Misspecified Recovery

• Author: JOSÉ A. SCHEINKMAN,LARS PETER HANSEN,JAROSLAV BOROVIČKA
• Published date: 01 December 2016
• Date: 01 December 2016
• DOI: http://doi.org/10.1111/jofi.12404

THE JOURNAL OF FINANCE •VOL. LXXI, NO. 6 •DECEMBER 2016

Misspecified Recovery

JAROSLAV BOROVI ˇ

CKA, LARS PETER HANSEN, and JOS ´

E A. SCHEINKMAN∗

ABSTRACT

Asset prices contain information about the probability distribution of future states

and the stochastic discounting of those states as used by investors. To better under-

stand the challenge in distinguishing investors’ beliefs from risk-adjusted discount-

ing, we use Perron–Frobenius Theory to isolate a positive martingale component

of the stochastic discount factor process. This component recovers a probability mea-

sure that absorbs long-term risk adjustments. When the martingale is not degenerate,

surmising that this recovered probability captures investors’ beliefs distorts inference

about risk-return tradeoffs. Stochastic discount factors in many structural models of

asset prices have empirically relevant martingale components.

ASSET PRICES ARE FORWARD LOOKING and encode information about investors’

beliefs. This leads researchers and policy makers to look at financial market

data to gauge the views of the private sector about the future of the macroecon-

omy. It has been known, at least since the path-breaking work of Arrow, that

asset prices reflect a combination of investors’ risk aversion and the probability

distributions used to assess risk. In dynamic models, investors’ risk aversion

is expressed by stochastic discount factors that include compensation for risk

exposures. In this paper, we ask what can be learned from Arrow prices about

investors’ beliefs. Data on asset prices alone are not sufficient to identify both

the stochastic discount factors and transition probabilities without imposing

additional restrictions. This additional information could be time-series evi-

dence on the evolution of the Markov state, or it could be information on the

market-determined stochastic discount factors.

In a Markovian environment, Perron–Frobenius Theory selects a single tran-

sition probability compatible with asset prices. This apparatus has been used in

previous research in at least two ways. First, Hansen and Scheinkman (2009)

∗Boroviˇ

cka is with New York University and NBER, Hansen is with the University of Chicago

and NBER, and Scheinkman is with Columbia University, Princeton University, and NBER. We

thank Fernando Alvarez, David Backus, Ravi Bansal, Anmol Bhandari, Peter Carr,Xiaohong Chen,

Ing-Haw Cheng, Mikhail Chernov, Kyle Jurado, Franc¸ois Le Grand, Stavros Panageas, Karthik

Sastry, Kenneth Singleton, Johan Walden, Wei Xiong, and the anonymous referees for useful

comments. Cynthia Mei Balloch provided excellent research assistance. Boroviˇ

cka acknowledges

financial support from New York University. Hansen’s research was supported in part by the

Becker–Friedman Institute and by the National Science Foundation, grant 0951576 to the DMUU:

Center for Robust Decision Making on Climate and Energy Policy. Scheinkman acknowledges

financial support from Columbia University and Princeton University. All authors declare no

conflict of interest related to this article.

DOI: 10.1111/jofi.12404

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2494 The Journal of Finance R

use Perron–Frobenius Theory to identify a probability measure that reflects

the long-term implications for risk pricing under rational expectations. We re-

fer to this probability as the long-term risk-neutral probability since use of this

measure renders the long-term risk-return tradeoffs degenerate. Hansen and

Scheinkman (2009) purposefully distinguish this constructed transition prob-

ability from the underlying time-series evolution. The ratio of the recovered to

the true probability measure manifests as a nontrivial martingale component

in the stochastic discount factor process. Second, Ross (2015) employs Perron–

Frobenius Theory to identify or “recover” investors’ beliefs. Interestingly, this

recovery does not impose rational expectations, and thus the resulting Markov

evolution could reflect investors’ subjective beliefs and not necessarily the ac-

tual time-series evolution.

