E C O N O M I C S T U D I E S 98
MARTIN ÅGREN
ESSAYS ON PROSPECT THEORY AND THE STATISTICAL MODELING OF
FINANCIAL RETURNS
MARTIN ÅGREN
ESSAYS ON PROSPECT THEORY AND THE
STATISTICAL MODELING OF FINANCIAL RETURNS
Department of Economics, Uppsala University
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ISBN 91-85519-05-7
ISSN 0283-7668
Doctoral dissertation presented to the Faculty of Social Sciences 2006
Abstract
ÅGREN, Martin, 2006, Essays on Prospect Theory and the Statistical Modeling of
Financial Returns; Department of Economics, Uppsala University, Economic Studies 98, 105 pp, ISBN 91-85519-05-7.
This thesis consists of three self-contained essays.
Essay 1 presents an empirical study of volatility spillover from oil prices to stock
markets within an asymmetric BEKK model. Using weekly data on the aggregate
stock markets of Japan, Norway, Sweden, the U.K., and the U.S., strong evidence
of volatility spillover is found for all stock markets but the Swedish one, where only
weak evidence is found. News impact surfaces show that, although statistically significant, the volatility spillovers are quantitatively small. The stock market’s own
shocks, which are related to other factors of uncertainty than the oil price, are more
prominent than oil shocks.
Essay 2 replicates the study of Benartzi and Thaler (1995), who suggest a behavioral explanation to the equity premium puzzle by myopic loss aversion. A technical
extension to their methodology is suggested where conditional heteroskedasticity is
incorporated when simulating returns, in place of the original temporal independence assumption. Swedish data is considered in addition to U.S. data. First, it is
found that myopic loss aversion can explain the U.S. equity premium over bonds,
although the obtained evaluation periods are somewhat shorter than a year. For
example, over the full U.S. sample period from 1926 to 2003, evaluation periods of
seven and ten months are found using the original and the new approach to simulating returns, respectively. Second, myopic loss aversion suggestively explains the
Swedish equity premium as well, which is new to the literature. Third, throughout
the analysis of both data sets, longer evaluation periods are obtained under conditional heteroskedasticity. The last result indicates that myopic loss-averse and,
in turn, cumulative prospect theory investors are sensitive to the distributional assumption made on returns.
Essay 3 relates cumulative prospect theory to the moments of returns distributions,
e.g. skewness and kurtosis, assuming returns are normal inverse Gaussian distributed. The normal inverse Gaussian distribution parametrizes the first- to forth-order
moments, making the investigation straightforward. Cumulative prospect theory
utility is found to be positively related to the skewness. However, the relation is
negative when probability weighing is set aside. This shows that cumulative prospect
theory investors display a preference for skewness through the probability weighting function. Furthermore, the investor’s utility is inverse hump-shape related to
the kurtosis. Consequences for portfolio choice issues are studied. The findings,
among others, suggest that optimal cumulative prospect theory portfolios are not
mean-variance efficient under the normal inverse Gaussian distribution.
Acknowledgements
First of all, I would like to express the deepest gratitude to my advisor, Annika
Alexius. Her guidance, encouraging support, and patience over the past years are
invaluable. The insightful comments on and suggestions to improve drafts have had
considerable impact on the quality of this thesis. I would also like to thank my
second advisor, Rolf Larsson, without whose statistical and mathematical expertise
I would never have finished this thesis. Both advisors have always found the time
to discuss my research problems for which I am most grateful.
Many others have contributed to this work. Andrei Simonov’s discussion and
comments at my final seminar have been very useful. The remarks and suggestions
of improvements by Anders Anderson at my licentiate seminar are most appreciated.
The numerous discussions with Anders Eriksson were helpful when writing Essay 2
and 3. I would also like to thank Johan Lyhagen for the comments on Essay 1.
The administrative staff has been very helpful with the practical details. I thank
Monica Ekström, Eva Holst, Katarina Grönvall, Åke Qvarfort, and all the rest for
contributing to the friendly atmosphere at the department as well. Furthermore,
the scholarship from Stiftelsen Bankforskningsinstitutet is gratefully acknowledged.
During the graduate studies, I have made many friends among the department
colleagues. Taking the bike to work each morning, I have looked forward not only
to digging into work, can you believe it, but also to the cheerful lunches and the
laughs during coffee breaks. Many thanks to Martin Söderström, Jovan Zamac, Pär
Holmberg, Jonas Lagerström, Christian Andersson, Fredrik Johansson, Erik Post,
Jon Enqvist, Jenny Nykvist, Mikael Elinder, Hanna Ågren, Andreas Westermark,
and all the rest for the great company. I have also had the privilege to share office
with Mikael Carlsson, Tobias Heldt, and Johan Söderberg at separate time periods.
A special thanks to Tobias for the joyful moments.
I believe that the productivity of working hours is positively related to the enjoyment of leisure time. Whether hiking in the mountains with Henrik Wåhlström,
Hugo Jansson, Kalle Alexandersson, and all the others, enjoying a good dinner with
all of Västeråsarna and Örebroarna, getting beaten in tennis by Tobias Bjöörn, rehearsing and performing with the members of Snerikekören and Västgöta manskör,
or engaging in all the other activities, I have really enjoyed the time off work. Does
this mean that my working hours have been extremely productive? Well, honestly,
I am not too sure about that, but relaxing from work has been essential for my
finishing this thesis. A huge thank you to all the friends, Henke in particular.
vii
Finally, my family has brought warmth and ongoing support through all the
years. My parents, Anders and Lena, have always been there for me, and I cannot
find words to thank them. Earlier this year, my brother Johan’s wife, Regina, had
a baby boy. I am so happy for them, and I cannot wait to see little Oscar again.
A sunny day in Uppsala, October 2006
Martin Ågren
viii
Contents
Introduction
Decision Under Risk: Prospect Theory . . . . . . . . . . . . . . . . . . . .
Oil Price Uncertainty and the Stock Market . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Does Oil Price Uncertainty Transmit to Stock Markets?
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Data Set . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Statistical Model . . . . . . . . . . . . . . . . . . . . . . .
3.1
Parameter Restrictions . . . . . . . . . . . . . . . .
3.2
Estimation . . . . . . . . . . . . . . . . . . . . . . .
3.3
Tests of Model Fitness . . . . . . . . . . . . . . . .
3.4
Testing for Volatility Spillover . . . . . . . . . . . .
4
Empirical Results . . . . . . . . . . . . . . . . . . . . . . .
4.1
Primary Estimations . . . . . . . . . . . . . . . . .
4.2
Second set of Estimations with Leaded Oil Price . .
5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Related Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Myopic Loss Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Cumulative Prospect Theory . . . . . . . . . . . . . . . . . . .
3.2
Explaining the Large Equity Premium with Myopic Loss Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Simulating Returns Distributions . . . . . . . . . . . . . . . . . . . .
4.1
Non-Parametric Bootstrap Approach . . . . . . . . . . . . . .
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4.2
Parametric Approach Using a GARCH Model . . . . . . . .
4.3
Differences in Simulated Distributions . . . . . . . . . . . . .
5
Application to Financial Data . . . . . . . . . . . . . . . . . . . . .
5.1
Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Are the Stock Returns Conditionally Heteroskedastic? . . . .
5.3
Comparing Evaluation Periods Obtained from the Two Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . .
5.5
What Drives the Results? . . . . . . . . . . . . . . . . . . .
6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Prospect Theory and Higher Moments
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2
Cumulative Prospect Theory . . . . . . . . . . . . . . . .
2.1
Value Function . . . . . . . . . . . . . . . . . . .
2.2
Probability Weighting . . . . . . . . . . . . . . .
2.3
Incorporating a Distributional Assumption . . . .
3
Normal Inverse Gaussian Distribution . . . . . . . . . . .
3.1
An Alternative Parameterization . . . . . . . . .
4
Utility in Relation to Distributional Characteristics . . .
4.1
Investor Utility with NIG Distributed Returns . .
4.2
Analysis Procedure . . . . . . . . . . . . . . . . .
4.3
Results . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Sensitivity Analysis . . . . . . . . . . . . . . . . .
5
Optimal Portfolio Choice with NIG Distributed Returns
5.1
Data Set . . . . . . . . . . . . . . . . . . . . . . .
5.2
Portfolio Choice Problem . . . . . . . . . . . . . .
5.3
Results . . . . . . . . . . . . . . . . . . . . . . . .
6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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104
Introduction
The thesis consists of three self-contained essays in financial economics. The first
essay, which also relates to macroeconomics, investigates the transmission of volatility from oil prices to the stock market. The second two essays concern behavioral
finance, and in particular prospect theory, which is a descriptive theory of individual
decision-making under risk. A common theme linking all three essays is the focus
on the statistical properties of financial returns, specifically their conditional and
unconditional moments. Since the second and third essays constitute the thesis’
main part, this introduction focuses mainly on prospect theory. The introduction
ends with a brief summary of the first essay.
Decision Under Risk: Prospect Theory
Expected utility has long been the dominant normative and descriptive model of
individual decision-making under risk. Most textbooks in economics introduce expected utility theory as the cornerstone framework. Von Neumann and Morgenstern
(1944) present the expected utility theorem, which states that if an individual’s behavior satisfies the four axioms of completeness, transitivity, continuity, and independence her preferences are representable by a utility function of expected utility
form. A large body of experimental research severely questions the theory as a
descriptive model of risky choice, arguing that observed individuals systematically
violate the axioms, and perform decision biases. Allais’ (1953) famous example,
also known as the Allais paradox, pioneers this research by showing that a change
in probabilities from 0.99 to 1 has larger impact on individual preferences than a
change from 0.10 to 0.11. Such intuitive behavior violates the independence axiom
of expected utility. In response to the experimental evidence, several non-expected
utility theories have been proposed, e.g., rank-dependent utility (Quiggin, 1982),
disappointment aversion (Gul, 1991), and prospect theory (Kahneman and Tversky,
1979; Tversky and Kahneman, 1992).
Prospect theory of Kahneman and Tversky (1979) is perhaps the most complete
model of the individual behavior observed in experimental settings. Over the last
2
Introduction
decades, prospect theory and its modified version cumulative prospect theory of
Tversky and Kahneman (1992) have been fruitfully applied in several research areas,
e.g., labor economics and macroeconomics (Camerer, 1998). It is in the context
of financial economics that the theory has found most prominence however. In
recognition of prospect theory and many other brilliant contributions to economics,
psychologist Daniel Kahneman was awarded The Bank of Sweden Prize in Economic
Science in Memory of Alfred Nobel in 2002.
Under cumulative prospect theory, investors derive utility by using a specific
value function, and by weighting probabilities non-linearly. The value function differs from standard concave utility functions in three main respects. First, utility is
derived from changes in wealth relative to a reference point, as opposed to final levels of wealth. This is not only motivated by experiments but through introspection,
since attributes such as brightness and temperature are often perceived relative to
earlier levels. Second, the value function is concave over gains, implying risk aversion, but convex over losses, reflecting a risk-seeking behavior in that domain. Third,
losses loom larger than gains do, which is referred to as loss aversion. Typically, the
displeasure of losses is about twice the pleasure of equally-sized gains. Conventional
risk aversion of traditional concave utility functions induce a sensitivity for losses
as well. What distinguishes loss aversion from conventional risk aversion, however,
is the abrupt change in marginal utility at the reference point. While loss aversion
causes for a kink in the value function so that individuals are severely averse even to
small-scale gambles, standard concave utility functions imply that people are close
to risk neutral over small stakes.
Panel A
Panel B
Panel C
1
0.5
τ(p)
0
−1
v(x)
u(W+x)
−0.5
−0.5
0.5
−1.5
−1
−2
0.5
1
W+x
1.5
−0.5
0
x
0.5
0
0
0.5
p
1
Panels A and B illustrate a standard utility function of expected utility theory, and the
cumulative prospect theory value function, respectively. Panel C shows the cumulative prospect
theory probability weighting function.
Panels A and B of the figure illustrate a standard utility function of expected
utility theory and the cumulative prospect theory value function, respectively. Current wealth, W , is standardized to one, and the risky outcome is denoted by x.
Notice that the value function is kinked by loss aversion (panel B).
Decision Under Risk: Prospect Theory
3
The figure illustrates the non-linear probability weighting of cumulative prospect
theory as well (panel C). Traditionally, linear probabilities (the dashed line) are assigned to each event when deriving utility. On the contrary, cumulative prospect
theory weights probabilities non-linearly (the solid line) so that small probabilities
are over-weighted, magnifying the tails of the distribution, and large ones are underweighted. Such a transformation helps to explain anomalies such as the Allais paradox, since probabilities of about 0.99, but not one, are extensively under-weighted,
while probabilities of 0.10 and 0.11 are almost equally affected. Furthermore, the
over-weighting of small probabilities explains why people buy lottery tickets, which
is dubious under standard theory.
Why should economists be interested in cumulative prospect theory as an alternative to expected utility? Although individual experiments show violations of
the expected utility theorem, the implications might be irrelevant for the aggregate investor’s behavior. However, considering the difficulties of standard theory
to explain even basic facts about, for instance, financial markets, it is only natural that economic theory turns to alternatives based on the experimental evidence.
Behavioral finance builds theoretical foundations for financial economics assuming
individual behavior is not fully rational but includes the ingredients suggested from
experiments.
Essay 2, Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH, concerns what is perhaps the most influential application of cumulative prospect theory.
Benartzi and Thaler (1995) (BT henceforth) argue that loss-averse agents attribute
lower risk to stocks if evaluated less often, since the probability of a fall in stock prices
decreases with a longer time horizon. The argument is formalized into a preferencescheme called myopic loss aversion, and the historical equity premium over bonds
is suggestively explained if agents evaluate their portfolios annually. The result is
intuitive, since most individual investors file their taxes and receive their financial
reports on a yearly basis. Consequently, BT propose a behavioral explanation to the
infamous equity premium puzzle of Mehra and Prescott (1985), which concerns the
inability to explain the large equity premium within a standard consumption-based
general equilibrium model.
To arrive at their twelve-month evaluation period result, BT consider a portfolio
of either one hundred percent stocks or one hundred percent bonds. The cumulative
prospect theory utility of each portfolio is derived at a stepwise increasing evaluation
period, which is reflected by the use of data at different frequencies, starting at
one month. For example, if the investor evaluates her portfolio every six months
the utility of holding a stock portfolio is derived using six-month data on stock
4
Introduction
returns. While the bond portfolio has larger utility than the stock portfolio at short
evaluation periods of, say, one month, the two portfolios are equally attractive when
evaluated annually. BT interpret this as an "equilibrium", where financial investors
are content with the risk-return relationship of stocks and bonds.
Essay 2 suggests a technical extension to BT’s methodology, and, also, replicates their study using both U.S. and Swedish data. Specifically, the distributional assumption made on returns is addressed. BT simulate returns distributions
using a non-parametric bootstrap procedure, which, implicitly, assumes that returns are temporally independent. Extensive empirical evidence severely questions
this assumption, showing that conditional volatilities are predictable. Therefore,
Essay 2 proposes a parametric approach to simulating returns where conditional
heteroskedasticity is incorporated. The method involves the temporal aggregation of generalized autoregressive conditionally heteroskedastic (GARCH) processes
(Bollerslev, 1986; Drost and Nijman, 1993).
The results show, first, that both the U.S. and the Swedish equity premiums can
be explained by myopic loss aversion, although the obtained evaluation periods are
occasionally shorter than a year. For instance, over the full U.S. sample period from
1926 to 2003, evaluation periods of seven and ten months are found using BT’s and
the proposed methods of simulating returns, respectively. The result that myopic
loss aversion can explain the Swedish equity premium is new to the literature, since
previous studies have considered only U.S. data.
Second, throughout the analysis of both data sets, longer evaluation periods are
obtained under conditional heteroskedasticity. This result indicates that the model
of myopic loss aversion is sensitive to the method used when simulating returns,
which relates directly to the shape of the returns distribution. Thus, the essay concludes that myopic loss-averse and, in turn, cumulative prospect theory investors
are sensitive to the distributional assumption made on returns. Plausibly, the distributional skewness is important.
