Higher Order Effects in Asset Pricing Models with Long‐Run Risks

• Author: OLE WILMS,WALTER POHL,KARL SCHMEDDERS
• Published date: 01 June 2018
• DOI: http://doi.org/10.1111/jofi.12615
• Date: 01 June 2018

THE JOURNAL OF FINANCE •VOL. LXXIII, NO. 3 •JUNE 2018

Higher Order Effects in Asset Pricing Models

with Long-Run Risks

WALTER POHL, KARL SCHMEDDERS, and OLE WILMS∗

ABSTRACT

This paper shows that the latest generation of asset pricing models with long-run risk

exhibit economically significant nonlinearities, and thus the ubiquitous Campbell-

Shiller log-linearization can generate large numerical errors. These errors translate

in turn to considerable errors in the model predictions, for example, for the mag-

nitude of the equity premium or return predictability. We demonstrate that these

nonlinearities arise from the presence of multiple highly persistent processes, which

cause the exogenous states to attain values far away from their long-run means with

nonnegligible probability. These extreme values have a significant impact on asset

price dynamics.

BANSAL AND YARON (2004)INTRODUCE TWO important innovations to the as-

set pricing literature. In particular, they put forward an economic mechanism

based on long-run risk to explain many empirical asset pricing facts, and they

demonstrate that the Campbell-Shiller (1988) log-linearization technique pro-

vides a simple method for analyzing such models. These two contributions

together have served as the foundation for a large literature on the ability of

long-run risk to solve empirical puzzles. By necessity,log-linearization neglects

higher order effects in asset pricing models. In this paper, we ask whether these

effects matter for the latest generation of long-run risk models. We show that

they do. More specifically, we demonstrate that, for many economically plau-

sible choices of parameters and exogenous processes, the errors introduced by

log-linearization can themselves be economically significant. In fact, for highly

persistent processes, as regularly used in the literature, the approximation

errors in moments can exceed 70%.

∗Walter Pohl is with Department of Finance, NHH Norwegian School of Economics. Karl

Schmedders is with Department of Business Administration, University of Zurich. Ole Wilms

is with Department of Finance, Tilburg University.This paper was previously circulated under the

title “Higher-Order Dynamics in Asset Pricing Models with Recursive Preferences.” We are grate-

ful to two anonymous referees, an anonymous Associate Editor, and the Editor, Stefan Nagel, for

thoughtful reviews of earlier versions. We are indebted to Nicole Branger,Lars Hansen, Ken Judd,

Martin Lettau, Malte Schumacher, Che-Lin Su, and particularly our discussant Alexis Akira Toda

for helpful discussions on the subject. We thank seminar audiences at the University of Zurich, the

Becker Friedman Institute, the University of M¨

unster, and various conferences for comments. We

are grateful to the copy editor, Brenda Priebe, for excellent editorial support. Ole Wilms gratefully

acknowledges financial support from the Z¨

urcher Universit¨

atsverein. The authors do not have any

conflicts of interest as per the Journal of Finance disclosure policy.

DOI: 10.1111/jofi.12615

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1062 The Journal of Finance R

Asset pricing models have become increasingly complex over the last three

decades. The first generation of such models, developed in the 1980s (Grossman

and Shiller (1981), Hansen and Singleton (1982), Mehra and Prescott (1985)),

proved inadequate in explaining key features of financial markets, such as

the high equity premium and the low risk-free rate. As the literature on asset

pricing has evolved and matured over time, researchers have added more com-

plex elements to their models such as incomplete markets (e.g., uninsurable

income risks), frictions (e.g., borrowing or collateral constraints), time-varying

risk aversion, and heterogeneous expectations. While these additional features

have had varying degrees of success, the new generation of long-run risk mod-

els (e.g., Bansal and Yaron (2004) or Hansen, Heaton, and Li (2008)), with

their interplay of persistent components in consumption growth and recursive

preferences, have had considerably more success in resolving long-standing

asset pricing puzzles. The key feature of these models is the combination of a

preference for the early resolution of risk together with highly persistent state

processes that potentially affect long-run model outcomes.

