A Generalized Earnings‐Based Stock Valuation Model with Learning

Date01 May 2017
Published date01 May 2017
The Financial Review 52 (2017) 199–232
A Generalized Earnings-Based Stock
Valuation Model with Learning
Gady Jacoby
University of Manitoba
Alexander Paseka
University of Manitoba
Yan W an g
Brock University
We present a stock valuation model in an incomplete-information environment in which
the unobservable mean of earnings growth rate (MEGR) is learned and price is updated contin-
uously. We calibrate our model to a market portfolio to empirically evaluate its performance.
Of the 8.84% total risk premium we estimate, the earnings growth premium is 4.57%, the short-
rate risk contributes 3.38%, and the learning-induced risk premium on the unknown MEGR
is 0.89% (a nontrivial 10% of the total risk premium). This result highlights the significant
learning effect on valuation,implying an additional risk premium in an incomplete-information
Keywords: asset pricing, incomplete information, filtering, equity risk premium
JEL Classifications: G12, G14
Corresponding author: Goodman School of Business, Brock University, St. Catharines, ON, Canada,
L2S 3A1; Phone: (905) 688 5550, ext. 4827; Fax: (905) 378 5723; E-mail: ywang11@brocku.ca.
We gratefully acknowledge helpful comments from two anonymous referees and the Editor, Srinivasan
Krishnamurthy, as well as from Albert S. Kyle, Nancy Anderson, and conference participants at the
Financial Management Association 2008 and 2009 annual meetings. Jacoby thanks the Bryce Douglas
Professorship in Finance and the Social Sciences and Humanities Research Council of Canada for their
financial support. All errors are our own.
C2017 The Eastern Finance Association 199
200 G. Jacoby et al./The Financial Review 52 (2017) 199–232
1. Introduction
Most traditional stock valuation models implicitly assume that market informa-
tion is complete and that agents know exactly how information (such as dividends,
income, or earnings cash flow) maps into prices.1However, substantial evidence
shows that market information is incomplete in the sense that parameter uncertainty
is ubiquitous and that agents suffer from uncertainty about state variables character-
izing financial markets.2
We extend the stock valuation model of Bakshi and Chen (2005) and Bakshi
and Chen (2008) (hereafter BC2005 and BC2008, respectively) from a complete-
information case to an incomplete-information environment with a learning process.
In our model, rational agents learn about the unknown state variable(mean ofearnings
growth rate [MEGR]) and update their estimate of stock prices continuously. We
show that the uncertainty about the MEGR declines over time due to learning, and the
posterior uncertainty induces a risk premium on the MEGR, beyond what is accounted
for in the complete-information case. We parameterize the learning risk premium on
the MEGR and show that its magnitude is inverselyrelated to the correlation between
the unobservable MEGR and the observable earnings growth process.
We calibrate our model to a market portfolio (the Standard & Poor [S&P]’s
500-stock index) to quantitatively evaluate its performance. Model parameters are
estimated using the maximum likelihood method with an embedded Kalman filter.
Our results show that the equity risk premium is composed of three components. First,
the short interest rate risk premium is 3.38%, which is 38% of equity risk premium
(8.84%). Second, the risk premium associated with earnings growth is 4.57%, and it is
responsible for 52% of equity risk premium. Finally,the MEGR-learning-induced risk
premium is 0.894%, accounting for 10% of equity risk premium, which is nontrivial
and revealing about the learning effect on the equity risk premium.
To illustrate the learning effect on valuation, we compute price-earnings (P/E)
ratios estimated based on our model and P/E ratios based on the BC2008 model,
and obtain a dynamic process of P/E differences over a typical learning cycle. We
observe that the P/E difference declines as uncertainty about MEGR is reduced and
that the difference converges to a steady-state level. In extreme cases, with a perfect
(positive or negative) correlation between the unknown MEGR and the observable
earnings growth information, the P/E difference converges to zero eventually. In all
other cases (with imperfect correlation), the P/E difference never vanishes regardless
of its learning horizon and speed of learning. The converged nonzero level of P/E
1Examples of classical complete information models include: Merton (1971, 1973), Samuelson (1969),
Breeden (1979), and others.
2See Pastor and Veronesi (2009a) for a comprehensive review of the literature on learning in financial
markets. Theoretically, Bayesian learning has been considered as an avenueto model how investors learn
about state variableswith incomplete information: for example, Williams (1977), Detemple (1986), Dothan
and Feldman (1986), Gennotte (1986), Timmerman (1993, 1996), Brennan (1997), and Feldman (2007).
G. Jacoby et al./The Financial Review 52 (2017) 199–232 201
difference implies unavoidable price error caused by unreduced ambiguity in the state
variable MEGR in an incomplete-information environment.
Theoretically, our model is closely related to those of BC2005 and BC2008,
with the exception of learning that is not a feature in the latter models. BC2005 and
BC2008 study stock valuation under complete information about stochastic earnings,
the MEGR, and the short interest rates. Since the MEGR is implicitly assumed to
be observable, BC2008 use the annual change in analyst earning-per-share (EPS)
forecasts as input data for the MEGR. The estimation errors in analyst forecasts
potentially impair the performance of their model. Diether, Malloy and Scherbina
(2002) show that there is a negative relation between stock returns and the dispersion
of analyst earnings forecasts, which implies that forecast errors affect pricing.
To avoidthis problem, we derive asset prices as a joint exercise in valuation and
learning about the unobservable MEGR. Some studies (e.g., Timmermann, 1993;
Pastor and Veronesi, 2003) model unobserved variables as constants. In such cases,
the model can neither characterize extra volatility nor reflect extra risk premium
associated with learning. In the limit of the steady state, learning effects disappear,
and the model converges to its complete-information version. In contrast, we model
the unobservable variable, MEGR, as a stochastic process as in Brennan and Xia
(2001) and Veronesi(2000), and the extra volatility and risk premium due to learning
are permanent.
Different from our model setup, Brennan and Xia (2001) and Veronesi (2000)
assume a constant interest rate. This assumption precludes them from examining the
learning effect on the short-rate elasticity of price-dividend ratios. Since the short-rate
portion of the equity premium is absent in these models, their model is limited to
studying the joint impact of both learning and short-rate risk on the equity premium.
On the empirical side, our paper is comparable to BC2008 in which they calculate
the composition of the equity risk premium implied by their 2005 model and claim
that the contribution of MEGR risk to the total equity risk premium is trivial at one
basis point. Therefore, BC2008 conclude that most of the risk premium is due to the
cash flow risk, whereas the MEGR and the short-rate risks contribute only a small
portion. To reexamine this issue, we calibrate our model to S&P 500 index data and
compute the composition of the risk premium to that of BC2008. Based on the sample
longer than that used in BC2008, we show that although the contribution of the cash
flow risk to the risk premium increases in longer samples, it remains substantially
below the levels reported by BC2008.
We emphasize that the assumption of perfect correlation between the innova-
tions of earnings growth and those of its mean is not borne out in the data. We show
that the disconnect between the assumption of perfect correlation (equivalently,com-
plete information as in BC2005 and BC2008) and the actual estimates compromises
the ability of the model to fit the equity risk premium. For example, in samples
that exclude the 2008 debt crisis period, the complete-information model (CIM) of
BC2008 underperforms our incomplete-information model (IIM) in terms of equity

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