Lecture 4: Campbell-Shiller Decomposition
Simon Gilchrist
Boston Univerity and NBER
EC 745
Fall, 2013
Campbell and Shiller decomposition
Introduce a log-linear approximation to the present-value
identity (prices = Present Discounted Value [PDV] of dividends).
Use this approximation to discuss the sources of stock price
volatility.
This formula has proven extremely useful in applied work,
because it is easy to use it in conjunction with linear time series
models (e.g. VARs). Also known as “Discount Rate - Cash Flow
Decompositions”
Present Discounted Values
Start from the definition of return:
Pt+1 + Dt+1
.
Rt+1 =
Pt
Rewrite this as “price equal PDV of dividends” by rewriting and
iterating forward:
Dt+1 + Pt+1
Rt+1
Dt+1
Dt+2
=
+
+ ....
Rt+1
Rt+1 Rt+2
∞
X
Dt+k
=
,
Rt,t+k
Pt =
k=1
where Rt,t+k is the realized return on the asset (not the risk-free
rate!!!) from t to t + k :
Rt,t+k = Rt+1 × ... × Rt+k .
Log Returns.
Use lowercase letters for logs: rt = log (1 + Rt ) ,
pt = log(Pt ), dt = log (Dt ) :
Compute log-return as:
rt+1 = log (Pt+1 + Dt+1 ) − log(Pt )
Dt+1
− pt
= pt+1 + log 1 +
Pt+1
= pt+1 − pt + log (1 + exp (dt+1 − pt+1 ))
Approximation
Do a first-order Taylor approximation to the function
f (x) = log (1 + exp(x)) around x = d − p where a denotes the
sample average value:
log (1 + exp (dt+1 − pt+1 )) ' k + (1 − ρ) (dt+1 − pt+1 ) ,
with:
1
1 + exp d − p
k = − log ρ + (1 − ρ) log(1/ρ − 1).
ρ =
This yields:
rt+1 = ρpt+1 − pt + k + (1 − ρ)dt+1 ,
pt = ρpt+1 + k + (1 − ρ)dt+1 − rt+1 .
PDV
Iterating forward yields:
pt =
X
k
+
ρj−1 ((1 − ρ) dt+j − rt+j ) ,
1−ρ
j≥1
=
X
k
+
ρj ((1 − ρ) dt+1+j − rt+1+j ) .
1−ρ
j≥0
This identity holds for any returns, prices and dividends ex-post.
(And thus ex-ante, if you take conditional expectations of this
equation.) It is an accounting identity without any behavorial
assumption.
Price-Dividend Ratio:
From the log-linearized return equation we have:
pt − dt = ρ(pt+1 − dt+1 ) + k + ∆dt+1 − rt+1 .
Iterating forward:
pt − dt =
X
k
+
ρj (∆dt+1+j − rt+1+j )
1−ρ
j≥0
Variation in the price-dividend ratio occurs because of variation
in dividend growth or discount factors.
Return innovations
Start with the return equation
rt+1 = ρpt+1 − pt + k + (1 − ρ)dt+1 ,
Now apply apply Et+1 − Et to both sides:
rt+1 − Et rt+1 = ρ (pt+1 − Et pt+1 ) + (1 − ρ) (dt+1 − Et dt+1 )
From the price-dividend expression
pt+1 − Et pt+1 = (Et+1 − Et )
X
ρj (∆dt+2+j − rt+2+j )
j≥0
+ dt+1 − Et dt+1
which implies:
rt+1 − Et rt+1 = ρ (Et+1 − Et )
X
j≥0
+ (dt+1 − Et dt+1 )
ρj (∆dt+2+j − rt+2+j )
Return innovations
We then have the expression
rt+1 − Et rt+1 = (Et+1 − Et )
X
− (Et+1 − Et )
X
ρs ∆dt+1+s
s≥0
ρs rt+1+s .
s≥1
An unexpectedly good stock return must occur because either the
current dividend went up, or expectations of future dividends go
up, or because expectations of future returns go down.
The first two terms are a standard “cash flow effect” and the
second is an expected return or risk premium effect: the price
goes up if the risk premium or risk-free interest rate go down.