Essay 3, Prospect Theory and Higher Moments, builds on the conclusion of Essay 2, and investigates the cumulative prospect theory utility dependence on the
moments of returns distributions, specifically the mean, the variance, the skewness, and the kurtosis. To do so, returns are assumed to follow the normal inverse
Gaussian distribution (Barndorff-Nielsen, 1997), which has the desirable property
of parameter-dependent first- to forth-order moments. Using this distribution and
a result of Eriksson, Forsberg, and Ghysels (2005), it is straightforward to analyze
utility as a function of a specific moment in isolation, i.e., without affecting the other
Decision Under Risk: Prospect Theory
5
moments. This makes the normal inverse Gaussian distribution highly suitable for
the investigation.
The findings of Essay 3 deepens the understanding of how cumulative prospect
theory relates to the higher-order moments of returns distributions, i.e., skewness
and kurtosis. Previous literature focuses mainly on the normality assumption, where
returns are fully characterized by the mean and the variance (Levy, De Giorgi, and
Hens, 2003; Levy and Levy, 2004; Barberis and Huang, 2005). The results show
that the skewness typically has a negative impact on utility when probabilities are
weighted linearly, but once probabilities are non-linearly transformed a clear preference for skewness appears. This shows that cumulative prospect theory investors
display a preference for skewness through the probability weighting function. Furthermore, utility is found to be positively related to the kurtosis when the investor
treats probabilities objectively, but inverse hump-shape related when probability
weighting is incorporated.
Essentially, loss aversion makes the investor sensitive to the probability of small
losses. On the contrary, probability weighting causes the investor to care more about
the probability of large losses. Thus, cumulative prospect theory investors prefer
lottery-type gambles with positively skewed outcomes, since there is the possibility
of a large gain at the risk of only a small loss. Loss-averse investors who treat
probabilities linearly are averse to such gambles however, since they incur a small
but almost sure loss.
What consequences do these findings have for portfolio choice? Essay 3 investigates this issue by optimizing the allocation to a risky and a relatively risk-free
asset, paying special interest to the implications of higher-order moments and, also,
to probability weighting, which have both received little attention in the previous
literature. Strong horizon effects in the investor’s asset allocation are found, i.e., the
portion to stocks progresses heavily as the horizon increases. This result is consistent
with related work, e.g., Aït-Sahalia and Brandt (2001). Moreover, it is suggested
that cumulative prospect theory optimal portfolios depart from mean-variance efficiency when returns follow the normal inverse Gaussian distribution. When taking
the higher moments into account, the investor typically places a larger weight in
stocks. Since higher moments are important to the cumulative prospect theory investor, the main priority is not mean-variance efficiency but a more complicated
preference-scheme including all first four moments.
6
Introduction
Oil Price Uncertainty and the Stock Market
Over recent years, oil prices have risen dramatically. During April of 2006, the
price per barrel Brent crude oil was in the neighborhood of (U.S.) $70, which is far
above the price of $20 during most of the 1990s. There are many plausible causing
factors, e.g., surging demand, tight supply, financial speculation, and the political
instability of the Middle East. A sensible forecast of future oil prices does not rule
oil any of these factors, and prices could possibly rise even more just as well as they
could fall. The only safe statement about oil prices is that they will continue to be
unstable. This is the starting point of the thesis’ first essay, which investigates the
transmission of oil price uncertainty to stock markets.
Hamilton (1983) presents an influential article on the effects of oil shocks on the
economy, showing that almost all U.S. recessions since the second world war have
been preceded by oil shocks. Following Hamilton (1983), a large body of research
has been presented, where the general consensus is that oil shocks are negatively
related to macroeconomic variables, e.g., industrial production and employment
(Mork, 1994). Furthermore, the impact of oil shocks on the economy appears to be
asymmetric, with positive oil shocks having larger effects than negative ones (Mork,
Olsen, and Mysen, 1994).
There exist some papers on the impact of oil shocks on the stock market, e.g.,
Jones and Kaul (1994), and Sadorsky (1999). Theoretically, stock prices equal
the discounted expectation of future cash-flows (dividends), which are likely to be
affected by macroeconomic movements and, in turn, oil shocks. Jones and Kaul
(1994) investigate this issue, and find evidence that the U.S. and Canadian aggregate
stock markets are rational in the sense that they fully account for oil shocks via the
effects on dividends.
While most of the previous work relates the level of oil price changes to the
level of stock returns, i.e. the first-order moment is analyzed, the first essay demonstrates a study of oil price and stock market volatility, i.e. the second-order moment
is considered. Similar investigations are conducted, e.g., by Schwert (1989) and
King, Sentana, and Wadhwani (1994). However, the essay here uses a substantially
different model.
Essay 1, Does Oil Price Uncertainty Transmit to Stock Markets?, conducts
an empirical investigation of volatility spillover from oil prices to stock markets.
The issue is studied empirically within a bivariate GARCH model, specifically, the
asymmetric BEKK model of Engle and Kroner (1995) and Kroner and Ng (1998).
Using aggregate stock market data representing Japan, Norway, Sweden, the U.K.,
and the U.S. over the sample period from week one of 1989 to week seventeen of 2005,
Oil Price Uncertainty and the Stock Market
7
strong evidence of volatility spillover is found for all economies but Sweden, where
only weak evidence is found. The results thus indicate that oil price uncertainty does
transmit to stock markets. News impact surfaces of Kroner and Ng (1998) display
small quantitative implications however. The stock market’s own shocks, which are
related to other factors of uncertainty than the oil price, are more prominent than
the effects of oil shocks. The essay improves our knowledge of how stock markets
link to oil prices.
8
Introduction
References
Aït-Sahalia, Y. and M. Brandt (2001), "Variable selection for portfolio choice",
Journal of Finance 56, 4, 1297-351.
Allais, M. (1953), "Le comportement de l’homme rationnel devant le risque, critiques des postulates et axiomes de l’ecole americaine", Econometrica 21, 503-46.
Barberis, N. and M. Huang (2005), "Stocks as lotteries: The implications of probability weighting for security prices", Working paper, Yale University.
Barndorff-Nielsen, O. (1997), "Normal inverse Gaussian distributions and stochastic
volatility modeling", Scandinavian Journal of Statistics 24, 1-13.
Benartzi, S. and R. Thaler (1995), "Myopic loss aversion and the equity premium
puzzle", Quarterly Journal of Economics 110, 73-92.
Bollerslev, T. (1986), "Generalized autoregressive conditional heteroskedasticity",
Journal of Econometrics 31, 307-27.
Camerer, C. (1998), "Prospect theory in the wild: Evidence from the field", in
D. Kahneman and A. Tversky, Choices, Values, and Frames, Cambridge University
Press, Cambridge.
Drost, F. and T. Nijman (1993), "Temporal aggregation of GARCH processes",
Econometrica 61, 4, 909-27.
Engle, R. and K. Kroner (1995), "Multivariate simultaneous generalized ARCH",
Econometric Theory 11, 122-50.
Eriksson, A., L. Forsberg, and E. Ghysels (2005), "Approximating the probability distribution of functions of random variables: A new approach", in A. Eriksson,
Essays on Gaussian Probability Laws with Stochastic Means and Variances, Ph.D.
dissertation thesis, Department of Information Science, Division of Statistics, Uppsala University.
Gul, F. (1991), "A theory of disappointment aversion", Econometrica 59, 3, 667-86.
Hamilton, J. (1983), "Oil and the macroeconomy since World War II", Journal
of Political Economy 91, 228-48.
Jones, C. and G. Kaul (1996), "Oil and the stock markets", Journal of Finance
51, 2, 463-91.
Kahneman, D. and A. Tversky (1979), "Prospect theory: An analysis of decision
under risk", Econometrica 47, 263-91.
References
9
King, M., E. Sentana, and S. Wadhwani (1994), "Volatility and links between national stock markets", Econometrica 62, 4, 901-33.
Kroner, K. and V. Ng (1998), "Modelling asymmetric comovements of asset returns", Review of Financial Studies 11, 4, 817-44.
Levy, H., E. De Giorgi, and T. Hens (2003), "Prospect theory and the CAPM:
A contradiction or coexistence?", Working paper, Institute for
Empirical Research in Economics, University of Zürich.
Levy, H and M. Levy (2004), "Prospect theory and mean-variance analysis", Review
of Financial Studies 17, 4, 1015-41.
Mehra, R. and E. Prescott (1985), "The equity premium puzzle", Journal of Monetary Economics 15, 145-61.
Mork, K. (1994), "Business cycles and the oil market (special issue)", Energy Journal 15, 15-37.
Mork, K., Ø. Olsen, and H. Mysen (1994), "Macroeconomic responses to oil price
increases and decreases in seven OECD countries", Energy Journal 15, 4, 19-35.
Sadorsky, P. (1999), "Oil price shocks and stock market activity", Energy Economics 21, 5, 449-69.
Schwert, G. (1989), "Why does stock market volatility change over time?", Journal
of Finance 44, 5, 1115-53.
Tversky, A. and D. Kahneman (1992), "Advances in prospect theory: Cumulative representation of uncertainty", Journal of Risk and Uncertainty 5, 297-323.
Quiggin, J. (1982), "A theory of anticipated utility", Journal of Economic Behavior
and Organization 3, 323-43.
von Neumann, J. and O. Morgenstern (1944), Theory of Games and Economic Behavior, Princeton University Press.
Essay 1
Does Oil Price Uncertainty
Transmit to Stock Markets?
1
Introduction
Understanding the links between financial markets is of great importance for a financial hedger, portfolio manager, asset allocator, or other financial analysts. The study
of volatility spillover from one market to another is a crucial part of this issue. There
exists a large literature on volatility spillover, and a variety of markets have been
considered, such as the equity, the bond, and the exchange rate markets. Karolyi
(1995) examines the short-run dynamics of returns and volatility between the U.S.
and Canadian stock markets. Kearney and Patton (2000) study how exchange rate
volatility transmits within the European monetary system prior to the unification of
currencies. Furthermore, Bollerslev, Engle, and Wooldridge (1988) model the conditional covariance of returns to bills, bonds, and stocks, and find that the covariances
are quite variable over time. This paper analyzes the conditional volatility of oil and
stock markets. Does oil price uncertainty transmit to stock markets? The issue is
studied empirically within a bivariate generalized autoregressive conditionally heteroskedastic (GARCH) model, specifically, the asymmetric BEKK model of Engle
and Kroner (1995) and Kroner and Ng (1998).
It is well documented that the conditional volatilities of stock market indices
change over time. Many researchers are intrigued by the causes for these changes,
and a large empirical literature exists where time series data on financial and macroeconomic variables are studied in relation to stock market data. Officer (1973) is
first to present evidence of a relationship between the market factor (aggregate stock
market) variability and business cycle fluctuations, as measured by industrial production. Schwert (1989) performs vector autoregressions and finds weak evidence
12
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
that macroeconomic volatility can predict stock market volatility. The volatility of
bond returns and the growth rates of the producer price index, the monetary base,
and industrial production, are used as macroeconomic variables. King, Sentana,
and Wadhwani (1994) employ a different approach and estimate a multivariate factor model, where comovements in stock return volatility are induced by the volatility
of a number of factors. Using data on not only the U.S. but on sixteen national stock
markets, King et al. (1994) try to identify the causes for stock volatility through
both "observable" factors, e.g. interest rates, industrial production and oil prices,
and "unobservable" factors, which reflect the influences on stock volatility that are
not captured by published statistics. Their results display little support for the
observable economic variables. Instead, King et al. (1994) argue that unobservable uncertainty contributes to the variability in stock returns, and, also, to the
comovements in stock volatility across national markets.
The current paper shifts focus from general macroeconomic variables to the oil
price, in analyzing the time-variation of stock volatility. The focus on oil is motivated by the large literature relating oil prices to the macroeconomy. Hamilton
(1983) presents an influential article, which shows that almost all U.S. recessions
since the second world war have been preceded by oil shocks. Mork (1994) surveys
the extensive literature on oil and the macroeconomy following Hamilton (1983),
and demonstrates a clear negative correlation between oil prices and aggregate measures of output or employment. Moreover, Hamilton (1985) argues that oil shocks
are exogenous events, since the causes can be attributed to historical events, e.g.,
the Iraq invasion of Kuwait in 1990. Since stock prices, in theory, equal the discounted expectation of future cash-flows (dividends), which are likely to be affected
by macroeconomic movements, they are possibly affected by oil shocks. Also, an
oil price increase acts like an inflation tax on consumption, reducing the amount of
disposible income for consumers. Non-oil producing companies face higher fix costs,
which are passed on to higher consumer prices. These effects decrease company
wealth, lowering their dividends.1
There exist a few papers that link oil prices to stock markets. Jones and Kaul
(1996) test whether stock markets are rational in the sense that they fully adjust
to the impact of oil shocks on dividends. Studying the U.S., Canadian, Japanese,
and U.K. stock markets, Jones and Kaul (1996) initially show that all the markets
respond negatively to oil shocks. A cash-flow valuation model is then applied, and
1
Recently, Rogoff (2006) surveys the literature on oil shocks and the global economy, and argues
that most oil consuming countries are less vulnerable to oil shocks than they were a few decades
ago. The greater energy efficiency is reported as one reason. Nevertheless, Rogoff (2006) stresses
that it would be very wrong to consider the oil-induced recessions as a thing of the past.
1. Introduction
13
evidence is found that U.S. and Canadian stock indices fully account for oil shocks
via the effects on dividends. In contrast, stock markets in Japan and the U.K.
display larger variation, following an oil shock, than can be explained by changes in
dividends. While Jones and Kaul (1996) use quarterly data, Huang, Masulis, and
Stoll (1996) consider daily data on the oil futures market and the stock market, and
estimate a vector autoregressive model. Evidence of a connection between oil futures
returns and oil stock returns is presented. There is no such support for aggregate
stock returns during the 1980s however. Sadorsky (1999) studies the impact of real
oil price shocks on real stock returns by estimating vector autoregressions, including
U.S. industrial production and short interest rates. The study separates positive
from negative oil shocks, and, contrary to Huang et al. (1996), presents evidence
that shocks to the oil price do affect aggregate stock returns. Moreover, the impact
appears to be asymmetric, since positive oil shocks are of large importance, whereas
negative ones have little or no effect. Basher and Sadorsky (2004), using a multifactor arbitrage pricing model, find strong evidence that oil price risk impacts returns
of emerging stock markets.2
Most of the previous work relates the level of oil price changes to the level of stock
returns, i.e., first-order moments are analyzed. The current paper demonstrates a
study of oil price and stock market volatility, i.e., second-order moments are considered. Although some papers address this issue, e.g., Schwert (1989) and King
et al. (1994), the present paper employs a substantially different model. The bivariate GARCH model specifies the conditional variances and covariance of oil price
changes and stock returns so that, for instance, volatility spillover can be tested for
in a simple manner.
Bollerslev et al. (1988) introduce multivariate GARCH (MGARCH) modeling,
and propose a general parameterization of the conditional covariance matrix called
VECH.3 The VECH model does not impose any restrictions on its parameters,
implying that the positive definiteness of the conditional covariance matrix is not
guaranteed. The model is also quite computer-intensive in estimation, relative to
other MGARCH models, because of its large number of parameters. To circumvent
these problems, Engle and Kroner (1995) present the BEKK specification of the
conditional covariance, and, later on, Kroner and Ng (1998) extend this model to
allow for asymmetry.4 The BEKK model is specified using quadratic forms, which
guarantees positive definiteness.
2
Other studies relating oil to stock markets include Sadorsky (2003) and Huang, Hwang, and
Peng (2005).
3
The name stems from its use of the vech-operator, which stacks the lower-triangular elements
of a square matrix into a vector.
4
The BEKK acronym stems from an unpublished paper by Y. Baba, R. Engle, D. Kraft, and
K. Kroner.
14
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
This paper adopts the asymmetric BEKK (ABEKK) model to examine if oil price
volatility transmits to stock market volatility. A bivariate VAR(2)-ABEKK model is
estimated using weekly returns on five aggregate stock market indices and a measure
of the oil world price.5 Parameter restrictions are imposed so that stock returns do
not affect oil prices, motivated by the proposed exogenity of oil shocks (Hamilton,
1985). The asymmetric effects of oil price shocks are motivated empirically by Mork,
Olsen, and Mysen (1994), studying macroeconomic variables, and, as previously
mentioned, Sadorsky (1999). Over the sample period from week one of 1989 to week
seventeen of 2005, strong evidence of volatility spillover is found for Japan, Norway,
the U.K., and the U.S. Weak evidence of volatility spillover is found for Sweden
over the sample period. Although the empirical results show that volatility spills
over from oil to stock markets, news impact surfaces, which illustrate the estimated
one-period ahead impact of an oil shock, reveal small quantative effects. The stock
market’s own shocks, which are related to other sources of stock market uncertainty
than the oil price, have more prominent implications.