In this paper we highlight the connection between these seemingly disparate

results. We make clear the special assumptions needed to guarantee that the

transition probabilities recovered using Perron–Frobenius Theory are equal to

the subjective transition probabilities of investors or to the actual probabili-

ties under rational expectations. We show that in some often-used economic

settings—with permanent shocks to the macroeconomic environment or in-

vestors endowed with recursive preferences—the recovered probabilities differ

from the subjective or actual transition probabilities, and we provide a cali-

brated workhorse asset pricing model that illustrates the magnitude of these

differences.

Section Iillustrates the challenge of identifying the correct probability mea-

sure from asset market data in a finite-state space environment. While the

finite-state Markov environment is too constraining for many applications, the

discussion in this section provides an overview of some of the main results in

this paper. In particular, we show that:

rthe Perron–Frobenius approach recovers a probability measure that ab-

sorbs long-term risk prices;

rthe density of the Perron–Frobenius probability relative to the physical

probability gives rise to a martingale component to the stochastic discount

factor process; and

runder rational expectations, the stochastic discount factor process used by

Ross (2015) implies that this martingale component is a constant.

Toplace these results in a substantive context, we provide prototypical exam-

ples of asset pricing models and show that a nontrivial martingale component

arises from (i) permanent shocks to the consumption process or (ii) continuation

value adjustments that appear when investors have recursive utilities.

In subsequent sections, we establish these insights in greater generality, a

generality rich enough to include many existing structural Markovian models

of asset pricing. The framework for this analysis, which allows for continuous

state spaces and a richer information structure, is introduced in Section II.

In Section III, we extend the Perron–Frobenius approach to this more general

setting. Provided we impose an additional ergodicity condition, this approach

Misspecified Recovery 2495

identifies a unique probability measure captured by a martingale component

to the stochastic discount factor process.

In Section IV, we discuss the consequences of using the probability mea-

sure recovered by the use of the Perron–Frobenius Theory when making infer-

ences on the risk-return tradeoff. The recovered probability measure absorbs

the martingale component of the original stochastic discount factor and thus

the recovered stochastic discount factor is trend stationary. Since the factors

determining long-term risk adjustments are now absorbed in the recovered

probability measure, assets are priced as if long-term risk prices were trivial.

This outcome is the reason we refer to the probability specification revealed by

the Perron–Frobenius approach as the long-term risk-neutral measure.

Section Villustrates the impact of a martingale component to the stochastic

discount factor in a workhorse asset pricing model that features long-run risk.

Starting in Section VI, we characterize the challenges in identifying subjec-

tive beliefs from asset prices. Initially we pose the fundamental identification

problem: data on asset prices can identify the stochastic discount factor only

up to an arbitrary strictly positive martingale, and thus the probability mea-

sure associated with a stochastic discount factor remains unidentified without

imposing additional restrictions or using additional data. We also extend the

analysis of Ross (2015) to this more general setting. By connecting to the results

in Section III, we demonstrate in Section VI that the martingale component to

the stochastic discount factor process must be identically equal to one for the

procedure in Ross (2015) to reveal the subjective beliefs of investors. Under

these beliefs, the long-term risk-return tradeoff is degenerate. One might won-

der whether the presence of a martingale component could be circumvented

in practice by approximating the martingale by a highly persistent station-

ary process. In Section VII, we show that, when we extend the state vector

to address this approximation issue, identification of beliefs becomes tenuous.

In Section VIII, we provide a unifying discussion of the empirical approaches

that quantify the impact of the martingale components to stochastic discount

factors when an econometrician does not use the full array of Arrow prices. We

also suggest other approaches that connect subjective beliefs to the actual time

series evolution of the Markov states. Section IX concludes.

I. Illustrating the Identification Challenge

There are various approaches for extracting probabilities from asset prices.

Risk-neutral probabilities (e.g., see Ross (1978) and Harrison and Kreps (1979))

and closely related forward measures, for instance, are frequently used in fi-

nancial engineering. More recently,Perron–Frobenius Theory has been applied

by Backus, Gregory, and Zin (1989), Hansen and Scheinkman (2009), and Ross

(2015) to study asset pricing, with the last two references featuring the con-

struction of an associated probability measure. Hansen and Scheinkman (2009)

and Ross (2015) have rather different interpretations of this measure, how-

ever. Ross (2015) identifies this measure with investors’ beliefs, while Hansen

and Scheinkman (2009) use it to characterize long-term contributions to risk