Bart F.C. Claassen August 2018
Abstract
I construct a framework that attempts to unify and rationalize historical patterns in return predictability in multiple asset classes. I consider an endowment economy in which a risk averse representative agent consumes N endogenous cash flows. The cash flow processes each resemble an asset class. The model features a covariance structure that varies endogenously due to market clearing effects. If there is a cash flow shock to one asset, investors want to rebalance. However, investors cannot all rebalance in the same direction, so prices and expected returns must adjust to ensure that markets clear. Consequently, if such adjustment happens slowly, expected returns respond slowly to cash flow news. I furthermore assess the model’s ability to rationalize historical patterns in return predictability by simulating the model and confronting the simulated data with historical U.S. data. The model features a limited number of parameters and yet is able to match many dynamic properties of financial markets. However, the model fails to match historical patterns in return predictability. I do nevertheless think that market clearing effects cannot be ignored; I argue that the model’s simplifying assumptions should be questioned instead. Therefore, I provide suggestions that could vindicate the model from its implied counterfactual patterns in return predictability.
JEL: G11, G12, G17
I. Introduction
Return predictability prevails in various asset classes. For instance, dividend-price ratios forecast stock returns, rent-house price ratios forecast housing returns, and bond yields forecast bonds returns. Research on return predictability has a long history, yet each study has focused on a single asset class. However, there are business cycle components and strong common elements in these relations between price ratios and expected returns. That is, for various risky asset classes expected returns are jointly high and prices are jointly low in bad economic times, when consumption, output, and investment are low – and vice versa. This notion suggests that return predictability between asset classes exhibits a common structure. Yet, even though there is a long history of research on return predictability, return predictability has only been studied within asset classes, one at a time. As a starter, we can pose the question how to best express the links between return predictability. Which factors that characterize return predictability should be included and excluded? That is, how can we shape a multivariate factor structure of expected returns?
Towards a factor structure of expected returns across asset classes, we need a framework to rationalize simultaneous return predictability in various asset classes. Within asset classes, these forecasts seem to carry similar information; for various risky asset classes expected returns are jointly high and prices are jointly low in recessions. Additionally, we can see in Figure 1 that portfolio weights slowly evolve but vary a lot over time, which suggests that investors are timing the market. Figure 1 depicts the relative composition of asset holdings in corporate equities, real estate, corporate bonds, and treasury securities in the U.S. from 1960 to 2017.
0.0 20.0 40.0 60.0 80.0 100.0 1960 1968 1976 1984 1992 2000 2008 2016
Corporate equities Real estate Corporate bonds Treasury securities
Figure 1. Aggregate composition of U.S. asset holdings 1960 – 2017. Holdings in percentages.
The portfolio weights are based upon the total values of outstanding assets. These values are the result from own computations on various Flow of Funds accounts from the Federal Reserve Economic database (FRED) within each asset class. The computed total values of total outstanding assets are converted to percentages, which can be interpreted as portfolio weights of a representative investor. A list of all the used accounts and an explanation of the computations can be found in Section V.
The asset compositions are expressed in percentages, such that they can be interpreted as portfolio weights of a representative investor. Henceforth, I will refer to these holdings as the “representative portfolio”. The portfolio weights appear to (slowly) mean revert around certain levels but they are not very stable. For instance, there are occasionally large shifts from bonds and treasury securities to stocks, as in the late 90’s, or from stocks and housing to treasuries, as in early 2008. Consequently, we cannot simply run multivariate regressions with price ratios to understand multivariate return predictability. Put differently, multivariate forecastability and consequent pricing implications can be quite different from those implied by univariate forecastability.
observations, a risk averse representative investor has a desire to collect cash flows from each asset; the representative investor does not want to put all eggs in one basket. I obtain closed form solutions for the return processes, the covariance structure of the returns, and the portfolio dynamics.
Even though the model has a closed form solution, the model is sufficiently flexible and rich to empirically validate it. To ascertain to what extent we should take the model’s implications w.r.t. return predictability seriously, I determine to what extent the model can match historical patterns in return predictability. I consider four broad asset classes: stocks, real estate, corporate bonds, and treasury securities, where the latter represent a risk-free asset. The model is calibrated on U.S. data from 1947 to 2017, by means of matching the first and second moments of the returns and yields of these asset classes, consumption growth, as well as on the correlation structure of the returns. I simulate the model and, subsequently, determine to what extent the simulated data match the patterns of return predictability in the historical data.
Imperfectly inelastic asset supply and market clearing conditions give rise to pricing dynamics that could have imperative implications for return predictability. Due to the market-clearing effects, the covariance structure of the returns varies for all asset classes, even if cash flow news pertains to other asset classes. If the value of some assets fall due to negative news, their expected returns have to rise and the expected returns on other assets have to fall to induce investors to hold all assets. The notion that market clearing has severe asset pricing implications is not new but, to the best of my knowledge, the model I propose is the first one that is sufficiently rich and flexible to confront it empirically and to confront it with multiple asset classes simultaneously. The model matches key moments of financial variables to a large extent but the simulated data exhibits counterfactual patterns in return predictability. I do think, nevertheless, that the model’s implications cannot be ignored since market clearing effects will be present in any equilibrium model that features multiple cash flows. Instead, I explain that the model’s assumptions or the Lucas tree approach should be challenged to vindicate the model from its counterfactual predictions.
Section II provides a review of the literature. The model is derived and explained in section III. The construction of the dataset that is used for the empirical validation is described in section V and the subsequent empirical validation is described and discussed in section VI. Section VII concludes.
II. Literature review
Attempts to predict asset returns go a long way back. Dow (1920) already tried to explain stock market returns with dividend ratios. The most seminal work on return predictability (in the stock market) is perhaps Campbell and Shiller (1988a), who find that dividend-price ratios predict equity returns well, particularly over long horizons. Since Campbell and Shiller (1988a), a plethora of research on additional predictors of stock market returns has emerged.1 Others have considered housing markets and found 1 e.g. see, Baker and Wurgler (2000), Bollerslev, Xu, and Zhou (2015), Boudoukh, Michaely, Richardson, and Roberts (2007),
that rent-house price ratios forecast housing returns well, again particularly over long horizons.2 Others
established that bond yields predict bond returns, not default (premia) or future yields.3
These articles, however, study asset markets in isolation; they do not consider asset class interlinkages and, thereby, might ignore lots of information. One might say that any investor who acknowledges insights from portfolio theory, knows (s)he is better off investing in multiple asset classes to reap diversification benefits. Furthermore, the time variation in expected returns has two important implications. On the one hand, time variation in expected returns implies that, at some point, investors find it optimal to substitute away from certain asset classes because they know that relatively high expected returns are offered in other asset classes. On the other hand, time-varying expected returns imply that investors will hedge state-variables (Cochrane, 2014). Moreover, Ilmanen, Maloney, and Ross (2014) find that the sensitivities of (risk-adjusted) risk premia to a variety of macroeconomic risks differ between asset classes.4 Thus, at
some point, investors will rebalance their portfolios to maintain their hedges.