Are the obtained results sensitive to the choice of data frequency? Since stock
markets respond quickly to economic uncertainty, it might be that volatility spills
over at a faster pace than first examined. Therefore, a second set of estimations
is carried out where the weekly oil price data is leaded one period. In this way,
volatility spillover is tested within the week instead of from one week to the next.
Evidence of volatility spillover is however not found, supporting the primary use of
weekly data. In all, the paper deepens our knowledge of how stock markets link to
oil prices.
Studying the oil price influence on stock markets is an interesting and important
issue, even more so recently when the world oil price has displayed great instability.
During April of 2006, the price of crude oil was in the neighborhood of (U.S.) $70
per barrel, which is well above the price of $20 during most of the 1990s. In a recent
survey of oil in the Economist, Vaitheeswaran (2005) proposes that the explanation
for the rise is that oil markets have seen an abnormal combination of tight supply,
surging demand, and financial speculation. One might also consider the unstable
political situation in the Middle East a candidate cause for the rise in oil prices.
With these aspects in mind, is the future price per barrel of oil expected to rise
even more? Not necessarily. The extensive list of oil projects financed by both
governments and private firms might lead to such a great supply that even high
demand economies, e.g. China, cannot prevent a future supply shock, leading to a
5
VAR(2) is an abbreviation for the second order vector autoregressive model.
2. Data Set
15
possible decline in prices.6 The only safe statement about oil prices in the future,
Vaitheeswaran (2005) argues, is that they will continue to be highly unstable.
The remaining part of the paper is organized as follows: Section 2 presents the
data set used. The statistical model is introduced in section 3, where estimation
and testing issues are discussed as well. Section 4 reports on the empirical results,
and section 5 concludes.
2
Data Set
National stock markets are most likely to be affected differently by oil shocks depending of the overall country-dependence on oil. For this reason it is important to
include a number of markets in the current analysis. I use a set of data consisting
of aggregate stock market indices representing five developed economies, namely
Japan, Norway, Sweden, the U.K., and the U.S., together with a measure of the
world oil price. Each index describes the overall performance of large-capitalization
firms in the respective country. Dividends are assumed reinvested at the end of each
period, and, hence, accounted for in the data. Furthermore, the price per barrel
Brent crude measures the world oil price.7 All data are at the weekly frequency
(last observation of the week), and cover the first week of 1989 through week seventeen of 2005, yielding a total of 852 observations. By using weekly data the study
is relieved from the noise of higher frequency data, e.g. daily or intraday, while
still capturing much of the information content of stock indices and oil prices, as
opposed to lower frequency data, e.g. monthly or quarterly. Moreover, the choice of
data frequency differs the current analysis from previous work within the literature,
where daily, monthly, as well as quarterly data have been considered.
The percentage change or return over one data period, denoted rit , is derived as
one hundred times the log-difference in prices over the period, i.e.,
rit = 100 × log
Pit
,
Pi,t−1
(1)
where Pit is the price level of market i at time t. Table 1 reports on summary statistics of the return data on all five stock indices and the oil price. All stock markets
but Japan have had a positive average weekly return over the sample period. Stan6
Recently, in May of 2005 the Baku-Tbilsi-Ceyhan pipeline was opened, bringing Caspian oil
to the world market. When its potential is fully operated in 2009, the pipeline will carry about
one million barrels of oil per day, or more than one percent of the world oil market.
7
The Brent blend is a light and sweet crude that ships from Sullom Voe in the Shetland Islands.
It serves as a benchmark for pricing oil from regions such as Europe, Africa, and the Middle East.
16
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
Table 1: Summary Statistics for Weekly Percentage Returns on Five National Stock
Market Indices and the Oil Price
Mean (%)
Max. (%)
Min. (%)
Std. dev. (%)
Skewness
Ex. kurtosis
JB
LBQ
LBQ2
ARCH LM
Japan
-0.0404
11.0
-10.4
2.72
0.0434
0.96
33.0*
(<0.001)
11.9
(0.156)
91.4*
(<0.001)
56.1*
(<0.001)
Norway
0.176
11.96
-18.21
2.704
-0.493
3.538
478*
(<0.001)
23.67*
(0.0026)
102*
(<0.001)
86.9*
(<0.001)
Sweden
0.206
17.05
-15.4
2.87
-0.108
3.01
323*
(<0.001)
30.6*
(<0.001)
102*
(<0.001)
65.4*
(<0.001)
U.K.
0.185
9.72
-8.50
2.04
-0.0626
1.63
95.1*
(<0.001)
5.64
(0.688)
79.8*
(<0.001)
64.5*
(<0.001)
U.S.
0.209
7.53
-12.3
2.13
-0.500
2.99
352*
(<0.001)
25.3*
(0.0014)
76.4*
(<0.001)
56.1*
(<0.001)
OIL
0.128
21.0
-32.7
5.21
-0.585
3.21
414*
(<0.001)
12.4
(0.133)
59.0*
(<0.001)
37.9*
(<0.001)
The table displays summary statistics for weekly returns on the aggregate stock markets of Japan
(S&P/TOPIX), Norway (BXLT), Sweden (SIXRX), the U.K. (FTSE350), and the U.S. (S&P500),
along with the price change of Brent crude oil. The sample period is from 1989:1 to 2005:17. JB is
the Jarque-Bera statistic under the null of normality. LBQ (LBQ2 ) is the univariate Ljung-Box Q
statistic for serial correlation in returns (squared returns). ARCH LM is the Lagrange multiplier
test of autoregressive conditional heteroskedasticity (Engle, 1982). All tests of correlation use
eight lags. p-values are in parentheses. * indicates significance at five percent level. Data source:
EcoWin Pro.
dard deviations are centered around 2.5 percent except for the oil price, which has
varied 5.21 percent. All data series display non-zero skewness and excess kurtosis,
leading to highly significant Jarque-Bera statistics, which indicate that the returns
are non-normally distributed. Moreover, results of the Ljung-Box Q test suggest
that serial correlations exists in the Norwegian, the Swedish, and the U.S. stock return data. Both the Ljung-Box Q test for squared returns and the ARCH Lagrange
multiplier test indicate strong presence of ARCH-structure in all data series.
Figures 1 and 2 illustrate the data. Figure 1 plots the weekly index/price level of
the six data series. Observe that the oil price was rather stable around $20 during
the 1990s apart from the spike in 1990-91 when the Gulf war commenced. Since the
1990s, the oil price level has shown more instability. The aggregate stock market
indices have risen during the sample period, with the exception of Japan. Figure
2 graphs the weekly percentage returns of all data series derived according to (1).
Notice that the stock return conditional volatilities are historically large, even during
the 1990s. The return series display volatility persistence (clustering) in accordance
with the previous statistical test results. It is, however, difficult to visually detect
2. Data Set
17
Figure 1: Weekly Indices of Five National Stock Markets and the Oil Price
JPN
NOR
1600
280
240
1400
200
1200
160
1000
120
800
80
600
40
90
92
94
96
98
00
02
04
90
92
94
OIL
96
98
00
02
04
98
00
02
04
98
00
02
04
SWE
60
500
50
400
40
300
30
200
20
100
10
0
0
90
92
94
96
98
00
02
04
90
92
94
UK
96
USA
3200
5000
2800
4000
2400
3000
2000
1600
2000
1200
1000
800
400
0
90
92
94
96
98
00
02
04
90
92
94
96
The figure plots the historical development of five aggregate stock markets representing Japan,
Norway, Sweden, the U.K., and the U.S., respectively; along with the price of Brent crude oil in
2005 U.S. dollars per barrel. The sample period covers week one of 1989 through week seventeen
of 2005.
18
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
Figure 2: Weekly Returns of Five National Stock Markets and Changes in the Oil
Price
JPN_R
NOR_R
12
15
8
10
5
4
0
0
-5
-4
-10
-8
-15
-12
-20
90
92
94
96
98
00
02
04
90
92
94
OIL_R
96
98
00
02
04
00
02
04
00
02
04
SW E_R
30
20
16
20
12
10
8
0
4
0
-10
-4
-20
-8
-30
-12
-40
-16
90
92
94
96
98
00
02
04
90
92
94
UK_R
96
98
USA_R
10
8
4
5
0
0
-4
-8
-5
-12
-10
-16
90
92
94
96
98
00
02
04
90
92
94
96
98
The figure plots historical returns (%) of five aggregate stocks representing Japan, Norway, Sweden,
the U.K., and the U.S., respectively; along with the price change of Brent crude oil. The sample
period covers week two of 1989 through week seventeen of 2005.
3. Statistical Model
19
any comovements in conditional volatility between oil and stock markets. I leave
this to statistical modeling and testing.
3
Statistical Model
Consider a bivariate sequence of data {rt }Tt=1 consisting of oil price changes and
stock market returns. The following statistical model is employed:
rt = μ + δrt−1 + πrt−2 + εt ,
εt =
1/2
Ht vt ,
(2)
(3)
and
Ht = C0 C + A0 εt−1 ε0t−1 A + B0 Ht−1 B + G0 η t−1 η 0t−1 G,
(4)
where εt is a 2 × 1 vector of residuals, vt is a 2 × 1 vector of standardized (i.i.d.)
residuals, Ht is the 2 × 2 conditional covariance matrix, ηt is a 2 × 1 asymmetric
term (defined subsequently), and μ, δ, π, C, A, B, and G are model parameter
matrices. The mean equation (2) is represented by a VAR(2) model. In this way,
any existing serial correlation in the return series is removed, which is crucial since
the parameter estimates of Ht would otherwise be biased. The conditional variancecovariance matrix of (4) is specified according to the ABEKK model of Kroner and
Ng (1998). Notice that the structure consists of quadratic forms, which secures the
positive definiteness of Ht . The statistical model of (2)-(4) is referred to as the
VAR(2)-ABEKK model.8
The ABEKK model includes an asymmetric term, η t = (η 1t , η 2t )0 , which elements
are defined as: η it = max[εit , 0], for oil price changes; and η it = min[εit , 0], for stock
returns. This specification of η t emphasizes on the effects of positive oil shocks and
negative stock returns. The latter emphasis is motivated by Glosten, Jagannathan,
and Runkle (1993).
3.1
Parameter Restrictions
To ensure that stock prices have no impact on oil prices, which is economically justifiable following Hamilton (1985), some restrictions are imposed on the parameter
matrices of (2) and (4).9 Explicitly, the restricted VAR(2)-ABEKK model has the
8
Bauwens, Laurent, and Rombouts (2006) survey the literature on MGARCH models.
The restricted model is supported statistically as well, since the parameters that are restricted
to zero display insignificant estimates following an unrestricted estimation.
9
20
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
following structure:
and
r1t = μ1 + δ 11 r1,t−1 + π 11 r1,t−2 + ε1t ,
(5)
r2t = μ2 + δ 21 r1,t−1 + δ 22 r2,t−1 + π 21 r1,t−2 + π 22 r2,t−2 + ε2t ,
(6)
Ã
ε1t
ε2t
!
=
Ã
h11,t h12,t
h12,t h22,t
!1/2 Ã
v1t
v2t
!
,
(7)
where
2 2
h11,t = c211 + a211 ε21,t−1 + b211 h211,t−1 + g11
η 1,t−1 ,
h12,t = c11 c12 +
a11 a12 ε21,t−1
(8)
+ a11 a22 ε1,t−1 ε2,t−1
+b11 b12 h11,t−1 + b11 b22 h12,t−1
+g11 g12 η 21,t−1 + g11 g22 η 1,t−1 η 2,t−1 ,
(9)
h22,t = c212 + c222 + a212 ε21,t−1 + a222 ε22,t−1 + 2a12 a22 ε1,t−1 ε2,t−1
+b212 h11,t−1 + b222 h22,t−1 + 2b12 b22 h12,t−1
2 2
2 2
+g12
η 1,t−1 + g22
η 2,t−1 + 2g12 g22 η 1,t−1 η 2,t−1 .
(10)
In (5) and (6), r1t (r2t ) represents the period t percentage change in oil (aggregate stock) prices according to (1). Stock returns do not affect oil price changes
in equation (5), but oil price changes do affect stock returns, as (6) shows. Moreover, the conditional variance of oil price changes, h11,t , is simply modeled by the
univariate GJR(1,1) model of Glosten et al. (1993), while the conditional variance
of stock returns, h22,t , and the conditional covariance, h12,t , are modeled with more
complexity. The ABEKK model allows, for instance, the conditional variance of
stock returns to depend on its own lagged conditional variance and lagged shocks,
the lagged conditional variance and lagged shocks of oil price changes, as well as
cross-terms. The parameter a12 in (10) captures the effect of an oil shock at t − 1
on the conditional variance of stock returns at t, and b12 measures the impact of
the oil price conditional variance on the one-period ahead conditional variance of
stock returns. The parameters of the ABEKK specification do not represent such
impacts directly however, since parameters are squared or cross-multiplied. This
implies that the interpretation of individual parameter estimates is not straightforward. Nevertheless, the statistical significance of the parameter estimates can be
investigated.
3. Statistical Model
3.2
21
Estimation
The bivariate restricted VAR(2)-ABEKK model is estimated using the quasi maximum likelihood (QML) method of Bollerslev and Wooldridge (1992). Given T
observations of rt = (r1t , r2t )0 , the following optimization is considered:
max log LT (θ) =
θ
T
X
lt (θ),
(11)
t=1
where LT is the sample likelihood function, θ is a vector of parameters,
1
lt (θ) = log(2π) − log |Ht | − ε0t H−1
t εt
2
(12)
is the conditional log-likelihood function for a bivariate normally distributed variable, and εt = (ε1t , ε2t )0 and vech(Ht ) = (h11,t , h12,t , h22,t )0 follow (7) and (8)-(10),
respectively. QML robust standard errors of the parameter estimates are derived to
account for the possibly false normality assumption.
To carry out the optimization of (11), the BFGS quasi-Newton method is applied
using the Matlab programming language.10 Since the parameter vector θ has a total
of 20 parameters (see equations (5)-(10)), the optimization is quite intricate and
sensitive to starting values. I use a number of simplex-algorithm iterations following
an initial guess of parameter values.11 This is found to help for convergence.
3.3
Tests of Model Fitness
To test the model’s fitness, the obtained estimated standardized residuals v̂t =
(v̂1t , v̂2t )0 are analyzed. These are derived as the inverse of the Cholesky decomposition of Ht times the estimated residual vector ε̂t , in line with (3). The statistical
model provides a good fit to the empirical data if a test of remaining serial correlation and ARCH-structure comes out insignificant. Two such tests are performed,
namely the multivariate Ljung-Box Q test, and a bivariate test based on the generalized method of moments (GMM).
Multivariate Ljung-Box Q Test
The multivariate Ljung-Box Q (MLBQ) test of Hosking (1980) is a test of serial
correlation.12 Under the null that v̂t is independent of v̂t−1 , ..., v̂t−K , where K is the
10
The unconstrained minimization routine fminunc is employed.
The Matlab function fminsearch is employed.
12
Box and Jenkins (1976) present the univariate Ljung-Box Q test.
11
22
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
maximum lag length, the test statistic
MLBQ = T (T + 2)
K
X
j=1
©
ª
1
−1 0
−1
tr C0j C00
C0j C00
,
T −j
(13)
P
0
, is derived.13 Applying the test to the squared
where C0j = T −1 Tt=j+1 v̂t v̂t−j
standardized residuals, v̂t2 , the MLBQ test provides a test for ARCH-effects too,
referred to as the MLBQ2 test. The statistic in (13) is χ2 distributed with 4(K − 2)
degrees of freedom. The lag length is arbitrarily set to K = 8, implying that serial
correlation up to eight weeks is examined.
