Generalized Disappointment Aversion, Long Run Volatility Risk and Asset Prices

TLDR: This paper proposed an asset pricing model where preferences display generalized disappointment and risk aversion for small gambles, and derived closed-form solutions for all returns moments and predictability of regressions.

Abstract: We propose an asset pricing model where preferences display generalized disappointment aversion (Routledge and Zin, 2009) and the endowment process involves long-run volatility risk. These preferences, which are embedded in the Epstein and Zin (1989) recursive utility framework, overweight disappointing results as compared to expected utility, and display relatively larger risk aversion for small gambles. With a Markov switching model for the endowment process, we derive closed-form solutions for all returns moments and predictability regressions. The model produces first and second moments of price-dividend ratios and asset returns and return predictability patterns in line with the data. Compared to Bansal and Yaron (2004), we generate: i) more predictability of excess returns by price-dividend ratios; ii) less predictability of consumption growth rates by price-dividend ratios. Differently from the Bansal and Yaron model, our results do not depend on a value of the elasticity of intertemporal substitution greater than one.

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Table 5: (LRR) Asset Prices and Predictability: Bansal and Yaron (2004) Endowment Process The entries of Panel A are model population values of asset prices. The expressions E [R−Rf ], E [Rf ]− 1 and E [P/D] are respectively the annualized equity premium, mean risk-free rate and mean price-dividend ratio. The expressions σ [R], σ [Rf ] and σ [D/P ] are respectively the annualized standard deviations of market return, risk-free rate and dividend-price ratio. Panels B, C and D show the R2 and the slope of the regression yt+1:t+12h = a (h) + b (h) ( D P ) t−11:t + ηt+12h (h), where y stands for excess returns, consumption growth and dividend growth respectively.

Figure 4: (RW) Sensitivity of Asset Prices to the Persistence of Consumption Volatility: KP and GDA. The figure displays population values of asset prices as functions of the persistence of consumption volatility. The expressions E [R−Rf ] and E [Pd/D] are respectively the annualized equity premium and mean price-dividend ratio. The expressions σ [R−Rf ] and σ [D/Pd] are respectively the annualized standard deviations of the equity excess return and the equity dividend-price ratio.

Table 1: Parameters of the Random Walk and the Long-Run Risks Markov Switching Models. The long-run risks model defined in (2.6) is calibrated with µx = 0.0015, φd = 2.5, νd = 6.5, φx = 0.975, νx = 0.038, √ µσ = 0.0072, φσ = 0.995, νσ = 0.62547 × 10−5 and ρ1 = 0.39985. In Panel A, we report the parameters of the two-state monthly Markov switching model of the form (3.1) such that µc,1 = µc,2 and µd,1 = µd,2. From the LRR in mean and volatility model, we set φx = 0 and νx = 0 to obtain a random walk model, and we adjust the other parameters when necessary such that consumption and dividend growth means, variances and covariance remain unchanged from the original model. The random walk model is then calibrated with µx = 0.0015, νd = 6.42322, √ µσ = 0.0073, φσ = 0.995, νσ = 0.62547 × 10−5 and ρ1 = 0.40434. In Panel A, we report the parameters of the four-state monthly Markov switching model of the form (3.1) that matches the full long-run risk model of Bansal, Kiku and Yaron (2007). In both panels, µc and µd are conditional means of consumption and dividend growths, ωc and ωd are conditional variances of consumption and dividend growths and ρ is the conditional correlation between consumption and dividend growths. P⊤ is the transition matrix across different regimes and Π is the vector of unconditional probabilities of regimes. Means and standard deviations are in percent.

Table 4: (RW) Asset Prices and Predictability: Robustness to Preference Parameters The entries of Panel A are model population values of asset prices. The expressions E [R−Rf ], E [Rf ]− 1 and E [P/D] are respectively the annualized equity premium, mean risk-free rate and mean price-dividend ratio. The expressions σ [R], σ [Rf ] and σ [D/P ] are respectively the annualized standard deviations of market return, risk-free rate and dividend-price ratio. Panels B, C and D show the R2 and the slope of the regression yt+1:t+12h = a (h) + b (h) ( D P ) t−11:t + ηt+12h (h), where y stands for excess returns, consumption growth and dividend growth respectively.

Figure 2: (RW) Equity Premium, Risk-Free Rate and Valuation Ratio, GDA The figure displays population values of asset prices. The expressions E [R−Rf ], E [Rf ]− 1 and E [P/D] are respectively the annualized equity premium, mean risk-free rate and mean pricedividend ratio. The parameter of risk aversion is set to γ = 2.5.

