Internet Appendix for “Dividend Dynamics and the Term Structure of Dividend Strips FREDERICO BELO, PIERRE COLLIN-DUFRESNE, and ROBERT S. GOLDSTEIN∗ Section I of this Internet Appendix reports the derivations of the long-run risk (Bansal, Kiku, and Yaron (BKY, 2012)) model. In addition, it reports the approximation errors in the baseline BKY calibration with large persistence of dividend volatility, and performs additional comparative statics analysis. Section II reports the derivations of the habit formation model (Campbell and Cochrane (CC, 1999)). Finally, Section III reports additional empirical analysis and robustness checks.
I.
Long-Run Risk Model
A. Identification of {z, κ0 , κ1 , A0 , Ax , Aσ } BKY specify log-consumption and log-dividend dynamics as driven by two persistent variables (xt , σt ): ∆ct+1 = µc + xt + σt ǫ˜c, t+1 ∆dt+1 = µd + ρd xt + νc σt ǫ˜c, t+1 + νd σt ˜ǫd, t+1 xt+1 = ρx xt + νx σt ǫ˜x, t+1 2 σt+1 = σ 2 + ρσ σt2 − σ 2 + νσ ǫ˜σ, t+1 .
(IA.1)
Epstein and Zin (1989) preferences imply that the log pricing kernel takes the form mt+1 = θ log δ −
θ ∆c + (θ − 1)rc,t+1 , ψ t+1
(IA.2)
Citation format: Belo, Frederico, Pierre Collin-Dufresne, and Robert S. Goldstein, Internet Appendix for “Dividend Dynamics and the Term Structure of Dividend Strips,” Journal of Finance [DOI STRING]. Please note: Wiley-Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing material) should be directed to the authors of the article. ∗
1
where rc is the return on the consumption claim, γ is relative risk aversion coefficient, 1−γ ψ is the elasticity of intertemporal substitution, and θ = 1−1/ψ .
For tractability, two approximations are made. The first is model-specific, and assumes that the log price-dividend ratio and the log price-consumption ratio are approximately affine in the state variables: V ≈ A0 + Ax xt + Aσ σt2 zt ≡ log Cc,t t V zd,t ≡ log Dd,t ≈ F0 + Fx xt + Fσ σt2 . t
(IA.3)
The second approximation (the Campbell-Shiller (1988) approximation) is mechanical, and approximates the log return rc ≡ log Rc on the consumption claim to be linear in the log price-dividend ratio (a similar approximation is made for the log stock return): Vc,t+1 + Ct+1 rc = log Vc,t Vc,t+1 + Ct+1 Ct+1 Ct = log Ct+1 Ct Vc,t zt+1 = log (1 + e ) + ∆ct+1 − zt ≈ log 1 + ez 1 + (zt+1 − z) + ∆ct+1 − zt z e z ≈ log 1 + e + zt+1 − z + ∆ct+1 − zt z 1+e ≈ κ0 + κ1 zt+1 − zt + ∆ct+1 , (IA.4) where κ1 = κ0
ez 1 + ez
(IA.5)
ez = log(1 + e ) − z 1 + ez κ1 = − log(1 − κ1 ) − κ1 log . 1 − κ1
z
(IA.6)
Plugging these approximations into the pricing kernel, and noting that (θ − 1 − ψθ ) = −γ, we find 2 mt+1 = θ log δ − γ∆ct+1 + (θ − 1)κ0 + (θ − 1)κ1 A0 + Ax xt+1 + Aσ σt+1 − (θ − 1)zt = θ log δ + (θ − 1)κ0 − γ µc + xt + σt ǫ˜c, t+1 + (θ − 1)κ1 A0 +(θ − 1)κ1 Ax ρx xt + νx σt ǫ˜x, t+1 + (θ − 1)κ1 Aσ σ 2 + ρσ σt2 − σ 2 + νσ ˜ǫσ, t+1 −(θ − 1)zt .
(IA.7)
2
Hence, Et mt+1 = θ log δ + (θ − 1)κ0 − γ (µc + xt ) + (θ − 1)κ1 A0 + (θ − 1)κ1 Ax ρx xt +(θ − 1)κ1 Aσ σ 2 + ρσ σt2 − σ 2 − (θ − 1)zt (IA.8)
and
where
mt+1 − Et mt+1 = −λc σt ǫ˜c, t+1 − λx σt ˜ǫx, t+1 − λσ νσ ˜ǫσ, t+1 ,
(IA.9)
λc ≡ γ λx ≡ (1 − θ)κ1 Ax νx λσ ≡ (1 − θ)κ1 Aσ .
(IA.10)
Hence, we have Vart mt+1 = λ2c σt2 + λ2x σt2 + λ2σ νσ2 .
(IA.11)
To solve the model, we need to identify the six parameters {z, κ0 , κ1 , A0 , Ax , Aσ }. We already have three equations: z e κ1 = (IA.12) 1 + ez z e z κ0 = log(1 + e ) − z 1 + ez 2 z = A0 + Aσ σ . (IA.13) The other three equations come from 1 = Et emt+1 +rc,t+1 .
Since mt+1 + rc,t+1 is normally distributed, this is equivalent to Note that
1 0 = Et mt+1 + rc,t+1 + Vart mt+1 + rc,t+1 . 2
mt+1 + rc,t+1
θ ∆c + θrc,t+1 ψ t+1 = θ log δ + θκ0 + (1 − γ) µc + xt + σt ǫ˜c, t+1 +θκ1 A0 + θκ1 Ax ρx xt + νx σt ǫ˜x, t+1 +θκ1 Aσ σ 2 + ρσ σt2 − σ 2 + νσ ˜ǫσ, t+1 − θzt .
(IA.14)
(IA.15)
= θ log δ −
3
(IA.16)
Hence,
Also,
Et mt+1 + rc,t+1 = θ log δ + θκ0 + (1 − γ) (µc + xt ) + θκ1 A0 + θκ1 Ax ρx xt +θκ1 Aσ σ 2 + ρσ σt2 − σ 2 − θzt . (IA.17) mt+1 + rc,t+1 − Et mt+1 + rc,t+1 = (1 − γ)σt ǫ˜c, t+1 + θκ1 Ax νx σt ǫ˜x, t+1 +θκ1 Aσ νσ ǫ˜σ, t+1 .
(IA.18)
Hence, Vart mt+1 + rc,t+1 = σt2 (1 − γ)2 + σt2 (θκ1 Ax νx )2 + (θκ1 Aσ νσ )2 .
(IA.19)
Plugging equations (IA.17) and (IA.19) into equation (IA.15) and collecting terms linear in xt , σt2 , and independent of those two state variables, we obtain the three conditions xt :
0 = (1 − γ) + θκ1 Ax ρx − θAx 1 1 0 = θκ1 Aσ ρσ − θAσ + (1 − γ)2 + (θκ1 Ax νx )2 2 2 0 = θ log δ + θκ0 + (1 − γ)µc + θκ1 A0 + θκ1 Aσ σ 2 (1 − ρσ ) 1 −θA0 + (θκ1 Aσ νσ )2 . 2
σt2 : indep :
Solving, we find
A0
1 ψ
(IA.20) 1 − κ1 ρx (1 − γ) 1 − ψ1 κ21 νx2 = 1+ 2(1 − κ1 ρσ ) (1 − κ1 ρx )2 1 1 θ 2 2 = log δ + κ0 + 1 − µc + κ1 Aσ σ (1 − ρσ ) + (κ1 Aσ νσ ) . (1 − κ1 ) ψ 2
Ax = Aσ
1−
To solve, we follow BKY by guessing a value for z, and then plugging this in to get κ0 and κ1 , and in turn {A0 , Ax , Aσ }. We then identify a new guess for z using z = A0 + Aσ σ 2 .
(IA.21)
We iterate until convergence. Once a fixed point has been found, the pricing kernel is identified, implying we can go forward to price the dividend claim. 4
B.