GMM Test
The GMM approach to testing the model fitness relies on a number of moment conditions. One advantage compared with the MLBQ test is that the GMM approach
tests for serial correlation and ARCH-effects simultaneously. Under the null of a correctly specified model, {v̂it }, {v̂it2 − 1}, and {v̂1t v̂2t } should be serially uncorrelated.
The following moment conditions should therefore hold:
E[(v̂it2
−
E[v̂it v̂i,t−k ] = 0, for i = 1, 2,
(14)
E[v̂it v̂j,t−k ] = 0, for (i, j) = (1, 2) and (2, 1),
(15)
2
1)(v̂i,t−k
− 1)] = 0, for i = 1, 2,
E[(v̂1t v̂2t )(v̂1,t−k v̂2,t−k )] = 0,
(16)
(17)
where k = 1, 2, ..., K. The model fitness is decided on by testing the seven moment
conditions (14)-(17) jointly using GMM.14 The test statistic is derived as
GMM = T · gT 0 ST −1 gT ,
(18)
where gT is a vector of sample counterparts to the moment conditions, and ST
is the corresponding sample variance-covariance matrix. The statistic (18) is χ2
distributed with 7K degrees of freedom. Again, I set the lag length to K = 8.
Two simple simulation studies are conducted to determine the size of the two
test of model fitness.15 There is a tendency for the GMM test to reject a true null
too often. This is however only a problem if one observes many rejections.
13
This presentation of the MLBQ test follows Hatemi-J (2004).
Ng (2000) carries out a similar GMM approach to testing model fitness.
15
The size of a statistical test equals the probability of rejecting a true null hypothesis.
14
4. Empirical Results
3.4
23
Testing for Volatility Spillover
Consider the statistical model’s expression for the conditional stock return variance
in (10). Oil price uncertainty transmits to stock volatility, h22,t , through three channels; via the symmetric shock, ε1,t−1 , the asymmetric shock, η 1,t−1 , or the conditional
oil price variance of the previous period, h11,t−1 . Thus volatility spillover is tested
via the corresponding parameter estimates of a12 , g12 , and b12 . There is evidence
of volatility spillover if a joint test of the three parameters being zero is rejected.
Formally, the null hypothesis
H0 : a12 = b12 = g12 = 0,
(19)
is tested by deriving both Wald and Likelihood ratio (LR) statistics. The Wald
test uses the obtained estimates of a12 , g12 , and b12 along with the corresponding
estimated variance-covarinace matrix, and a Wald statistic is derived in the usual
way. The LR test compares the maximum likelihood of the unconstrained estimation
with the one obtained when the constraint (19) is enforced.16
4
Empirical Results
This section presents the results of the statistical model estimation. The obtained
parameter estimates are analyzed, and news impact surfaces of Kroner and Ng (1998)
are presented. Such graphical illustrations show the impacts of an oil shock and a
stock price shock on, e.g., the one-period ahead conditional stock price volatility,
holding all past conditional variances and covariances constant at their unconditional
averages.17 In this way, the magnitude of the impact of an oil shock on conditional
stock volatility is illustrated. Moreover, a second set of estimations is carried out,
where oil prices a leaded one period, to test for within-the-week effects of volatility
spillover.
4.1
Primary Estimations
Table 2 summarizes the bivariate restricted VAR(2)-ABEKK estimation results.18
Panel A presents the conditional mean parameter estimates. The estimated conditional stock return intercepts, μ2 , are all positive and significantly different from
16
To read more on the Wald and Likelihood ratio statistics, see, e.g., Hamilton (1994).
The news impact surface is a direct multivariate extension of Engle and Ng’s (1993) univariate
news impact curve.
18
Full estimation results including QML standard errors and p-values are presented in tables
A.1-A.5 in the appendix.
17
24
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
Table 2: Restricted VAR(2)-ABEKK Estimation Results
μ1
μ2
δ 11
δ 21
δ 22
π 11
π 21
π 22
c11
c12
c22
a11
a12
a22
b11
b12
b22
g11
g12
g22
max L
MLBQ
MLBQ2
GMM
Wald
LR
Japan
Norway
Sweden
U.K.
U.S
Panel A: Conditional mean estimates
0.0907
0.0699
0.0736
0.0771
0.0683
-0.1047
0.1843*
0.2553*
0.1844*
0.2769*
-0.0247
-0.0166
-0.0047
-0.0306
0.0061
-0.0176
0.0103
-0.0009
-0.0067
-0.0156
-0.0528
0.0719
0.0914*
-0.0178
-0.1377*
-0.0150
-0.0198
-0.0211
-0.0147
-0.0250
0.0156
-0.0132
-0.0196
-0.0028
-0.0382*
0.0082
0.1049*
0.0555
0.0441
0.0505
Panel B: Conditional variance-covariance estimates
0.7938*
0.8428*
0.8599*
0.7871*
0.8425*
0.1182
0.7017*
0.6269*
-0.0434
0.1038
0.6603*
-0.0072
0.0001
0.2984*
-0.0001
0.3127*
-0.3529*
0.3637*
0.3338*
0.3470*
0.0049
-0.0367
-0.0462
-0.0388*
0.0003
0.1768*
0.1628
0.0841
0.0433
0.0919
0.9254*
-0.9184*
-0.9150*
0.9271*
-0.9172*
0.0145
-0.1489*
0.0001
0.0200*
0.0147
0.9136*
0.9140*
0.8971*
0.9417*
0.9731*
-0.2563*
0.1763
0.1627
-0.1681
0.2141
0.0908*
-0.0541
0.0774
0.0225
-0.0996*
-0.2980*
0.3303*
0.5295*
-0.3691*
-0.2394*
-4539.24
-4493.88
-4524.26 -4270.38
-4280.92
Panel C: Tests of model fitness
26.88
28.47
39.34
39.63
34.07
(0.525)
(0.440)
(0.076)
(0.071)
(0.199)
38.12
31.05
23.57
25.62
29.56
(0.096)
(0.315)
(0.704)
(0.594)
(0.385)
51.53
61.05
67.28
68.86
73.36
(0.645)
(0.299)
(0.144)
(0.116)
(0.060)
Panel D: Tests of volatility spillover
17.57*
26.97*
12.07*
15.68*
67.95*
(<0.001)
(<0.001)
(0.007)
(0.001)
(<0.001)
10.50*
21.52*
4.81
10.81*
15.04*
(0.015)
(<0.001)
(0.186)
(0.013)
(0.002)
The table summarizes the bivariate restricted VAR(2)-ABEKK estimation results. Variable 1 (2)
refers to oil (stocks). QML standard errors are used to determine parameter-estimate significance.
p-values are in parentheses. * indicates significance at five percent level.
4. Empirical Results
25
zero at the five percent level except for Japan, where the estimated intercept equals
-0.1047 and insignificant. Oil price changes have conditional mean equations with
insignificant intercepts throughout. Evidence of stock return serial correlation is
found for Norway, Sweden, and the U.S, as was previously suggested by the significant LBQ statistics of table 1. The Sweden and U.S. estimations give significant
estimates of δ 22 , which indicates serial correlation over one period, and the Norway estimation results in a significant estimate of π 22 , suggesting a correlation over
two periods. Furthermore, the estimate of π 21 is negative and significant for the
U.S. alone, implying that positive oil shocks have a significant negative impact on
one-week ahead U.S. stock returns. The result for the U.S. supports previous work
by Jones and Kaul (1996) and Sadorsky (1999), although they use quarterly and
monthly data, respectively.
The stock return serial correlations are successfully removed by the VAR(2)
model, as the insignificant MLBQ statistics in panel C show. Considering, also,
the insignificant MLBQ2 and GMM statistics in the panel, the statistical model
provides an overall good fit to the data. Panel B reports on estimates of the conditional variance-covariance parameters. Notice that the a11 and b11 parameters of
conditional oil price volatility are significant throughout, but that the parameter
representing asymmetric oil price volatility, g11 , is only significant in the Japan estimation. Consequently, oil price volatility is conditionally heteroskedastic, however
without displaying asymmetric effects to oil price increases and decreases in general.
Evidence of time-persistence in conditional stock market volatility is reported
on by the significant estimates of b22 across all the five regressions. Rarely any
of the estimated symmetric shock parameters, a22 , come out significant, while the
estimates of the asymmetric terms, g22 , are significant across every stock market.
Thus the results support the finding of Glosten et al. (1993), and many others, who
present empirical evidence of asymmetric stock market volatility, i.e., negative stock
price shocks cause for larger swings in the next period’s conditional variance than
positive ones do.
The parameters of volatility spillover from oil price changes to stock returns, a12 ,
b12 , and g12 , are significant for some economies and insignificant for others. Oil price
shocks have significant symmetric effects on only the U.K. conditional stock price
volatility. Indications that the Japanese and U.S. aggregate stock markets respond
asymmetrically to oil shocks are shown via the respective significant estimates of
g12 . Different from the other economies, the conditional stock price volatilities of
Sweden and Norway display no significant signs of responses to oil shocks. For
Norway, evidence of time-persistence between the conditional oil price volatility
26
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
and the one-period ahead conditional stock volatility, measured by b12 , is presented
however. This is also the case for the U.K. stock market.
Although the parameters of volatility spillover are not significant overall, the tests
of volatility spillover, reported on in panel D of table 2, show significant evidence of
such across all national stock markets. The Wald and LR statistics derived under
the null in (19) are greater than ten and significant across all countries but Sweden,
where the LR statistic is insignificant at 4.81. The Wald statistic for Sweden is
significant though. Hence, the results show strong evidence of volatility spillover
for Japan, Norway, the U.K., and the U.S., but only weak evidence for Sweden.
Moreover, the significant g12 estimates of Japan and the U.S. suggest that oil prices
have asymmetric volatility spillover effects on the stock markets of these economies.
Figure 3: News Impact Surfaces for Japan
Panel A: Stock Variance, t
Panel B: Oil Variance, t
100
100
50
50
0
20
0
stock shock, t−1 −20 −20
20
0
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
Panel C: Covariance, t
Panel D: Correlation, t
100
1
0
0
−100
20
0
stock shock, t−1 −20 −20
20
0
oil shock, t−1
20
−1
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
20
0
oil shock, t−1
The figure shows the news impact of shocks at time t − 1 on the variances, covariance, and
correlation at time t using Japanese data.
Table 2 presents evidence of volatility spillovers but gives no information about
their size. How large impacts do the oil price volatility spillovers have on the aggregate stock markets? Figures 3-7 illustrate news impact surfaces for each respective
estimation. The graphs show the implied conditional variances (panels A and B), the
implied conditional covariances (panels C), and the implied conditional correlations
(panels D) following last period’s shocks, with all previous conditional variances and
4. Empirical Results
27
Figure 4: News Impact Surfaces for Norway
Panel A: Stock Variance, t
Panel B: Oil Variance, t
100
100
50
50
0
20
0
20
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
0
stock shock, t−1 −20 −20
Panel C: Covariance, t
Panel D: Correlation, t
50
1
0
0
−50
20
0
stock shock, t−1 −20 −20
20
0
oil shock, t−1
−1
20
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
20
0
oil shock, t−1
The figure shows the news impact of shocks at time t − 1 on the variances, covariance, and
correlation at time t using Norwegian data.
Figure 5: News Impact Surfaces for Sweden
Panel A: Stock Variance, t
Panel B: Oil Variance, t
200
100
100
50
0
20
0
stock shock, t−1 −20 −20
20
0
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
Panel C: Covariance, t
Panel D: Correlation, t
50
1
0
0
−50
20
0
stock shock, t−1 −20 −20
20
0
oil shock, t−1
20
−1
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
20
0
oil shock, t−1
The figure shows the news impact of shocks at time t − 1 on the variances, covariance, and
correlation at time t using Swedish data.
28
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
Figure 6: News Impact Surfaces for the U.K.
Panel A: Stock Variance, t
Panel B: Oil Variance, t
100
100
50
50
0
20
0
stock shock, t−1 −20 −20
20
0
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
Panel C: Covariance, t
Panel D: Correlation, t
50
1
0
0
−50
20
0
stock shock, t−1 −20 −20
20
0
oil shock, t−1
20
−1
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
20
0
oil shock, t−1
The figure shows the news impact of shocks at time t − 1 on the variances, covariance, and
correlation at time t using U.K. data.
Figure 7: News Impact Surfaces for the U.S.
Panel A: Stock Variance, t
Panel B: Oil Variance, t
40
100
20
50
0
20
0
stock shock, t−1 −20 −20
20
0
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
Panel C: Covariance, t
Panel D: Correlation, t
20
1
0
0
−20
20
0
stock shock, t−1 −20 −20
20
0
oil shock, t−1
20
−1
20
0
0
oil shock, t−1stock shock, t−1 −20 −20
20
0
oil shock, t−1
The figure shows the news impact of shocks at time t − 1 on the variances, covariance, and
correlation at time t using U.S. data.
4. Empirical Results
29
covariances held constant at their unconditional averages. Specifically, panels A of
each figure present the impact of oil shocks and stock shocks on the one-period ahead
conditional stock variances. The figures show that, although significant spillovers
were previously presented, the impacts of oil shocks on stock volatility are quite
minor in comparison to the effect that stock returns’ own shocks have on volatility.
For example, studying panel A of figure 3, negative shocks to the stock price cause
the stock volatility to increase quite dramatically. For oil shocks however, the news
impact surface shows that a ten percent decrease in the oil price has barely any
effect on the Japanese stock price volatility. A ten percent increase in the oil price
has small effects on the Japanese aggregate stock variance, illustrating the suggested
asymmetric volatility spillover. For the U.S., where an asymmetry was suggested
as well, panel A of figure 7 indicates that positive oil shocks decrease the conditional stock variance. This result is strange, since one expects positive oil shocks to
increase, and not decrease, stock volatility.
Panels B of figures 3-7 show that, because of the ABEKK model parameter restrictions oil price volatility is only affected by its own shocks. There is no clear
indication of asymmtric responses to positive and negative oil shocks, confirming the
previous results of table 2 except, perhaps, for the Japan estimation, where the g11
estimate was indeed significant. Moreover, panels C and D display the news impacts
on the conditional covariances and the conditional correlations. These illustrations
show how oil shocks and stock shocks affect the one-period ahead conditional covariance and correlation, respectively, between stock price returns and oil price changes.
Negative oil shocks and positive stock shocks cause for a clearly positive next-period
correlation in the Norway estimation, as panel D of figure 4 shows. However, the
relation is the opposite for the rest of the analyzed economies, where negative oil
shocks and positive stock shocks cause for a negative next-period correlation.
4.2
Second set of Estimations with Leaded Oil Price
The previous subsection presents strong evidence of volatility spillover from oil price
changes to most of the analyzed aggregate stock markets. Since weekly data is
considered, oil price volatility spills over to stock markets from one week to the
next. One could, however, argue that the flow of oil price uncertainty is faster than
a week. Do stock markets actually move "simultaneously" with the uncertainty of
oil prices, i.e., within the same week?19 To answer this question, a second set of
estimations is considered where oil prices are leaded one period. Such a modification
19
Within-the-week volatility spillovers are henceforth referred to as simultaneous or contemporaneous ones.
30
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
alters the presentation of the statistical model in (5)-(10), which becomes
and
r1t = μ1 + δ 11 r1,t−1 + π 11 r1,t−2 + ε1t ,
(20)
r2t = μ2 + δ 21 r1t + δ 22 r2,t−1 + π 21 r1,t−1 + π 22 r2,t−2 + ε2t ,
(21)
Ã
ε1t
ε2t
!
Ã
h11,t h12,t
h12,t h22,t
!1/2 Ã
v1t
v2t
!
,
(22)
2 2
h11,t = c211 + a211 ε21,t−1 + b211 h211,t−1 + g11
η 1,t−1 ,
(23)
=
where
h12,t = c11 c12 +
a11 a12 ε21t
+ a11 a22 ε1t ε2,t−1
+b11 b12 h11,t + b11 b22 h12,t−1
+g11 g12 η 21t + g11 g22 η 1t η 2,t−1 ,
(24)
h22,t = c212 + c222 + a212 ε21t + a222 ε22,t−1 + 2a12 a22 ε1t ε2,t−1
+b212 h11,t + b222 h22,t−1 + 2b12 b22 h12,t−1
2 2
2 2
+g12
η 1t + g22
η 2,t−1 + 2g12 g22 η 1t η 2,t−1 ,
(25)
where r1t (r2t ) represents conditional oil price changes (stock returns).
