Predictive Regressions: A Present Value Approach
Jules van Binsbergen and Ralph Koijen
Sept 25, 2007
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Introduction
Most studies have relied on log approximations to derive thepresent value relationship between the Price-Dividend ratio,and future expected returns and dividend growth
Empirically, the Price-Dividend ratio seems to predict returnsbut not dividend growth
van Binsbergen and Koijen want to avoid the log linearapproximation framework, instead develop their own toanalyze these predictive relationships
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Results
Develop a framework that does not rely on the logapproximation, Price-Dividend ratio is an exact linear functionof expected returns and expected dividend growth rates
Find that Price-Dividend ratio predicts both future returnsand dividend growth, R2 of 16% and 8% respectively.
Find evidence of a transient and persistent component of theexpected dividend growth rate, R2 rise to 18% and 16%
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
The Model
Price-Dividend Ratio: PDt = PtDt
Returns: Rt+1 = Pt+1
Pt−Dt
Expected returns: µ∗t = Et [Rt+1]− 1
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Unobserved States
Do not observe the expected return sequence {µ∗t } orsequence of expected dividend growth rate {gt}
How one models these sequences is extremely important!
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Unobserved States
Model Dividend Growth as:
Dt+1
Dt= (1 + gt)(1 + εDt+1)
Define µt such that:
1 + gt
1 + µ∗t= 1 + gt − µt
Introduce two new variables, gt and µt such that:
gt = γ0 + gt
µt = δ0 + µt
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Unobserved States: Dynamics
Model these as non linear equations:
gt+1 = γ11 + γ0 − δ0
1 + γ0 − δ0 + gt − µtgt + εgt+1
µt+1 = δ11 + γ0 − δ0
1 + γ0 − δ0 + gt − µtµt + εµt+1
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Unobserved States: Dynamics
Re-label as in the paper:
gt+1 = γ1t gt + εgt+1
µt+1 = δ1t µt + εµt+1
Where:γ1t = γ1λt
δ1t = δ1λt
λt =α
α + gt − µt
α = 1 + γ0 − δ0
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Correlation
The covariance between innovations in expected dividendgrowth and expected return is:
Cov(εgt+1, εµt+1) = σgµ
The covariance between innovations in unexpected dividendgrowth and expected return is:
Cov(εDt+1, εµt+1) = σDµ
For identification, the covariance between innovations inunexpected and expected dividend growth is zero:
Cov(εDt+1, εgt+1) = σDg = 0
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Price-Dividend Ratio
The Price-Dividend Ratio can be written as:
PDt = A + B1µt + B2gt
WhereA = (1− α +
σDµα
1− δ1α + σDµ)−1
B1 =−A
1− δ1α + σDµ
B2 =A + B1σDµ
1− γ1α
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Why do variables move?
Changes in the dividend growth, returns, or the price dividendratio can be attributed to changes in expected returns,expected growth rates, or unexpected growth rates
Expected change in PDt is:
Et [PDt ]− PDt = B1(δ1t − 1)µt + B2(γ1t − 1)gt
Unexpected change in price dividend ratio is:
PDt+1 − Et [PDt+1] = B1εµt+1 + B2ε
gt+1
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Unexpected Returns
Unexpected return is:
Rt+1 − (1 + µ∗t ) = ht
[B1ε
µt+1 + B2ε
gt+1 + Et [PDt+1]εDt+1
+B1(εdt+1εµt+1 − σDµ) + B2ε
Dt+1ε
gt+1
]Where:
ht =1 + gt
PDt − 1
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Implications for Variance of Returns
If we assume that (εgt+1, εµt+1, ε
Dt+1) are jointly normal, then
the conditional variance of returns is:
σ2R,t = h2
t
[B2
1σ2µ + B2
2σ2g + (Et [PDt+1])2σ2
D + B21 (σ2
Dσ2µ + 2σ2
Dµ)
+B22σ
2gσ
2D + B1B2σ
2gµ + B1Et [PDt+1]σD,µ + B1B2σ
2Dσgµ
]
The conditional covariance between expected returns andunexpected returns is:
Covt(Rt+1 − Et [Rt+1], εµt+1) = ht(B1σ2µ + B2σµg + Et [PDt+1]σµd)
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Data
Annual data, 1946-2005
Use total payout data as in some previous studies
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Empirical Exercise
First, want to characterize small sample performance ofestimator given constant dividend growth and compare theirmaximum likelihood estimator to that of conventionalpredictive regressions
PV model: PDt = A + B1µt ⇒ L(Θ; ∆DT ,PDT )
Standard Predictive Regression: Rt+1 = α + βPDt + ηt+1
Can link the two as follows:
δ0 = α + βA1 = βB1
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Empirical Exercise
Generate 10000 series of 60 observations, given some set ofparameters
Estimate model
Look at distribution of estimators
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Results
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Results
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Estimation of Full Model
Transition Equations:
gt+1 = γ1t gt + εgt+1
µt+1 = δ1t µt + εgt+1
Measurement equations:
PDt = A + B1µt + B2gtDt+1
Dt= (1 + δ0 + gt) + (1 + δ0 + gt)εDt+1)
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Estimating the Model
Want to evaluate the likelihood:
L(y t ; Θ) =T∏
t=1
l(yt |y t−1; Θ)
L(y t ; Θ) =T∏
t=1
∫ ∫l(yt |y t−1, εtg , g0,Θ)l(εtg , g0|y t−1,Θ)dεtgdg0
This is hard in a non linear world!
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Estimating the Model
To evaluate the likelihood function use 1 of:
Unscented Kalman Filter
Particle Filter
Then maximize the likelihood and bootstrap for standard errors.
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Estimation of Model
Can reduce the dimension by subbing out one of the transitionequations to get:
gt+1 = γ1t gt + εgt+1
Measurement equations:
PDt+1 = A + δ1t [PDt − A− B2gt ] + B2γ1t gt + B1εµt+1 + B2ε
gt+1
Dt+1
Dt= (1 + δ0 + gt) + (1 + δ0 + gt)εDt+1)
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Results
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Expected Returns
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Expected Dividend Growth Rates
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Decomposing Unexpected Returns: First Order Effects
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Decomposing Unexpected Returns: Second Order Effects
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Variance Decomposition
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Decomposing PDt
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Multiple Growth Rates
Want to allow for two growth rates
gt = γ0 + g1t + g2t
git = γit git + εgi ,t+1
Now the price dividend ratio
PDt = A + B1µt + B2g1t + B3g2t
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Results
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Variance Decomposition
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Price Dividend Decomposition
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Unexpected Returns
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Test for Persistent Growth Rate
Perform a likelihood ratio test, LR = 2(L1 − L0)
H0 : γ1 = σg1 = ρµg1 = 0 gives p value of 0.087
H0 : δ1 = σµ = ρµg1 = ρDµ = 0 is rejected
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach
Conclusion
Developed a closed form present value model with timevarying expected dividend growth rates and expected returns
Find that the Price-Dividend ratio can predict both returnsand dividend growth rates
Decompose dividend growth rates and find a transient andpersistent component
Jules van Binsbergen and Ralph Koijen Predictive Regressions: A Present Value Approach