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.
0
500
sp500
1000
1500
S&P 500
1960q1
1970q1
1980q1
1990q1
date
2000q1
2010q1
-150
-100
-50
0
50
100
S&P 500 and value-weighted market return
1960q1
1970q1
1980q1
1990q1
2000q1
date
ret_sp500
vwxlretd
2010q1
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:
Dt
e
+ εt
Rt,t+k
=α+β
Pt
e
where Rt,t+k
is the realized cumulative return over k periods.
k
β
tstat
R2
1y
3.83
2.47
0.07
2y
7.42
3.13
0.11
3y
11.57
4.04
0.18
4y
15.81
4.35
0.20
1y
3.39
2.18
0.06
2y
6.44
2.74
0.09
3y
9.99
3.58
0.15
4y
13.54
3.83
0.17
-40
-20
0
20
40
Returns (2yr cumulative): Actual vs Predicted
1960q1
1970q1
1980q1
1990q1
2000q1
date
vw8xlretd_8
Fitted values
2010q1
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
Et Σ∞
i=0 β U (Ct+i ),
0<β<1
subject to intertemporal budget constraint
St+1 + Bt+1 = R̃t St + Rtf Bt + Wt − Ct
We also have the no-ponzi scheme conditions.
Comments
St and Bt are endogenous choice variables.
f
Returns R̃t and Rtf are stochastic stationary processes with Rt+1
known at time t. R̃t+1 realized at time t + 1.
Euler equations:
Optimal portfolio choices imply
U 0 (Ct ) = Et R̃t+1 βU 0 (Ct+1 )
f
Since Rt+1
is non-stochastic we have
f
U 0 (Ct ) = Rt+1
Et βU 0 (Ct+1 )
Rearranging we have:
1 = Et R̃t+1
βU 0 (Ct+1 )
U 0 (Ct )
f
1 = Rt+1
Et
βU 0 (Ct+1 )
U 0 (Ct )
Risk Neutrality:
Constant U 0 (C)
Euler equations imply:
f
Et R̃t+1 = Rt+1
General framework
Euler equation implies
Et {Mt+1 Rt+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 0 (Ct+1 )
U 0 (Ct )
Implications
For stocks
Pt = Et {Mt+1 Xt+1 }
where Xt+1 = Pt+1 + Dt+1 and Rt+1 =
Xt+1
Pt .
Assume risk-neutrality then
Pt = Et
X
β s Dt+s
s=1
Let β = 1/(1 + r) and suppose dividends are a random walk
with drift where:
Et Dt+s = (1 + g)s Dt
then
Pt
(1 + g)
=
Dt
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
n
where Rt+1
=
PtI
n
Xt+1
Ptn
Xn
1
Mt+1 t+1
n
Pt (1 + πt+1 )
=1
is the nominal return, 1 + πt+1 =
is the price-index (e.g. CPI)
I
Pt+1
PtI
and
Consumption-Based Asset Pricing:
Equating the Euler equations gives:
f
Rt+1
Et
βU 0 (Ct+1 )
βU 0 (Ct+1 )
=
E
R̃
t t+1
U 0 (Ct )
U 0 (Ct )
Rearranging:
Et R̃t+1 −
f
Rt+1
βU 0 (Ct+1 )
βU 0 (Ct+1 )
Et
= −COVt R̃t+1 ,
U 0 (Ct )
U 0 (Ct )
Risk Premium
From Euler equation for risk-free asset
Et
βU 0 (Ct+1 )
f
= 1/Rt+1
U 0 (Ct )
Therefore:
f
Et R̃t+1 − Rt+1
f
Rt+1
= −COVt
βU 0 (Ct+1 )
R̃t+1 ,
U 0 (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) =
C 1−γ
1−γ
The Euler equations are:
−γ
Ct−γ = Et R̃t+1 βCt+1
f
−γ
Ct−γ = Rt+1
Et βCt+1
An approximation to the Euler equation:
Let xt+1 = ln(Ct+1 ) − ln(Ct ), r̃t+1 = ln(R̃t+1 ), the Euler
equation becomes:
f
1 = Rt+1
βEt exp(−γxt+1 )
1 = βEt exp(−γxt+1 + r̃t+1 )
Assume that consumption growth and asset returns are jointly
log-normally distributed:
2
2
σx,t+1 , σx,r,t+1
xt+1
xt+1
∼N
,
2
2
r̃t+1
r̄t+1
σx,r,t+1
, σr,t+1
If x ∼ N (x, σx2 ) then X = exp(x) is log-normally distributed
with
1
E(X) = exp(x + σ 2 )
2
Risk premium with log-normal distribution
The Euler equations becomes
1
1 = β exp −γxt+1 + r̄t+1 + var(−γxt+1 + rt+1 )
2
1
f
1 = β exp −γxt+1 + rt+1 + var(−γxt+1 )
2
Implications
Take logs and equate these equations:
1
1
var(−γxt+1 ) − var(−γxt+1 + r̃t+1 )
2
2
1 2
= − σr + γcov(x, r̃)
2
f
r̄t+1 − rt+1
=
Let r̄t+1 = E(log R̃t+1 )) then log E(R̃t+1 ) = r̄t+1 + 12 σr2
f
log Et Rt+1 − log Rt+1
= γcorr(x, r̃)σx σr
Quantitative implications:
The equity premium is:
f
log Et Rt+1 − log Rt+1
= γcorr(x, r̃)σx σr
In US data, σr = 0.167, σx = 0.036, corr(x, r̃) = 0.4 so
f
If γ = 1 we have log ERt+1 − log Rt+1
= 0.24%.
f
If γ = 10 we have log ERt+1 − log Rt+1 = 2.4%
f
If γ = 25 we have log ERt+1 − log Rt+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:
f
rt+1
= − log β + γxt+1 −
γ2
var(xt+1 )
2
Suppose β = 0.999, x = 0.015, σx = 0.036 then we need
f
γ = 0.6 to obtain rt+1
= 1%.
f
If γ = 10 we have rt+1
= 22%
f
If γ = 25 we have rt+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.