Lecture 1: Asset pricing and the equity premium puzzle Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013
Overview Some basic facts. Study the asset pricing implications of household portfolio choice. Consider the quantitative implications of a second-order approximation to asset return equations. Reference: Mehra and Prescott (JME, 1984)
Some Facts Stock returns: Average real return on SP500 is 8% per year Standard error is large since σ ( E ( R )) = σ ( R ) T ) √ Stock returns are very volatilie: σ ( R ) = 17% per year. Stock returns show very little serial correlation ( ρ = 0 . 08 quarterly data, -0.04 annual data). Bond returns: The average risk free rate is 1% per year (US Tbill - Inflation) The risk free rate is not very volatile: σ ( R ) = 2% per year but is persistent ( ρ = 0 . 6 in annual data) leading to medium-run variation. These imply that the equity premium is large – 7% per year on an annual basis.
S&P 500 1500 1000 sp500 500 0 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 date
S&P 500 and value-weighted market return 100 50 0 -50 -100 -150 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 date ret_sp500 vwxlretd
Recent data (1970-2012) SP500 Return: Mean = 5.96, Std.Dev = 25.4 Value weighted excess return: mean = 1.56, Std. Dev = 37.54
Return predictability Cambpell and Shiller (and many others) consider the following regression: t,t + k = α + β D t R e + ε t P t where R e t,t + k is the realized cumulative return over k periods. k 1y 2y 3y 4y 1y 2y 3y 4y β 3.83 7.42 11.57 15.81 3.39 6.44 9.99 13.54 tstat 2.47 3.13 4.04 4.35 2.18 2.74 3.58 3.83 R 2 0.07 0.11 0.18 0.20 0.06 0.09 0.15 0.17
Returns (2yr cumulative): Actual vs Predicted 40 20 0 -20 -40 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 date vw8xlretd_8 Fitted values
Comments Returns appear to be predictable: High current price relative to dividends predicts low future returns. Other variables also have predictive power: CAY, term premium, short-term nominal interest rate (Fed model). Does this violate asset-pricing theory? Econometric issues: overlapping data and standard error corrections, robustness to sample. Data mining? Not much out-of-sample forecasting power.
Cross-sectional evidence Small firms have high returns on average (size premium) Firms with low Tobins’ Q (low book/market) have higher returns on average (value premium) Firms with high recent returns tend to have high returns in near future (momemtum anomaly)
Setup: Household makes portfolio choices chooses to maximize i =0 β i U ( C t + i ) , E t Σ ∞ 0 < β < 1 subject to intertemporal budget constraint S t +1 + B t +1 = ˜ R t S t + R f t B t + W t − C t We also have the no-ponzi scheme conditions.
Comments S t and B t are endogenous choice variables. Returns ˜ R t and R f t are stochastic stationary processes with R f t +1 known at time t . ˜ R t +1 realized at time t + 1 .
Euler equations: Optimal portfolio choices imply U ′ ( C t ) = E t ˜ R t +1 βU ′ ( C t +1 ) Since R f t +1 is non-stochastic we have U ′ ( C t ) = R f t +1 E t βU ′ ( C t +1 ) Rearranging we have: βU ′ ( C t +1 ) 1 = E t ˜ R t +1 U ′ ( C t ) βU ′ ( C t +1 ) 1 = R f t +1 E t U ′ ( C t )
Risk Neutrality: Constant U ′ ( C ) Euler equations imply: E t ˜ R t +1 = R f t +1
General framework Euler equation implies E t { M t +1 R t +1 } = 1 where M t +1 is pricing kernel and R t +1 is the return. Euler equation implies pricing kernel depends on consumption: M t +1 = β U ′ ( C t +1 ) U ′ ( C t )
Implications For stocks P t = E t { M t +1 X t +1 } where X t +1 = P t +1 + D t +1 and R t +1 = X t +1 P t . Assume risk-neutrality then � β s D t + s P t = E t s =1 Let β = 1 / (1 + r ) and suppose dividends are a random walk with drift where: E t D t + s = (1 + g ) s D t then P t = (1 + g ) D t r − g
Implications For risk-free one-period bond that pays one unit of consumption tomorrow: P t = E t { M t +1 } where R t +1 = 1 P t Nominal claims: X n � � 1 t +1 E t M t +1 = 1 P n (1 + π t +1 ) t X n P I where R n t +1 t +1 t +1 = is the nominal return, 1 + π t +1 = and P n P I t t P I t is the price-index (e.g. CPI)
Consumption-Based Asset Pricing: Equating the Euler equations gives: βU ′ ( C t +1 ) βU ′ ( C t +1 ) R f = E t ˜ t +1 E t R t +1 U ′ ( C t ) U ′ ( C t ) Rearranging: βU ′ ( C t +1 ) � R t +1 , βU ′ ( C t +1 ) � � � E t ˜ R t +1 − R f ˜ E t = − COV t t +1 U ′ ( C t ) U ′ ( C t )
Risk Premium From Euler equation for risk-free asset βU ′ ( C t +1 ) = 1 /R f E t t +1 U ′ ( C t ) Therefore: � � E t ˜ R t +1 − R f � R t +1 , βU ′ ( C t +1 ) � t +1 ˜ = − COV t R f U ′ ( C t ) t +1
Implications: If the risky return covaries positively with tomorrow’s consumption, C t +1 , then the LHS is positive and the asset return bears a positive premium over the risk free rate. If the risky return covaries negatively with tomorrow’s consumption then the LHS is negative and the asset return bears a negative premium over the risk free rate. Intuition: assets whose returns have a negative covariance with consumption provide a hedge against consumption risk. Households are willing to accept a lower expected return since these assets provide insurance against low future consumption.