By now it is well-established expected returns vary over the business cycle and that their predictability is intimately linked to economic conditions. For instance, Cochrane (2017) argues that most (consumption-based) asset pricing models capture the notion that expected returns increase during recessions and fall during booms, and Muir (2017) shows that risk premia are significantly and substantially higher during recessions than in more tranquil times. Furthermore, Cochrane (2011) finds that the Lettau and Ludvigson (2001) consumption-wealth ratio forecasts short term stock returns, but not long term stock returns. Additionally, the consumption-wealth ratio adds a lot of volatility to short term predicted returns, whereas the predicted volatility of long term returns by the dividend-price ratio is rather low. This notion suggests that there exists a term structure of expected returns in which more volatile variations in expected returns at short horizons can be contributed to business cycles (Cochrane, 2011). However, Golez and Koudijs (2017) find that return predictability with dividend yields is related to the business cycle as well: both the dividend-to-price ratio and subsequent returns tend to increase in recessions. Additionally, Bork and Møller (2016) and Gargano, Pettenuzzo, and Timmermann (2017) show that incorporating macro-factors into, respectively, bond price factor models and house price factor models significantly and substantially increases the accuracy of return predictions. Gargano, Pettenuzzo, and Timmermann furthermore find that the degree of predictability in bond returns rises during recessions. Thus, these links between macroeconomic indicators and expected returns prevail simultaneously in multiple asset classes.
These notions furthermore suggest that predictive variables capture, among others, business cycle conditions and market expectations. Investors require high expected risk premia to hold risky assets at the bottom of recessions. We see low prices for risky assets, which are accompanied by the higher returns that the market requires and expects. On the one hand, this notion implies that variables that forecast one return do forecast another. On the other hand, we see that macroeconomic state variables forecast returns at short horizons as well. How can we best express and interpret these inter-linkages between asset classes? That is, how can we shape a multivariate factor structure of time-varying expected returns?
2 e.g. see, Campbell, Davis, Gallin, and Martin (2009), Case and Shiller (1990), and Engsted and Pedersen (2015). 3 e.g. see, Fama (1986) and Goyal and Welch (2003)
A natural way to proceed towards the characterization of a multivariate factor structure is to augment the regressions with additional variables that reflect the investor’s information set. By now it is well established that dividend-price ratios forecast stock returns, rent-house price ratios forecast returns in the housing market, and bond yields forecast bond returns. One could run regressions of the returns with the predictors of the other assets (cf. Cochrane, 2011), but that is uninformative. First, given that there is a strong common factor between the predictors, we would have collinearity issues that would yield imprecise estimates of the regression coefficients. Hence, inference will be difficult and, perhaps, misleading. Second, what would it mean that one price ratio can forecast the return of another asset class? For instance, most research on return predictability starts with a decomposition of returns into a cash flow growth component (e.g. dividend growth), a bubble component, and the price component (e.g. dividend-price ratio and rent-house price ratio). The latter typically predicts returns, which implies that price variation does not come from changes in (expected) future cash flows or price bubbles. Thus, we see a univariate structure in which prices (scaled by dividends) forecast returns. But how should we interpret results of regressions in which the price ratios of other asset classes might or might not predict returns? Third, there is a plethora of additional factors that might forecast returns. Besides dividend-price ratios, we can also include other variables that reflect the investor’s information set. We cannot include all of them; which ones should we exclude, and why? The bottom line is that multivariate forecastability and consequent price implications can be quite different from univariate forecastability.
Towards a factor structure of expected returns, we need a suitable framework to understand multivariate return predictability. Specifically, we need a framework that features multiple sources of cash flows that can resemble various asset classes. There exists various work on asset pricing models that feature multiple sources of cash flows. These studies build upon the Lucas (1978) tree model by incorporating multiple trees, including Bick (1990), Cochrane, Longstaff, and Santa-Clara (2007), He and Leland (1993), Martin (2013), and Santos and Veronesi (2005). Cochrane et al. (2007) and He and Leland (1993) consider two trees, but we need a framework that features more than two cash flows. Moreover, the framework Cochrane et al. (2007) is too simplistic for empirical applications. Cochrane et al. (2007) make many simplifying assumptions that benefit their purely theoretical study of multiple cash flows. Martin (2013) and Santos and Veronesi (2005) also derived frameworks that feature N cash flows. However, the model of Martin (2013) is unfit for empirical applications and the model of Santos and Veronesi (2005) is only fit to study multiple stock returns and cross-sectional pricing implications. These models can, nevertheless, be used as a stepping stone towards a characterization of a factor structure of time-varying expected returns that spans various asset classes.
III. The model
with Bick (1990) and He and Leland (1993), nevertheless, I make assumptions on the investors’ preferences and the consumption growth process and subsequently derive said viable equilibrium N-dimensional asset price process.
As Cochrane et al. (2007) and Martin (2013), I consider a general equilibrium setting where market clearing conditions give rise to interesting asset pricing dynamics. The model I propose is, nonetheless, less restrictive than the one proposed by Cochrane et al. (2007) and Martin (2013), because their goal is purely theoretical inquiry into the market-clearing mechanism. For instance, one of the assets dominates the other in the long run because they assume a fixed supply of assets and they put constraints on the model’s parameters that render it unfit to confront it with the real data. Next to that, Cochrane et al. (2007) assume counterfactual i.i.d. dividend growth and log utility to obtain explicit solutions for the price levels and many processes. In my model, asset supply adjusts so that no asset will dominate in the long run. Additionally, dividend growth is endogenous and not i.i.d. and risk aversion can be scaled to match observed risk premia. By imposing little restrictions on parameters and processes, it is more difficult to obtain explicit solutions for various processes and prices. The upshot of this flexibility is that the model lends itself for empirical applications.
The model derivation consists of three steps. First, I characterize the dynamics of the asset price and return process. Second, I introduce investors’ preferences. Third, and last, the former two are combined to solve for the closed-form equilibrium price process. The details of the derivation are presented in Appendix A.
A. Asset price processes
Let Pi(t) denote the price of asset i at time t. There are N assets of which the prices are collected in
the N-vector P (t) that has typical element Pi(t), i ∈ {1, . . . , N }. Let ¯z(t) denote a K-dimensional Wiener
process, with independent elements ¯zj(t), j ∈ {1, . . . , K}, on the probability space (Ω,P,F ), K ≥ N. The
probability measure P comprises real probabilities, not risk-neutral probabilities. Let furthermore ∆ be a given N × K matrix of constants, which has rows δi: 1 × K and ∀i : ||δi||= 1. Then, ∆ characterizes
the correlated N-dimensional vector Wiener process z(t) (e.g. see, Øksendal, 2003, p. 269-270) . That is, z(t) := ∆¯z(t) : N × 1 and Ψ := ∆∆0
: N × N is the local (or instantaneous) correlation matrix of z(t). Additionally, note that zi(t) = δiz(t)¯ or, equivalently,
dzi(t) = δid ¯z(t), (1)
where d ¯z(t) := (d ¯z1(t), . . . , d ¯zN(t)) 0
. Since ||δi||= 1, zi(t) are also standard (i.e. unit variance) Wiener
processes, a property that will be exploited throughout the paper.
Let dR(t) denote the N-vector of instantaneous returns, with typical element dRi(t), and di(t) the
and coupon payments on bonds. In this section I will refer to them as dividend rates, nonetheless. Asset prices evolve according to the following N-dimensional stochastic differential equation (SDE):
dR(t) := dP1(t) P1(t) .. . dPN(t) PN(t) + d1(t) P1(t) .. . dN(t) PN(t) dt := µ1(W (t)) .. . µN(W (t)) dt + σ1(W (t)) 0 . . . 0 0 σ2(W (t)) . . . 0 .. . ... . .. ... 0 0 . . . σN(W (t)) δ11 . . . δ1N .. . . .. ... δN 1 . . . δN N d ¯z1(t) .. . d ¯zN(t) , (2)
where µi and σi are, respectively, the drift and diffusion terms of the return on asset i, and W (t) the
wealth of a representative investor. The vectors µ(.), σ (.) : R++→ RN++ are to be determined functions
of the state variable W (t). Let µ(W (t)) := (µ1(W (t)) , . . . , µN(W (t))) 0
, d ¯z(t) := (d ¯z1(t), . . . , d ¯zN(t)) 0
, and
Σ (W (t)) := diag {σ1(W (t)) , . . . , σN(W (t))}. Then, (2) can be written as
dR(t) = µ (W (t)) dt + Σ (W (t)) ∆d ¯z, (3)
and Ω(W (t))dt := Σ(W (t))Ψ Σ(W (t))dt is the local covariance matrix of the returns.