Complex models generally require numerical solution techniques. Bansal

and Yaron (2004) show that a simple linearized solution method based on the

Campbell-Shiller (1988) present-value relation works well for their original

model because the log price-dividend ratio in the model is approximately a lin-

ear function of the underlying shocks. Additionally,the linearization procedure

has the attractive property of lending itself to a simple analysis of the economic

impact of different shocks. This property is particularly appealing as it allows

the researcher to draw conclusions about parameter dependencies and eco-

nomic mechanisms in the model. Accordingly,a large group of researchers have

followed Bansal and Yaron (2004) and used the log-linearization technique to

solve asset pricing models with recursive preferences (e.g., Bollerslev,Tauchen,

and Zhou (2009), Bansal, Kiku, and Yaron(2010), Koijen et al. (2010), Constan-

tinides and Ghosh (2011), Drechsler and Yaron(2011), Bansal, Kiku, and Yaron

(2012), Beeler and Campbell (2012), Bansal and Shaliastovich (2013), Bansal

et al. (2014), and Segal, Shaliastovich, and Yaron (2015), among others). Ex-

amining this strand of literature, it is difficult to find studies that do not rely

on the Campbell-Shiller (1988) approach1—it has become the standard method

for solving asset pricing models with long-run risk.

To better match features of the data, the tendency in the long-run risk liter-

ature has been toward both higher persistence and greater model complexity.

This development suggests that it is time to take stock of whether the Campbell-

Shiller (1988) approximation is still appropriate. To address this question, we

first identify the factors that can make higher order dynamics matter for the

benchmark long-run risk model with stochastic volatility. We show that, for

1Prominent exceptions that do not rely on log-linearized solutions are the studies by Croce, Let-

tau, and Ludvigson (2015) and Collin-Dufresne, Johannes, and Lochstoer (2016). Kaltenbrunner

and Lochstoer (2010) use log-linearized solutions to infer qualitative model implications but use

value function iteration for their quantitative results. They approximate the value function with

Chebyshev polynomials.

Higher Order Effects in Asset Pricing Models with Long-Run Risks 1063

highly persistent processes, these effects do matter. Economically, when shocks

are highly persistent, agents can spend longer periods trapped in undesirable

parts of the state space, which can have a large impact on investor behavior.The

problem is particularly severe when both long-run risk and stochastic volatility

are persistent. In that case, the interaction between long-run risk and stochas-

tic volatility becomes especially significant when the long-run growth rate is

low but the volatility of the growth rate is high, which in turn creates signif-

icant deviations from linearity. As a side effect, this interaction leads to large

errors in predictability regressions using simulated data: the log-linearized so-

lutions overstate the predictability of returns and understate the predictability

of dividends by the log price-dividend ratio, relative to a higher precision so-

lution. Interestingly, the persistence is a problem only in the presence of a

strong preference for the early resolution of risk. However, without a prefer-

ence for the early resolution of risk, it is difficult to generate a high equity

premium.

To show that this nonlinearity matters for real models, and is not just an

intellectual exercise, we examine the consequences of ignoring nonlinear dy-

namics in the Bansal and Yaron (2004) model as well as five other recent stud-

ies, namely, the newly calibrated version of Bansal, Kiku, and Yaron (2012),

the extensive calibration study of Schorfheide, Song, and Yaron (2018), the

volatility-of-volatility model of Bollerslev, Xu, and Zhou (2015), and the studies

on real and nominal bonds of Koijen et al. (2010) and Bansal and Shalias-

tovich (2013). We show that the higher order dynamics not captured by the

Campbell-Shiller (1988) approximation can be large and economically signifi-

cant. For example, for the calibration of Bansal, Kiku, and Yaron (2012), which

is an improved calibration of the original Bansal and Yaron (2004) model (in

particular, with greater persistence in shocks to stochastic volatility), we find

that the log-linearization introduces economically significant errors in model

predictions. The equity premium, for instance, is overestimated by about 100

bps and the relative error in the volatility of the price-dividend ratio exceeds

20%. The true amount of return predictability is approximately half of what

log-linearization would suggest.

In the Bansal, Kiku, and Yaron (2012) model, most of the nonlinearities

are introduced by the pricing of equities, and using a linearized solution for

the stochastic discount factor does not lead to economically significant errors.

This approach does not generally work, however; for other model specifications

analyzed in this paper, we find significant errors in the stochastic discount

factor. Therefore, linearized solutions should not be used for any of the return

equations without carefully testing its accuracy for the specific asset pricing

model under consideration.

Schorfheide, Song, and Yaron (2018) perform a Bayesian estimation of the

model using the same approximation and find evidence of higher persistence

for long-run risk. In this case, we find approximation errors as large as 70% for

some key model moments for the 95% quantiles of the state persistence param-

eters (for the median estimates they are significantly lower). When persistence

is very high, log-linearization can actually invert the slope of the yield curve in