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Journal ArticleDOI

Arbitrage-free models of stocks and bonds

01 Dec 2013-Social Science Research Network (New York, NY: Federal Reserve Bank of New York)-

Abstract: A small but ambitious literature uses affine arbitrage-free models to estimate jointly U.S. Treasury term premiums and the term structure of equity risk premiums. Within this approach, this paper identifies the parameter restrictions that are consistent with a simple dividend discount model, extends the cross-section to Germany and France, averages across multiple observable-factor and market prices of risk specifications, and considers alternative samples for parameter estimation. The results produce intuitive trajectories for both sets of premiums given standard samples starting from July 1993. However, the decomposition of nominal U.S. Treasury yields, but not long-run equity risk premiums, is sensitive to data beyond 2008, which raises some questions about the net effects of unconventional monetary policy measures. Nonetheless, the rotation from sharp inversion during the financial crisis to an upward-sloping term structure of equity risk premiums more recently, with modest readings at the front end, is not inconsistent with some net moderation in required compensation for equity risk in the United States.
Topics: Equity risk (66%), Risk premium (55%), Dividend discount model (53%), Arbitrage (53%), Bond (52%)

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This paper presents preliminary findings and is being distributed to economists
and other interested readers solely to stimulate discussion and elicit comments.
The views expressed in this paper are those of the author and are not necessarily
reflective of views at the Federal Reserve Bank of New York or the Federal
Reserve System. Any errors or omissions are the responsibility of the author.
Federal Reserve Bank of New York
Staff Reports
Arbitrage-Free Models of Stocks and Bonds
J. Benson Durham
Staff Report No. 656
December 2013

Arbitrage-Free Models of Stocks and Bonds
J. Benson Durham
Federal Reserve Bank of New York Staff Reports, no. 656
December 2013
JEL classification: G10, G12, G13, G15
Abstract
A small but ambitious literature uses affine arbitrage-free models to estimate jointly U.S.
Treasury term premiums and the term structure of equity risk premiums. Within this approach,
this paper identifies the parameter restrictions that are consistent with a simple dividend discount
model, extends the cross-section to Germany and France, averages across multiple observable-
factor and market prices of risk specifications, and considers alternative samples for parameter
estimation. The results produce intuitive trajectories for both sets of premiums given standard
samples starting from July 1993. However, the decomposition of nominal U.S. Treasury yields,
but not long-run equity risk premiums, is sensitive to data beyond 2008, which raises some
questions about the net effects of unconventional monetary policy measures. Nonetheless, the
rotation from sharp inversion during the financial crisis to an upward-sloping term structure of
equity risk premiums more recently, with modest readings at the front end, is not inconsistent
with some net moderation in required compensation for equity risk in the United States.
Key words: equity risk premium, Treasury term premium, affine arbitrage-free models
_________________
Durham: Federal Reserve Bank of New York (e-mail: jbenson.durham@ny.frb.org). With no implication,
the author thanks seminar participants at the Federal Reserve Bank of New York as well as Tobias Adrian,
Michael Bauer, Wolfgang Lemke, and Matthew Raskin for discussions and comments. The views
expressed in this paper are those of the author and do not necessarily reflect the position of the Federal
Reserve Bank of New York or the Federal Reserve System.

1
1. Introduction
Term and the equity risk premiums are arguably the most critical unobservable variables
in finance. Central bankers attempt to disentangle from yield curves and stock prices investors’
expectations for the real economy and inflation as well as their perceptions and attitudes toward
various risks, perhaps especially so and with increasing difficulty amid unconventional policy
measures that possibly work through portfolio rebalancing or perhaps other transmission
mechanisms. Investors estimate these premiums to gauge expected returns and compensation for
bearing uncertainty, from the risk-free to the yardstick risky asset class. Along with active views
on returns as well as variance and covariance, such estimates comprise the required inputs to
quantitative portfolio optimization (e.g., Black and Litterman, 1992). Despite these strong
motivations, a surprisingly small practitioner or even academic literature addresses the joint
dynamics of stock and government bond prices in a comprehensive arbitrage-free framework.
Starting from a few promising exceptions (e.g., Bekaert and Grenadier, 2001; d’Addona
and Kind, 2006; Koijen et al., 2010; Mamaysky, 2002; Lemke and Werner, 2009; Adrian et al.,
2013), this study endeavors to extend this inclusive approach.
1
Regarding theory, the following
amends Gaussian affine term structure equity models (GATSEMs) to specify the conditions
under which arbitrage-free and dividend discount models (DDMs) of stock prices are consistent.
With respect to empirics, the analyses cover cases besides the U.S., including Germany and
France, and the underlying factors incorporate a simple macro-finance approach to include
variables such as survey-based expectations of one-year-ahead inflation, real GDP growth, and
budget deficits. Beyond an extension of the cross-section, and in addition to sample and bond
maturity selection, the sensitivity analyses address model uncertainty with respect to the
1
Not all of these studies permit time-varying risk premiums (e.g., d’Addona and Kind, 2006) or produce stock
prices that are exponentially affine functions of the state variables (e.g., Bekeart and Grenadier, 2001). The methods
described below most closely follow Mamayksy (2002) and, in particular, Lemke and Werner (2009).