VAR model
Hence, stock price volatility can come from either volatility of
future dividends or volatility of expected future returns. Which
of these terms contribute more to volatility empirically?
Fit a VAR to dividend growth and returns and use the VAR to
compute the implied decomposition:



  d 
∆dt+1
∆dt
εt+1
 rt+1  = A(L)  rt  +  εrt+1  ,
xt+1
xt
εxt+1
Iterating on this VAR one can compute the forecast Et ∆dt+1+j
and then the revision of the forecast.
Companion form:
Let
yt = (∆dt+1 , rt+1 , xt+1 )0
and define the 3px1 vector:
zt = [yt ..yt−p ]0
Then zt follows an AR1 process:
zt+1 = Azt + νt+1
with
νt = ε0t 0..0 ,
Eεt ε0t = Σ.
Present values with companion form:
The expected PDV can be computed as:
X
X
Et
β j zt+j =
β j Aj zt = (I − βA)−1 zt
j≥0
j≥0
Premultiplying by the correct row vector to obtain the forecast of
the PV of a component of z.
Innovations to PDV
Using this PDV we have
X
(Et+1 − Et )
β j zt+j = zt + β(I − βA)−1 zt+1 − (I − βA)−1 zt
j≥0
= zt + β(I − βA)−1 (Azt + εt+1 )
− (I − βA)−1 zt
The terms involving zt cancel so that
X
(Et+1 − Et )
β j zt+j = β(I − βA)−1 εt+1 .
j≥0
Results
Dividends are difficult to predict but returns are predictable as
we saw before.
This implies that changes in discount rates account for all of the
volatility of revisions to returns.
Since the risk-free interest rate does not move much in the data,
it means the changes in expected returns are mainly changes in
risk premia.
Implication – we need to study models with time-varying risk
premia
Campbell-Cochrane, which has time-varying risk aversion.
Stochastic volatility in consumption growth which leads to higher
risk aversion when consumption volatility is high vs low.
A Simple Model of Predictability:
Stochastic processes:
xt = bxt−1 + δt
rt+1 = xt + εrt+1
∆dt+1 = εdt+1
Model Solution:
Price-dividend ratio:
dt − pt = Et
∞
X
ρj−1 (rt+j − dt+j ) =
j=1
Returns:
Rt+1
xt
1 − ρb
Pt+1 Dt+1 /Dt
= 1+
Dt+1
Pt /Dt
Log-linearization:
rt+1 = ρ (pt+1 − dt+1 ) + ∆dt+1 − (pt − dt )
Prices:
∆pt+1 = − (dt+1 − pt+1 ) + (dt − pt ) + ∆dt+1
VAR representation:
1
δt+1
1 − ρb
ρ
d
= (1 − ρb) (dt − pt ) + εt+1 −
δt+1
1 − ρb
ρ
d
= (1 − b) (dt − pt ) + εt+1 −
δt+1
1 − ρb
dt+1 − pt+1 = b (dt − pt ) +
rt+1
∆pt+1
Implications for volatility:
Assume ρ = 0.96 then estimated coefficient of returns on
dividend price ratio is 1 − ρb where b is autocorrelation of
returns. This implies b = 0.9.
Assume D/P = 0.04. Returns as a function of levels:
rt+1 =
1 − ρb Dt
+ εrt+1
D/P Pt
If b = 0, a 1% rise in the D/P ratio implies that returns must rise
25%.
If b = 0.9 then 1 − b = 0.1 and 1 − ρb = 0.14 and a 1% rise in
D/P implies prices and returns rise by 2.5 and 3.4% respectively.
If b = 0.96 then a 1% rise in D/P implies prices and returns rise
by 1% and 2% respectively. (This seems like upper-bound on
persistence).
Result: iid dividend growth and persistent price-dividend ratio
implies returns must respond more than one for one to the
dividend-price change.
Volatility
Also
1
σ (x)
1 − ρb
= 7.4σ (x)
σ (d − p) =
Dividend-price ratio should be much more volatile than returns.
Persistence in returns implies that small variation in returns can
have a large effect on the dividend-price ratio – similar to
Gordon growth formula
P =
D
r−g