Risk-Free Rate The risk-free rate at date t is determined from the pricing kernel via Rf−1 (xt , σt ) ≡ e−rf (xt ,σt ) = Et [emt+1 ] 1
2
= e−r0 − ψ xt −rσ σt ,
(IA.22)
where we have defined −r0 ≡ θ log δ + (θ − 1)κ0 − γµc + (θ − 1)κ1 A0
−rσ
1 +(θ − 1)κ1 Aσ σ 2 (1 − ρσ ) − (θ − 1)A0 + λ2σ νσ2 2 1 2 1 2 ≡ λ + λ − (θ − 1)Aσ (1 − κ1 ρσ ). 2 c 2 x
(IA.23)
C. The Dividend Claim As noted previously, dividend dynamics are specified as ∆dt+1 = µd + ρd xt + νc σt ˜ǫc, t+1 + νd σt ǫ˜d, t+1 . We assume that the log price-dividend ratio is approximately V zd,t ≡ log Dd,t ≈ F0 + Fx xt + Fσ σt2 . t
(IA.24)
(IA.25)
We also approximate the log stock return via
rd ≈ κ0d + κ1d zd,t+1 − zd,t + ∆dt+1 ,
(IA.26)
where κ1d = κ0d
ez d 1 + ez d
(IA.27)
ez d = log(1 + e ) − z d 1 + ezd κ1d = − log(1 − κ1d ) − κ1d log . 1 − κ1d
zd
(IA.28)
Because the parameters {z, κ0 , κ1 , A0 , Ax , Aσ } have already been identified, here all we need to identify is {z d , κ0d , κ1d , F0 , Fx , Fσ }. Again, we already have three equations: the definitions of {κ0d , κ1d } and the self-consistent condition z d = F0 + Fσ σ 2 . 5
(IA.29)
The last three conditions are determined from the following first-order condition:
Note that
1 0 = Et mt+1 + rd,t+1 + Vart mt+1 + rd,t+1 . 2
(IA.30)
mt+1 + rd,t+1 = θ log δ + (θ − 1)κ0 + κ0d − γ µc + xt + σt ˜ǫc, t+1 + (θ − 1)κ1 A0 +κ1d F0 + [(θ − 1)κ1 Ax + κ1d Fx ] ρx xt + νx σt ˜ǫx, t+1 +[(θ − 1)κ1 Aσ + κ1d Fσ ] σ 2 + ρσ σt2 − σ 2 + νσ ǫ˜σ, t+1 −(θ − 1)zt − zd,t + µd + ρd xt + νc σt ˜ǫc, t+1 + νd σt ˜ǫd, t+1 .
(IA.31)
Hence, Et mt+1 + rd,t+1 = θ log δ + (θ − 1)κ0 − γ (µc + xt ) + (θ − 1)κ1 A0 + [(θ − 1)κ1 Ax + κ1d Fx ] ρx xt +[(θ − 1)κ1 Aσ + κ1d Fσ ] σ 2 + ρσ σt2 − σ 2
−(θ − 1)zt + κ0d + κ1d F0 − zd,t + µd + ρd xt .
(IA.32)
Also,
mt+1 + rd,t+1 − Et mt+1 + rd,t+1 = −γσt ˜ǫc, t+1
+ [(θ − 1)κ1 Ax + κ1d Fx ] νx σt ǫ˜x, t+1 +[(θ − 1)κ1 Aσ + κ1d Fσ ]νσ ǫ˜σ, t+1 +νc σt ǫ˜c, t+1 + νd σt ǫ˜d, t+1 .
(IA.33)
This implies 2 Vart mt+1 + rd,t+1 = σt2 (νc − γ)2 + νd2 σt2 + (θ − 1)κ1 Aσ + κ1d Fσ νσ2 2 +σt2 νx2 (θ − 1)κ1 Ax + κ1d Fx . (IA.34)
Plugging equations (IA.32) and (IA.34) into equation (IA.30) and collecting terms
6
linear in xt , σt2 and independent of the state vector, we find the three equations x1t : 0 = −γ + (θ − 1)κ1 Ax ρx + κ1d Fx ρx − (θ − 1)Ax − Fx + ρd (IA.35) 1 σt2 : 0 = (θ − 1)κ1 Aσ + κ1d Fσ ρσ − (θ − 1)Aσ − Fσ + (νc − γ)2 2 2 1 2 1 2 + νx (θ − 1)κ1 Ax + κ1d Fx + νd 2 2 const : 0 = θ log δ + (θ − 1)κ0 − γµc + (θ − 1)κ1 A0 + (θ − 1)κ1 Aσ + κ1d Fσ σ 2 (1 − ρσ ) − (θ − 1)A0 + κ0d 2 νσ2 +κ1d F0 − F0 + µd + (θ − 1)κ1 Aσ + κ1d Fσ . 2
Simplifying, this gives Fx = Fσ
F0
ρd −
1 ψ
(IA.36) (1 − κ1d ρx ) 1 1 = (θ − 1)(κ1 ρσ − 1)Aσ + [(θ − 1)κ1 Ax + κ1d Fx ]2 νx2 1 − κ1d ρσ 2 2 ν 1 (IA.37) + (νc − γ)2 + d 2 2 1 1 θ = log δ − µc − (θ − 1) (κ1 Aσ νσ )2 + κ1d Fσ σ 2 (1 − ρσ ) 1 − κ1d ψ 2 2 ν (IA.38) +κ0d + µd + σ (κ1d Fσ − λσ )2 . 2
These results correct some typos in the appendix of BKY. Combining equations (IA.22) and (IA.26), we express the log excess stock return as rd,t+1 − rf,t = κ0d + κ1d zd,t+1 − zd,t + ∆dt+1 − rf,t . (IA.39) Further, we define the log expected excess return as h i r d,t = log Et e(rd,t+1 −rf,t ) = κ0d + κ1d F0 + κ1d Fx ρx xt + κ1d Fσ σ 2 (1 − ρσ ) + ρσ σt2 − F0 − Fx xt − Fσ σt2 + µd 1 +ρd xt − r0 − xt − rσ σt2 ψ 1 1 1 + (κ1d Fx νx )2 σt2 + (κ1d Fσ νσ )2 + σt2 (νc2 + νd2 ). (IA.40) 2 2 2
This simplifies to r d,t = νσ2 κ1d Fσ λσ + νx κ1d Fx λx + νc λc σt2 . 7
(IA.41)
We also define the stock volatility as r h i h i2 σd,t = log Et e2(rd,t+1 −rf,t ) − log Et e(rd,t+1 −rf,t ) q = (κ1d Fσ νσ )2 + σt2 νc2 + νd2 + (κ1d Fx νx )2 .
(IA.42)
To get the term structure of dividends, define
) G(j,T (t, dt , xt , σt ) = Et ejdT d ) G(j,T (t − 1, dt−1 , xt−1 , σt−1 ) = Et−1 ejdT . d
(IA.43)
Because the state vector dynamics are affine, it is well known (e.g., Duffie and Kan (1996)) that the solution is of the form 2
) G(j,T (t, dt , xt , σt ) = ejdt +F0,j (n)+Fx,j (n) xt +Fσ,j (n) σt d
2 jdt−1 +F0,j (n+1)+Fx,j (n+1) xt−1 +Fσ,j (n+1) σt−1
) G(j,T (t − 1, dt−1 , xt−1 , σt−1 ) = e d
(IA.44) ,(IA.45)
where we have defined n = (T − t) as the number of periods between dates t and T . Note that for the solution to be self-consistent at (t = T ), we must have the boundary conditions F0,j (0) = 0, Fx,j (0) = 0, and Fσ,j (0) = 0. By the law of iterated expectations, we have jd ) G(j,T (t − 1, d , x , σ ) = E Et e T t−1 t−1 t−1 t−1 d ) = Et−1 G(j,T (t, dt , xt , σt ) . d
(IA.46)
Plugging equations (IA.44) and (IA.45) into equation (IA.46), performing the expectation, and then collecting terms linear in xt , σt2 and independent of the state vector, we obtain the three recursive equations Fx,j (n + 1) = ρx Fx,j (n) + jρd νσ2 2 F0,j (n + 1) = F0,j (n) + Fσ,j (n)σ (1 − ρσ ) + jµd + Fσ,j (n) 2 1 2 2 1 2 2 Fσ,j (n + 1) = ρσ Fσ,j (n) + νx Fx,j (n) + j νc + νd2 . 2 2 2
(IA.47)
Setting j = 1, we can determine the term structure of dividend expected growth rates over horizon n via 1 log Et edT −dt n 1 = F0,1 (n) + Fx,1 (n)xt + Fσ,1 (n)σt2 . n
gd,n =
8
(IA.48)
Similarly, using both j = 2 and j = 1, we define the term structure of dividend volatilities via s 2 i 1 h σd,n = log Et e2(dT −dt ) − log Et edT −dt (IA.49) n 1 = F0,2 (n) − 2F0,1 (n) + Fx,2 (n) − 2Fx,1 (n) xt n 1 2 2 + Fσ,2 (n) − 2Fσ,1 (n) σt 1 2 2 1 = F0,2 (n) − 2F0,1 (n) + Fσ,2 (n) − 2Fσ,1 (n) σt , n
where the last step follows because
Fx,2 (n) = 2Fx,1 (n) 2ρd 1 − ρnx . = 1 − ρx
D.
(IA.50)
Dividend Strips The date t claim to the dividend edT paid out at date T is defined as h Pn i VdT (t, dt , xt , σt ) = Et e( i=1 mt+i )+dT .
(IA.51)
Note that from the law of iterated expectations we have h Pn i VdT (t − 1, dt−1 , xt−1 , σt−1 ) = Et−1 e( i=0 mt+i )+dT h h ii Pn = Et−1 Et emt +( i=1 mt+i )+dT = Et−1 emt VdT (t, dt , xt , σt ) .
(IA.52)
Again, since the state vector dynamics are affine, the solution is of the form 2
VdT (t, dt , xt , σt ) = edt +F0 (n)+Fx (n)xt +Fσ (n)σt
2 dt−1 +F0 (n+1)+Fx (n+1)xt−1 +Fσ (n+1)σt−1
VdT (t − 1, dt−1 , xt−1 , σt−1 ) = e
(IA.53) .