We can interpret the K-dimensional vector Wiener processes as a collection of K fundamental shocks that drive the assets’ returns. The matrix ∆, then, characterizes the relative exposure of each financial assets to the shock and, since zi(t)has unit variance, the local “volatility matrix” Σ governs the assets’
respective volatilities. Let us furthermore assume that Ψ is a positive definite matrix for the general case that K ≥ N . This assumption implies that none of the assets is redundant.
B. Consumers and preferences
An infinitely-lived representative agent has access only to information that is collected in the sigma-algebra Ft= σ {P (s) : 0 ≤ s ≤ t}. The representative agent maximizes
EFt "Z ∞ t e−ρ(s−t)u(c(s))ds # , u(c(t)) = c(t) 1−γ 1 − γ , (4) where EF
t [.] := E [.|Ft], ρ ∈ (0,1) denotes the rate of time preference, c(t) the consumption rate, and u(c(t)) is the felicity function with the parameter of relative risk aversion γ ∈ R+. The Ft-adapted N-vector a(t) := (a1(t), . . . , aN(t))0 describes the trading strategy, where ai(t)denotes the (dollar) amount invested in asset i at time t. Then, total wealth evolves according to
dW (t) =W (t)r(t) + a(t)0[µ(t) − ιr(t)] − c(t) dt + a(t)0Σ (W (t))dz(t), (5)
Next, I make the following assumption about consumption growth.
Assumption 1 (The aggregate consumption process). The aggregate consumption process is given by
dc(t) c(t) = µcdt + σ 0 cdz(t), (6) where σc := σc,1, . . . , σc,N 0
, and µc, σc,i are given constants, where µc > 0, σc,i ≥0, and ι0σc > 0. Log consumption growth has constant local variance σ0
cΨ σcdt.
Assumption 1 implies that log consumption growth is i.i.d. in the model, which in line with empirical evidence that log consumption growth is (very close to) i.i.d. (e.g. see, Santos and Veronesi, 2005). Innovations in the consumption rate depend on the fundamental shocks, as do the financial assets. Mechanically, consumption growth is proportional to the shocks to the return process since both depend on the correlated Wiener processes. That is to say, if the economy receives a shock, the financial assets will change in value and the investor will – next to possibly adjusting the portfolio weightsa/W – alter the
consumption pattern.
Next, with the specification for consumption growth and the utility function in hand, we can deduce the short rate process. The representative agent discounts with the stochastic discount factor Λ(t) := e−ρt
u0(c).
By Itô’s Lemma we have
dΛ(t) Λ(t) = − ρ + γµc− 1 2γ (γ + 1) σ 0 cΨ σc dt − γσc0dz. (7)
The short-rate satisfies (Cochrane, 2009, p. 29)
r(t)dt = − Et " dΛ(t) Λ(t) # , (8) so that we know r = ρ + γµc−1 2γ (γ + 1) σ 2 c, (9) where σ2 c := σ 0
cΨ σc. The short-rate, thus, is constant.5
Let us define the (consumer’s) value function:
V (W (t); t) = Et "Z ∞ t e−ρ(s−t)u (c(s)) ds # . (11)
5 For any utility function and consumption process, the SDF satisfies Λ = e−ρt
u0(c(t))and so dΛ(t) Λ(t) = −ρdt + c(t)u00(c(t)) u0(c(t)) dc(t) c(t) + 1 2 c(t)2u000(c(t)) u0(c(t)) dc(t) c(t) !2 . (10)
From here onwards, I will drop both the arguments of functions and the timing if no confusion can arise. The value function implies the following Hamilton-Jacobi-Bellman (HJB) equation:
0 = max {c,a} u(c) − ρV + Vt+ VW[W r + a 0 (µ − ιr) − c] +1 2VW Wa 0 Ωa, (12)
where, for any variables x and y, Vx:=∂V/∂xand Vxy:=∂2V/∂x∂y. The first order conditions are: c : c = V −1 γ W , (13a) a : a = − VW VW WΩ −1 (µ − ιr) . (13b)
C. A viable equilibrium price process
In equilibrium prices adjust such that the representative agent consumes the dividends:
c(t) = N
X
i=1
Di(t), (14)
where Di(t) := ni(t)di(t) is the total dividend of asset i and ni denotes the asset supply of asset i. Then,
by Itô’s Lemma we know that total dividends evolve according to the SDE
dDi Di = µD,i c Di dt + σc,i c Di δid ¯z, (15)
where the coefficients µD,i are given constants and satisfy µc=
PN
i=1µD,i. Thus, the growth in the assets’
cash flows are functions of the inverted consumption shares. The total value of an asset, Si(t), satisfies
Si(t) = ni(t)Pi(t). (16)
Asset supply evolves according to
dni(t) = gi(.)ni(t)dt, (17)
where gi(.) : R → R is a yet to be determined function for each asset i. Then, by applying Itô’s Lemma we
Additionally, the portfolio is self-financing iff dW (t) = N X i=1 dSi(t), (19)
for all t ≥ s (Duffie, 2010). Equation (19) pins down the growth rates of the asset supply gi(.). To determine gi(.), the equilibrium return process dR(t) has to be solved for first.
When the market clears, ∀i : ai(s) ≡ Si(s)and both the Value function and the volatility matrix Σ
depend only on the state variable wealth, W (s), and the model’s parameters, ∀s ≥ t.6 The solution to the HJB equation (12) is: V (W (t)) = k 1 − γW (t) 1−γ, k =" ρ γ+ γ − 1 γ r + 1 2 (µ − ιr)0Ω−1(µ − ιr) γ !#−γ . (20)
In equilibrium, we have that the scalar (µ − ιr)0
Ω−1(µ − ιr)is constant and
σc= (Σ Ψ )
−1(µ − ιr)
γ . (21)
The drift of excess returns (µ − ιr) and the covariance matrix Ω can be time varying, nevertheless. In fact, we will see later that they both are time varying indeed.
This solution has several implications. Firstly, it follows from the solution to the HJB and (13a) that the representative agent consumes a fixed fraction of wealth:
c(t) = θW (t), (22) where θ :=" ρ γ + γ − 1 γ r + 1 2 (µ − ιr)0Ω−1(µ − ιr) γ !# . (23)
Note that θ is a constant, so that wealth and consumption correlate perfectly. By applying Itô’s Lemma to (22), we know that wealth evolves according to
dW = µcW dt + σ 0
cW ∆d ¯z. (24)
Secondly, note that the solution is non-degenerate iff 0 < c(s) < W (s), ∀s ≥ t, which imposes the following restriction on the parameters:
0 < θ < 1. (25)
6 If the Value function or Σ would depend on t, the dynamic programming problem would be degenerate since we have an
Third, note that the equilibrium condition (14) can be written as c W = N X i=1 Di W = N X i=1 Si W Di Si = N X i=1 αi di Pi , (26)
and since c = θW , we have
θ = N X i=1 αi di Pi . (27)
Thus, θ equals the weighted average of the yieldsdi/Pi.