Notice the slight differences of model (20)-(25) compared with the previous one
of (5)-(10). Simultaneous effects of oil shocks onto conditional stock returns (21)
and conditional stock volatility (25) are now considered.
Table 3 summarizes the bivariate restricted VAR(2)-ABEKK model estimation
results when oil prices are leaded one week. Panel A reports on the mean equation
parameter estimates, where evidence of serial correlation in Norwegian, Swedish, and
U.S. stock returns is shown, just like in the previous estimations. Moreover, evidence
of positive mean spillover from oil prices to the Norwegian stock index is implied
by the significant estimate of δ 21 . This suggests that the Norwegian aggregate stock
market responds positively to a contemporaneous oil shock.
Panel B of table 3 presents the conditional variance-covariance parameter estimates, and panel D reports on the tests of volatility spillover. The most striking
differences compared with the estimation results without leading the oil price is
that the two tests of volatility spillover now come out unanimously significant for
the estimation with the U.S. stock market only. The previous evidence of volatility
spillovers for the other stock markets are no longer present, which suggests that
only U.S. stocks vary contemporaneously with oil price variations. However, panel
C presents a significant GMM statistic of overall model fitness for the U.S. estima-
4. Empirical Results
31
Table 3: Restricted VAR(2)-ABEKK Leaded Oil Price Estimation Results
Japan
Norway
Sweden
U.K.
U.S
Panel A: Conditional mean estimates
μ1
0.0568
0.1273
0.0389
0.0992
0.0704
-0.0882
0.1737*
0.2585*
0.1760*
0.2312*
μ2
-0.0071
-0.0084
-0.0199
-0.0187
-0.0164
δ 11
δ 21
0.0289
0.0927*
0.0034
0.0032
-0.0135
-0.0408
0.0634
0.0892*
-0.0190
-0.1354*
δ 22
-0.0170
-0.0244
-0.0243
-0.0189
-0.0298
π 11
-0.0144
0.0097
-0.0001
0.0006
-0.0084
π 21
0.0291
0.0980*
0.0519
0.0387
0.0730*
π 22
Panel B: Conditional variance-covariance estimates
c11
0.8163*
0.8879*
0.9004*
0.8056*
0.7905*
c12
-0.0629
-0.0001
0.1534
-0.0991
-0.1314
0.7540*
0.7283*
0.5707*
-0.3684*
-0.2155
c22
0.3514*
0.3363*
0.3773*
0.3497*
0.3634*
a11
0.0295
0.0447
-0.0181
-0.0321
-0.0311
a12
0.0830
-0.0983
-0.1373
0.0288
-0.0425
a22
0.9198*
0.9164*
0.9127*
0.9208*
0.9226*
b11
b12
-0.0006
-0.0139
0.0042
0.0117
0.0191
0.9079*
0.9017*
0.9049*
0.9373*
0.9500*
b22
-0.1800
0.2379*
-0.0933
-0.1806
0.0902
g11
0.1300
-0.0485
0.0384
0.0041
-0.0511*
g12
0.4359*
-0.4279* -0.4534*
0.3934*
0.3204*
g22
max L
-4545.64 -4486.89 -4521.09 -4270.87 -4285.27
Panel C: Tests of model fitness
23.98
27.38
43.23*
36.57
47.39*
MLBQ
(0.683)
(0.498)
(0.033)
(0.129)
(0.012)
37.81
41.35*
28.32
23.80
30.62
MLBQ2
(0.102)
(0.050)
(0.448)
(0.692)
(0.334)
58.05
59.20
79.01*
63.59
75.23*
GMM
(0.400)
(0.360)
(0.023)
(0.227)
(0.044)
Panel D: Tests of volatility spillover
10.62*
4.74
3.67
4.09
11.98*
Wald
(0.014)
(0.192)
(0.300)
(0.251)
(0.008)
1.95
4.03
3.51
3.55
8.25*
LR
(0.584)
(0.258)
(0.320)
(0.314)
(0.041)
The table summarizes the bivariate restricted VAR(2)-ABEKK model estimation results when oil
price changes are leaded one period. Variable 1 (2) refers to oil (stocks). QML standard errors
are used to determine parameter-estimate significance. p-values are in parentheses. * indicates
significance at five percent level.
32
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
tion, rejecting the null of a good model fit, which weakens the interpretation of the
significant volatility spillover. Generally, there is no empirical evidence of simultaneous volatility spillovers from oil prices to stock markets, making the result that
volatility spills over from one week to the next more robust.
5
Conclusions
The paper conducts an empirical investigation of volatility spillover from oil prices
to stock markets. The statistical model includes a parameterization of the conditional variance-covariance of oil price changes and stock returns, specifically the
asymmetric BEKK model. Parameter restrictions are imposed so that stock returns
cannot affect oil prices. Aggregate stock market data representing Japan, Norway,
Sweden, the U.K., and the U.S. are used. Over the sample period from week one
of 1989 to week seventeen of 2005, strong evidence of volatility spillover is found
for Japan, Norway, the U.K., and the U.S. Weak evidence of volatility spillover is
found for Sweden. Although results of significant volatility spillovers are obtained,
news impact surfaces display small quantative implications. The stock market’s own
shocks, which are related to other factors of uncertainty than the oil price, are more
prominent than the effects of oil shocks.
The paper also examines whether volatility spillovers occur simulateneously, i.e.,
within-the-week instead of from one week to the next, as in the primary examination.
To do so, the oil price is leaded one week, and a second set of estimations is carried
out. However, no strong evidence of contemporanous volatility spillovers are found.
In all, the paper improves our knowledge of how stock markets link to oil prices.
References
33
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Box, G. and G. Jenkins (1976), Time Series Analysis: Forecasting and Control,
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Engle, R. (1982), "Autoregressive conditional heteroskedasticity with estimates of
the variances of United Kingdom inflation", Econometrica 50, 987-1007.
Engle, R. and K. Kroner (1995), "Multivariate simultaneous generalized ARCH",
Econometric Theory 11, 122-50.
Engle, R. and V. Ng (1993), "Measuring and testing the impact of news on volatility", Journal of Finance 48, 1749-78.
Glosten, L., R. Jagannathan, and D. Runkle (1993), "Relationship between the
expected value and the volatility of the nominal excess return on stocks", Journal
of Finance 48, 1779-1801.
Hamilton, J. (1983), "Oil and the macroeconomy since World War II", Journal
of Political Economy 91, 228-48.
Hamilton, J. (1985), "Historical causes of postwar oil shocks and recessions", Energy
Journal 6, 97-116.
Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton,
New Jersey.
Hatemi-J, A. (2004), "Multivariate tests for autocorrelation in the stable and unstable VAR models", Economic Modelling 21, 4, 661-83.
Hosking, J. (1980), "The multivariate portmanteau statistic", Journal of American Statistical Association 75, 371, 602-8.
Huang, B.-N., M. Hwang, and H.-P. Peng (2005), "The asymmetry of the impact
34
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of oil shocks on economic activities: An application of the multivariate threshold
model, Energy Economics 27, 455-76.
Huang, R., R. Masulis, and H. Stoll (1996), "Energy shocks and financial markets", Journal of Futures Markets 16, 1, 1-27.
Jones, C. and G. Kaul (1996), "Oil and the stock markets", Journal of Finance
51, 2, 463-91.
Karolyi, G. (1995), "A multivariate GARCH model of international transmission
of stock returns and volatility: The case of the United States and Canada", Journal
of Business and Economic Statistics 13, 11-25.
Kearney, C. and A. Patton (2000), "Multivariate GARCH modelling of exchange
rate volatility transmission in the European Monetary System", Financial Review
41, 29-48.
King, M., E. Sentana, and S. Wadhwani (1994), "Volatility and links between national stock markets", Econometrica 62, 4, 901-33.
Kroner, K. and V. Ng (1998), "Modelling asymmetric comovements of asset returns", Review of Financial Studies 11, 4, 817-44.
Mork, K. (1994), "Business cycles and the oil market (special issue)", Energy Journal 15, 15-37.
Mork, K., Ø. Olsen, and H. Mysen (1994), "Macroeconomic responses to oil price
increases and decreases in seven OECD countries", Energy Journal 15, 4, 19-35.
Ng, A. (2000), "Volatility spillover effects from Japan and the U.S. to the PacificBasin", Journal of International Money and Finance 19, 207-33.
Officer, R. (1973), "The variability of the market factor of New York Stock Exchange", Journal of Business 46, 434-53.
Rogoff, K. (2006), "Oil and the global economy", Manuscript, Harvard University.
Sadorsky, P. (1999), "Oil price shocks and stock market activity", Energy Economics 21, 5, 449-69.
Sadorsky, P. (2003), "The macroeconomic determinants of technology stock price
volatility", Review of Financial Economics 12, 191-205.
Schwert, G. (1989), "Why does stock market volatility change over time?", Journal
of Finance 44, 5, 1115-53.
Vaitheeswaran, V. (2005), "Oil in troubled waters", Economist, April 30th.
Appendix
35
Appendix
Table A.1: Restricted VAR(2)-ABEKK Estimation Results for Japan
μ1
μ2
δ 11
δ 21
δ 22
π 11
π 21
π 22
c11
c12
c22
a11
a12
a22
b11
b12
b22
g11
g12
g22
max L
Est
0.0907
-0.1047
-0.0247
-0.0176
-0.0528
-0.0150
0.0156
0.0082
0.7938*
0.1182
0.6603*
0.3127*
0.0049
0.1768*
0.9254*
0.0145
0.9136*
-0.2563*
0.0908*
-0.2980*
-4539.24
QML S.E.
0.1565
0.0860
0.0388
0.0163
0.0379
0.0348
0.0160
0.0333
0.1888
0.1513
0.1729
0.0807
0.0209
0.0519
0.0228
0.0076
0.0260
0.0676
0.0255
0.0573
p-value
0.5626
0.2234
0.5243
0.2802
0.1635
0.6667
0.3292
0.8060
<0.001
0.4347
<0.001
<0.001
0.8154
<0.001
<0.001
0.0564
<0.001
<0.001
<0.001
<0.001
The table reports on bivariate restricted VAR(2)-ABEKK estimation results using oil price changes
and Japanese aggregate stock returns. QML standard errors and p-values are presented. * indicates
significance at five percent level.
36
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
Table A.2: Restricted VAR(2)-ABEKK Estimation Results for Norway
μ1
μ2
δ 11
δ 21
δ 22
π 11
π 21
π 22
c11
c12
c22
a11
a12
a22
b11
b12
b22
g11
g12
g22
max L
Est
0.0699
0.1843*
-0.0166
0.0103
0.0719
-0.0198
-0.0132
0.1049*
0.8428*
0.7017*
-0.0072
-0.3529*
-0.0367
0.1628
-0.9184*
-0.1489*
0.9140*
0.1763
-0.0541
0.3303*
-4493.88
QML S.E.
0.0988
0.0826
0.0364
0.0166
0.0375
0.0221
0.0151
0.0350
0.2032
0.1809
0.1191
0.0997
0.0309
0.0944
0.0240
0.0356
0.0299
0.2020
0.0286
0.0688
p-value
0.4798
0.0259
0.6491
0.5353
0.0554
0.3716
0.3816
0.0028
<0.001
<0.001
0.9519
<0.001
0.2341
0.0850
<0.001
<0.001
<0.001
0.3829
0.0588
<0.001
The table reports on bivariate restricted VAR(2)-ABEKK estimation results using oil price changes
and Norwegian aggregate stock returns. QML standard errors and p-values are presented. *
indicates significance at five percent level.
Appendix
37
Table A.3: Restricted VAR(2)-ABEKK Estimation Results for Sweden
μ1
μ2
δ 11
δ 21
δ 22
π 11
π 21
π 22
c11
c12
c22
a11
a12
a22
b11
b12
b22
g11
g12
g22
max L
Est
0.0736
0.2553*
-0.0047
-0.0009
0.0914*
-0.0211
-0.0196
0.0555
0.8599*
0.6269*
0.0001
0.3637*
-0.0462
0.0841
-0.9150*
0.0001
0.8971*
0.1627
0.0774
0.5295*
-4524.26
QML S.E.
0.1731
0.0874
0.0427
0.0170
0.0379
0.0350
0.0150
0.0363
0.2066
0.0732
0.0030
0.0783
0.0262
0.1179
0.0222
0.0342
0.0145
0.1803
0.0420
0.0446
p-value
0.6708
0.0036
0.9122
0.9562
0.0161
0.5472
0.1937
0.1261
<0.001
<0.001
0.9867
<0.001
0.0789
0.4759
<0.001
0.9992
<0.001
0.3671
0.0657
<0.001
The table reports on bivariate restricted VAR(2)-ABEKK estimation results using oil price changes
and Swedish aggregate stock returns. QML standard errors and p-values are presented. * indicates
significance at five percent level.
38
Essay 1. Does Oil Price Uncertainty Transmit to Stock Markets?
Table A.4: Restricted VAR(2)-ABEKK Estimation Results for the U.K.
μ1
μ2
δ 11
δ 21
δ 22
π 11
π 21
π 22
c11
c12
c22
a11
a12
a22
b11
b12
b22
g11
g12
g22
max L
Est
0.0771
0.1844*
-0.0306
-0.0067
-0.0178
-0.0147
-0.0028
0.0441
0.7871*
-0.0434
0.2984*
0.3338*
-0.0388*
0.0433
0.9271*
0.0200*
0.9417*
-0.1681
0.0225
-0.3691*
-4270.38
QML S.E.
0.1658
0.0648
0.0403
0.0131
0.0356
0.0354
0.0117
0.0341
0.2101
0.0661
0.0954
0.1015
0.0179
0.0527
0.0251
0.0052
0.0163
0.1787
0.0311
0.0589
p-value
0.6420
0.0045
0.4479
0.6071
0.6178
0.6779
0.8144
0.1964
<0.001
0.5112
0.0018
0.0010
0.0303
0.4116
<0.001
<0.001
<0.001
0.3471
0.4691
<0.001
The table reports on bivariate restricted VAR(2)-ABEKK estimation results using oil price changes
and U.K. aggregate stock returns. QML standard errors and p-values are presented. * indicates
significance at five percent level.
Appendix
39
Table A.5: Restricted VAR(2)-ABEKK Estimation Results for the U.S.
μ1
μ2
δ 11
δ 21
δ 22
π 11
π 21
π 22
c11
c12
c22
a11
a12
a22
b11
b12
b22
g11
g12
g22
max L
Est
0.0683
0.2769*
0.0061
-0.0156
-0.1377*
-0.0250
-0.0382*
0.0505
0.8425*
0.1038
-0.0001
0.3470*
0.0003
0.0919
-0.9172*
0.0147
0.9731*
0.2141
-0.0996*
-0.2394*
-4280.92
QML S.E.
0.1556
0.0638
0.0394
0.0134
0.0377
0.0360
0.0136
0.0359
0.1806
0.0723
0.0013
0.0750
0.0197
0.0563
0.0204
0.0288
0.0081
0.1210
0.0149
0.0667
p-value
0.6608
<0.001
0.8766
0.2761
<0.001
0.4864
0.0050
0.1603
<0.001
0.1516
0.9948
<0.001
0.9866
0.1026
<0.001
0.6084
<0.001
0.0722
<0.001
<0.001
The table reports on bivariate restricted VAR(2)-ABEKK estimation results using oil price changes
and U.S. aggregate stock returns. QML standard errors and p-values are presented. * indicates
significance at five percent level.
Essay 2
Myopic Loss Aversion, the Equity
Premium Puzzle, and GARCH
1
Introduction
Imagine having the opportunity to flip a coin and win either $50 or $100, a bet with
an expected value of $75. How much would you be willing to pay for such a bet
to take place? Following the results of Mehra and Prescott (1985), an individual
is so averse to the uncertainty of this gamble that she would think it is worth only
$51.21. Their study, however, does not concern gambles of this kind, but focuses
on the risk-return relationship of stocks to less risky assets, such as bonds. Stocks
have outperformed bonds to a large extent over the past century. Using a standard
consumption-based general equilibrium model, Mehra and Prescott (1985) find that
in order to explain the large equity risk premium a relative risk aversion in excess
of 30 is needed. Theoretical arguments and estimates from various studies, such as
Arrow (1971) and Friend and Blume (1975), suggest that this parameter should be
in the neighborhood of three. Mehra and Prescott (1985), finding the model implied
level of risk aversion unreasonably high, announce the existence of an anomaly they
call the equity premium puzzle (EPP).