(IA.54)
The final conditions are F0 (0) = 0, Fx (0) = 0, and Fσ (0) = 0. Plugging equations (IA.53) and (IA.54) into equation (IA.52), performing the expectation, and then collecting terms independent of the state vector and linear in (xt , σt2 ), 9
we obtain the three recursive equations xt :
Fx (n + 1) = −γ + (θ − 1)κ1 Ax ρx − (θ − 1)Ax + ρd + Fx (n)ρx 1 σt2 : Fσ (n + 1) = (θ − 1)κ1 Aσ ρσ − (θ − 1)Aσ + Fσ (n)ρσ + (νc − λc )2 2 2 ν 1 + (Fx (n)νx − λx )2 + d 2 2 indep : F0 (n + 1) = θ log δ + (θ − 1)κ0 − γµc + (θ − 1)κ1 A0 +(θ − 1)κ1 Aσ σ 2 (1 − ρσ ) − (θ − 1)A0 + µd ν2 +F0 (n) + Fσ (n)σ 2 (1 − ρσ ) + σ (Fσ (n) − λσ )2 . 2 To obtain the dividend strip returns, define the date t one-period gross return on a strip that matures at date T via r˜T ˜T R ≡ e t+1 t+1
VdT (t + 1, dt+1 , xt+1 , σt+1 ) = . VdT (t, dt , xt , σt )
(IA.55)
The log expected excess returns for the dividend strips are h T i r(VdT (t)) = log Et e(r˜t+1 −rf,t )
= µd + ρd xt + F0 (n − 1) + Fx (n − 1)ρx xt + Fσ (n − 1) σ 2 (1 − ρσ ) + ρσ σt2 1 −F0 (n) − Fx (n)xt − Fσ (n)σt2 − r0 − xt − rσ σt2 ψ 1 1 1 + (Fx (n − 1)νx )2 σt2 + (Fσ (n − 1)νσ )2 + σt2 (νc2 + νd2 ). (IA.56) 2 2 2
This simplifies to r(VdT (t)) = νσ2 Fσ (n − 1)λσ + (νc λc + νx Fx (n − 1)λx ) σt2 . We also define dividend strip volatility as r h i h T i2 T −r 2(r˜t+1 r˜t+1 −rf,t ) T ) ( f,t σ(Vd (t)) = log Et e − log Et e q = (Fx (n − 1)νx )2 σt2 + (Fσ (n − 1)νσ )2 + σt2 (νc2 + νd2 ).
E.
(IA.57)
(IA.58)
Modified BKY Model
Since the modified BKY model for EBIT is structurally equivalent to the BKY model for dividends, we omit replicating the same proofs and instead jump to the new results. 10
E.1. Leverage Dynamics The claim to EBIT is approximated as 2
Vy (yt , xt , σt ) ≈ eyt +U0 +Ux xt +Uσ σt .
(IA.59)
Here, we assume that at all dates t the firm issues riskless debt that matures at date (t + dt) with present value equal to 2
B(ℓt , yt , xt , σt ) = eℓt +yt +U0 +Ux xt +Uσ σt ≈ eℓt Vy (yt , xt , σt ). We interpret eℓt ≈
B(ℓt ,yt ,xt ,σt ) Vy (yt ,xt ,σt )
(IA.60)
as the leverage of the firm. Since it is riskless, the firm
must pay erf (xt ,σt ) B(ℓt , yt , xt , σt ) at date (t + 1). It does so by issuing at this time debt with face value B(ℓt+1 , yt+1 , xt+1 , σt+1 ), with all residual cash flows paid out as dividends. As such, dividends D(t + 1) paid out at date (t + 1) are D(t + 1) = B(ℓt+1 , yt+1 , xt+1 , σt+1 ) − erf (xt ,σt ) B(ℓt , yt , xt , σt ) + eyt+1 . (IA.61) We specify the dynamics of log-leverage as ℓt+1 = ℓ + ρℓ ℓt − ℓ + ρℓx xt + ρℓσ (σt2 − σ 2 ) − νyc σt ˜ǫc, t+1 − νy σt ǫ˜y, t+1 −Ux νx σt ˜ǫx, t+1 − Uσ νσ ǫ˜σ, t+1 .
(IA.62)
Hence, dividends follow the endogenously determined process h 2 2 2 2 yt D(t + 1) = e eℓ+ρℓ (ℓt −ℓ)+ρℓx xt +ρℓσ (σt −σ )+µy +ρy xt +U0 +Ux ρx xt +Uσ (σ (1−ρσ )+ρσ σt )(IA.63) i 1 2 2 −e(ℓt +U0 +Ux xt +Uσ σt +r0 + ψ xt +rσ σt ) + e(µy +ρy xt +νyc σt ǫ˜c, t+1 +νy σt ˜ǫy, t+1 ) ≡ D1 (t + 1) − D2 (t + 1) + eyt+1 . E.2.
(IA.64)
Dividend Strips
Here we provide a closed-form expression for the price of dividend strips, defined as h Pn i T ( i=1 mt+i ) Vd (t, ℓt , yt , xt , σt ) = Et e D(T ) h Pn i = Et e( i=1 mt+i ) (D1 (T ) − D2 (T ) + eyT ) . (IA.65)
The price of dividend strips is a sum of three terms, each of which can be expressed in an exponential-affine form. The first term is h Pn i T Vd,1 (t, ℓt , yt , xt , σt ) = Et e( i=1 mt+i ) B(ℓT , yT , xT , σT ) . (IA.66) 11
The second term is h Pn i T Vd,2 (t, ℓt , yt , xt , σt ) = Et e( i=1 mt+i ) B(ℓT −1 , yT −1 , xT −1 , σT −1 )erf (xT −1 ,σT −1 ) h Pn−1 h ii mT +rf (xT −1 ,σT −1 ) ( i=1 mt+i ) = Et e B(ℓT −1 , yT −1 , xT −1 , σT −1 )ET −1 e h Pn−1 i = Et e( i=1 mt+i ) B(ℓT −1 , yT −1 , xT −1 , σT −1 ) T −1 = Vd,1 (t, ℓt , yt , xt , σt ).
(IA.67)
The third term is the claim to the EBIT strip, VyT (t, yt , xt , σt ). From the law of iterated expectations, the first term satisfies h Pn i T Vd,1 (t − 1, ℓt−1 , yt−1 , xt−1 , σt−1 ) = Et−1 e( i=0 mt+i ) B(ℓT , yT , xT , σT ) h h ii P mt +( n mt+i ) i=1 = Et−1 Et e B(ℓT , yT , xT , σT ) h i T = Et−1 emt Vd,1 (t, ℓt , yt , xt , σt ) . (IA.68) Because the state vector dynamics are affine, we know the solution is of the form 2
T Vd,1 (t, ℓt , yt , xt , σt ) = eyt +F0 (n)+Fℓ (n)ℓt +Fx (n)xt +Fσ (n)σt
Fx (n+1)xt−1 +Fσ (n+1)σt2−1
T Vd,1 (t−1, ℓt−1 , yt−1 , xt−1 , σt−1 ) = eyt−1 +F0 (n+1)+Fℓ (n+1)ℓt−1 e
(IA.69) . (IA.70)
The final conditions are F0 (0) = U0 , Fℓ (0) = 1, and Fx (0) = Ux , Fσ (0) = Uσ . Plugging equations (IA.69) and (IA.70) into equation (IA.68), performing the expectation, and then collecting terms independent of the state vector and linear in (ℓt , xt , σt2 ),
12
we obtain the four recursive equations ℓt :
Fℓ (n + 1) = ρℓ Fℓ (n)
xt :
Fx (n + 1) = −γ + (θ − 1)κ1 Ax ρx − (θ − 1)Ax + ρy +Fℓ (n)ρℓx + Fx (n)ρx
σt2 :
Fσ (n + 1) = (θ − 1)κ1 Aσ ρσ − (θ − 1)Aσ + Fℓ (n)ρℓσ νy2 +Fσ (n)ρσ + (1 − Fℓ (n))2 2 1 + (Fx (n)νx − Fℓ (n)Ux νx − λx )2 2 2 1 + νyc − νyc Fℓ (n) − λc 2
indep :
F0 (n + 1) = θ log δ + (θ − 1)κ0 − γµc + (θ − 1)κ1 A0 +(θ − 1)κ1 Aσ σ 2 (1 − ρσ ) − (θ − 1)A0 + µy +F0 (n) + Fℓ (n)ℓ(1 − ρℓ ) − Fℓ (n)ρℓσ σ 2 ν2 +Fσ (n)σ 2 (1 − ρσ ) + σ (Fσ (n) − Uσ Fℓ (n) − λσ )2 . 2
To obtain the dividend strip returns, we define the gross excess return on dividend strips as ˜T R d,t+1 Rf (t)
= =
T Vd,t+1 T R (t) Vd,t f
1 T Vd,t Rf (t)
!
h i Vd1T (t + 1) − Vd2T (t + 1) + VyT (t + 1) .