Fourth, and last, from the first order condition (13b) and the solution to the HJB, we know that the portfolio weights satisfy
α := a
W = Ω
−1(µ − ιr)
γ . (28)
Subsequently, we know from (21) that
σc= Σ a
W. (29)
Since the volatility matrix Σ(W (t)) is diagonal, (29) implies that Σ(W (t)) has elements
σi= σc,i W
ai, i ∈ {1, . . . , N }, (30)
on the diagonal.
The above results pin down the return process.
Proposition 1 (The equilibrium asset price process). In equilibrium, the asset price processes are given by
dR(t) = ιrdt + γI (σc) {I (α)} −1
Ψ σcdt + I (σc) {I (α)} −1
∆d ¯z, (31)
where I(x) := diag(x) for any vector x. Equivalently, each excess return, dRi−rdt, is given by: dRi(t) − rdt = γσc,iψiσc W (t) ni(t)Pi(t) dt + σc,iδi W (t) ni(t)Pi(t) d ¯z, (32)
where ψidenotes row i of Ψ . The local covariance matrix of the returns, Ωdt, has elements ωij= ωji= ρijσc,iσc,j W (t)2 ai(t)aj(t) dt = ρijσc,iσc,j αi(t)αj(t) dt, i, j ∈ {1, . . . , N }. (33)
represents risk premia that satisfy the classical asset pricing result that securities that covary more with consumption growth must deliver higher risk premia. Second, there is a market clearing effect. If one asset rises in value, investors want to substitute away from the others. For instance, if the dividend rate increases, bonds and real estate become relatively less attractive. However, unless supply, ni(t), adjusts
perfectly, not everyone can substitute away from bonds and real estate; the average investor should hold the market portfolio. Therefore, when the value of one asset increases, the expected returns of the other assets have to increase to induce investors to hold them. Equivalently, prices of other assets fall. The implications of this result will be further explained when the model is fully characterized.
Next, consider the asset supply dynamics. For reasons that will become clear below, I assume the following about the growth rates gi(.)of the asset supplies.
Assumption 2 (Growth rates of the asset supplies). The growth rates gi(t) of the asset supplies ni(t) are given by gi(t) = µc−σ2 c −γσc,iψiσc W (t) Si(t) + ψiσcW (t) Si(t) +di(t) Pi(t) −r + κ α¯i αi(t) −1 ! , ∀i ∈ {1, . . . , N }, (34)
where the specifications of the parameters ¯αi and κ will be explained below.
These specifications of gi(t) imply dynamics for the total asset values, S, that satisfy the self-financing
condition (19). Furthermore, asset supply is endogenous and stochastic, but adapted to Ft.
Proposition 1, assumption 2, and an application of Itô’s Lemma to the portfolio weightsSi/W give rise
to the following result.
Proposition 2 (Portfolio dynamics). The portfolio weights αi(t)evolve according to
dαi= κ ( ¯αi−αi) dt + σc,iδi−αiσ 0
c∆ d ¯z, (35)
where ¯αi is the target portfolio weight of asset i and κ ∈ R+the speed of reversion to this target. Thus, ¯αi are restricted to PN
i=1α¯i= 1, where ∀i : ¯αi∈(0, 1).
Since the portfolio weights have to sum to one, the self-financing condition requires that the changes in the portfolio weights sum to zero in all periods. Note that at the start of the planning period
N X i=1 dαi= κ N X i=1 ¯ αi− N X i=1 αi dt + N X i=1 σc,iδi− σ 0 c∆ N X i=1 αi d ¯z = 0, (36) since PN
i=1α¯i=PNi=1αi= 1and PNi=1σc,iδi= σc0∆. Since (36) holds at the start of the planning period,
PN
i=1αi= 1 and PNi=1dαi= 0hold for each subsequent period as well. These portfolio dynamics satisfy
portfolio weights do not become negative. The assumed growth rate processes of the asset supplies, gi(t),
gives rise to the mean reverting portfolio weights; asset supply adjusts such that the representative agent attains the target portfolio weights in the long run.
This specification of the portfolio dynamics is rather flexible. Santos and Veronesi (2005) propose a different set of assumptions to obtain tractable formulas for prices and returns. They start with defining a vector SDE for the dynamics of the consumption shares,Di/c, because that allows them to derive an explicit
solution for the price level and because the model is calibrated on, amongst other, the consumption shares. However, to ensure that the portfolio weights sum to one and no single asset will dominate in the long run, they have to impose severe constraints on both the correlation structure of the returns as well as on
N × N parameters that pin down the drift-terms of the vector SDE that characterizes the dynamics of the
The model’s main equations are exhibited in Table 1 below.
Table 1. The model’s main equations
Representative agent Wealth dW (t) = µcW (t)dt + σ 0 cW (t)∆d ¯z(t) (T.1) Consumption c(t) = θW (t) (T.2) dc(t) c(t) = µcdt + σ 0 c∆d ¯z(t) (T.3) Portfolio weights αi(t) =W (t)ai(t) = ni(t)Pi(t) W (t) (T.4) dαi(t) = κ ( ¯αi−αi(t)) dt + σc,iδi−αi(t)σ 0 c∆ d ¯z(t) (T.5) Market
Returns on risky assets dRi(t) = rdt + γσc,iψiσcniW (t)(t)Pi(t)dt + σc,iδiniW (t)(t)Pi(t)d ¯z(t) (T.6)
Short rate r = ρ + γµc−12γ(γ + 1)σc2 (T.7)
Total dividends dDi(t) = µD,ic(t)dt + σc,ic(t)δid ¯z(t) (T.8)
Dividends di(t) =Dnii(t)(t) (T.9)
Total asset value Si(t) = ni(t)Pi(t) (T.10)
dSi(t) Si(t) = dRi(t) − di(t) Pi(t)dt + gi(t)dt (T.11) Asset supply dni(t) = gi(t)ni(t)dt (T.12) gi(t) = µc−r − σc2+dPii(t)(t)− σc,i(γ−1)ψiσc αi(t) + κ α¯ i αi(t) −1 (T.13) There are N assets and there are K shocks in ¯z, K ≥ N. The correlation matrix Ψ = ∆∆0
: N × N, where Ψ has elements ρij and rows ψi = (ρi1, . . . , ρiN) and ∆ : N × K has elements δij and rows
δi= (δi1, . . . , δiN); the volatility matrix Σ(W (t)) = diag
σc,1W (t)S1(t), . . . , σc,NSW (t)N(t)
, and; the covariance matrix Ω(W (t)) has elements Ωij= ρijσc,iσc,j W (t)
2
Si(t)Sj(t), i,j = 1,...,N. The constant θ := ρ + (γ −
1)µc−12γσc2.