2
selection of observable factors as well as alternative market price(s) of risk parameter
restrictions, two subjects of notable uncertainty in the existing literature.
In general, the estimates of government bond term and equity risk premiums are
consistent with common priors regarding magnitudes and trends starting from July 1993 (e.g.,
Lemke and Werner, 2009). However, and germane to calibration of Gaussian affine term
structure models (GATSMs) in general (Durham, 2013b), the results raise questions regarding
the sensitivity of long-run U.S. term premium estimates, but apparently not long-run equity risk
premiums, to data through the aftermath of the global financial crisis. In addition to the elevated
level of required long-run equity returns over the past few years, the prospect of higher term
premiums than consensus estimates suggest perhaps raise questions about the net impact of
unconventional monetary policy measures, such as large scale asset purchases (LSAPs), on risky
financial asset prices, counterfactuals aside (e.g., Bernanke, 2010; Stein, 2012). On the other
hand, the recent slope of the U.S. term structure of required equity returns is indeed positive,
notably from moderated levels at the front end and in clear contrast to the sharp inversion
observed at the height of the financial crisis, a finding that is not inconsistent with meaningful
accommodation in broader financial conditions amid aggressive policy innovations. In addition,
these results also highlight some additional issues in the cross section of major equity markets.
For example, recent term structures of equity premiums are sharply downward-sloping for
Germany and France, in contrast to the hump-shape in the U.S., which could reflect varying
perceptions of monetary policy transmission or nearer-term idiosyncratic risks, such as lingering
investor uncertainty that stems from EMU.
Section 2 briefly reviews recent joint estimations of bond and stock market dynamics.
Section 3 extends the framework and identifies the parameter restrictions that impose

3
consistency between a simple DDM and this arbitrage-free approach. Section 4 describes the
general research design and outlines the two-step estimation procedure. Section 5 describes the
general results, and Section 6 includes some implications for the contemporary environment.
Section 7 concludes.
2. A Brief Review of the Model
Contrary to most equity risk premium measures,
2
GATSEMs (e.g., Mamaysky, 2002;
Lemke and Werner, 2009) afford an ex-ante time-varying term structure as opposed to a fixed
estimate over an undefined investment horizon. Similar to Gaussian affine term structure models
(GATSMs) based on Vasicek (1977), the n factors, denoted by the
1n
vector, X, follow a
mean-reverting process, as in
1
1
t
tt
X a X

(1)
where a is an
1n
vector,
is an
nn
matrix,
is an
nn
matrix, and
is an
1n
normally distributed vector of shocks, as in
~ . . ., 0,i i d N I
. Also, the dynamics of the
nominal pricing kernel or stochastic discount factor, M, follow
''
11
1
exp
2
t t t t t t
Mr




(2)
where r is the instantaneous nominal risk-free short rate,
3
an affine function of the underlying
factors following
'
01tt
rX


(3)
where
0
is a scalar, and
1
is an
1n
vector. The market price of risk,
, is an
1n
vector
and a linear function of the state variables following
2
For recent surveys of equity risk premium estimates that do not adopt an arbitrage-free approach, see Mehra (2008)
and Hammond et al. (2011).
3
Lemke and Werner (2009) delineate the real rate and inflation processes in the models.