The equity premium puzzle: Assume CRRA: U ( C ) = C 1 − γ 1 − γ The Euler equations are: C − γ = E t ˜ R t +1 βC − γ t t +1 C − γ = R f t +1 E t βC − γ t t +1
An approximation to the Euler equation: r t +1 = ln( ˜ Let x t +1 = ln( C t +1 ) − ln( C t ) , ˜ R t +1 ) , the Euler equation becomes: 1 = R f t +1 βE t exp( − γx t +1 ) 1 = βE t exp( − γx t +1 + ˜ r t +1 ) Assume that consumption growth and asset returns are jointly log-normally distributed: � x t +1 �� x t +1 � σ 2 σ 2 � � x,t +1 , �� x,r,t +1 ∼ N , σ 2 x,r,t +1 , σ 2 r t +1 ˜ r t +1 ¯ r,t +1 If x ∼ N ( x, σ 2 x ) then X = exp( x ) is log-normally distributed with E ( X ) = exp( x + 1 2 σ 2 )
Risk premium with log-normal distribution The Euler equations becomes � r t +1 + 1 � 1 = β exp − γx t +1 + ¯ 2 var ( − γx t +1 + r t +1 ) � t +1 + 1 � − γx t +1 + r f 1 = β exp 2 var ( − γx t +1 )
Implications Take logs and equate these equations: 1 2 var ( − γx t +1 ) − 1 r t +1 − r f ¯ = 2 var ( − γx t +1 + ˜ r t +1 ) t +1 − 1 2 σ 2 = r + γcov ( x, ˜ r ) r t +1 = E (log ˜ R t +1 ) ) then log E ( ˜ r t +1 + 1 2 σ 2 Let ¯ R t +1 ) = ¯ r log E t R t +1 − log R f t +1 = γcorr ( x, ˜ r ) σ x σ r
Quantitative implications: The equity premium is: log E t R t +1 − log R f t +1 = γcorr ( x, ˜ r ) σ x σ r In US data, σ r = 0 . 167 , σ x = 0 . 036 , corr ( x, ˜ r ) = 0 . 4 so If γ = 1 we have log ER t +1 − log R f t +1 = 0 . 24% . If γ = 10 we have log ER t +1 − log R f t +1 = 2 . 4% If γ = 25 we have log ER t +1 − log R f t +1 = 6 . 0%
Additional Implications If return variance and consumption variance are constant, excess return is unpredictable. If consumption growth iid and C t = D t , price-dividend ratio is a constant.
Quantitative implications: Risk free rate The risk free rate is: t +1 = − log β + γx t +1 − γ 2 r f 2 var ( x t +1 ) Suppose β = 0 . 999 , x = 0 . 015 , σ x = 0 . 036 then we need γ = 0 . 6 to obtain r f t +1 = 1% . If γ = 10 we have r f t +1 = 22% If γ = 25 we have r f t +1 = 78% This is opposite to equity-premium puzzle – we need very low γ to match risk-free rate.
Additional implications for risk free rate: If consumption growth is iid and homoskedastic, then risk free rate is constant. Risk free rate is high when expected consumption growth is high (intertemporal substitution). Risk free rate is low when conditional consumption volatility is low (precautionary savings). Consumption growth is close to iid but risk-free rates are persistent.
Recommend
More recommend