The model’s deep parameters are ρ,γ,{µD,i}, {σc,i}, { ¯αi}, κ, and {δij}. Note that {δij} defines Ψ which
has elements ρij. However, one can work the other way around by postulating {ρij} and, for instance,
deduce ∆ = {δij} by a Cholesky decomposition of Ψ , which requires that ∆ is N × N.7 Additionally, using
the equilibrium results for (µ − ιr) and Ω, I have simplified the expression for θ to
θ = ρ + (γ − 1) µc− 1 2γσ 2 c . (37) D. Intuition
We can deduce additional pricing dynamics at this point. When asset supply is somewhat inelastic, the aforementioned market clearing effects will play a role. Jensen’s inequality implies, however, that it is
difficult, and most probably impossible, to obtain an analytical expression for the unconditionally expected risk premia, since the risk premia dRi(t) − rdt are convex functions of the portfolio weights αi(t)on the
domain (0,1). This convexity implies that when the value of an asset, Si, declines (relative to total wealth),
its expected return increases more than it falls when the value of an asset increases. Furthermore, this effect is more pronounced for assets that have a relatively low value in the first place. Additionally, when the value of an asset falls, the excess returns volatility increases, and it increases more when an asset’s value falls than it decreases when an asset value increases. Thus, the financial assets exhibit a leverage effect: when prices fall and expected returns increase due to negative cash flow news, volatility increases substantially. Note furthermore that if one asset increases in value – for instance, due to a positive dividend shock – this increase lowers the share of the other assets, which increases their subsequent expected returns. However, as investors (slowly) rebalance their portfolios, returns will typically display temporary episodes of momentum as they slowly mean revert after a substantial cash flow shock. This is mechanically similar to the momentum effect in Cochrane et al. (2007). Furthermore, since the portfolio weights are mean reverting, Proposition 1 implies that returns are mean-reverting as well. The existence of return predictability suggests that there is mean-reversion in (stock) returns, even though the evidence for it is weak (e.g. see Cochrane (2009, p. 415-424) for an elaborate discussion on mean-reversion in light of return predictability). For one thing, mean-reversion is at best very slow, which suggests that κ should be very low.
The simple local covariance structure of the returns delivers interesting asset pricing results as well. First, even though the correlation structure is time-invariant, Ω will be time varying almost surely since
W (t)and, possibly, a(t) are time varying. Since Proposition 1 implies that both µ − ιr and Ω are time
varying, the equilibrium portfolio weights (28) imply that the representative investor is timing the market. That is to say, in times of high excess returns or low volatility, the representative investors allocates more wealth to otherwise more risky assets. As in Cochrane et al. (2007), we can see movements in an asset’s prices and volatilities even though cash flows news about that asset has been absent. Thus, there will be temporary spikes in the contemporaneous covariance structure, or “contagion”, on top of the common factors (governed by Ψ ) in the cash flow news. Even if correlation is absent, i.e. Ψ = I, cash flow news that pertains to one asset class, will induce price changes in all asset classes.
Let us now consider a case of market polarization. That is, let us imagine a situation in which some assets sharply decline in value relative to some safe asset. We can see from the specification of Ω that the return on the market portfolio has the same local variance as consumption growth; α0
Ωα = σc0Ψ σc.
When, for instance, stocks and houses fall more in value than bonds, say, the variance of stocks and houses spikes. Wealth evolves i.i.d. but its constituents do not. Furthermore, the variances are inverse functions of the portfolio weights and, thus, decreasing and convex functions of the portfolio weights. The effects of market polarization are, thus, stronger in bad economic times, when some assets fall more then others and momentum builds up. This mechanism reconciles the findings of Chordia and Shivakumar (2002), who show that profits to momentum strategies can be explained by a set of lagged macroeconomic variables that capture business cycle conditions. These links between macroeconomic indicators and expected returns, as documented by Chordia and Shivakumar (2002), thus, have to prevail simultaneously in multiple asset classes due to the effects of market clearing.
IV. Empirical validation
To assess the model’s ability to rationalize patterns in return predictability, I use the Campbell and Shiller (1988a) framework. First, I show how this framework can be applied to the stock market. Second, I argue that this framework can be applied to the real estate market as well. Third and last, I derive a similar framework for the corporate bonds market. Since the short-rate is time-invariant, the model’s implied yield curve is flat. Though counterfactual, this constant short-rate permits the explicit solution. Since U.S. government bonds are virtually risk free, the government bonds are assumed to earn the short-rate. Since the short-rate is fixed, we will not consider a decomposition returns on government bonds and the yield curve.
A. The stock market
Let Dt denote the dividend received by the investor at time t, Pt the stock price at time t. Then, the one
period returns can de defined as
Rst+1:=Pt+1+ Dt+1
Pt
, (38)
which is rewritten into
Rst+1= 1 Dt 1 Dt Pt+1+ Dt+1 Pt = P t+1 Dt+1+ 1 D t+1 Dt Pt Dt . (39)
Let lower-case letter denote natural logarithms of their uppercase counterparts, i.e. xt= ln (Xt), pdt:= pt−dt= ln (Pt/Dt), and ∆ : xt7→xt−xt−1for any variable xt, i.e. ∆ is the backward first difference operator.
Then, by taking logarithms (39) can be rewritten into,
rt+1s = ln1 + epdt+1+ ∆d
Next, (40) is linearized around the pointP/D= epd: rt+1s ≈ln1 + epd+ e
pd
1 + epd(pdt+1−pd) + ∆dt+1−pdt
= λs+ φspdt+1+ ∆dt+1−pdt, (41)
or, equivalently, we can write
pdt≈λs−rt+1s + ∆dt+1+ φspdt+1 (42) where φs:= epd 1 + epd = 1 D P + 1 and λs:= ln (1 +P/D) − φspd.
One can take, for instance, the time series average of pdt as the point of approximation pd. Forward
iteration of (42) gives dpt ≈ −λs k X j=1 φj−1s + k X j=1 φj−1s rt+js − k X j=1 φj−1s ∆dt+j+ φksdpt+k, (43) where dpt:= dt−pt= −pdt.
These equations deliver both an important message as well as an important question. Since dividend price ratios vary, dividend price ratios must forecast (long term) returns, (long term) dividend growth, or a rational “bubble” of increasing prices, φk
sdpt+k. The question, then is, how much variation in the
dividend price ratios can be explained by each of the three sources? To answer that question, we can consider the following VAR equation:
k X j=1 φj−1s rt+js = ar+ bs,kdpt+ εt+ks (44a) k X j=1 φj−1s ∆dt+j= ad+ bd,kdpt+ εt+kd (44b) dpt+k= adp+ bdp,kdpt+ εt+kdp. (44c)
One should, however, be careful with interpreting the results of the estimation of this VAR equation. Too understand this point, note that (43) holds ex-ante since it also holds ex-post. By taking conditional expectations on both sides of (43) we have:
where Et[.] := Et[.| It]and It denotes the investors’ information set. We attempt to understand prices,
the LHS variable, as a function of changes in (expected) returns, (expected) changes in dividends, or (expected) price bubbles. The VAR has scaled prices (scaled by dividends) on the RHS so that the forecast errors are orthogonal to the forecasting variable, dpt.
Equation (43) imposes some restrictions on the VAR equations. These restrictions hold because (43) is the result of manipulations of the definition of returns. First, note that equation (43) tells us that
var (dpt) ≈ cov dpt, k X j=1 φj−1s rt+j −cov dpt, k X j=1 φj−1s ∆dt+j + φkscov (dpt, dpt+k) ⇐⇒ 1 ≈ cov dpt,Pkj=1φ j−1 s rt+j var (dpt) − cov dpt,Pkj=1φ j−1 s ∆dt+j var (dpt) + φkscov (dpt, dpt+k) var (dpt) = bs,k−bd,k+ φksbdp,k, (46)
where bs,k, bd,k, and bdp,k are defined by (44a)-(44c). Since the coefficients sum to 1, they tell us how much
variation in prices can be attributed to each source. Second, we can subsequently note from (44a)-(44c) that
εst+1= −φsε dp
t+1+ εt+1d . (47)
This result can be used as an identification assumption for the VAR estimators. Third, forward iteration of the Vector Autoregression (VAR) equations (44a)-(44c) yield
lim k→∞ bs,k= bs,1 1 − φsbdp,1, k→∞lim bd,k= bd,1 1 − φsbd,1 and limk→∞ φ k sbdp,k→∞= 0, (48)
where the last equality follows from assuming that the dividend price ratio is AR(1) stationary; |bdp,1|< 1.