Economists have been struggling to solve the puzzle for more than twenty years
now. Plausible explanations have been studied with varying success. As originally
stated, Mehra and Prescott’s (1985) model relies on three assumptions on individual
behavior and asset market structure. First, individual preferences are explained
by a power utility function over consumption in an expected utility framework.1
Second, the market is complete, meaning that every element of risk can be diversified,
1
The most common power utility function is u(C) =
aversion equal to γ.
C 1−γ
1−γ ,
which has a constant relative risk
42
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
and, third, the market is free of transaction costs. Although all three assumptions
have been debated when trying to solve the puzzle, the literature on the choice of
individual preferences is most profound. Attempts in explaining the EPP within
this line of research include generalized expected utility of Epstein and Zin (1991),
and habit formation of Constantinides (1990).2
Behavioral finance presents an alternative explanation from those of the traditional framework. Benartzi and Thaler (1995) (BT henceforth) suggest a clarification of the EPP that incorporates several experimentally observed behavioral
concepts, the most important ones being loss aversion and mental accounting. Loss
aversion is the individual tendency to be more sensitive to losses than to gains, and
is the main ingredient of Kahneman and Tversky’s (1979, 1992) prospect theory.
This descriptive utility theory can explain some of the economic anomalies that
neoclassical expected utility theory cannot. Mental accounting refers to how people
are affected by information feedback on their portfolio. BT argue that loss-averse
investors will find a risky portfolio even more risky if they evaluate it frequently.
Evaluating the portfolio less "myopically" will reduce the risk.3 The two concepts together form a preference scheme BT call myopic loss aversion (MLA). Using monthly
data on aggregate stock returns and five-year bond returns in the United States over
the historical period 1926 to 1990, BT derive the prospect theory utility of holding
an all-stocks and an all-bonds portfolio at various evaluation periods (information
feedback frequencies). The large observed equity premium can be explained by
MLA preferences if financial investors have an evaluation period of approximately
twelve months, since the two portfolio’s have equal utilities at this evaluation period.
BT conclude that their result is intuitive, since most individual investors file their
taxes on a yearly basis, and receive reports from their brokers, mutual funds and
retirement accounts annually.
The present paper takes a closer look at the proposed MLA solution to the
EPP. Specifically, the distributional assumption made on stock returns is addressed.
When deriving the prospective utility of holding a portfolio at a specific evaluation
period, the portfolio returns distribution needs to be simulated. BT use a nonparametric bootstrap technique in this simulation, implying that any existing serial
correlation in returns is removed by construction. In fact, their method implicitly
assumes that stock returns are independent over time. Fama and French (1988),
among others, point out the existence of mean reversion (negative serial correlation)
in stock returns, and one might consider whether neglecting this could have affected
2
3
Kocherlakota (1996) and Siegel and Thaler (1997) present general literature overviews.
To read more on mental accounting, see Thaler (1985) and Thaler (1999).
1. Introduction
43
BT’s results. As BT claim though, the findings of mean reversion are only trivial over
short periods of evaluation (up to one year), and should not be a concern. The claim
that there exists a predictable component in average stock returns over longer time
horizons has, also, been criticized by, e.g., Lamoureux and Zhou (1996). However,
is the assumption of temporal independence justified? Although the first moment
might be uncorrelated, what about the second? In the finance literature it is broadly
accepted that stock returns display variations in volatility over time, i.e., return
volatility tends to cluster, bringing time periods of frequent large swings, and other
periods of calm and low volatility in returns. Such conditional heteroskedasticity
affects the unconditional returns distribution, making its shape different from the one
obtained under temporal independence. Will a simulation approach incorporating
conditional heteroskedasticity still support BT’s proposed explanation to the EPP?
How is the evaluation period consistent with the historical equity premium affected
by conditional heteroskedasticity? The current paper investigates these issues.
Relaxing the temporal independence assumption, the paper introduces a parametric approach to simulating returns distributions in the model of BT. The method
takes the stock return’s varying volatility-structure into account by estimating a generalized autoregressive conditionally heteroskedastic (GARCH) process from which
the distribution of returns is simulated. Since various evaluation periods, i.e., aggregations of data to different frequencies are considered, I follow Drost and Nijman’s
(1993) work on temporal aggregation of GARCH processes. Drost and Nijman
(1993) prove that the set of symmetric (weak) GARCH(p,q) processes is closed
under temporal aggregation. Thus when aggregating high-frequency data generated by, say, a GARCH(1,1) process, the obtained lower frequency data generating
process is also GARCH(1,1) but with a new set of parameter values. Drost and Nijman (1993) present formulas for deriving these low-frequency parameters using the
corresponding high-frequency ones. Aggregated low-frequency returns can then be
simulated from this aggregated GARCH process, and conditional heteroskedasticity
is preserved under aggregation.
The study considers the U.S. equity premium, and the Swedish one as well.
Campbell (2002) shows that the EPP exists not only in the U.S. but in other
economies, among them Sweden. Both data sets consist of monthly aggregate stock
and long-term bond returns covering 1926 to 2003 for the U.S., and 1919 to 2003 for
Sweden. Following BT, the analysis is carried out by calculating the prospect theory
utilities of an all-stocks and an all-bonds portfolio as a function of the evaluation
period. Tversky and Kahneman (1992) estimates of the prospect theory parameters are employed, although parameter variations are considered in the subsequent
44
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
sensitivity analysis. When simulating returns, both the non-parametric bootstrap
approach, following BT, and the proposed parametric method incorporating conditional heteroskedasticity are considered.
Over the full U.S. sample of data, evaluation periods of seven and ten months
are obtained when using the non-parametric bootstrap and the parametric GARCH
approach, respectively. These evaluation periods are smaller than the twelve-month
counterpart reported by BT. The twelve-month evaluation period is not replicated
over BT’s sample period of 1926 to 1990 either. When simulating returns under
temporal independence using this subsample, an evaluation period of six months is
obtained. One far reached explanation for the difference is the use of different data
sets; the current study uses data from Ibbotson Associates, while BT use CRSP data.
Perhaps more likely however, the interpretation of BT’s non-parametric bootstrap
simulation technique could somehow be mistaken, although I have no reason to
believe this to be the case. Nonetheless, obtaining a six-month evaluation period
instead of twelve months suggests that the MLA model is sensitive to the method
used when simulating portfolio returns distributions.
A problem with GARCH approach occurs when the full Swedish data sample
is considered, and analysis using this approach is made on two subsamples of data
instead.4 The non-parametric approach produces a twelve-month evaluation period over the full sample however, in line with BT, which suggests that MLA can
explain the large Swedish equity premium. This result is new to the literature,
since previous studies have not applied MLA preferences to the EPPs of non-U.S.
economies. Furthermore, longer evaluation periods are obtained under conditional
heteroskedasticity when studying the two subsamples of data. For example, over
the period from July 1961 to December 2003, evaluation periods of ten and fifteen
months are obtained using the bootstrap and GARCH methods, respectively.
Throughout the analysis of both data sets, an overall longer evaluation period
is found to be consistent with the observed equity premium when the GARCH
approach to simulating returns is used compared with the bootstrap approach. The
result suggests that the MLA investor finds stocks to be more risky when return
volatility is time-varying. Also, it further indicates that the MLA model is sensitive
to the procedure applied when simulating returns, which, in turn, indicates that
MLA investors are sensitive to the shape of the returns distribution. Therefore, the
two simulation techniques are analyzed concerning how the first four unconditional
moments evolve as the evaluation period increases. Plausibly, the skewness of the
returns distribution is important for the loss-averse investor, which is intuitive since
4
See section 5.3.
2. Related Research
45
loss aversion induces an asymmetric preference over gains and losses. Furthermore,
although somewhat different evaluation periods are obtained, they can be considered
to be of similar magnitude as the twelve-month counterpart reported by BT, thus
supporting their result.
The rest of the paper is outlined as follows: Section 2 presents some related research. Section 3 introduces the MLA framework, where prospect theory is central,
and discusses the intuition behind its solution to the EPP. In section 4, BT’s approach to simulating returns distributions is explained. The section also introduces
the paper’s contribution to this simulation concerning GARCH processes. Section
5 presents an application to financial data, where results are reported on and discussed. Section 6 concludes.
2
Related Research
BT influence a large body of related research. Barberis, Huang, and Santos (2001)
introduce loss aversion in a consumption-based general equilibrium model. They find
that loss aversion alone does not produce a large enough equity premium however,
and, therefore, extend the model to incorporate the importance of prior outcomes,
another idea from psychology. A loss is seen as less painful if it comes after a period
of gains, while it hurts badly if it follows subsequent losses. Using this extended
model, Barberis et al. (2001) report an equity premium of historical magnitude at
reasonable parameter values.
Disappointment aversion of Gul (1991) is another set of preferences related to
loss aversion and prospect theory that has been applied to the EPP. Similar to
loss aversion, a disappointment averse agent has an asymmetric preference for outcomes. One main difference is that the reference point from which gains and losses
are derived is endogenous in the model. Loss aversion usually sets current wealth as
the (exogenous) reference point. Ang, Bekaert, and Liu (2005) employ disappointment aversion in studying the equity premium, and find that a reasonable level of
disappointment aversion is consistent with the historical U.S. equity premium.
Durand, Lloyd and, Wee Tee (2004) investigate BT’s methodology, just like I do
in this paper. To determine the "equilibrium" evaluation period, BT use a graphical
inspection of the crossing-point of two lines, which represent the respective all-stocks
and all-bonds portfolio utilities at different evaluation periods. Durand et al. (2004)
rely on statistical tests to determine the crossing-point, e.g., the Wilcoxon singlerank test, and argue that BT’s analysis is not robust to the modification. However,
one can criticize Durand et al. (2004) in the way they interpret BT’s method
46
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
of sampling lower frequency returns. They claim that low-frequency returns are
sampled over data clusters, which implies that a number of return observations can
overlap, in turn implying that returns are not assumed independent. I believe this
is an erroneous interpretation.
Related work concerns experimental studies of MLA preferences among individual investors as well. This research involves both students (Thaler, Tversky,
Kahneman, and Schwartz, 1997) and professional traders (Haigh and List, 2004) in
an individual setting as well as under market conditions (Gneezy, Kapteyn, and Potters, 2003). The major result is that MLA is an observed behavior among financial
investors.
3
Myopic Loss Aversion
MLA combines mainly two experimentally observed behavioral concepts, namely loss
aversion and mental accounting. Prospect theory of Kahneman and Tversky (1979)
and its modified version cumulative prospect theory of Tversky and Kahneman
(1992) incorporate loss aversion. Cumulative prospect theory extends the original
version by making it possible to derive the utility of gambles of more than two
outcomes, and by incorporating first-order stochastic dominance. Before discussing
the intuition behind BT’s proposed explanation to the EPP, cumulative prospect
theory is presented.
3.1
Cumulative Prospect Theory
In line with empirical evidence, cumulative prospect theory has the following key
elements: reference dependence; utility is derived through comparing the outcome
with a reference level of outcome, loss aversion; losses loom larger than gains do,
risk-seeking; while being risk-averse over pure gains individuals are risk-seeking over
pure losses, and non-linear probabilities; outcome probabilities are not weighted
linearly but are non-linearly transformed.
Following Tversky and Kahneman (1992), the cumulative prospect theory utility
(henceforth prospective utility) U of a lottery L with outcomes {xi }ni=1 and corresponding probabilities {pi }ni=1 is derived as
U(L) =
n
X
i=1
π(pi ) · v(xi ),
(1)
where n is the total number of outcomes, π(·) is a function that transforms prob-
3. Myopic Loss Aversion
47
Figure 1: The Value Function
0.5
value
0
λ=1, γ=1
λ=2.25, γ=0.6
λ=2.25, γ=0.88
−0.5
−1
−0.5
0
return
0.5
The figure displays the cumulative prospect theory value function for varying parameter values.
abilities, and v(·) is a value function. This value function depends on changes in
outcomes rather than absolute levels, as in traditional expected utility. Assuming a
zero reference point, the value function is defined as
v(x) =
(
if x ≥ 0
xγ
,
γ
−λ(−x) if x < 0
(2)
where γ reflects the degrees of risk aversion over gains and risk-seeking over losses,
and λ is the loss aversion parameter.5 Tversky and Kahneman (1992) estimate the
parameters in (2) through laboratory experiments to γ̂ = 0.88 and λ̂ = 2.25, which
will be used throughout the application.6 These estimates result in an "S-shaped"
value function kinked at the origin, as figure 1 shows.
The probability transformation function π(·) uses the whole cumulative distribution function as an argument. Ranking all outcomes in increasing order from −m
to n, where m and n + 1 are the numbers of strictly negative and positive outcomes,
5
Tversky and Kahneman (1992) actually consider two separate parameters for risk aversion
over gains and risk-seeking over losses, respectively, but as the estimates of the two are equal, I
choose to unify the parameters.
6
The reference point will be zero.
48
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
Figure 2: The Probability Weighting Function
weighted probability, w(p)
1
0.8
0.6
0.4
τ = 1.00
τ = 0.69
τ = 0.61
0.2
0
0
0.2
0.4
0.6
probability, p
0.8
1
The figure displays the cumulative prospect theory probability weighting function at different
parameter values.
respectively, the function has the following form:
where
⎧
⎪
w(p−m )
⎪
⎪
⎪
⎨ w(p + ... + p ) − w(p + ... + p )
−m
i
−m
i−1
π(pi ) =
⎪
w(pi + ... + pn ) − w(pi+1 + ... + pn )
⎪
⎪
⎪
⎩
w(pn )
w(p) =
(pτ
pτ
.
+ (1 − p)τ )1/τ
if
i = −m
if −m < i < 0
,
if 0 ≤ i ≤ n − 1
if
i=n
(3)
(4)
The functional form of w(·) in (4) is an attempt in describing the Allais-type
behavior violating the expected utility theorem.7 Figure 2 presents the weighting
function w(·). Instead of weighting probabilities linearly in an objective way, the
probability of gains and losses are weighted subjectively. When τ < 1 in (4), the
transformation π(·) over-weights small cumulative probabilities, and under-weights
moderate to large ones. Tversky and Kahneman (1992), conducting individual experiments, estimate the parameter τ to 0.61 in the domain of gains, and 0.69 in
the domain of losses. An estimate of τ equal to one produces linear weights so that
probabilities are treated objectively. Moreover, the probability transformation π(·)
should not be mistaken for a probability measure, since the weighted probabilities
necessarily do not sum up to one.
7
Allais (1953) challenges the expectation principle by showing that a change in probabilities
from 0.99 to 1 has larger impact on preferences than a change from 0.10 to 0.11.
3. Myopic Loss Aversion
3.2
49
Explaining the Large Equity Premium with Myopic Loss
Aversion
BT use cumulative prospect theory in their proposed MLA solution to the EPP, and
emphasize the role of loss aversion for the individual decision-maker. They stress an
important implication the preference scheme brings to portfolio evaluation periods,
i.e., the mental accounting loss-averse investors perform. To further understand
the core of their reasoning, loss aversion and mental accounting are well illustrated
by the famous example of Samuelson (1963). In the example, Samuelson offers a
colleague of his a fifty-fifty chance of wining $200 or loosing $100. The colleague
turns the bet down, saying that he would feel a $100 loss more than a $200 gain, but
expresses the willingness to take on one hundred such bets. This exemplifies loss
aversion and, also, the kind of mental accounting it can imply. To see why, assume
that Samuelson’s colleague has the following piece-wise linear value function over
changes in wealth:
(
x if x ≥ 0
v(x) =
.
(5)
2.5x if x < 0
Considering the value function in (5) and, for simplicity, objective probabilities,
the prospective utility of the single bet equals 0.5 · 200 + 0.5 · 2.5 · (−100) = −25.