(IA.71)
The expectation has three terms: 2 2 Et Vd1T (t + 1) = eyt +µy +ρy xt +F0 (n−1)+Fℓ (n−1)[ℓ+ρℓ (ℓt −ℓ)+ρℓx xt +ρℓσ (σt −σ )] 2 2 2 ×eFx (n−1)ρx xt +Fσ (n−1)[σ +ρσ (σt −σ )]
×e
×e
σ2 t 2
ν2 σ 2
2 +ν 2 1−F (n−1) (νyc ] y )[ ℓ
2
[Fσ (n−1)−Uσ Fℓ (n−1)]
13
+
2
σ2 t 2
νx2 [Fx (n−1)−Ux Fℓ (n−1)]
2
2 2 Et Vd2T (t + 1) = eyt +µy +ρy xt +F0 (n−2)+Fℓ (n−2)[ℓ+ρℓ (ℓt −ℓ)+ρℓx xt +ρℓσ (σt −σ )] 2 2 2 ×eFx (n−2)ρx xt +Fσ (n−2)[σ +ρσ (σt −σ )]
×e
×e
σ2 t 2
ν2 σ 2
2 +ν 2 1−F (n−2) (νyc ] y )[ ℓ
2
[Fσ (n−2)−Uσ Fℓ (n−2)]
+
σ2 t 2
νx2 [Fx (n−2)−Ux Fℓ (n−2)]
2
2
h i 2 2 Et VyT (t + 1) = eyt +µy +ρy xt +U0 (n−1)+Ux (n−1)ρx xt +Uσ (n−1)[σ (1−ρσ )+ρσ σt ] 2 σ 2 + 1 (U (n−1)ν )2 + 1 σ 2 (ν 2 +ν 2 ) σ σ yc y 2 2 t t
1
×e 2 (Ux (n−1)νx )
The second moments are h 2 i Et Vd1T (t + 1) = e2yt +2µy +2ρy xt +2F0 (n−1)
2 2 ×e2Fℓ (n−1)[ℓ+ρℓ (ℓt −ℓ)+ρℓx xt +ρℓσ (σt −σ )] 2 2 2 ×e2Fx (n−1)ρx xt +2Fσ (n−1)[σ +ρσ (σt −σ )] 2
2 2 2 2 2 ×e2σt (νyc +νy )[1−Fℓ (n−1)] +2σt νx [Fx (n−1)−Ux Fℓ (n−1)] 2 2 ×e2νσ [Fσ (n−1)−Uσ Fℓ (n−1)]
Et
h
Vd2T (t + 1)
2 i
2
= e2yt +2µy +2ρy xt +2F0 (n−2) 2 2 ×e2Fℓ (n−2)[ℓ+ρℓ (ℓt −ℓ)+ρℓx xt +ρℓσ (σt −σ )] 2 2 2 ×e2Fx (n−2)ρx xt +2Fσ (n−2)[σ +ρσ (σt −σ )] 2
2 2 2 2 2 ×e2σt (νyc +νy )[1−Fℓ (n−2)] +2σt νx [Fx (n−2)−Ux Fℓ (n−2)] 2 2 ×e2νσ [Fσ (n−2)−Uσ Fℓ (n−2)]
2 T Et Vy (t + 1) = e2yt +2µy +2ρy xt +2U0 (n−1)+2Ux (n−1)ρx xt ×e2Uσ (n−1)[σ 2
2 (1−ρ )+ρ σ 2 σ σ t
2
2
2
]
2
2
2
2
×e2Ux (n−1)νx σt +2Uσ (n−1)νσ +2σt (νyc +νy )
14
2
Et Vd1T (t + 1) Vd2T (t + 1) = e2yt +2µy +2ρy xt +F0 (n−1)+F0 (n−2) 2 2 ×e(Fℓ (n−1)+Fℓ (n−2))[ℓ+ρℓ (ℓt −ℓ)+ρℓx xt +ρℓσ (σt −σ )] ×e(Fx (n−1)+Fx (n−2))ρx xt +(Fσ (n−1)+Fσ (n−2))[σ σ2 t 2
2 2 ×e (νyc +νy )[2−Fℓ (n−1)−Fℓ (n−2)]
×e
ν 2 σ2 x t 2
2 +ρ σ
(σt2 −σ2 )]
2
[Fx (n−1)+Fx (n−2)−Ux (Fℓ (n−1)+Fℓ (n−2))]
ν2
σ ×e 2 [Fσ (n−1)+Fσ (n−2)−Uσ (Fℓ (n−1)+Fℓ (n−2))]
2
2
h i T T Et Vd1 (t + 1) Vy (t + 1) = e2yt +2µy +2ρy xt +F0 (n−1)+U0 (n−1)
2 2 ×eFℓ (n−1)[ℓ+ρℓ (ℓt −ℓ)+ρℓx xt +ρℓσ (σt −σ )]
×e(Fx (n−1)+Ux (n−1))ρx xt +(Fσ (n−1)+Uσ (n−1))[σ σ2 t 2
2
2 +ρ σ
(σt2 −σ2 )]
ν 2 σ2
x t 2 2 ×e (νyc +νy )[2−Fℓ (n−1)] + 2 [Fx (n−1)+Ux (n−1)−Ux Fℓ (n−1)] ν2 2 σ ×e 2 [Fσ (n−1)+Uσ (n−1)−Uσ Fℓ (n−1)]
2
h i Et Vd2T (t + 1) VyT (t + 1) = e2yt +2µy +2ρy xt +F0 (n−2)+U0 (n−1)
2 2 ×eFℓ (n−2)[ℓ+ρℓ (ℓt −ℓ)+ρℓx xt +ρℓσ (σt −σ )]
×e(Fx (n−2)+Ux (n−1))ρx xt +(Fσ (n−2)+Uσ (n−1))[σ σ2 t 2
2
ν 2 σ2
2 +ρ σ
(σt2 −σ2 )]
x t 2 2 ×e (νyc +νy )[2−Fℓ (n−2)] + 2 [Fx (n−2)+Ux (n−1)−Ux Fℓ (n−2)] ν2 2 σ ×e 2 [Fσ (n−2)+Uσ (n−1)−Uσ Fℓ (n−2)] .
2
E.3. Equity Returns To derive the relation between returns on the EBIT claim and returns on the dividend claim, we solve for the exact solution in continuous time, and then use the solution as an approximation for our discrete time-setting. In particular, we notionally express the
15
returns to enterprise value, the stock, and the risk-free bond via dVy + δV Vy dt ≡ ery − 1 = µV dt + σV dz Vy dVd + δS Vd dt ≡ erd − 1 = µS dt + σS dz Vy dB = r dt. B Defining leverage via
B Vy
(IA.72)
≡ eℓ , standard results generate the relations µV − r = (1 − eℓ ) [µS − r] σV
= (1 − eℓ )σS .
(IA.73)
From their definitions, we can express the log returns via 1 2 ry = µV − σV dt + σV dz 2 1 2 rd = µS − σS dt + σS dz. 2
(IA.74)
Combining the last four equations, we find that we can relate the log stock return rd to the enterprise value return ry via σV2 1 1 rd − rf = ry − rf + . (IA.75) 1− 1 − eℓ 2 1 − eℓ Importantly, the stochastic components are related via 1 rd |stoch = r . y stoch 1 − eℓ E.4.
(IA.76)
Bounding Leverage
To guarantee that leverage remains within empirically observed limits, we assume that there are asset sales when leverage moves above the upper threshold, and that the funds raised are used to repurchase debt. All of this is accomplished at “fair market values,” so these asset sales have no impact on firm value. For example, assume that a firm finds itself with leverage higher than some specified maximum value Lmax , which we set to 65%. At this time, enterprise value and the value of debt satisfy 2
Vy (yt , xt , σt ) ≈ eyt +U0 +Ux xt +Uσ σt
2
B(ℓt , yt , xt , σt ) = eℓt +yt +U0 +Ux xt +Uσ σt . 16
(IA.77)
The firm acts to lower leverage from Lt ≡ eℓt to Lmax . It does so by selling off a fraction of its EBIT-generating machine. In particular, the firm sells off the fraction that reduces EBIT from Yt ≡ eyt to (Yt − ∆Yt ). Because the value of the EBIT claim is linear in Yt , it follows that the value of this sale is 2
Salet = ∆Yt eU0 +Ux xt +Uσ σt .
(IA.78)
Because the funds raised are used to reduce the debt, the new enterprise value and new debt value are 2
Vy,new (yt , xt , σt ) ≈ (Yt − ∆Yt ) eU0 +Ux xt +Uσ σt 2 Bnew (ℓt , yt , xt , σt ) ≈ Yt eℓt − ∆Yt eU0 +Ux xt +Uσ σt .
(IA.79)
Hence, we determine the size of ∆Yt by choosing it so that the new leverage is equal to the maximum: Bnew (ℓt , yt , xt , σt ) Vy,new (yt , xt , σt ) ℓ Yt e t − ∆Yt = . Yt − ∆Yt
Lmax ≡
(IA.80)
Solving for ∆Yt , we find ∆Yt = Yt
eℓt − Lmax 1 − Lmax
(IA.81)
.