These equations can be used to derive long-term implications of return predictability.
B. The real estate market
We can easily apply the framework of the stock market to the real estate market: replace dividends by rents and stock prices by real estate prices, and take exactly the same steps for the derivation. Similar to an investor in the stock market collecting dividends, an investor in the real estate market collects rents. That is to say, we can ask if high rent-price ratios signal low returns on the real estate market, rising rents, or a rational “bubble” in the real estate market?
Let Ht denote the price of a housing portfolio at time t and Lt the rents that can be collected at time t. Then, we consider the housing returns
Rht+1:=Ht+1+ Lt+1
Ht
where it is assumed that the rents are reinvested. A decomposition of returns in the real estate market is obtained by substituting house prices for stock prices and rents for dividends into the framework of the previous subsection. Instead of dividend-price ratios, we can now consider rent-price ratios. This ratio can be decomposed as follows:
lht ≈ −λ h k X j=1 φj−1h + k X j=1 φj−1h rt+jh − k X j=1 φj−1h ∆lt+j+ φkhlht+k, (50)
where lht:= lt−ht, φh≈1/(1+elh)is a coefficient of approximation, and λh:= ln
1 + ehl−φhhl. As for stocks, the time-series average of lht could be taken as the point of approximation lh. The VAR equation
for the housing market is as follows:
k X j=1 φj−1h rt+jh = ah,k+ bh,klht+ εht+k (51a) k X j=1 φj−1h ∆lt+j= al,k+ bl,klht+ εt+kl (51b) lht+k= alh,k+ blh,klht+ εt+klh , (51c)
where ∆lt the log rent growth. Again, the coefficients sum to 1:
1 = bh,k−bl,k+ φkhblh,k. (52)
C. Corporate bonds
Since we consider broad asset classes, we consider one bond index rather than various bond returns that differ in their rating. The analysis, therefore, differs from Cochrane and Piazzesi (2005) and Fama (1986), who consider various bond returns with different ratings. They use default premia – proxied by differences in bond yields with different ratings – to decompose the relation between yields (the predictor), returns, and defaults. We will consider the default probability itself. Cochrane and Piazzesi (2005) show that bond returns can be very well predicted by bond yields of bonds with different ratings and maturities. That is to say, we can take one bond yield to predict returns of a bond index.
Specifically, we will consider an analogy of corporate bonds returns and yields w.r.t. equity returns and the dividend price ratio. The corporate bonds pay a coupon C and is traded at price Bt at time t.
When a firm defaults on its payments, the coupon is not paid and future coupons will not be paid as well, so that the bonds become worthless. The variable ˜Dt+1 denotes the fraction of the firms that pays, so the
so Eth ˜Dt+1
i
= (1 − Pt) under the assumption that defaults are independent. The return on the bond, then,
is defined as Rct+1:=(C + Bt+1) ˜Dt+1 Bt = 1 +Bt+1 C ˜Dt+1 Bt C = 1 +Y1 t+1 ˜Dt+1 1 Yt , (53)
where Yt:=C/Bt is the bond yield. By taking logarithms of both sides of (53) we obtain rt+1c = ln 1 + 1
Yt+1
!
+ ˜dt+1+ yt. (54)
Next, the return is approximated around the point Y :
rt+1c ≈ln 1 + 1 Y + 1 1 + Y " ln 1 Yt+1 ! −ln 1 Y # + ˜dt+1+ yt = λc−φcyt+1+ ˜dt+1+ yt, (55)
which is rewritten to:
yt≈ −λc+ rt+1c − ˜dt+1+ φcyt+1, (56)
where φc:=1+Y1 and λ := ln(1 + 1/Y ) + φcln (Y ) .Because (56) holds ex-post, it also holds ex-ante, so yt≈ −λc+ Etrt+1c − Eth ˜dt+1
i
+ φcEt[yt+1]
≈ −λc+ Etrc
t+1 + ln (1 − Pt) + φcEt[yt+1], (57)
where the second approximate equality holds if Pt and vart(Pt)are relatively small.
Equation (57) tells us that a higher yield implies higher returns, a higher probability of default, or increases in future yields. To assess the predictability of corporate bond returns, we consider the following VAR equation: k X j=1 φj−1c rt+jc = ac,k+ bc,kyt+ εct+k (58a) k X j=1 φj−1c pst+j= aps+ bps,kyt+ εpst+k (58b) yt+k = ay+ by,kyt+ ε y t+k, (58c)
where pst:= ln (1 − Pt), the log of the “survival probability”. As for housing and stocks, the coefficients
must add up to one:
V. Data
The model will be confronted with U.S. data from 1947 to 2017. All nominal series converted to real series with the GDP deflator Gross Domestic Product: Chain-type Price Index (GDPCTPI) obtained from Federal
Reserve Economic Data (FRED), the database maintained by the St. Louis Federal Reserve. A. Aggregate economy
Per capita consumption growth has been computed by collecting data from FRED. Consumption growth is calculated as the innovations in the sums of Personal Consumption Expenditures: Nondurable Goods (BEA:PCNDA) and Personal Consumption Expenditures: Services (BEA:PCESVA). I have adjusted for popula-tion growth with Populapopula-tion (BEA:B230RC0A052N).
The portfolio weights are compiled from the Flow of Fund accounts from FRED. Since the return series are from the U.S. I only consider the holdings by U.S. domestic investors. The value of the outstanding corporate equity are computed by taking Nonfinancial corporate business; corporate equities;
liability, Level (Z1.FL103164103.Q) net of Nonfinancial corporate business; mutual fund shares; asset, Level
(Z1.FL103064203.Q) and Rest of the world; U.S. corporate equities; asset, Level (ROWCESQ027S). The value of the real estate market is computed by taking the sum of Households and nonprofit organizations; real
estate at market value, Level (Z1.FL155035005.Q), Nonfinancial corporate business; real estate at market value, Level (Z1.FL105035005.Q), and Nonfinancial noncorporate business; real estate at market value, Level
(Z1.FL115035005.Q). Real estate holdings of the Rest of the World are not reported. The value of the corporate bonds market is approximated by Nonfinancial Corporate Business; Credit Market Instruments;
Liability and Financial Business; Credit Market Instruments; Liability, Level (Z1.FL264004005.Q) net of Rest of the World; Credit Market Instruments; Asset, Level (WCMITCMAHDNS). The value of the Treasury
securities equal Federal Government; Treasury Securities; Liability (Z1.FL313161105.Q) net of Rest of the world;
Treasury securities; asset, Level (Z1.ROWTSEQ027S). The sum of the market value of each asset class, then,
are taken as the total asset value which, in turn, is used to compute the shares. The Flow of Fund data consists of annual observations from 1947 to 1953 and of quarterly observations from 1953Q1 onwards. The quarterly data are annualized by taking the annual averages of the weights.
B. Stocks
The return on the S&P500 index are month-end values for the period 1947–2017 reported by The Center
for Research in Security Prices (CRSP). Dividends are the 12-month moving sums of dividends paid on the
S&P500 index. Dividend data from 1947 to 1987 are from Robert Shiller’s website and from 1988 onwards from the S&P500 corporation.