Figure 7: (LRR) Sensitivity of Excess Return and Growth Rates Predictability to the Persistence of Expected Consumption Growth: KP and GDA. The figure shows the population R2 of the monthly regression yt+1:t+h = a (h)+ b (h) ( D Pd ) t−11:t + ηt+h (h) for horizons corresponding to one year (h = 12), three years (h = 36) and five years (h = 60). The variable y stands for excess returns R − Rf and consumption growth ∆c. The R2 is plotted as a function of the persistence of expected consumption growth.

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Institut d’Economie Industrielle (IDEI) – Manufacture des Tabacs
Aile Jean-Jacques Laffont – 21, allée de Brienne – 31000 TOULOUSE – FRANCE
Tél. + 33(0)5 61 12 85 89 – Fax + 33(0)5 61 12 86 37 – www.idei.fr – contact@cict.fr
June, 2010
n° 636
“Generalized Disappointment
Aversion, Long-Run Volatility Risk
and Asset Prices”
Marco BONOMO, René GARCIA,
Nour MEDDAHI and Roméo TÉDONGAP

Generalized Disappointment Aversion, Long-Run
Volatility Risk and Asset Prices
∗
Marco Bonomo
Graduate School of Economics,
Getulio Vargas Foundation
Ren´e Garcia
Edhec Business School
Nour Meddahi
Toulouse School of Economics
(GREMAQ, IDEI)
Rom´eo T´edongap
Stockholm School of Economics
June 2010
Abstract
We propose an asset pricing model where preferences display generalized disappointment
aver sion (Routledge and Zin, 2009 ) and the endowment process involves long-run volatility
risk. These preferences, which are embedded in the Epstein and Zin (1989) recursive util-
ity framework, overweight disappointing results as compared to expected utility, and dis play
relatively larger risk aversion for s mall gambles. With a Markov switching model for the en-
dowment process, we derive closed-form solutions for all returns mo ments a nd predictability
regress ions. The model produces first and second moments of price-dividend ratios and asset
returns and return predictability patterns in line with the da ta. Compared to Bansal and
Yaron (2004), we generate: i) mor e predictability of excess returns by price-dividend ratios; ii)
less predicta bility o f co nsumption growth rates by price-dividend ratios. Differently from the
Bansal and Yaron model, our results do not depend on a value of the elas ticity of intertemporal
substitution greater than one.
Keywords: Equilibrium Asset Pricing, Disappointment Aversion, Long-run Risks, Volatility Risk,
Equity Premium, Risk-free Rate Puzzle, Predictability of returns
JEL Classification: G1 , G12, G11, C1, C5
∗
Address for correspondence: Edhec Business School, 393, Promenade des Anglais, BP 3 116, 0620 2 Nice
Cedex 3. Email: rene.garcia@edhec.edu. An earlier version of this paper has been circulated and presented
at various seminars and conferences under the title ”An Analytical Framework for Assessing Asset Pricing
Models and Predictability”. We thank Ravi Bansal, John Campbell, Lars Hansen, and Moto Yogo for useful
comments a nd participants at the CIREQ and CIRANO Conference in Financial Econometrics (May 2006,
Montreal), seminar at the Universit´e de Cergy (May 2006), Confere nce in Financial Econometrics (CREST,
Paris, May 2006), North America n Summer Meetings 20 07 (Duke University, Raleigh), Luso Brazilian
Finance Meeting (2009), European Meeting of the Econometric Society (2009), Latin American Meeting
of the E conometric Society (2009), Brazilian Econometric Meeting (2009), lunch seminar at the New York
Federal Reserve (2010), and Financial Econometrics Workshop at the Fields Institute (2010). We also thank
three referees and the editor for their useful and c onstructive comments. The first author acknowledges the
research fellowship from CNPq. The second and third authors are research fellows of CIRANO and CIREQ.
The second author thanks the Network of Ce ntres of Excellence for financial support. The third author
benefited from the financial support of the chair ”March´e des risques et cr´eation de valeur” Fonda tion
du risque/SCOR and from the European Community’s Seventh Framework Programme (FP7/2007-2013)
Grant Agreement no. 23 0589.

