NBER WORKING PAPER SERIES
DOES THE FAILURE OF THE EXPECTATIONS HYPOTHESIS
MATTER FOR LONG-TERM INVESTORS?
Antonios Sangvinatsos
Jessica A. Wachter
Working Paper 10086
http://www.nber.org/papers/w10086
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
November 2003
The authors are grateful for helpful comments from Utpal Bhattacharya, Michael Brennan, John Campbell,
Jennifer Carpenter, Qiang Dai, Ned Elton, Blake LeBaron, Anthony Lynch, Lasse Pedersen, Matthew
Richardson, Luis Viceira, an anonymous referee, and seminar participants at Brandeis University, the
Norwegian School of Economics and Business, the Norwegian School of Management, University of
California at Los Angeles, University of Indiana at Bloomington, the Spring 2003 NBER Asset Pricing
meeting and the 2003 CIRANO Conference on Portfolio Choice. The views expressed herein are those of the
authors and not necessarily those of the National Bureau of Economic Research.
©2003 by Antonios Sangvinatsos and Jessica A. Wachter. All rights reserved. Short sections of text, not to
exceed two paragraphs, may be quoted without explicit permission provided that full credit, including ©
notice, is given to the source.
Does the Failure of the Expectations Hypothesis Matter for Long-Tern Investors?
Antonios Sangvinatsos and Jessica A. Wachter
NBER Working Paper No. 10086
November 2003
JEL No. G1
ABSTRACT
We consider the consumption and portfolio choice problem of a long-run investor when the term
structure is affine and when the investor has access to nominal bonds and a stock portfolio. In the
presence of unhedgeable inflation risk, there exist multiple pricing kernels that produce the same
bond prices, but a unique pricing kernel equal to the marginal utility of the investor. We apply our
method to a three-factor Gaussian model with a time-varying price of risk that captures the failure
of the expectations hypothesis seen in the data. We extend this model to account for time-varying
expected inflation, and estimate the model with both inflation and term structure data. The estimates
imply that the bond portfolio for the long-run investor looks very different from the portfolio of a
mean-variance optimizer. In particular, the desire to hedge changes in term premia generates large
hedging demands for long-term bonds.
Antonios Sangvinatsos
Stern School of Business
New York University
44 West Fourth Streeet
Suite 9-190
New York, NY 10012-1126
asangvin@stern.nyu.edu
Jessica A. Wachter
Stern School of Business
New York University
44 West Fourth Streeet
Suite 9-190
New York, NY 10012-1126
and NBER
jwachter@stern.nyu.edu
1 Introduction
The expectations hypothesis of interest rates states that the premium on long-term bonds over
short-term bonds is constant over time. According to this hypothesis, there are no particularly
good times to invest in long-term bonds relative to short-term bonds, nor are there particularly
bad times. Long-term bonds will always offer the same expected excess return.
1
While the expectations hypothesis is theoretically appealing, it has consistently failed in U.S.
postwar data. Fama and Bliss (1987) and Campbell and Shiller (1991), among others, show that
expected excess returns on long-term bonds (term premia) do vary over time, and moreover, it
is possible to predict excess returns on bonds using observables such as the forward rate or the
term spread. This paper explores the consequences of the failure of the expectations hypothesis for
long-term investors.
We estimate a three-factor affine term structure model similar to that proposed in Dai and Sin-
gleton (2002a) and Duffee (2002) that accounts for the fact that excess bond returns are predictable.
We then solve for the optimal portfolio for an investor taking this term structure as given. Bond
market predictability will clearly affect the mean-variance efficient portfolio, but the consequences
for long-horizon investors go beyond this. Merton (1971) shows that when investment opportu-
nities are time-varying, a mean-variance efficient portfolio is generally sub-optimal. Long-horizon
investors wish to hedge changes in the investment opportunity set; depending on the level of risk
aversion, the investor may want more or less wealth when investment opportunities deteriorate than
when they improve. As we will show, investors gain by hedging time-variation in the term premia.
Thus the investor’s bond portfolio looks different from that dictated by mean-variance efficiency.
Despite the obvious importance of bonds to investors, as well as the strength of the empirical
findings mentioned above, recent literature on portfolio choice has focused almost exclusively on
predictability in stock returns. As shown by Fama and French (1989) and Campbell and Shiller
(1988), the price-dividend ratio predicts excess stock returns with a negative sign. Based on this
finding, a number of studies (e.g. Balduzzi and Lynch (1999), Barberis (2000), Brandt (1999),
Brennan, Schwartz, and Lagnado (1997), Campbell and Viceira (1999), Liu (1999) and Wachter
1
The expectations hypothesis, as we refer to it, should be distinguished from the pure expectations hypothesis
which states that term premia are not just constant but equal to zero. Cox, Ingersoll, and Ross (1981) examine
variants of the pure expectations hypothesis in the context of continuous-time equilibrium theory, and find that they
are inconsistent with each other, and that several imply arbitrage opportunities (see however Longstaff (2000a)).
Campbell (1986) shows that these inconsistencies do not occur with the more general expectations hypothesis, which
does not require term premia to be zero. In fact, it is the expectations hypothesis, as opposed to the pure expectations
hypothesis, which is typically examined in the empirical literature (see Bekaert and Hodrick (2001) for a discussion
of recent empirical work testing the expectations hypothesis).