In terms of the state variable yt , we find eynew = eyt − ∆Y,
(IA.82)
ynew = log (eyt − ∆Y ) .
(IA.83)
implying that
An analogous argument holds for asset purchases at a lower leverage threshold Lmin , which we set to 25%. In particular, if Lt < Lmin , then the firm purchases EBITgenerating machinery at cost 2
P urchaset = ∆Yt eU0 +Ux xt +Uσ σt .
17
(IA.84)
The funds raised to make this purchase are obtained by issuing debt. Hence, the new enterprise value and debt value are 2
Vy,new (yt , xt , σt ) ≈ (Yt + ∆Yt ) eU0 +Ux xt +Uσ σt 2 Bnew (ℓt , yt , xt , σt ) ≈ Yt eℓt + ∆Yt eU0 +Ux xt +Uσ σt .
(IA.85)
Hence, we determine the size of ∆Yt by choosing it so that the new leverage is equal to Lmin : Bnew (ℓt , yt , xt , σt ) Vy,new (yt , xt , σt ) ℓ Yt e t + ∆Yt . = Yt + ∆Yt
Lmin ≡
(IA.86)
Solving for ∆Yt , we find ∆Yt = Yt
Lmin − eℓt 1 − Lmin
.
(IA.87)
In terms of the state variable yt , we find eynew = eyt + ∆Y,
(IA.88)
ynew = log (eyt + ∆Y ) .
(IA.89)
implying that
F.
Additional Results on the Long-Run Risk Model F.1. Approximation Errors in the BKY Calibration
In the manuscript we closely follow the calibration of BKY because it matches several empirical asset pricing moments better than the original Bansal and Yaron (BY, 2004) calibration. However, we changed the persistence of the volatility process from ρσ = 0.999 used in BKY to ρσ = 0.995, because the accuracy of the approximation they use to solve the model deteriorates quickly when the persistence parameter ρσ is pushed above 0.995. Figure IA.1 reports the results from this analysis. 18
F.2. Comparative Statics: The Role of the Economic State BBK show that the price of dividends, scaled by the current level of dividends, predicts dividend strip returns and this predictability is more pronounced for shortterm strips than for the market as a whole. Here, we perform comparative statics with respect to the main driver of time-varying risk premia (i.e., stochastic volatility) and show that the model is qualitatively consistent with this result. In the BY framework, the stock price is a decreasing function of volatility, and excess returns are an increasing function. To examine the impact of the economic state, in Figure IA.2 we plot the term structure of excess returns for dividend strips when the variance is one standard deviation above its mean. We see that short-horizon dividend strips increase their excess return by an additional 2.7%, whereas the excess returns on longer-horizon dividend strips also increase, but to a lesser extent. This result is mostly consistent with the findings of BBK that the price of dividends, scaled by the current level of dividends, predicts dividend strip returns, and that this predictability is more pronounced for short-term strips than for the market as a whole.
II.
Habit Model
The pricing kernel, market price of risk, and EBIT dynamics follow 1 mt+1 = −rf − θt2 − θt ǫ˜m,t+1 2 θt+1 = θ (1 − ρθ ) + ρθ θt − σθ ǫ˜m,t+1 ∆yt+1 = µy + νym ǫ˜m, t+1 + νy ǫ˜y, t+1 . We approximate the log of enterprise value to EBIT via V zy,t ≡ log Yy,t ≈ U0 − Uθ θt . t
(IA.90) (IA.91) (IA.92)
(IA.93)
We also approximate the log return to the EBIT claim via
ry,t+1 ≈ κ0y + κ1y zy,t+1 − zy,t + ∆yt+1 ,
19
(IA.94)
where yt = log Yt and κ1y = κ0y
ez y 1 + ez y
(IA.95)
ezy = log(1 + e ) − z y 1 + ez y κ1y = − log(1 − κ1y ) − κ1y log . 1 − κ1y
zy
(IA.96)
We need to identify {z y , κ0y , κ1y , U0 , Uθ }. Again, we already have three equations: the definitions of {κ0y , κ1y } and the self-consistent condition z y = U0 − Uθ θ.
(IA.97)
The last two conditions are determined from the following first-order condition:
Note that
1 0 = Et mt+1 + ry,t+1 + Vart mt+1 + ry,t+1 . 2
(IA.98)
1 Et mt+1 + ry,t+1 = −rf − θt2 + κ0y + κ1y U0 − κ1y Uθ θ (1 − ρθ ) 2 −κ1y Uθ ρθ θt − U0 + Uθ θt + µy . Furthermore, we have mt+1 + ry,t+1 − Et mt+1 + ry,t+1 =
νym + κ1y Uθ σθ − θt ǫ˜m,t+1 + νy ǫ˜y,t+1 . (IA.99)
Collecting terms linear and independent of θt , we find the two conditions 0 = −κ1y Uθ ρθ + Uθ − κ1y Uθ σθ − νym
1 1 0 = −rf + κ0y + κ1y U0 − κ1y Uθ θ (1 − ρθ ) − U0 + µy + νy2 + κ21y Uθ2 σθ2 2 2 1 2 + νym + νym κ1y Uθ σθ . 2 Simplifying, this gives νym Uθ = 1 − κ1y ρθ − κ1y σθ 1 1 1 U0 = −rf + κ0y − κ1y Uθ θ (1 − ρθ ) + µy + νy2 + κ21y Uθ2 σθ2 1 − κ1y 2 2 1 2 + νym + νym κ1y Uθ σθ . 2 20
A. Enterprise Value Return The log-linear approximation implies that the excess return on enterprise value can be expressed as ry,t+1 − rf
= κ0y + κ1y zy,t+1 − zy,t + ∆yt+1 − rf .
(IA.100)
It is convenient to decompose the previous equation in terms of the predicted and unexpected components: Et ry,t+1 − rf = κ0y + κ1y U0 − κ1y Uθ θ(1 − ρθ ) + ρθ θt
ry,t+1 − rf − Et ry,t+1 − rf
=
−U0 + Uθ θt + µy − rf κ1y Uθ σθ + νym ǫ˜m,t+1 + νy ˜ǫy,t+1 .
(IA.101)
We define the log expected excess return on the EBIT claim as h i r y,t = log Et e(ry,t+1 −rf ) = κ0y + κ1y U0 − κ1y Uθ θ(1 − ρθ ) + ρθ θt − U0 + Uθ θt + µy − rf 2 1 1 (IA.102) + νym + κ1y Uθ σθ + νy2 . 2 2
This simplifies to
ry,t = νym θt
B.
1 − κ1y ρθ 1 − κ1y ρθ − κ1y σθ
.
We also define the volatility on the EBIT claim as r h i h i σy,t = log Et e2(ry,t+1 −rf ) − 2 log Et e(ry,t+1 −rf ) q 2 = κ1y Uθ σθ + νym + νy2 .
(IA.103)
(IA.104)
Term Structure of EBIT
Because log-EBIT follows an arithmetic Brownian motion process, the term structure of dividend expected growth rates is flat over all horizons n: 1 log Et eyT −yt n 1 2 = µy + νym + νy2 2
gy,n =
21
∀n.
(IA.105)
Similarly, the term structure of dividend volatilities is also flat over all horizons: s 1 σy,n = log Et e2(yT −yt ) − 2 log Et [eyT −yt ] n r 2 2 = νym + νy ∀n. (IA.106)
C. EBIT Strips The date t claim to the EBIT strip eyT paid out at date T is defined as h Pn i VyT (t, yt , θt ) = Et e( i=1 mt+i )+yT .
Note that from the law of iterated expectations we have h Pn i VyT (t − 1, yt−1 , θt−1 ) = Et−1 e( i=0 mt+i )+yT h h ii P mt +( n mt+i )+yT i=1 = Et−1 Et e h i = Et−1 emt VyT (t, yt , θt ) .
(IA.107)
(IA.108)
Again, since the state vector dynamics are affine, the solution is of the form VyT (t, yt , θt ) = eyt +U0 (n)−Uθ (n)θt
(IA.109)
VyT (t − 1, yt−1 , θt−1 ) = eyt−1 +U0 (n+1)−Uθ (n+1)θt−1 .
(IA.110)
The final conditions are U0 (0) = 0 and Uθ (0) = 0. Plugging equations (IA.109) and (IA.110) into equation (IA.108), performing the expectation, and then collecting terms independent of the state vector and linear in (θt ), we obtain the two recursive equations θt : indep :
Uθ (n + 1) = Uθ (n) (ρθ + σθ ) + νym
(IA.111)
U0 (n + 1) = −rf + µy + U0 (n) − Uθ (n)θ(1 − ρθ ) σ2 1 2 + νy2 + νym + θ Uθ2 (n) + νym σθ Uθ (n). (IA.112) 2 2
To obtain the EBIT strip returns, define the date t one-period gross return on a strip that matures at date T via T
r˜ ˜T R ≡ e y,t+1 y,t+1
=
VyT (t + 1, yt+1 , θt+1 ) VyT (t, yt , θt ) 22
.