C. Real estate
The updated data on the house price index are from Robert Shiller’s website.8 I harmonize the dataset so
that the model is calibrated on time series that span the same time period. The total value of collected rents are only available from 1960 onwards. However, I do not want to drop the observations from the other time series. Therefore, I infer the collected rents from 1947 to 1960 by extrapolating them from the available time series. The rents have been extrapolated with Consumer Price Index for All Urban Consumers:
Rent of primary residence (CUUR0000SEHA) from FRED as the instrument.
The data on the real estate market is converted to annual data, since the time series of the real estate market consist quarterly data while the time series of the other asset classes consist of yearly data. As Cochrane (2011), I assume that prices equal the values in the first quarter and that the rents that determine the annual price are the rents of that quarter and the last three quarters of the previous year. I compute an unweighted average of those rents.
D. Corporate bonds
Corporate bond returns on long-term bonds from 1947 to 2017 are from Ibbotson’s Stocks, Bonds, Bills and
Inflation Yearbook. The corporate Bond Yields on AAA-rated bonds from 1919 to 2017 are from FRED.
The default rates are from Giesecke, Longstaff, Schaefer, and Strebulaev (2011).9 However, the default
rates are only available up to 2011. Therefore, I only use data from 1947 to 2011 for the corporate bond regressions.
E. Treasury bills
The yield on 3-month treasury bills serve as a proxy for the risk-free rate, i.e. the short rate. Treasury-bill rates from 1960 to 2017 are the 3- Month Treasury Bill: Secondary Market Rate (TB3MS) from FRED.
8 http://www.econ.yale.edu/~shiller/
VI. Results
The SDEs in Table 1 are approximated by stochastic difference equations. The difference equations and the simulation algorithm are described in Appendix B. The model is calibrated with the aim to match key unconditional first and second moments of the historical data. The key unconditional first and second moments of both the historical data and the simulated data are exhibited in Table 2 below.
Table 2. Unconditional moments: data v.s. simulation
Data: 1947 - 2017 Simulation
mean in % sd % Autocorr. mean in % sd % Autocorr.
∆ln(c) 1.8 1.6 0.21 1.7 6.0 -0.01 r 0.6 2.3 0.56 1.9 0.0 -rs 6.9 16.5 0.01 6.8 16.0 0.02 rh 6.6 3.9 0.61 5.9 8.7 -0.03 rc 2.5 9.4 0.08 2.2 10.4 0.07 D P 3.3 1.4 0.91 3.7 2.1 0.99 L H 5.1 0.6 0.96 4.7 0.7 0.99 Y 2.7 1.7 0.64 2.2 0.9 0.98 αs 18.6 5.1 0.92 19.5 4.6 0.87 αh 58.2 6.7 0.97 57.8 5.6 0.88 αc 13.7 5.1 0.98 13.2 2.5 0.84 αg 9.6 3.8 0.99 9.5 1.0 0.84
The parameters are calibrated as follows: ρ = 0.046, γ = 10, κ = 0.14, ρii = 1, ρ1j = 0, j = 2,3,4, ρ23 = −0.16, ρ24 = 0.12, ρ34 = 0.09, µD =
(0.0008, 0.0016, 0.0122, 0.0025)0, σc = (0, 0.012, 0.026, 0.025), ¯α1 = 0.0960, ¯α2 =
0.1360, ¯α3= 0.5810, ¯α4= 0.1820. These parameter values imply that θ = 0.0384,
r = 0.0194, µc= 0.0171, and σc= 0.0600. The degree of approximation, ∆t, equals
0.001.
I have appended the values of the calibrated parameters below Table 2. In the simulation, parameters indexed by i = 1 denote treasuries, i = 2 corporate bonds, i = 3 real estate, and i = 4 stocks; e.g. ¯α4is the
target portfolio weights of stocks. The target portfolio weights equal their historical averages from the data. Given these weights, the σc,i’s for stocks and corporate bonds are set such that the simulated volatilities
of stock returns and bonds returns match the empirical ones. The correlations between the risky assets more or less equal the correlation of the returns in the data, but have been slightly altered to match the empirical returns. Subsequently, σc,3 has been set such to match the returns and volatilities. For real
too high.10 Yet, I do not aim to resolve the equity premium puzzle, nor the interest-rate puzzle.11 The
constants in the drift terms of the total dividend processes, µD,i, are set such that the unconditional means
of the simulated yields match their empirical counterparts. Additionally, the growth rate of corporate bonds supply and the expected corporate bond yield have been calibrated such that the implied average coupon rate is constant. Because the expected corporate yield can only be approximated (see Appendix B), I have simulated only 600 years of data. The speed of reversion for the portfolio weights, κ, equals 0.14, as this value ensures that the portfolio weights remain positive.
The simulated consumption growth rate, 1.7%, closely matches the empirical growth rate, 1.8%. However, the volatility of simulated consumption growth is 4.4% too high. The simulated risk-free rate is 1.3% higher than the empirical real interest rate. Any lower value would, however, lower θ and, thus, the weighted average of the dividend yields; see (27). For the returns on stocks and bonds, both the means and volatilities are closely matched. Given that the observed volatility of housing returns is likely to be far too low, I consider the difference between the simulated volatility of real estate returns and the observed volatility to be impertinent. Its return is closely matched, nevertheless.
Next, let us consider the yields. The means of the yields are closely matched, but their volatilities are a little too low. Note that the simulated bond return and yield have to be about the same, as otherwise the bond-price would explode or implode. We can see that the simulated bond yield, 2.2%, is about equal to the return, 2.2%. More importantly, the simulated yields are rather persistent as we can derive from their high serial correlations. Note that the observed yields exhibit a high serial correlation as well. This persistence is important, because only persistent variables can generate long-term forecastability (Cochrane, 2009).
The means of the simulated portfolio weights are virtually equal to their simulated counterparts, but their volatilities are too low. Most likely, the speed of mean reversion, κ, is too high at 0.14, so that the weights do not divert far from their targets. Moreover, since κ also regulates the speed of mean-reversion in the asset returns, this value is perhaps too high given that mean reversion in stock returns is very slow.
10 Kroencke (2017) argues the empirical volatility of consumption growth is too low because available consumption data has
been filtered. He shows that the consumption volatility of unfiltered consumption volatility might be as high as 4.07%, which is still 2% lower than the simulated volatility.
Table 3 exhibits the correlations of the key financial variables.
Table 3. Correlation between key financial variables: data vs. simulation
Data: 1947 – 2017 ∆ln(c) r rs rh rc DP LH Y P S ∆ln(c) 1.00 r 0.44 1.00 rs 0.38 0.26 1.00 rh 0.40 0.38 0.12 1.00 rc 0.18 0.04 0.15 -0.19 1.00 DP 0.11 0.14 0.28 0.14 -0.04 1.00 LH 0.34 0.27 0.17 0.38 -0.14 0.55 1.00 Y 0.22 0.29 0.14 0.01 0.41 -0.03 -0.09 1.00 Simulation ∆ln(c) r rs rh rc DP LH Y P S ∆ln(c) 1.00 r 0.00 0.00 rs 0.53 0.00 1.00 rh 0.84 0.00 0.06 1.00 rc 0.19 0.00 0.16 -0.12 1.00 DP -0.04 0.00 -0.05 -0.02 0.01 1.00 LH -0.03 0.00 0.04 -0.07 0.04 -0.92 1.00 Y 0.07 0.00 0.04 0.06 0.05 0.01 -0.21 1.00
The correlations between consumption growth and the asset returns are adequate for stocks and bonds. Stocks exhibit a correlation of 0.38 with consumption growth in the historical data and 0.53 in the simulation and for bonds these are, respectively, 0.18 and 0.19. However, the correlation between simulated consumption growth and stock returns is 0.44 too high vis-a-vis the simulated data. The correlation between the yields and consumption growth is negligible in the simulations, but the empirical yields exhibit substantial positive correlation with consumption growth.