Citations
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Posted Content
Fernando Duarte1, Carlo Rosa1Institutions (1)
Abstract: 1. INTRODUCTION The equity risk premium--the expected return on stocks in excess of the risk-free rate--is a fundamental quantity in all of asset pricing, both for theoretical and practical reasons. It is a key measure of aggregate risk-aversion and an important determinant of the cost of capital for corporations, the savings decisions of individuals, and budgeting plans for governments. Recently, the equity risk premium (ERP) has also moved to the forefront as a leading indicator of the evolution of the economy, a potential explanation for jobless recoveries, and a gauge of financial stability. (1) In this article, we estimate the ERP by combining information from twenty prominent models used by practitioners and featured in the academic literature. Our main finding is that the ERP has reached heightened levels. The first principal component of all models--a linear combination that explains as much of the variance of the underlying data as possible--places the one-year-ahead ERP in June 2012 at 12.2 percent, above the 10.5 percent reached during the financial crisis in 2009 and at levels similar to those in the mid- and late 1970s. From June 2012 to the end of our sample in June 2013, the ERP has changed little, despite substantial positive realized returns. It is worth keeping in mind, however, that there is considerable uncertainty around these estimates. In fact, the issue of whether stock returns are predictable is still an active area of research. (2) Nevertheless, we find that the dispersion in estimates across models, while quite large, has been shrinking, potentially signaling increased agreement even when the models differ substantially from one another and use more than one hundred different economic variables. In addition to estimating the level of the ERP, we investigate the reasons behind its recent behavior. Because the ERP is the difference between expected stock returns and the risk-free rate, a high estimate can be the result of expected stock returns being high or risk-free rates being low. We conclude that the ERP is high because Treasury yields are unusually low. Current and expected future dividend and earnings growth play a smaller role. In fact, expected stock returns are close to their long-run mean. One implication of a bond-yield-driven ERP is that traditional indicators of the ERP like the price-dividend or price-earnings ratios, which do not use data from the term structure of risk-free rates, may not be as good a guide to future excess returns as they have been in the past. As a second contribution, we present a concise and coherent taxonomy of ERP models. We categorize the twenty models into five groups: predictors that use historical mean returns only, dividend discount models, cross-sectional regressions, time-series regressions, and surveys. We explain the methodological and practical differences among these classes of models, including the diverse assumptions and data sources that they require. 2. THE EQUITY RISK PREMIUM: DEFINITION Conceptually, the ERP is the compensation that investors require to make them indifferent at the margin between holding the risky market portfolio and a risk-free bond. Because this compensation depends on the future performance of stocks, the ERP incorporates expectations of future stock market returns, which are not directly observable. At the end of the day, any model of the ERP is a model of investor expectations. One challenge in estimating the ERP is that it is not clear what truly constitutes the market return and the risk-free rate in the real world. In practice, the most common measures of total market return are based on broad stock market indexes, such as the S&P 500 or the Dow Jones Industrial Average, which do not include the whole universe of traded stocks and miss several other components of wealth such as housing, private equity, and nontradable human capital. Even if we restricted ourselves to all traded stocks, we would still have several choices to make, such as whether to use value or equal-weighted indexes, and whether to exclude penny or infrequently traded stocks. …

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Abstract: Several studies that use affine term structure models (ATSMs) or survey data suggest that subdued nominal U.S. Treasury yields during the global financial crisis and its aftermath primarily reflected exceptionally low, if not negative, term premiums as distinct from depressed anticipated short rates. However, this literature pays little attention to the length of time market participants anticipated low term premiums to prevail, as captured by the ?forward? or ?expected? term premium over a given horizon, distinct from the ?spot? term premium. Besides the implications for investors at the back end of the term structure, this issue relates to recent policy-related studies that argue that the persistence of interest rate shocks affects real outcomes. Unlike the consensus inference on low spot term premiums, the evidence on expected term premiums is somewhat mixed. Some ATSMs suggest that expected term premiums did drop substantially along with spot measures after 2007, but the simple survey-based estimate reported here notably indicates the opposite.

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References
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Oldrich A Vasicek1Institutions (1)
Abstract: The paper derives a general form of the term structure of interest rates. The following assumptions are made: (A.1) The instantaneous (spot) interest rate follows a diffusion process; (A.2) the price of a discount bond depends only on the spot rate over its term; and (A.3) the market is efficient. Under these assumptions, it is shown by means of an arbitrage argument that the expected rate of return on any bond in excess of the spot rate is proportional to its standard deviation. This property is then used to derive a partial differential equation for bond prices. The solution to that equation is given in the form of a stochastic integral representation. An interpretation of the bond pricing formula is provided. The model is illustrated on a specific case.

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