(IA.113)
T We can decompose r˜y,t+1 − rf into its predictable and unpredictable components via h i T Et r˜y,t+1 − rf = µy + U0 (n − 1) − Uθ (n − 1) θ(1 − ρθ ) + ρθ θt
h i h i T T r˜y,t+1 − rf − Et r˜y,t+1 − rf =
−U0 (n) + Uθ (n)θt − rf . νym + σθ Uθ (n − 1) ǫ˜m,t+1 + νy ˜ǫy,t+1 .
The log expected excess returns for the EBIT strips are T r˜y,t+1 −rf T r(Vy (t)) = log Et e
= µy + U0 (n − 1) − U0 (n) − Uθ (n − 1) θ(1 − ρθ ) + ρθ θt 1 1 (IA.114) +Uθ (n)θt − rf + (σθ Uθ (n − 1) + νym )2 + νy2 . 2 2
This simplifies to r(VyT (t)) = θt σθ Uθ (n − 1) + νym .
(IA.115)
We also define dividend strip volatility as s T T 2 r˜y,t+1 −rf r˜y,t+1 −rf T log Et e − 2 log Et e σ(Vy (t)) = =
D.
q
2
+ νy2 .
(IA.116)
Vy (yt , θt ) ≈ eyt +U0 −Uθ θt .
(IA.117)
σθ Uθ (n − 1) + νym
Leverage Dynamics Recall from equation (IA.93) that
Here, we assume that at all dates t the firm issues riskless debt that matures at date (t + dt) with present value equal to B(ℓt , yt , θt ) = eℓt +yt +U0 −Uθ θt ≈ eℓt Vy (yt , θt ). B(ℓ ,y ,θ )
(IA.118)
t t t We interpret eℓt ≈ V (y as the leverage of the firm. Since it is riskless, the firm must y t ,θt ) rf pay e B(ℓt , yt , θt ) at date (t + 1). It does so by issuing at this time debt with face value
23
B(ℓt+1 , yt+1 , θt+1 ), with all residual cash flows paid out as dividends. As such, dividends D(t + 1) paid out at date (t + 1) are D(t + 1) = B(ℓt+1 , yt+1 , θt+1 ) − erf B(ℓt , yt , θt ) + eyt+1 ≡ D1 (t + 1) − D2 (t + 1) + eyt+1 .
(IA.119)
We specify log-leverage dynamics via
E.
ℓt+1 = ℓ + ρℓ ℓt − ℓ + ρℓθ (θt − θ) − νym ˜ǫm, t+1 − νy ǫ˜y, t+1 − Uθ σθ ǫ˜m, t+1 . (IA.120)
Dividend Strips
Here we provide a closed-form expression for the price of dividend strips, defined as h Pn i T ( i=1 mt+i ) Vd (t, ℓt , yt , θt ) = Et e D(T ) h Pn i = Et e( i=1 mt+i ) (D1 (T ) − D2 (T ) + eyT ) . (IA.121)
The price of dividend strips is the sum of three terms, each of which can be expressed in an exponential-affine form. The first term is h Pn i T Vd,1 (t, ℓt , yt , θt ) = Et e( i=1 mt+i ) B(ℓT , yT , θT ) .
(IA.122)
The second term is
i h Pn T Vd,2 (t, ℓt , yt , θt ) = Et e( i=1 mt+i ) B(ℓT −1 , yT −1 , θT −1 )erf h Pn−1 m +r i ( i=1 mt+i ) = Et e B(ℓT −1 , yT −1 , θT −1 )ET −1 e T f h Pn−1 i = Et e( i=1 mt+i ) B(ℓT −1 , yT −1 , θT −1 ) T −1 = Vd,1 (t, ℓt , yt , θt ).
(IA.123)
The third term is the claim to the EBIT strip, VyT (t, yt , θt ). From the law of iterated expectations, the first term satisfies h Pn i T Vd,1 (t − 1, ℓt−1 , yt−1 , θt−1 ) = Et−1 e( i=0 mt+i ) B(ℓT , yT , θT ) h h ii Pn = Et−1 Et emt +( i=1 mt+i ) B(ℓT , yT , θT ) h i T (t, ℓt , yt , θt ) . (IA.124) = Et−1 emt Vd,1 24
Because the state vector dynamics are affine, we know the solution is of the form T Vd,1 (t, ℓt , yt , θt ) = eyt +F0 (n)+Fℓ (n)ℓt −Fθ (n)θt T Vd,1 (t − 1, ℓt−1 , yt−1 , θt−1 ) = eyt−1 +F0 (n+1)+Fℓ (n+1)ℓt−1 −Fθ (n+1)θt−1 .
(IA.125) (IA.126)
The final conditions are F0 (0) = U0 , Fℓ (0) = 1, and Fθ (0) = Uθ . Plugging equations (IA.125) and (IA.126) into equation (IA.124), performing the expectation, and then collecting terms independent of the state vector and linear in (ℓt , θt ), we obtain the three recursive equations ℓt :
Fℓ (n + 1) = ρℓ Fℓ (n)
θt :
Fθ (n + 1) = −ρℓθ Fℓ (n) + ρθ Fθ (n) + νym −Fℓ (n) νym + Uθ σθ + Fθ (n)σθ
indep :
(IA.127)
F0 (n + 1) = −rf + µy + F0 (n) + Fℓ (n)ℓ(1 − ρℓ ) − ρℓθ θFℓ (n) νy2 −(1 − ρθ )θFθ (n) + [1 − Fℓ (n)]2 2 2 1 + νym − Fℓ (n)νym − Fℓ (n)Uθ σθ + Fθ (n)σθ . 2
To obtain the return on dividend strips, we define the gross excess return on dividend strips as ˜T R d,t+1 Rf
=
T Vd,t+1 T R Vd,t f
1 T Vd,t Rf
=
!
h i T T T Vd1 (t + 1) − Vd2 (t + 1) + Vy (t + 1) .
(IA.128)
The expectation has three terms: Et Vd1T (t + 1) = eyt +µy +F0 (n−1)+Fℓ (n−1)[ℓ+ρℓ (ℓt −ℓ)+ρℓθ (θt −θ)]−Fθ (n−1)[θ(1−ρθ )+ρθ θt ] 2 νy
×e
2
!
[1−Fℓ (n−1)]
2
+ 12 [νym −νym Fℓ (n−1)−Uθ σθ Fℓ (n−1)+Fθ (n−1)σθ ]
2
Et Vd2T (t + 1) = eyt +µy +F0 (n−2)+Fℓ (n−2)[ℓ+ρℓ (ℓt −ℓ)+ρℓθ (θt −θ)]−Fθ (n−2)[θ(1−ρθ )+ρθ θt ] 2 νy
×e
2
!
[1−Fℓ (n−2)]
2
+ 12 [νym −νym Fℓ (n−2)−Uθ σθ Fℓ (n−2)+Fθ (n−2)σθ ]
2 h i νy 1 2 Et VyT (t + 1) = eyt +µy +U0 (n−1)−Uθ (n−1)[θ(1−ρθ )+ρθ θt ]+ 2 (Uθ (n−1)σθ +νym ) + 2 .
25
2
The second moments are h 2 i Et Vd1T (t + 1) = e2yt +2µy +2F0 (n−1)+2Fℓ (n−1)[ℓ+ρℓ (ℓt −ℓ)+ρℓθ (θt −θ)] 2
×e−2Fθ (n−1)[θ(1−ρθ )+ρθ θt ]+2νy [1−Fℓ (n−1)] 2 ×e2[νym −νym Fℓ (n−1)−Uθ σθ Fℓ (n−1)+Fθ (n−1)σθ ] 2
Et
Et
h
T
2 i
= e2yt +2µy +2F0 (n−2)+2Fℓ (n−2)[ℓ+ρℓ (ℓt −ℓ)+ρℓθ (θt −θ)]
2
= e2yt +2µy +2U0 (n−1)−2Uθ (n−1)[θ(1−ρθ )+ρθ θt ]+2(Uθ (n−1)σθ +νym )
Vd2 (t + 1)
T
Vy (t + 1)
×e−2Fθ (n−2)[θ(1−ρθ )+ρθ θt ] 2 2 2 ×e2νy [1−Fℓ (n−2)] +2[νym −νym Fℓ (n−2)−Uθ σθ Fℓ (n−2)+Fθ (n−2)σθ ] 2 +2ν 2 y
Et Vd1T (t + 1) Vd2T (t + 1) = e2yt +2µy +F0 (n−1)+F0 (n−2)+(Fℓ (n−1)+Fℓ (n−2))[ℓ+ρℓ (ℓt −ℓ)+ρℓθ (θt −θ)] ν2 y 2
!
[2−Fℓ (n−1)−Fℓ (n−2)]
2
×e 2 [2νym −(νym +Uθ σθ )(Fℓ (n−1)+Fℓ (n−2))+σθ (Fθ (n−1)+Fθ (n−2))]
2
−(Fθ (n−1)+Fθ (n−2))[θ(1−ρθ )+ρθ θt ]+
×e
1
h i Et Vd1T (t + 1) VyT (t + 1) = e2yt +2µy +F0 (n−1)+U0 (n−1)+Fℓ (n−1)[ℓ+ρℓ (ℓt −ℓ)+ρℓθ (θt −θ)] −(Fθ (n−1)+Uθ (n−1))[θ(1−ρθ )+ρθ θt ]+
×e
ν2 y 2
!