Table 4 exhibits the results of the Campbell-Shiller like regressions for the three asset classes under consideration. For each asset class I have put the results of the regressions with the historical data on the left and the results of simulated data on the right. As in Cochrane (2011), the standard errors have been omitted for the sake of clarity. Since the coefficients should be close to either zero or one and they can be interpreted as relative attributions of variation, contending that zeros should be ones and vice-versa is futile. Additionally, I have simulated 600 years of data, so that the standard errors will be very small for the simulated data.
Table 4. Return predictability: data vs. simulation
Stocks Data: 1947 – 2017 Simulation k bs,k bd,k φksbdp,k bs,k bd,k φksbdp,k 1 0.10 -0.01 0.89 0.00 -0.28 0.70 15 0.91 0.02 0.07 -0.02 -0.65 0.23 ∞ 0.88 -0.09 0.00 -0.01 -0.93 0.00 Real estate Data: 1947 – 2017 Simulation k bh,k bl,k φkhblh,k bh,k bl,k φkhblh,k 1 0.10 0.03 0.93 -0.01 -0.06 0.95 15 0.67 0.26 0.61 -0.07 -0.65 0.40 ∞ 1.38 0.39 0.00 -0.15 -1.15 0.00 Corporate bonds Data: 1947 – 2011 Simulation k bc,k bps,k φkcby,k bc,k bps,k φkcby,k 1 0.08 0.00 0.90 -0.02 -0.09 0.93 15 0.70 -0.04 0.11 -0.23 -0.90 0.34 ∞ 0.82 -0.02 0.00 -0.36 -1.34 0.00
Let us first consider the historical data. The yields are persistent, as they should be. We can see this persistence from “bubble coefficients” in the rightmost columns. For stocks, the bubble component equals 0.89, for real estate 0.93, and for corporate bonds 0.90 at the 1-year horizon. At the 15-year horizon, however, most variation in yields corresponds to changes in expected returns. Bubbles seem a little more persistent in the real estate market. In the long run (i.e. when k → ∞) all variation in prices can be contributed to changes in expected returns. If prices are high, relative to the cash flows, then either one of the following must hold (Cochrane, 2009, p. 360):
1. Investors expect returns to be low in the future. Future cash flows are, then, discounted at a lower rate, which results in higher prices.
3. Investors expect a forever increase in prices, which provides adequate returns even in the absence of cash flows.
If the dividend-price ratio is low, for instance, either prices must decline, dividends must grow, or the price-dividend ratio must grow explosively. The results suggest that for each asset class the yield shock is essentially a pure “discount-rate shock”: an increase in the yields correspond to rises in expected returns with little to no change in expected cash flows.
Now let us consider the model’s implied patterns for return predictability. In the model, the yields are persistent too. The model predicts, however, that investors expect that cash flows rise in the long-run. To understand that erratic patterns, let us consider Figure 2, which exhibits the first 100 years of the simulated time-series for the key financial variables of the risky assets. Note that the prices, exhibited in panel 2a, move almost one-for-one with cash flows, exhibited in panel 2b. Even though the yields, exhibited in panel 2c, are rather persistent, their movements are not accompanied by similar movements in the returns, exhibited in panel 2d. Additionally, the dividend-price ratio is rather stable. The dividend-price ratio is, thus, persistent but mainly because prices and dividends move very similar. Consequently, the model suggests that a change in the dividend price ratio corresponds to a “cash-flow shock”, which is incorporated rather quickly.
The model generates counterfactual patterns in return predictability, but this does not necessarily mean that market clearing effects cannot generate return predictability. On the one hand, recall that supply has to be somewhat inelastic to generate return predictability, since expected returns are inverse functions of the portfolio weights. Intuitively, after a negative cash flow shock, the value of the risky asset should decline. Investors will, moreover, want to move away from that risky asset. Because the average investor should hold the market portfolio, the value of the asset must decrease even further until the expected return has increased to the point where the average investor is willing to hold the risky asset again. When this adjustment happens slowly and, thus, low prices forecast long-term returns. The assets’ supplies are, apparently, too elastic; mechanically, gi(t)varies too much. On the other hand, the
model’s lack of business cycle patterns is a shortcoming as well. Recall that there is a strong business cycle component in stock returns, that most empirical evidence suggests that returns vary a lot because the ability/willingness to bear risk varies over the business cycle, which suggests that the market Sharpe ratio is countercyclical. The model does not feature these business cycle effects and the model’s market Sharpe ratio, p(µ − ιr)0
Ω−1(µ − ιr), is constant. The constant market Sharpe ratio implies that the investment
0 50 100 Years 0 200 400 600 P t Stocks 0 50 100 Years 50 100 150 200 H t Real estate 0 50 100 Years 6 8 10 12 14 16 B t Corporate bonds (a) Prices 0 50 100 Years 0 5 10 15 20 25 D t Stocks 0 50 100 Years 2 4 6 8 10 L t Real estate 0 50 100 Years 0 0.1 0.2 0.3 0.4 0.5 Corporate bonds (b) Cash flows 0 50 100 Years 0 0.01 0.02 0.03 0.04 0.05
Stocks: dividend-price ratio
0 50 100 Years 0 0.01 0.02 0.03 0.04 0.05 Rent-price ratio 0 50 100 Years 0 0.01 0.02 0.03 0.04 0.05 Y t
Corporate bond yield
(c) Yields 0 50 100 Years -0.2 0 0.2 0.4 rs t Stocks 0 50 100 Years -0.2 0 0.2 0.4 rh t Real estate 0 50 100 Years -0.2 0 0.2 0.4 rc t Corporate bonds (d) Returns
VII. Conclusion
The existing literature on the return predictability has a long history, yet each piece has focused on a single asset class. I attempt to provide a unifying framework for return predictability. I have proposed a relatively simple general equilibrium framework that illustrates the role of market clearing effects in return predictability. In contrast to existing work, the model can capture a rich (conditional) covariance structure of returns and puts less severe restrictions on asset supply dynamics. I have calibrated the model to four asset classes: stocks, real estate, corporate bonds, and treasury securities. Specifically, I have calibrated the model on the first and second moments of the returns and price ratios in these asset classes, as well as the first and second moments of consumption growth and the risk-free rate. The model can match many of these moments.
Whereas the model can match many intratemporal asset pricing dynamics, the model fails to match well-established return predictability dynamics. I do not conclude that we can ignore market-clearing dynamics, nonetheless. For instance, the postulated dividend growth, asset supply, and consumption growth dynamics yield counterfactual patterns in the return predictability. Although the assumption that consumption growth is i.i.d. is not at odds with the empirical evidence that consumption growth is (close to) i.i.d., the asset pricing implications of non i.i.d. consumption growth can be extensive. Market clearing dynamics will still be present in models that feature other cash flow and asset supply processes or non i.i.d. processes for consumption growth. Furthermore, the model features only one state variable and does not feature a production economy, so that the model is simple to implement. The model might, however, be too simple; return predictability might be more intimately related to frictions in the real or financial economy or to non-standard preferences of investors than the model suggests. For instance, the term structure is not flat and return predictability appears to be intimately linked to business cycle conditions. That is to say, I conjecture that the model’s inability to match historical patterns in return predictability should be attributed to inadequate assumptions or the endowment economy approach, not to the market clearing mechanism.