1
(2002a)) document gains from timing the stock market based on the price-dividend ratio, and from
hedging time-variation in expected stock returns. One result of this literature is that when investors
have relative risk aversion greater than one, hedging demands dictate that their allocation to stock
should increase with the horizon. A natural question to ask is whether the same mechanism is
at work for bond returns. Just as stock prices are negatively correlated with increases in future
risk premia on stocks, bond prices are negatively correlated with increases in future risk premia on
bonds.
2
This intuition suggests that time-variation in risk premia would cause the optimal portfolio
allocation to long-term bonds to increase with horizon.
In the case where the investor allocates wealth between a long and a short-term bond, we show
that this intuition holds. Hedging demands induced by time-variation in risk premia more than
double the investor’s allocation to the long-term bond. Moreover, we find large horizon effects. The
investor with a horizon of twenty years holds a much greater percentage of his wealth in long-term
bonds than an investor with a horizon of ten years. In the case of multiple long-term bonds, the
mean-variance efficient portfolio often consists of a long and short position in long-term bonds. This
occurs because of the high positive correlation between bonds of different maturities implied by
the model and found in the data. Hedging demands induced by time-varying risk premia generally
make the allocation to long-term bonds more extreme. We find that following a myopic strategy
and, in particular, failing to hedge time variation in risk premia carries a high utility cost for the
investor.
Our framework generalizes previous studies of portfolio choice when real interest rates vary over
time and there is inflation. Brennan and Xia (2002) and Campbell and Viceira (2001) estimate a
two-factor Vasicek (1977) term structure model and determine optimal bond portfolios. Both of
these studies assume that risk premia on bonds and stocks are constant.
3
Our study also relates to
that of Campbell, Chan, and Viceira (2002) who estimate a vector-autoregression (VAR) including
the returns on a long-term bond, a stock index, the dividend yield and the yield spread. Campbell
et al. derive an approximate solution to the optimal portfolio choice problem when asset returns
are described by the VAR. The advantage of the VAR approach is that it captures predictability in
2
We consider U.S. government bonds that are not subject to default risk. Nonetheless, we use risk premia and
term premia interchangeably, as we do not take a stand on the source of the premia.
3
Other work on bond returns and portfolio choice includes Brennan and Xia (2000) and Sorensen (1999), who
assume that interest rates are Vasicek, and Liu (1999) and Schroder and Skiadas (1999) who assumes general affine
dynamics. These studies assume that bonds are indexed, or equivalently, that there is no inflation. Xia (2002)
examines the welfare consequences of limited access to nominal bonds under a Vasicek model. Wachter (2002b)
shows under general conditions that as risk aversion approaches infinity, the investor’s allocation approaches 100%
in a long-term indexed bond. None of these papers explore the consequences of bond return predictability.
2
bond and stock returns in a relatively simple way. The disadvantage is that the term structure is not
well-defined; it is necessary to assume that the investor only has access to those bonds included in
the VAR. Moreover, estimating bond returns using a VAR gives up the extra information resulting
from the no-arbitrage restriction on bonds, namely that bonds have to pay their (nominal) face
value when they mature.
Rather than modeling bond return predictability using a VAR, we follow the affine bond pricing
literature (e.g. Dai and Singleton (2000, 2002a) and Duffee (2002)) and specify a nominal pricing
kernel.
4
The drift and diffusion of the pricing kernel is driven by three underlying factors which
follow a multivariate Ornstein-Uhlenbeck process. The price of risk is a linear function of the state
variables. Thus the model is in the “essentially affine” class proposed by Duffee (2002), and shown
by Dai and Singleton (2002a) to capture the pattern of bond predictability in the data.
As a necessary step to showing the implications of affine term structure models for investors,
we show how parameters of the inflation process can be jointly estimated with term structure
parameters. This joint estimation produces a series for expected inflation that explains 37% of the
variance of realized inflation. This result has implications not only for portfolio choice problems,
but for the estimation of term structure models more generally.
The remainder of the paper is organized as follows. Section 2 describes the general form of an
economy where nominal bond prices are affine, and there exists equity and unhedgeable inflation.
Section 3 derives a closed-form solution for optimal portfolio choice when the investor has utility
over terminal wealth and over intermediate consumption. When inflation is introduced, the pricing
kernel that determines asset prices is not unique; from the point of view of the investor it is not
well-defined. As He and Pearson (1991) show, there is a unique pricing kernel that gives the
marginal utility process for the investor.
5
We derive a closed-form expression for this pricing kernel
when incompleteness results from inflation. This expression holds regardless of the form of the
term structure. Section 4 uses maximum likelihood to estimate the parameters of the process, and
demonstrates that the model provides a good fit to term structure data and to inflation. Section 5
discusses the properties of the optimal portfolio for the parameters we have estimated and calculates
utility costs resulting from sub-optimal strategies.
4
For recent surveys of this literature, see Dai and Singleton (2002b) and Piazzesi (2002).
5
Liu and Pan (2002) also associate the pricing kernel in the economy with the pricing kernel for the investor. In
Liu and Pan’s model markets are complete, so a unique pricing kernel exists.
3
