Lecture 1: Asset pricing and the equity premium puzzle Simon - - PowerPoint PPT Presentation

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

500 1000 1500 sp500 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 date

S&P 500 and value-weighted market return

• 150
• 100
• 50

50 100 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: Re

t,t+k = α + β Dt

Pt + εt where Re

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 R2 0.07 0.11 0.18 0.20 0.06 0.09 0.15 0.17

Returns (2yr cumulative): Actual vs Predicted

• 40
• 20

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

• n 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 EtΣ∞

i=0βiU(Ct+i),

0 < β < 1 subject to intertemporal budget constraint St+1 + Bt+1 = ˜ RtSt + Rf

t Bt + Wt − Ct

We also have the no-ponzi scheme conditions.

Comments

St and Bt are endogenous choice variables. Returns ˜ Rt and Rf

t are stochastic stationary processes with Rf t+1

known at time t. ˜ Rt+1 realized at time t + 1.

Euler equations:

Optimal portfolio choices imply U ′(Ct) = Et ˜ Rt+1βU ′(Ct+1) Since Rf

t+1 is non-stochastic we have

U ′(Ct) = Rf

t+1EtβU ′(Ct+1)

Rearranging we have: 1 = Et ˜ Rt+1 βU ′(Ct+1) U ′(Ct) 1 = Rf

t+1Et

βU ′(Ct+1) U ′(Ct)

Risk Neutrality:

Constant U ′(C) Euler equations imply: Et ˜ Rt+1 = Rf

t+1

General framework

Euler equation implies Et {Mt+1Rt+1} = 1 where Mt+1 is pricing kernel and Rt+1 is the return. Euler equation implies pricing kernel depends on consumption: Mt+1 = β U ′(Ct+1) U ′(Ct)

Implications

For stocks Pt = Et {Mt+1Xt+1} where Xt+1 = Pt+1 + Dt+1 and Rt+1 = Xt+1

Pt .

Assume risk-neutrality then Pt = Et

• s=1

βsDt+s Let β = 1/(1 + r) and suppose dividends are a random walk with drift where: EtDt+s = (1 + g)sDt then Pt Dt = (1 + g) r − g

Implications

For risk-free one-period bond that pays one unit of consumption tomorrow: Pt = Et {Mt+1} where Rt+1 = 1 Pt Nominal claims: Et

• Mt+1

Xn

t+1

P n

t

1 (1 + πt+1)

• = 1

where Rn

t+1 = Xn

t+1

P n

t

is the nominal return, 1 + πt+1 =

P I

t+1

P I

t

and P I

t is the price-index (e.g. CPI)

Consumption-Based Asset Pricing:

Equating the Euler equations gives: Rf

t+1Et

βU ′(Ct+1) U ′(Ct) = Et ˜ Rt+1 βU ′(Ct+1) U ′(Ct) Rearranging:

• Et ˜

Rt+1 − Rf

t+1

• Et

βU ′(Ct+1) U ′(Ct) = −COVt

• ˜

Rt+1, βU ′(Ct+1) U ′(Ct)

Risk Premium

From Euler equation for risk-free asset Et βU ′(Ct+1) U ′(Ct) = 1/Rf

t+1

Therefore:

• Et ˜

Rt+1 − Rf

t+1

• Rf

t+1

= −COVt

• ˜

Rt+1, βU ′(Ct+1) U ′(Ct)

Implications:

If the risky return covaries positively with tomorrow’s consumption, Ct+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) = C1−γ 1 − γ The Euler equations are: C−γ

t

= Et ˜ Rt+1βC−γ

t+1

C−γ

t

= Rf

t+1EtβC−γ t+1

An approximation to the Euler equation:

Let xt+1 = ln(Ct+1) − ln(Ct), ˜ rt+1= ln( ˜ Rt+1), the Euler equation becomes: 1 = Rf

t+1βEt exp(−γxt+1)

1 = βEt exp(−γxt+1 + ˜ rt+1) Assume that consumption growth and asset returns are jointly log-normally distributed: xt+1 ˜ rt+1

• ∼ N

xt+1 ¯ rt+1

• ,

σ2

x,t+1,

σ2

x,r,t+1

σ2

x,r,t+1, σ2 r,t+1

• If x ∼ N(x, σ2

x) then X = exp(x) is log-normally distributed

with E(X) = exp(x + 1 2σ2)

Risk premium with log-normal distribution

The Euler equations becomes 1 = β exp

• −γxt+1 + ¯

rt+1 + 1 2var(−γxt+1 + rt+1)

• 1 = β exp
• −γxt+1 + rf

t+1 + 1

2var(−γxt+1)

Implications

Take logs and equate these equations: ¯ rt+1 − rf

t+1

= 1 2var(−γxt+1) − 1 2var(−γxt+1 + ˜ rt+1) = −1 2σ2

r + γcov(x, ˜

r) Let ¯ rt+1 = E(log ˜ Rt+1)) then log E( ˜ Rt+1) = ¯ rt+1 + 1

2σ2 r

log EtRt+1 − log Rf

t+1 = γcorr(x, ˜

r)σxσr

Quantitative implications:

The equity premium is: log EtRt+1 − log Rf

t+1 = γcorr(x, ˜

r)σxσr In US data, σr = 0.167, σx = 0.036, corr(x, ˜ r) = 0.4 so

If γ = 1 we have log ERt+1 − log Rf

t+1 = 0.24%.

If γ = 10 we have log ERt+1 − log Rf

t+1 = 2.4%

If γ = 25 we have log ERt+1 − log Rf

t+1 = 6.0%

Additional Implications

If return variance and consumption variance are constant, excess return is unpredictable. If consumption growth iid and Ct = Dt, price-dividend ratio is a constant.

Quantitative implications: Risk free rate

The risk free rate is: rf

t+1 = − log β + γxt+1 − γ2

2 var(xt+1) Suppose β = 0.999, x = 0.015, σx = 0.036 then we need γ = 0.6 to obtain rf

t+1 = 1%.

If γ = 10 we have rf

t+1 = 22%

If γ = 25 we have rf

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.