Since its prospective utility is negative, the bet is rejected. But what about a game
of two bets? The attractiveness of this gamble will depend heavily on the mental
accounting of the problem. If the two bets are treated separately the game has
double the unattractiveness. However, if the two bets are compounded into a single
bet, it will have positive expected utility, and will be accepted by the colleague.8 As
it turns out, compounding any number of bets greater than one will be favorable for
the colleague so long as he does not have to monitor the separate bets being played.
Moreover, Samuelson (1963) proves in a theorem that if an individual turns down a
bet at every level of wealth, accepting a multiple gamble contradicts expected utility
maximization. Thus the behavior of Samuelson’s colleague is inconsistent with the
traditional theory.
A parallel to the above example is a loss-averse investor choosing between stocks
and bonds. In this setup, the evaluation period is crucial for the investor’s attitude
to the risk of the investment. If the decision-maker evaluates her portfolio on a
daily basis a portfolio consisting of stocks will be unattractive, since stock returns
go down almost as often as they go up from day to day, and losses are mentally
doubled. On the other hand, consider a long evaluation period of, say, ten years.
8
The compounded lottery has outcome-probability-set {$400, 0.25; $100, 0.50; -$200, 0.25}
with prospective utilty equal to 25.
50
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
The investor can rest assured that stocks most surely will increase in value every
ten years. Hence, a stock portfolio can be an unattractive investment if evaluated
often, but an attractive one over longer evaluation periods. Low-risk bond portfolios
are not affected by this phenomenon to the same extent, since they do not display
as frequent losses. Moreover, it is important to distinguish an investor’s evaluation
period from her investment horizon. Although the planning horizon may be five
years, the time between portfolio evaluations can be twelve months.
The above argument brings two questions to mind. First, how loss-averse are
financial investors? Second, assuming a reasonable level of loss aversion, how short
an evaluation period will be in accordance with the observed equity risk premium?
BT use the estimate of loss aversion provided by Tversky and Kahneman (1992),
i.e. 2.25, and find that an investor with cumulative prospect theory preferences
will be indifferent between an all-stocks portfolio and an all-bonds portfolio if the
evaluation period is twelve months. At this interval in between evaluations there
is an equilibrium, where investors are content with the risk-return relationship of
stocks and bonds.9 Thus BT argue that MLA preferences are consistent with the
historical magnitude of equity premium, since it compensates the investor for her
fear of stock portfolio losses as well as her myopic way of evaluating the portfolio.
4
Simulating Returns Distributions
To determine the evaluation period that makes a loss-averse investor indifferent
between the returns of stocks and bonds, BT derive the prospective utilities of
holding these assets at various lengths in between evaluations. Different frequencies
of data are used to reflect these evaluation periods. If the agent evaluates her
portfolio every six months her utility of holding a stock portfolio is derived using sixmonth data on stock returns. To apply equation (1), the possible portfolio outcomes
and corresponding probabilities need to be determined at each data frequency. With
a historical data set at hand, this involves simulating distributions of returns at
different frequencies. Such simulations can be performed in different ways. BT use
a non-parametric bootstrap approach.
9
The evaluation period where stocks and bonds are associated with equal utilities is at times
referred to as the equilibrium evaluation period. This equilibrium should not be mistaken for a
general equilibrium.
4. Simulating Returns Distributions
4.1
51
Non-Parametric Bootstrap Approach
Using the high-frequency monthly data, an n-month return is simulated by, first,
drawing n returns at random (with replacement), and, second, deriving the lowfrequency return as if the n returns were consecutive. Let us say the three monthly
returns x1 , x2 and x3 are drawn. Using these high-frequency returns, the lowfrequency three-month return is calculated as (1 + x1 )(1 + x2 )(1 + x3 ) − 1. The
procedure is performed 100,000 times to obtain a smooth n-month return distribution.10 A histogram over the data is then derived, using an interval size of choice, so
that one can associate the possible returns (midpoint of every histogram interval)
with a specific probability (the frequency of returns in each interval divided by the
total number of returns).11 For example, BT use an interval size of twenty, which
constructs a distribution over twenty portfolio outcomes. Hence, the risky stock
portfolio investment is seen as a gamble over twenty predetermined outcomes with
specific probabilities. Equation (1) is thus directly applicable.
Using historical data on any portfolio, it is straightforward to derive its prospective utility at an evaluation period of choice. One only needs to decide on the
level of loss aversion and how often the investor evaluates her portfolio, i.e., what
frequency of data to use. Since the non-parametric bootstrap method constructs
low-frequency data by drawing returns at random, any existing serial correlation
is removed. Implicitly, the observations are assumed independent. A direct way
to produce low-frequency returns without removing a serial dependence would be
to derive the actual n-month returns. However, such a method does not produce
sufficiently many observations to obtain a smooth distribution of returns. Close
to 100,000 observations are needed for this purpose, motivating the use of simulation techniques. By introducing a parametric approach, this paper relaxes the
independence assumption of BT, and produces smooth returns distributions where
conditional heteroskedasticity is preserved.
4.2
Parametric Approach Using a GARCH Model
In the finance literature, it is broadly accepted that financial time series, e.g., exchange rates and stock market returns display volatilities that vary over time. One
way of modeling the fluctuations is by fitting the data to a GARCH model, which
10
When n = 1, drawing 100,000 returns with replacement will produce a large number of equal
returns, since there is a limit to the sample size. As n increases the number of possible combinations
increases, and a smoother (more continuous) set of low-frequency returns is obtained.
11
The interval size for the histograms is fifty throughout the application. The results are not
depentent on the interval size however.
52
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
parametrizes the conditional variance. Bollerslev (1986) introduces the generalization of ARCH models, originally presented by Engle (1982). Furthermore, Drost
and Nijman (1993) study the temporal aggregation of GARCH processes, and prove
that the set of symmetric (weak) GARCH(p,q) processes is closed under temporal aggregation.12 Thus when aggregating high-frequency data generated by, say,
a GARCH(1,1) process, the obtained lower frequency data generating process is
also GARCH(1,1) but with a new set of parameter values. Formulas for deriving
these low-frequency parameters using the corresponding high-frequency ones are
presented.
Temporal Aggregation of GARCH Processes
In the application, the high-frequency dynamics is modeled using a GARCH(1,1)
model with Gaussian innovations. This data generating process is the most widely
used GARCH model in the finance literature, and captures the conditional heteroskedasticity of financial time series well. The unconditional data distribution can
be shown to have fatter tails than the normal distribution, which is a commonly
observed feature of empirical stock returns distributions.
Letting {xt }t∈Z denote the return series, the GARCH (1,1) model takes on the
following form:
xt = c + ut ,
p
ut =
ht · vt ,
ht = ψ + βht−1 +
(6)
αu2t−1 ,
where {vt } is an independent identically distributed Gaussian sequence with zero
mean and unit variance, and (c, ψ, β, α)0 gathers the model’s parameters. The
conditional variance of the innovation ut is given by ht . The restrictions ψ > 0,
β ≥ 0, and α ≥ 0 are sufficient to ensure the non-negativity of ht . Also, the
restriction α + β < 1 is needed for ht to be covariance-stationary.13
Letting h(m)t denote the conditional variance of the aggregated (low-frequency)
series {u(m)t }, Drost and Nijman (1993) show that h(m)t has GARCH(1,1) dynamics
h(m)t = ψ(m) + β (m) h(m)t−1 + α(m) u2(m)t−1 ,
(7)
where ψ(m) , β (m) and α(m) are the aggregated GARCH(1,1) parameters. Following
12
Drost and Nijman (1993) point out that in applied work one assumes that the result holds
for strong GARCH processes too.
13
To read more on GARCH processes, see Hamilton (1994).
4. Simulating Returns Distributions
53
Drost and Nijman (1993), these low-frequency parameters are given by
1 − (β + α)m
,
1 − (β + α)
= (β + α)m − β (m) ,
ψ(m) = mψ
(8)
α(m)
(9)
where β (m) ∈ (0, 1) is the solution to the quadratic equation
β (m)
1+
β 2(m)
=
a(β, α, κu , m)(β + α)m − b(β, α, m)
,
a(β, α, κu , m){1 + (β + α)2m } − 2b(β, α, m)
(10)
with
(1 − β − α)2 (1 − β 2 − 2βα)
(κu − 1){1 − (β + α)2 }
{m − 1 − m(β + α) + (β + α)m }{α − βα(β + α)}
, (11)
+4
1 − (β + α)2
a(β, α, κu , m) = m(1 − β)2 + 2m(m − 1)
and
b(β, α, m) = {α − βα(β + α)}
1 − (β + α)2m
.
1 − (β + α)2
(12)
Notice that the unconditional kurtosis, κu , is included in equation (11). It is
derived as
1 − (β + α)2
,
(13)
κu = κξ
1 − (β + α)2 − (κξ − 1)α2
where κξ = 3 denotes the kurtosis of the standard Gaussian innovations {vt }.
How Are the Formulas Applied?
Using the temporal aggregation of GARCH processes framework, it is possible to
keep the serial dependence in the data variance when aggregating observations. To
simulate a distribution for returns at an aggregate level, one starts by estimating
the high-frequency GARCH(1,1) model in (6). The aggregated GARCH parameters
are derived using the above formulas (8) - (13). Low-frequency innovations, u(m)t ,
p
are simulated using equation (7) together with u(m)t = h(m)t · vt . After adding
the aggregate mean return to these innovations, the set of low-frequency returns
is complete, and the simulated low-frequency returns distribution can finally be
derived using the same histogram procedure as in the non-parametric bootstrap
approach, making equation (1) directly applicable.14 Thus, the GARCH approach
is an alternative to the bootstrap approach to simulating returns distributions at
14
The aggregate m-month mean return is derived as (1 + η)m − 1, where η denotes the mean of
the monthly return series.
54
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
various data frequencies. The important difference is that the GARCH procedure
takes the conditional heteroskedasticity of financial data into account.
4.3
Differences in Simulated Distributions
What do the simulated distributions look like? Figure 3 exemplifies distributions
for 3-, 12-, and 24-month returns using both the non-parametric bootstrap (panels
A1-A3), as well as the GARCH approach (panels B1-B3). Monthly U.S. stock
returns over the sample period from January 1926 to December 2003 are used (see
table 1). Quite naturally, the distributions display larger means and become more
outspread with a greater aggregation, irrespective of the method used. Notice that
the distributions estimated with the non-parametric approach tend to display a
larger (positive) skewness with a greater aggregation. With the GARCH approach
the distributions are symmetric by construction. By visual inspection it is difficult
to detect any differences in kurtosis across the two approaches, although there might
be such. Overall, the two methods of generating stock returns distributions show
changes in, possibly, all four unconditional moments as the aggregation increases.
Figure 3: Estimated Stock Returns Distributions for 3-, 12-, and 24-Month Returns
Panel A1: 3−month
Panel A2: 12−month
Panel A3: 24−month
0.2
0.2
0.2
0.1
0.1
0.1
0
−1
0
1
2
0
−1
Panel B1: 3−month
0
1
2
0
−1
Panel B2: 12−month
0.2
0.2
0.1
0.1
0.1
0
1
2
0
−1
0
1
2
1
2
Panel B3: 24−month
0.2
0
−1
0
0
−1
0
1
2
The figure illustrates approximated distributions of financial stock returns. The non-parametric
bootstrap approach is employed in panels A1-A3, while panels B1-B3 use the GARCH approach.
U.S. aggregate stock returns covering 1926:1 - 2003:12 are used. The histogram interval size is
twenty.
The …gure presents summary statistics of nominal returns on the U.S. and the Swedish aggregate stock markets over various sample periods. The U.S.
data is provided by Ibbotson Associates, and the Swedish data stems from an updated version of Frennberg and Hansson’s (1992) data set. EP stands for
equity premium.
U.S.
U.S.
Sweden
Sweden
Sweden
1926:1 - 2003:12 1926:1 - 1990:12 1919:1 - 2003:12 1919:1 - 1961:6 1961:7 - 2003:12
Stocks Bonds Stocks Bonds Stocks Bonds Stocks Bonds Stocks Bonds
Mean (%)
0.99
0.46
0.97
0.40
0.92
0.56
0.62
0.39
1.23
0.74
Max. (%)
42.56
15.23
42.56
15.23
27.58
16.47
19.39 16.47 27.58
7.51
Min. (%)
-29.73
-9.82
-29.73
-8.41
-27.12 -10.79 -27.12 -10.79 -21.49
-5.93
Std. dev. (%)
5.62
2.27
5.86
2.20
4.87
1.91
4.01
2.09
5.58
1.70
Skewness
0.39
0.68
0.46
1.00
-0.11
0.30
-0.52
0.70
-0.04
-0.30
Kurtosis
12.45
8.10
12.49
9.55
6.19
12.57
8.91
15.34
4.74
5.68
Annual EP (%)
6.55
7.06
4.41
2.80
6.04
No. obs.
936
780
1020
510
510
Table 1: Summary Statistics for U.S. and Swedish Monthly Returns
4. Simulating Returns Distributions
55
56
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
5
Application to Financial Data
This section returns to the proposed MLA solution to the puzzling magnitude of
historical equity premium. Recall that the investor’s preferences have two main
factors of risk: loss aversion and portfolio evaluation myopia. Just as BT, the
relationship between the levels of these risks and the equity premium is analyzed.
5.1
Data Sets
Campbell (2002) shows that the EPP exists not only in the U.S. but in several
other economies, among them Sweden. I consider two sets of data, representing the
U.S. and Sweden. In this way, the proposed explanation of BT is analyzed not only
focusing on the originally investigated U.S. equity premium, but another one as well.
Figure 4: U.S. Monthly Stock Returns Covering 1926 to 2003
40
30
return
20
10
0
−10
−20
200
400
600
observation no. (time)
800
The table displays U.S. monthly stock returns (%) over the sample period from January 1926 to
December 2003. Data source: Ibbotson Associates.
The U.S data is from Ibbotson Associates, while the Swedish data set is an
updated version of the one presented by Frennberg and Hansson (1992). Following
BT, long-term bond returns are considered when studying the equity premium as
opposed to Treasury bills, since bonds are the closest substitutes to stocks for the
long-term investor. Both data sets thus consist of monthly returns of a long-term
government bond, along with monthly aggregate stock returns.15 Dividends are
15
Mehra and Prescott (1985) originally analyze the equity premium over Treasury bills, but
5. Application to Financial Data
57
Figure 5: Swedish Monthly Stock Returns Covering 1919 to 2003
20
return
10
0
−10
−20
200
400
600
800
observation no. (time)
1000
The table displays Swedish monthly stock returns (%) over the sample period from January 1919
to December 2003. Data source: Frennberg and Hansson (1992).
included in the stock returns, assumed reinvested at the end of each period, which
is important since dividends are a part of the utility of holding a stock portfolio.
Furthermore, BT argue that in a descriptive model, nominal returns are the ones
that matter for the investor, since they are given most prominence in annual reports.
For this reason, nominal returns are studied, although real returns are considered
in the subsequent sensitivity analysis. The U.S. data sample period covers January
1926 to December 2003, yielding a total of 936 observations, and the Swedish one
covers January 1919 to December 2003, which yields 1020 observations.
Table 1 reports on summary statistics. Over the full sample period, the U.S. aggregate stock market has risen by 0.99 percent per month on average. Considering
the monthly average bond return of only 0.46 percent, stocks have outperformed
bonds quite substantially. The annual equity premium is 6.55 percent, which is
of similar magnitude as in several other economies. The sample period BT use,
i.e., January 1926 to December 1990, is studied here as well, and the annual equity premium is slightly larger at 7.06 percent. Furthermore, over the full sample
period, larger standard deviations are reported for stock returns compared with
bond returns; 5.62 and 2.27 percent, respectively. Observe, also, that stock returns
are mildly positively skewed, with a skewness of 0.39. The empirical kurtosis is
Campbell (2002), for instance, shows that the EPP is just as severe when bond returns are considered.
58
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
Table 2: Results of GARCH(1,1) Estimations and ARCH LM tests
c
ψ
β
α
ARCH
U.S.
1926:12003:12
1.094
(<0.001)
0.693
(<0.001)
0.860
(<0.001)
0.119
(<0.001)
220.1
(<0.001)
U.S.