[2−Fℓ (n−1)]
×e 2 [2νym −(νym +Uθ σθ )Fℓ (n−1)+σθ (Fθ (n−1)+Uθ (n−1))] 1
2
2
h i Et Vd2T (t + 1) VyT (t + 1) = e2yt +2µy +F0 (n−2)+U0 (n−1)+Fℓ (n−2)[ℓ+ρℓ (ℓt −ℓ)+ρℓθ (θt −θ)] −(Fθ (n−2)+Uθ (n−1))[θ(1−ρθ )+ρθ θt ]+
×e
ν2 y 2
!
[2−Fℓ (n−2)] 2
×e 2 [2νym −(νym +Uθ σθ )Fℓ (n−2)+σθ (Fθ (n−2)+Uθ (n−1))] . 1
26
2
III.
Additional Empirical Results
In the main text, we investigate the properties (variance ratios) of aggregate dividends of publicly and privately traded firms (that is, using Flow of Funds data). To help establish the robustness of the analysis, here we investigate the properties of three alternative measures of aggregate dividends using data from publicly traded firms only. The analysis here confirms that the downward slope of the term structure of aggregate dividend volatility is a robust feature of the data, and we explicitly link this feature of the data to the negative serial correlation of the aggregate dividend series. In addition, we report the following additional empirical analysis and robustness checks to support the economic mechanisms highlighted in the main text: (i) We document a strong positive link between the aggregate leverage ratio and the ratio of bondholders’ and stockholders’ payout (which captures the co-integration residual between the two payout variables); (ii) We show that leverage predicts future excess returns with a positive slope (the risk premium on equity is high when leverage is high); and (iii) We show that the net (which excludes short-term assets such as cash), not just gross as reported in the main text, aggregate leverage ratio of public and privately traded firms is also stationary.
A. Data As in the main text, we perform the analysis using annual data to avoid the seasonality in dividend payments. The use of an annual dividend series implies that we need to take a stance on how dividends received within a particular year are reinvested. We consider two alternative reinvestment strategies. In the first strategy, we assume the monthly dividends are reinvested in the aggregate stock market. As in van Binsbergen and Koijen (2010), we refer to this dividend series as market-invested dividends. This measure of dividends is by far the most common in the dividend growth and return forecasting literature.1 In the second strategy, we invest the monthly dividends in cash and obtain a time series of annual dividends that we call cash-invested dividends. As shown by van Binsbergen and Koijen (2010) and Chen (2009), the two dividend series have different time-series properties in the post-war sample period. We obtain the data for the two dividend series from Long Chen’s webpage (the data are used in Chen (2009), and we extend the data to 2010 by replicating Chen’s procedure). We use this data set because it covers a long sample period from 1873 to 2008, thus covering the pre-Center for Research in Security Prices (CRSP) period. 27
Focusing on this long sample allows us to obtain more robust results. To construct the two dividend series, Chen (2009) combines the pre-CRSP data compiled by Schwert (1990) with the data from the CRSP (NYSE/Amex/NASDAQ) value-weighted market portfolio at the monthly frequency. We refer the reader to Chen (2009) for additional details on the construction of the two dividend series. In addition to the previous two dividend series, we investigate a third alternative measure of dividends that includes share repurchases. The data for this alternative dividend series are available from Motohiro Yogo’s webpage (the data are used in Gomes, Kogan, and Yogo (2009), and cover a relatively shorter sample period from 1927 to 2007). Examining this alternative definition of dividends is motivated by the observation that firms have increased the fraction of payouts to shareholders via repurchase programs compared to dividends in recent years (Fama and French (2001), Grullon and Michaely (2002)). Still, as discussed in Lettau and Ludvigson (2005), large firms with high earnings have continued to increase traditional dividend payouts over time (DeAngelo, DeAngelo, and Skinner (2004)). The impact on aggregate dividends is therefore unclear. To show that our main findings are not altered by adjusting dividends to account for share repurchase activity since 1971, we consider a dividend series augmented with equity repurchases using Compustat’s statement of cash flows. All nominal quantities are deflated by the consumer price index (CPI), which is available from Robert Shiller’s webpage.
B.
Dividend Variance Ratios of Publicly Traded Firms
The first two panels in Table IA.I reports the VR test results for the three definitions of aggregate dividends of publicly traded firms. Consistent with the results reported in the main text, the table shows that dividends do not follow a random walk. The VR test statistic decreases strongly with horizon for the three alternative dividend measures, that is, the variance of dividend growth is significantly smaller at long horizons than at short horizons. At a fundamental level, the finding that the dividend variance decreases with horizon reflects negative serial correlation in the dividend growth series. To show this formally, we consider a simple econometric approach based on a linear regression. Specifically, we investigate if past values of dividend growth help predict future dividend growth by P running a regression of the form ∆dt = a + 5i=1 bi ∆dt−i . The results reported in Table IA.II show that past values of dividend growth help predict future dividends. The slope 28
coefficients on the lagged values of dividend growth are negative. It is this pattern of negative autocorrelation that drives the decreasing pattern of dividend volatility across horizon.
C. Leverage, Dividends, and Bondholders Payout The main economic mechanism in the model ties the aggregate firm leverage ratio to the aggregate bondholder and stockholder payouts. The stationarity of the leverage ratio implies stationarity of the bondholder and stockholder payouts (i.e., these two payouts should be co-integrated). Thus, the aggregate leverage ratio should be related to the co-integration residual of the bondholders’ and stockholders’ payouts. Here, we provide empirical evidence for this link in the data. Figure IA.3 plots the time series of the aggregate leverage ratio and the aggregate bondholder-shareholder payout ratio (this ratio is naturally closely related to the cointegration between the two payouts). Both variables are standardized (mean zero and unit standard deviation) to facilitate the analysis. The two series are strongly positively contemporaneously correlated, with a correlation of +50%. Importantly, the two series also share a strong common cycle (low frequency correlation). Thus, this figure shows that when the ratio of bondholder to stockholder payout is below the mean (here, below zero), the leverage ratio tends to be below its mean as well (here, below zero). Given mean-reversion in the leverage ratio, we expect to see leverage increase in the future if it is currently below the mean. As leverage increases, the bondholder-shareholder payout ratio increases as well, consistent with Figure IA.3.
D.
Leverage and Time-Varying Risk Premia According to the discussion in the main text, time-varying leverage implies time-
variation in the aggregate risk premium (return on equity). In this section, we provide empirical support for this additional prediction of the model(s). We confirm empirically the prediction that leverage forecasts future excess returns with a positive slope. To examine the relationship between leverage and risk premiums, we run standard short- and long-horizon predictive regressions (e.g., Fama and French (1989), and Lettau and Ludvigson (2002)). The dependent variable in the predictive regression is the T -year cumulated log aggregate stock market return, in which T is the forecast horizon ranging from one year to 20 years. Specifically, we run a long-horizon forecasting regression of
29
the form ΣTh=1 yt+h = a + bLevt + εit ,
(IA.129)
where yt = rst − rf t , rst is the log aggregate stock market return, rf t is the log riskfree rate, and Levt is the current value of the log aggregate leverage ratio. For each horizon T = 1, ..., 20, we report the estimated slope associated with leverage, the corresponding t-statistic, and the regression R2 . In computing the t-statistic of the slope coefficient, we use standard errors corrected for autocorrelation per Newey and West (1987) with lag equal to three years plus the overlapping period, and a GMM correction for heteroskedasticity. Table IA.III reports the results of the long-horizon predictability regressions. The results show that leverage forecasts future excess returns (risk premia) with a positive slope. Thus, periods with high leverage are periods with high risk premia. The regression R2 increases from 4.6% at the one-year horizon to 29.3% at the 20-year horizon.
E.
Stationarity of Gross and Net Leverage In the main text we show that the aggregate gross leverage ratio is stationary. Here,
we show that aggregate net leverage (which excludes short-term assets such as cash) exhibits similar stationary behavior. See also Wright (2004) for a similar analysis (our results are an update of Wright’s analysis). Figure IA.4 plots the time series of the aggregate gross and net leverage ratio of public and private traded firms (Flow of Funds data). Although the level of the two series is different (net leverage is naturally smaller), both series appear to be equally stationary, consistent with the main mechanism in the model.