1 Introduction
The Consumption-based Capital Asset Pricing Model (CCAPM) has recently been revived
by models of long-run risks (LRR) in mean and vo la t ility
1
. Bansal and Yaron (2004) explain
several asset market stylized facts by a model with a small long-run predictable component
driving consumption and dividend growth and persistent economic uncertainty measured
by consumption volatility, together with recursive preferences that separate risk aversion
from intertemporal substitution (Epstein and Zin, 1989; Kreps and Por teus, 1 978). These
preferences play a crucial role in the long-run risks model. In a canonical expected utility
risk, only short-run risks are compensated, while long-run risks do not carry separate risk
premia. With Kreps-Porteus preferences, long-run r isks earn a positive risk premium as
long as investors prefer early resolution of uncertainty. Routledge and Zin (2009) introduced
recently preferences that exhibit generalized disappointment aversion (GDA) and showed
that they can generate a large equity premium and countercyclical risk aversion. Compared
with expected utility, GDA overweights outcomes below a threshold set at a fraction of the
certainty equivalent of future utility. Disappointment aversion (Gul, 1991) sets the threshold
at the certainty equiva lent.
Despite the economic appeal to link expected consumption growth to asset prices, the
existence of a long-run risk component in expected consumption growth is a source of debate.
If a very persistent predictable compo nent exists in consumption growth, as proposed by
Bansal and Yaron (2004), it is certainly hard to detect it as consumption appears very much
as a random wa lk in the data
2
. Moreover, this slow mean-reverting component has the
counterfactual implication of making consumption growth predictable by the price-dividend
ratio. There is less controversy about the persistence in consumption growth volatility.
Bansal and Yaron (2004) show that the variance ratios of the absolute value of r esiduals
from regressing current annual consumption growth on five lags increase gradually up to
10 years, suggesting a slow-moving predictable variation in this measure of consumption
growth volatility. Calvet and Fisher (2007) find empirical evidence of volatility shocks of
much longer duration than in Bansal and Yaron (2004), creating the potential of a more
important contribution of volatility risk in explaining a sset pricing stylized facts.
1
Another featured approach is the rare dis aster model of Barro (2 006). The extensive literature about
the equity premium puzzle and other puzzling features of asset markets are reviewed in a collection of essays
in Mehra (2008). See also Campbell (2000, 2003), Cochrane and Hansen (1992), Kocherlakota (1996), and
Mehra and Presc ott (2003).
2
See in particular Campbell (2003). Bans al (200 7) cites several studies that provide empirical support
for the existence of a long-r un component in consumption. Bansal, Gallant and Tauchen (2007) and Bansal,
Kiku and Yaron (2007a) test the LRR in mean and volatility model using the efficient and generalized
method of moments, respectively. Hansen, Heaton and Li (20 08) and Bansal, Kiku and Yaron (2007a,
2009) pre sent evidence for a long-run component in consumption growth using multivariate analysis.
1

In this paper, we revisit the LRR model with GDA preferences. In Bansal, Kiku and
Yaron (2007b), the presence of a slow mean-reverting long-run component in the mean
of consumption and dividend growth series, coupled with Kreps-Porteus preferences, is
essential to achieve an equity premium commensurate with historical data
3
. Given the
debate about the nature of the consumption process, we start by restricting t he LRR model
to a r andom walk model with LRR in volatility to investigate whether persistent fluctuations
in economic uncertainty are sufficient, with GDA preferences, to explain observed asset
pricing stylized facts.
This benchmark model reproduces asset pricing stylized facts and predictability patterns
put forward in the previous literature. The equity premium and t he risk-free rate are very
closely matched, as well as the volatility of the price-dividend ratio and of returns. The
price-dividend ratio predicts excess returns at various horizons even though consumption
and dividend growth rates are assumed to be unpredictable.
The intuition becomes clear fro m the simplest representation of GDA preferences, where
the only source of risk aversion is disappointment aversion (the utility function is otherwise
linear with a zero curvature parameter and an infinite elasticity of intertemporal substitu-
tion). With these simple preferences, the stochastic discount factor has only two values in
each state of the economy at time t. The SDF for disappointing outcomes is ϕ times the
SDF for non-disappointing o utcomes, where ϕ − 1 > 0 is the extra weight given by disap-
pointment averse preferences to disappointing outcomes. This could give rise to a sizable
negative covariance between the pricing kernel and the return o n a risky asset, making the
risk premium substantia l.
More generally, the SDF has an infinite number of outcomes with a kink at the point
where future utility is equal to a given fraction of the certainty equivalent. When volatility
of consumption growth is persistent, an increase in volatility increases the volatility of f uture
utility. A more volatile future utility increases the probability of disappo inting o utcomes,
making the SDF more volatile. Since both consumption and dividends share the same
stochastic volatility process, an increase in volatility will increase the negative covariance
between the SDF and the equity r eturn, implying a substantial increase in the stock risk
premium.
If volatility is persistent, as it is the case in the long- run volatility risk model we assume,
this will result in persistent and predictable conditional exp ected returns. As argued by
Fama and French (1988) , such a process for expected returns generates mean reversion in
asset prices. Therefore, the price-dividend ratio today should be a good predictor of returns
3
Although a persistent volatility would also incr e ase the equity premium with Kre ps-Porteus preferences,
it would do it only in the presence of this fir st source of LRR.
2

