and Prescott use real returns on 90-day Prime Commercial Paper before 1920, 90-day US Government Treasury Certificates for 1920 to 1930 and 90-day US Government Treasury bills for 1931 to 1978 to proxy the risk free rate. Campbell et al. use real return on 6-month commercial paper as a proxy to the risk free rate. Nominal terms are converted to real terms by consumption deflator series. Mehra µ σ and Prescott report change in level while Campbell et al. report change in log. (⋅) and (⋅) are φ λ the mean and standard deviation of a lognormal random variable. (⋅) and (⋅) are the mean and λ standard deviation of a normal random variable. g ,(⋅) is the covariance between the log consumption growth, which is normal by assumption, and another normal random variable. Low case letter denotes log and upper case denotes level of a variable.

Table 2. Estimated Moments of the Intertemporal Marginal Rate of Substitution (IMRS) Based on CRRA Power Utility Function, Point Estimates of the Unconditional Mean of Incentive Adjustment Due to Pessimism, Point Estimates of the Preference for Robustness, Fractions of the Empirical Equity Premium Priced by the IMRS, and Fractions of Empirical Equity Premium Attributable to Pessimism for the Coefficient of Relative Risk Aversion Between 2 to 8 and Subjective Discount Factor Being 0.99, 0.95 and 0.90

Panel 1. β =0.99

2 3 4 5 6 7 8

0.95858 0.94477 0.93217 0.92072 0.91039 0.90114 0.89295 EIN σ I 0.06295 0.09319 0.12283 0.15202 0.18091 0.20966 0.23840 ω θ E ( t ; ) 0.05090 0.04728 0.04382 0.04049 0.03728 0.03419 0.03119 1.3649e- 8.0717e- 4.7568e- 2.7916e- 1.6301e- 1/θ 0.00387 0.00023 005 007 008 009 010

PR1 0.89656 0.84507 0.79373 0.74253 0.69149 0.64059 0.58983

PR2 0.10344 0.15493 0.20627 0.25747 0.30851 0.35941 0.41017

Panel 2. β =0.95

2 3 4 5 6 7 8

0.91985 0.90660 0.89450 0.88352 0.87360 0.86473 0.85687 EIN σ I 0.060407 0.089426 0.11787 0.14588 0.17360 0.20119 0.22877 ω θ E ( t ; ) 0.04884 0.04537 0.04205 0.03885 0.03578 0.03281 0.02993 1.4224e- 8.4116e- 4.9571e- 2.9092e- 1.6988e- 1/θ 0.00403 0.00024 005 007 008 009 010 PR1 0.89656 0.84507 0.79373 0.74253 0.69149 0.64059 0.58983

PR2 0.10344 0.15493 0.20627 0.25747 0.30851 0.35941 0.41017

Panel 3. β =0.90

2 3 4 5 6 7 8

0.87144 0.85889 0.84742 0.83702 0.82762 0.81922 0.81177 EIN σ I 0.05723 0.08472 0.11166 0.13820 0.16447 0.19060 0.21673 ω θ E ( t ; ) 0.04627 0.04299 0.03984 0.03681 0.03389 0.03108 0.02836 1.5014e- 8.8789e- 5.2325e- 3.0708e- 1.7931e- 1/θ 0.00426 0.00025 005 007 008 009 010

PR1 0.89656 0.84507 0.79373 0.74253 0.69149 0.64059 0.58983

PR2 0.10344 0.15493 0.20627 0.25747 0.30851 0.35941 0.41017

σ ω θ Note: EIN and I are determined using (6a) and (6b) respectively. E ( t ; ) is computed with θ (7). 1/ is approximated by (8). PR1 and PR2 are calculated according to definitions (10a) and (10b) respectively. All figures are done with summary statistics in panel B of table 1.

Table 3. Lower Bounds of the Unconditional Mean of Incentive Adjustment Due to Pessimism and Distance Between the Corresponding Point Estimates as Fractions of the Lower Bounds, and Bounds of the Mean Equity Premium for the Coefficient of Relative Risk Aversion Between 2 to 8 and Subjective Discount Factor Being 0.99, 0.95 and 0.90

Panel 1. β =0.99

γ 2 3 4 5 6 7 8 ω θ E ( t ; )min 0.04801 0.04177 0.03571 0.02981 0.02401 0.01830 0.01264

dEω 0.06016 0.13199 0.22693 0.35843 0.55262 0.86791 1.46700

| ER e |max 0.06481 0.06764 0.07050 0.07340 0.07636 0.07940 0.08252

Panel 2. β =0.95

2 3 4 5 6 7 8 ω θ E ( t ; )min 0.04607 0.04008 0.03427 0.02860 0.02304 0.01756 0.01213

dEω 0.06016 0.13199 0.22693 0.35843 0.55262 0.86791 1.46700

| ER e |max 0.06481 0.06764 0.07050 0.07340 0.07636 0.07940 0.08252

Panel 3. β =0.90

2 3 4 5 6 7 8 ω θ E ( t ; )min 0.04365 0.03797 0.03247 0.02710 0.02183 0.01664 0.01149

dEω 0.06016 0.13199 0.22693 0.35843 0.55262 0.86791 1.46700

| ER e |max 0.06481 0.06764 0.07050 0.07340 0.07636 0.07940 0.08252

ω θ

Note: E ( t ; )min is determined using definition (11). dE(ω) is calculated according to ω θ definitions (12) and values of E ( t ; ) from table 2. | ERe |max is computed with definition (13) ω θ and values of E ( t ; ) from table 2. All other figures are based on the summary statistics in panel A of table 1.