30
REFERENCES Bansal, Ravi, Dana Kiku, and Amir Yaron, 2012, An empirical evaluation of the long-run risks model for asset prices, Critical Finance Review 1, 183–221. Bansal, Ravi, and Amir Yaron, 2004, Risks for the long run: A potential resolution of asset pricing puzzles, Journal of Finance 59, 1481–1509. Campbell, John, and Robert Shiller, 1988, The dividend-price ratio and expectations of future dividends and discount factors, Review of Financial Studies 1, 195–228. Chen, Long, 2009, On the reversal of dividend and return predictability: A tale of two periods, Journal of Financial Economics 92, 128–151. Cochrane, John H., 2008, The dog that did not bark: A defense of return predictability, Review of Financial Studies 21, 1533–1575. DeAngelo, Harry, Linda DeAngelo, and Douglas J. Skinner, 2004, Are dividends disappearing? Dividend concentration and the consolidation of earnings, Journal of Financial Economics 72, 425–456. Duffie, Darrell, and Rui Kan, 1996, A yield-factor model of interest rates, Mathematical Finance 6, 379–406. Epstein, Larry G., and Stanley E. Zin, 1989, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica 57, 937–969. Fama, Eugene F., and Kenneth R. French, 1989, Business conditions and expected returns on stocks and bonds, Journal of Financial Economics 25, 23–49. Fama, Eugene F., and Kenneth R. French, 2001, Disappearing dividends: Changing firm characteristics or lower propensity to pay, Journal of Financial Economics 60, 3–43. Gomes, Jo˜ao, Leonid Kogan, and Motohiro Yogo, 2009, Durability of output and expected stock returns, Journal of Political Economy 117, 941–86. 31
Grullon, Gustavo, and Roni Michaely, 2002, Dividends, share repurchases, and the substitution hypothesis, Journal of Finance 57, 1649–1684. Lettau, Martin, and Sydney Ludvigson, 2002, Time-varying risk premia and the cost of capital: an alternative implication of the q theory of investment, Journal of Monetary Economics 49, 31–66. Lettau, Martin, and Sydney Ludvigson, 2005, Expected returns and expected dividend growth, Journal of Financial Economics 76, 583–626. Lettau, Martin, and Stijn Van Nieuwerburgh, 2008, Reconciling the return predictability evidence, Review of Financial Studies 21, 1607–1652. Lo, Andrew W., and A. Craig MacKinlay, 1988, Stock market prices do not follow random walks: Evidence from a simple specification test, Review of Financial Studies 1, 41–66. Newey, Whitney K., and Kenneth D. West, 1987, A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica 55, 703–708. Schwert, G. William, 1990, Indexes of u.s. stock prices from 1802 to 1987, Journal of Business 63, 399–426. van Binsbergen, Jules H., and Ralph Koijen, 2010, Predictive regressions: A presentvalue approach, Journal of Finance 65, 1439–1471. Wright, Stephen, 2004, Measures of stock market value and returns for the US nonfinancial corporate sector, 1900-2002, Review of Income and Wealth 50, 561–584.
32
Notes 1
A noncomprehensive list of studies that use this measure of dividends includes Lettau and Ludvigson (2005), Cochrane (2008), and Lettau and Van Nieuwerburgh (2008).
33
Table IA.I Dividend Variance Ratios In the main article, the dividend variance ratio test demonstrates that dividend volatility drops significantly with horizon in the data. This table reports the variance ratios of three alternative definitions T of dividends of publicly traded firms. σD,i are the per-year standard deviation of the growth rate of variable dividends, for dividend volatility definition i = 1, 2. VR is the standard variance ratio using the formula in Lo and MacKinlay (1988). The data for dividend definitions 1 and 2 are annual from 1873 to 2010, and the data for dividend definition 3 are annual from 1927 to 2007.
1
T σD,1 T σD,2 VR
T σD,1 T σD,2
VR
T σD,1 T σD,2
VR
2
Horizon (years) 4 6 8
Dividend definition 1: 15.67 15.12 12.94 11.16 15.95 15.31 12.15 10.47 1.00 0.93 0.70 0.52 Dividend definition 13.29 13.89 12.34 10.53 13.41 13.80 11.82 10.17 1.00 1.09 0.88 0.63
Diff 1 − 10
Diff 1 − 20
Market-invested dividends 9.39 8.16 7.87 7.15 7.51 9.01 7.66 7.24 6.91 8.28 0.37 0.29 0.28 0.28 −
8.52 9.04 −
10
15
20
2: Cash-invested dividends 8.98 7.62 7.62 6.69 8.71 7.22 7.17 6.8 0.48 0.36 0.38 0.32
5.68 6.20 −
6.60 6.62 −
Dividend definition 3: With equity repurchases 13.65 14.15 13.64 10.65 7.61 7.27 6.87 5.41 6.39 13.45 14.26 14.04 10.60 7.25 7.11 7.16 5.20 6.35 1.00 1.09 1.04 0.55 0.34 0.32 0.32 0.21 −
8.24 8.25 −
34
Table IA.II Dividend Autocorrelation / Predictability This table presents predictability regressions of real dividend growth on lagged values of dividend growth P5 (∆dt = a + i=1 bi ∆dt−i ). The data for dividend definitions 1 and 2 are annual from 1930 to 2010, and the data for dividend definitions 3 and 4 are annual from 1873 to 2008.
a
Slope [t]
b1 b2 b3 b4 b5 R2 Flow of Funds: Public and Private Firms
Dividend 1: Cash dividends 7.74 −0.43 −0.36 −0.33 −0.16 −0.08 2.84 −2.22 −2.26 −2.82 −0.99 −0.53
13.27
Dividend 2: Cash dividends + equity repurchases Slope 6.58 −0.09 −0.24 −0.15 −0.11 −0.05 0.05 [t] 2.01 −0.79 −2.28 −1.16 −0.74 −0.54 CRSP+Schwert: Public Firms
Slope [t]
Dividend 3: Market-invested dividends 5.99 −0.27 −0.26 −0.14 −0.18 −0.25 2.78 −2.54 −2.75 −1.12 −1.75 −1.79
8.63
Slope [t]
3.69 2.31
Dividend 4: Cash-invested dividends 0.06 −0.30 −0.06 −0.17 −0.07 0.64 −2.11 −0.40 −0.75 −0.59
5.16
35
Table IA.III Leverage and Time-Varying Risk Premia This table reports results from the following long-horizon predictability regression: ΣTh=1 yt+h = a + bLevt + εit , where yt = rst − rf t , rst is the log aggregate stock market return, rf t is the log risk-free rate, and Levt is the current value of the log aggregate leverage ratio. T is the forecast horizon in years. The table reports the OLS estimate of the slope coefficient associated with Levt , Slope, the Newey-West (1987) corrected t-statistic, [t], and the adjusted R2 . The sample is annual from 1930 to 2010.
Regressor Levt
Slope [t] R2
Forecast horizon (years) 1 2 4 6 8 10 15 20 0.20 0.38 0.54 0.61 0.79 1.03 1.39 1.64 2.67 2.59 2.27 2.18 2.50 2.76 2.73 2.44 4.62 10.72 16.05 15.71 19.50 24.73 26.69 29.34
36
ρ =0.999 σ ρσ=0.995
0.2
Fractional Error
Fractional Error
0.25
0.15 0.1 0.05 0 −0.015 −0.01 −0.005
0
0.25
ρσ=0.999
0.2
ρσ=0.995
0.15 0.1 0.05 0
0.005 0.01
x
0
0.005
σ
0.01
t
Figure IA.1. Approximation errors. The figures report the fractional error between two estimates for the stock price for the cases ρσ = 0.995 and ρσ = 0.999: one estimate directly estimates the stock price Vd,t via equation (IA.25) and the other estimates the stock price via the sum of dividend strips P T Vd,t via equation (IA.53). The first figure sets σt = σ and then reports the fractional error as a function of x. The second figure sets xt = 0 and then reports the fractional error as a function of σt .
37
Dividend Strip Excess Returns 14
Baseline Calibration High Volatility n
35 30
Excess Return: r
Standard Deviation: σd,n
Dividend Strip Volatility
25 20 15 10
12 10 8 6 4 2
5 0
Baseline Calibration High Volatility
50
100
150
200
250
Horizon (months)
0
50
100
150
200
250
Horizon (months)
Figure IA.2. Impact of economic state. The left panel shows the term structure of dividend volatilities, and the right panel shows the term structure of excess returns on dividend strips, in the baseline calibration of the modified BY framework evaluated at the long-run mean of volatility, and in the baseline calibration with the current volatility evaluated at one standard deviation above its mean.
38
Leverage Ratio=Debt/(Debt+Equity) Bondholder Payout/Stockolder Payout
4
Standardized Value
3 2 1 0 −1 −2 −3 1930
1940
1950
1960
1970
1980
1990
2000
2010
Figure IA.3. Leverage ratio and bondholders’ and stockholders’ payouts. This figure provides a time-series plot of the aggregate leverage ratio and the ratio of bondholders’ payout to stockholders’ payout. Both variables are standardized (mean zero and unit standard deviation). The sample is annual from 1930 to 2010.
39
Aggregate Leverage Ratio (in %)
Gross Leverage Ratio Net Leverage Ratio
80
60
40
20
0 1930
1940
1950
1960
1970
1980
1990
2000
2010
Figure IA.4. Gross versus net leverage ratio. This figure provides a time-series plot of the aggregate gross and net leverage (exclude short-term assets) ratio. The sample is annual from 1930 to 2010.
40