over several future periods.
Bansal and Yaron (2004), as most recent models, rely on par ameter calibration for con-
sumption and dividend processes as well as preferences to derive a sset pricing implications
from the model. The solution technique to solve for asset valuation ratios is based on
loglinear approximations. Since the GDA utility is non-differentiable at the kink where
disappointment sets in, one cannot rely on the same approximation techniques to solve the
model. In this paper, we propose a methodology t hat provides an analytical solution to the
LRR in mean and volatility model with GDA preferences and a fortiori with Kreps-Porteus
preferences, yielding formulas fo r the asset valuation ratios in equilibrium. The key to t his
analytical solution is to use a Markov switching process for consumption and dividends
that matches the LRR specifications. In addition, we report analytical formulas for the
population moments of equity premia as well as for the coefficients and R
2
of predictability
regressions that have been used to assess the ability of asset pricing models to reproduce
stylized facts.
Thanks to our analytical formulas, we ar e able to conduct a thorough comparative anal-
ysis between models by varying extensively the preference and endowment parameters. We
produce graphs that exhibit the sensitivity of asset pricing statistics or predictability regres-
sions R
2
to key parameters such as the persistence in volatility or expected consumption
growth. This provides a very useful tool to measure the robustness of model implications.
We consider in particular the value of the elasticity of intertemporal substitution, which
has been a source of lively debate. Bansal, Kiku and Yaron (2009) r eport empirical evidence
in favor of a value g r eater than 1
4
, but Beeler and Campbell (2009), as well as Hall (1988)
and Campbell (1999 ) , estimate an elasticity of intertempo ral substitution below 1
5
. One
important aspect of our model is that the elasticity of intertemporal substitution value of o ne
is not pivotal for r eproducing asset pricing stylized facts. Moment fitting and predictability
results remain intact with values of t he elasticity of intertemporal substitution below 1. The
main effect of setting the elasticity of intertempo r al substitution below 1 is, of course, an
increase in the level and volatility of the risk-free rate, but these moments remain in line
with the data.
We also conduct a sensitivity analysis with respect to the specification of risk preferences.
We investigate the simplest specification among disappointment averse preferences. We
4
They cite Hansen and Singleton (1982) and Attanasio and Weber (1989), Bans al, Kiku and Yaron
(2007a ), and Hansen, Heaton, Lee and Roussanov (2007), among others.
5
Bansal and Yaron (2 004) also argue tha t in the presence of time-va rying volatility, there is a severe
downward bias in the point estimates of the elas ticity of intertemporal substitution. While the argument is
correct in principle, Beeler and C ampbell (2009) simulate the Bansal a nd Yaron (2004) model a nd report
no bias if the ris kless interest rate is used as an instrument. They confirm the presence of a bias (nega tive
estimate of the e lasticity of intertempo ral substitution) when the equity return is used and attribute it to
a weak instrument problem.
3

Frequently asked questions

Q1. What are the contributions in this paper?

The authors propose an asset pricing model where preferences display generalized disappointment aversion ( Routledge and Zin, 2009 ) and the endowment process involves long-run volatility risk. 

Q2. What is the mechanism at play in the recursive utility model?

The economic mechanism at play is an endogenous variation in the probability of disappointment in the representative investor’s intertemporal consumption-saving problem that underlies the asset-pricing model. 

Q3. What is the importance of the persistence of volatility in the kreps-porte?

It is really for the volatility of the dividend-price ratio that the persistence of volatility is very important, since it is decreasing fast as the value of φσ is approaching 0.9. 

Q4. How do they determine the level of effective risk aversion of investors?

Since the disappointment aversion parameters α and κ interact with γ to determine the level of effective risk aversion of investors, the authors certainly need to lower γ. 

Q5. What is the method used to produce population moments from the model?

The methodology used in Beeler and Campbell (2009) to produce population moments from the model rests on solving a loglinear approximate solution to the model and on a single simulation run over 1.2 million months (100,000 years). 

Q6. What is the key for reproducing the predictability of consumption growth?

For the GDA preferences that the authors advocate in this paper, the persistence in the volatility of consumption growth φσ is key for reproducing the predictability stylized facts but results are not as sensitive to this persistence as they are with Kreps-Porteus preferences for the persistence of expected consumption growth. 

Q7. What is the main source of long-run risk?

Their benchmark endowment process had only one of the two sources of long-run risks proposed by Bansal and Yaron (2004): the volatility risk. 

Q8. What are the predictability regressions for the US economy?

The authors also consider several predictability regressions by the price-dividend ratio, for excess returns, consumption growth and dividend growth. 

Q9. What is the implication of the slow mean-reverting component in the consumption growth model?

this slow mean-reverting component has the counterfactual implication of making consumption growth predictable by the price-dividend ratio. 

Q10. What is the probability of disappointment in a state?

The probability of disappointment may change with the state even with DA preferences, generating predictable time-variation in returns. 

Q11. What is the certainty equivalent of the Kreps-Porteus preferences?

When α is equal to one, R becomes the certainty equivalent corresponding to expected utility while Vt represents the Kreps-Porteus preferences.