1926:11990:12
1.086
(<0.001)
0.774
(<0.001)
0.857
(<0.001)
0.121
(<0.001)
185.2
(<0.001)
Sweden
1919:12003:12
1.066
(<0.001)
0.572
(<0.001)
0.812
(<0.001)
0.175
(<0.001)
75.9
(<0.001)
Sweden
1919:11961:6
1.075
(<0.001)
0.486
(0.005)
0.803
(<0.001)
0.170
(<0.001)
43.5
(<0.001)
Sweden
1961:72003:12
1.093
(<0.001)
1.949
(0.013)
0.761
(<0.001)
0.188
(<0.001)
34.4
(<0.001)
The table presents parameter estimates from a GARCH(1,1) estimation using percentage returns
on the U.S. and Swedish aggregate stock markets over different sample periods. ARCH is the
ARCH Lagrange multiplier test statistic using 12 lags. p-values are given in parentheses.
substantial at 12.45.
Historically, the stock market has dominated the bond market in Sweden too,
with average monthly returns of 0.92 and 0.56 percent for the aggregate stock and
bond markets, respectively. Over the full sample period, the annual equity premium
is 4.41 percent. The stock and bond return standard deviations are 4.87 and 1.91
percent, respectively. Moreover, the empirical stock returns distribution displays
only slight negative skewness at -0.11, and a large kurtosis of 6.19.
5.2
Are the Stock Returns Conditionally Heteroskedastic?
Since a model of conditional heteroskedasticity is employed, it is natural to analyze
the data for this characteristic. Figures 4 and 5 graph the stock return time series
for the U.S. and Sweden, respectively, over their full sample periods. The volatility
seems to cluster over time, showing periods of frequent large swings, and other
periods of calm and low volatility in returns. Table 2 reports on ARCH Lagrange
multiplier test statistics of Engle (1982), which are significant throughout, indicating
the presence of time-varying volatility. The table, also, presents estimation outputs
from fitting the GARCH(1,1) model in (6) to the stock return series. Indeed, the
GARCH(1,1) parameter estimates are significant throughout, which supports the
conditional heteroskedasticity of stock returns.
5. Application to Financial Data
59
Figure 6: Prospective Utility as a Function of the Evaluation Period Using U.S.
Data
prospective utility
Panel A: 1926:1 − 2003:12
stocks using GARCH
stocks using bootstrap
bonds using bootstrap
0.1
0.05
0
2
4
6
8
10
12
evaluation period (months)
14
16
18
16
18
prospective utility
Panel B: 1926:1 − 1990:12
stocks using GARCH
stocks using bootstrap
bonds using bootstrap
0.1
0.05
0
2
4
6
8
10
12
evaluation period (months)
14
The figure shows prospective utility as a function of the investor’s portfolio evaluation period
using U.S. data. The portfolio either consists of one hundred percent stocks or one hundred
percent long-term bonds. Two approaches to simulating returns distributions are considered; the
non-parametric bootstrap approach, and the parametric approach using a GARCH model. Panels
A and B consider different data sample periods.
5.3
Comparing Evaluation Periods Obtained from the Two
Approaches
Figure 6 presents portfolio prospective utility as functions of the evaluation period
using U.S. data. The portfolio consist of either one hundred percent stocks or one
hundred percent bonds. The point where the respective function lines cross gives the
evaluation period at which the investor finds the two portfolios equally attractive.
When simulating stock returns distributions at different data frequencies, the nonparametric bootstrap approach as well as the parametric GARCH approach are
employed. Thus the stock portfolio utility over an increasing evaluation period is
represented by two lines in the figure.
60
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
Panel A presents the results using the full U.S. sample period from 1926 to
2003. With the bootstrap procedure, the equilibrium evaluation period is about
seven months, while the GARCH simulation approach produces an approximate
evaluation period of ten months. Thus the GARCH approach produces a longer
evaluation period than the bootstrap counterpart, i.e., when temporal independence
is set aside and time-varying volatility is incorporated, the equilibrium evaluation
period increases.
BT analyze the monthly sample period from 1926 to 1990 for which panel B
presents the results. The obtained evaluation periods are six months using the
bootstrap method, and ten months using the GARCH approach. Again, the longer
evaluation period is produced under conditional heteroskedasticity. The procedure I
employ when simulating returns distributions non-parametrically does not replicate
the twelve-month evaluation period of BT however. Since the U.S. market is studied
over the same sample period as BT consider, an evaluation period of six months
is quite unexpected. One far reached explanation for the difference is the use of
different data sets. Although both measure U.S. aggregate stock and long-term
bond returns, the data set used in the current study is from Ibbotson Associates,
while BT consider CRSP data. Perhaps more likely however, the applied nonparametric bootstrap simulation techniques differ somehow. Although I have no
reason to believe that the interpretation of BT’s simulation procedure presented in
section 4.1 to be incorrect, the possibility exists that some part of the interpreted
procedure is different from the one BT use.16 Obtaining an evaluation period of six
instead of twelve months suggests that prospective utility, and the MLA model, is
sensitive to the exact method used when simulating returns.
Next, the MLA framework is applied to the Swedish equity premium. A problem
with the GARCH method of simulating returns distributions arises when the full
data sample is considered however. The estimated GARCH(1,1) model has an implied unconditional kurtosis that is negative at -2.2, which does not make sense, and
makes temporal aggregation impossible.17 This indicates that the GARCH model
is misspecified, which could be caused by a structural break in the eighty-five year
data sample. To resolve the problem in a simple way, the sample is split into two
equally-sized parts covering 1919:1 to 1961:6 and 1961:7 to 2003, and each subsample is studied separately. Over the subsamples, the implied unconditional kurtosis
from estimating the GARCH(1,1) model come out positive, causing no problems for
16
It is unclear whether BT compound returns continuously or not. The results come out the
same irrespectively though.
17
Kurtosis is defined as the forth moment about the mean devided by the squared variance, and
is non-negative by construction.
5. Application to Financial Data
61
Figure 7: Prospective Utility as a Function of the Evaluation Period Using Swedish
Data
prospective utility
Panel A: 1919:1 − 2003:12
0.15
stocks using bootstrap
bonds using bootstrap
0.1
0.05
0
5
10
15
evaluation period (months)
20
prospective utility
Panel B: 1919:1 − 1961:6
0.08
0.06
0.04
stocks using GARCH
stocks using bootstrap
bonds using bootstrap
0.02
0
−0.02
5
10
15
evaluation period (months)
20
prospective utility
Panel C: 1961:7 − 2003:12
0.2
stocks using GARCH
stocks using bootstrap
bonds using bootstrap
0.1
0
5
10
15
evaluation period (months)
20
The figure shows prospective utility as a function of the investor’s portfolio evaluation period
using Swedish data. The portfolio either consists of one hundred percent stocks or one hundred
percent long-term bonds. Two approaches to simulating returns distributions are considered; the
non-parametric bootstrap approach, and the parametric approach using a GARCH model. Panels
A-C consider different data sample periods.
62
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
temporal aggregation. Moreover, although the GARCH simulation method cannot
be applied to the full sample period of data, analysis is still carried out using the
non-parametric bootstrap approach.
Table 1 reports on summary statistics over the two subsamples of Swedish data.
Over the first half, stocks have outperformed bonds to an extent measured by an
annual equity premium of 2.8 percent, while the second half presents a corresponding
equity premium of just over six percent. Table 2 presents tests of ARCH structure
in the stock returns as well as GARCH(1,1) estimation outputs, which statistically
indicate the presence of time-variation in stock return conditional volatility over
both subsamples.
Figure 7 presents portfolio prospective utility as functions of the evaluation period using the Swedish data. Over the full sample, the bootstrap approach produces
an evaluation period of twelve months, as panel A shows. The result is consistent
with BT, and suggests that the EPP of Sweden can be explained by MLA preferences with a yearly evaluation period. The result is new to the literature, since
previous studies have not applied MLA preferences to non-U.S. equity premiums.
Over the subsample from 1919:1 to 1961:6, panel B displays an equilibrium evaluation period of about sixteen months using the bootstrap method, and nineteen
months using the GARCH approach. Furthermore, panel C graphs the functions
over the second half of the Swedish full sample period: 1961:7 to 2003:12. Evaluation periods of ten and fifteen months for the respective bootstrap and GARCH
approaches can be elicited. Thus the results using the Swedish data show that
evaluation periods are quite variable and depend on the data sample considered.
They, also, support the previous finding when U.S. data was used, that the GARCH
approach to simulating returns produces a longer evaluation period than the corresponding one obtained using the bootstrap method.
To summarize, the results using the U.S. and Swedish data produce varying
evaluation periods depending of the sample studied. Nevertheless, the obtained
evaluation periods can be considered to be of similar magnitude as the twelve-month
counterpart reported by BT, and thus their result is supported. Furthermore, the
evaluation periods produced under conditional heteroskedasticity are longer than the
ones produced under temporal independence throughout. The prospective utility
maximizer needs a longer period between evaluations to be content with the equity
premium when returns are simulated with time-varying volatility. Since a longer
evaluation period is needed, stocks are perceived as more risky under conditional
heteroskedasticity. Interestingly, this result further indicates that prospective utility,
and the MLA model, is sensitive to the simulation procedure applied, which relates
to the distributional assumption made on stock returns.
5. Application to Financial Data
5.4
63
Sensitivity Analysis
The previous analysis used Tversky and Kahneman’s (1992) estimates of the parameters of (2) and (4), i.e., λ = 2.25, γ = 0.88, τ = 0.61 over gains, and τ = 0.69 over
losses. Are the obtained results sensitive to changes in these estimates? By varying
the parameter values it seems that the parameter of loss aversion, to begin with, has
a strong influence on prospective utility. Using the U.S. data over the full sample,
a loss aversion equal to three, i.e. λ = 3, raises the obtained evaluation periods
considerably from seven and ten months, when λ = 2.25, to thirteen and sixteen
months using the non-parametric and parametric methods of simulating returns,
respectively. The increase in evaluation period is intuitive, since it compensates
for the additional risk perceived at a higher level of loss aversion. The risk-return
relationship of stocks and bonds is thus kept at equilibrium.
The parameter of risk aversion over gains and risk-seeking over losses, γ, does not
seem to have any strong impacts on prospective utility. Altering the value function
from having γ = 0.88 to γ = 1, i.e. the function becomes piece-wise linear, affects
the equilibrium evaluation periods by only one month or so.
The weighting function parameter τ shows of some importance though. When
altered to τ = 1, implying that probabilities are treated objectively, the obtained
evaluation periods change from seven and ten months to five and five months, using the bootstrap and GARCH approaches, respectively. For one, when τ = 1,
both approaches to simulating returns result in smaller evaluation periods needed
to match the historical equity premium, indicating that stocks are perceived as less
risky. For the other, the two approaches produce approximately equal evaluation
periods under objective probabilities. Interestingly, this is not only the case when
(λ, γ) = (2.25, 0.88), but is a general result that holds irrespective of the values of
λ and γ so long as probabilities are treated objectively.18 Thus, when probabilities
are objective, the MLA investor becomes less sensitive to the distributional shape
of portfolio returns.
Another sensitivity analysis involves the use of real instead of nominal returns.
Do the previously obtained results change when using data in real terms? The
answer is no. For instance, when considering the U.S. data in real terms over the full
sample period, evaluation periods of about five and nine months are obtained, using
the bootstrap and GARCH approaches, respectively.19 These evaluation periods do
18
This result is also found using the Swedish data.
Figures A.1 and A.2 in the appendix present prospective utility as functions of the evaluation
period when using real data, for both the U.S. and Sweden.
19
64
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
not differ much from the corresponding ones obtained when nominal returns are
employed, i.e., seven and ten months.
BT consider comparing the prospective utility of an all-stocks portfolio with an
all-bills portfolio as well, i.e., short-term Treasury bills are considered as the alternative investment to stocks instead of long-term bonds. Such an analysis does change
the evaluation periods to some extent, e.g., over the U.S. subsample of 1926 to 1990,
eleven- and fourteen-month evaluation periods are obtained when using the nonparametric and parametric approaches, respectively.20 The eleven-month evaluation
period obtained under temporal independence is thus in line with BT. Recall that a
six-month evaluation period was produced under temporal independence when using
bonds. Since BT’s approximate yearly evaluation period is replicated when bills are
assumed to be the alternative investment, perhaps the long-term bond data used
here differs from the corresponding data used by BT. This could be the cause for the
previously reported different result. Furthermore, overall longer evaluation periods
are obtained when using bills compared with bonds in the analysis. Bills are thus
considered to be more attractive than bonds under MLA preferences. The portfolio
evaluation period needs to be less frequent in order to make stocks as appealing as
the alternative investment.
5.5
What Drives the Results?
One main result of this paper is that the MLA model seems to be sensitive to
how returns distributions are simulated. When using a simulation technique that
incorporates conditional heteroskedasticity rather than simulating under temporal
independence, the evaluation periods consistent with the historical equity premiums
in the U.S. and Sweden increase. What are the distributional differences in simulated
returns causing for this result? Figure 3 exemplifies simulated distributions using the
two approaches. The histograms indicate that the unconditional moments, which
describe the distribution’s shape, change as the evaluation period increases.
Figure 8 makes it more clear what distributional changes actually occur. Panels
A-D present the first four unconditional moments of the simulated returns distributions as functions of the evaluation period when using both the bootstrap and
the GARCH simulation methods. The full sample of monthly U.S. aggregate stock
returns is used. Panel A shows that the mean evolves similarly for the two methods.
With a longer evaluation period, the increase in the variance is steeper when the
bootstrap approach is considered compared with the GARCH approach, as panel B
shows. This would suggest that stocks are perceived as more risky under temporal
20
Figures A.3 and A.4 in the appendix show prospective utility as functions of the evaluation
period when Treasury bills are considered as the alternative investment to stocks.
5. Application to Financial Data
65
Figure 8: Unconditional Moments as a Function of the Evaluation Period
Panel A: Mean
Panel B: Variance
0.08
GARCH
bootstrap
0.15
0.1
0.04
0.05
0.02
5
10
15
5
Panel C: Skewness
12
0.6
10
0.4
8
GARCH
bootstrap
10
15
Panel D: Kurtosis
0.8
0.2
GARCH
bootstrap
0.06
GARCH
bootstrap
6
4
0
5
10
15
5
10
15
The figure shows the unconditional moments mean, variance, skewness, and kurtosis of simulated
returns over an increasing evaluation period. Both simulation methods; the non-parametric bootstrap and the parametric GARCH, are considered. U.S. aggregate stock returns covering 1926:1
to 2003:12 are employed.
independence, which was however not found in the previous investigation. Therefore, the changes in the skewness and the kurtosis, shown in panels C and D, must
play a part in the way the MLA investor perceives the risk of a stock investment,
making stocks more favorable under temporal independence than under conditional
heteroskedasticity. Panel C shows that the skewness of the bootstrap-simulated
distribution rises with an increasing evaluation period, while the GARCH method
leaves the skewness at practically zero everywhere, which is not surprising since the
GARCH(1,1) is symmetric. Since the prospective utility maximizer suffers from loss
aversion, the increasing skewness is possibly the factor that dominates the effect
of a more progressive variance, making the risky investment more attractive under
temporal independence than under conditional heteroskedasticity. However, such a
reasoning is speculative, and further research on the relationship between prospective utility and the distributional skewness is needed. Moreover, the kurtosis falls
when the evaluation period increases across both methods, as panel D shows, with
somewhat larger levels of kurtosis when using the bootstrap method overall.
66
6
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
Conclusions
The paper replicates the study of BT, who suggest an explanation to the EPP by
MLA preferences, and, furthermore, considers a technical extension to their methodology. Specifically, the distributional assumption made on returns is addressed.
When simulating returns distributions, conditional heteroskedasticity is incorporated through a GARCH model in place of the temporal independence assumption
of BT. Moreover, Swedish data is considered in addition to U.S. data, which further
extends BT’s analysis.
Over the full U.S. sample of data, evaluation periods of seven and ten months
are obtained when using the non-parametric bootstrap approach, which assumes
temporal independence in returns, and the GARCH approach, respectively. These
evaluation periods are smaller than the twelve-month counterpart reported by BT.
Furthermore, the twelve-month evaluation period of BT is not replicated when simulating returns under temporal independence, although a U.S. data set covering BT’s
sample period of 1926 to 1990 is used. Instead, an evaluation period of six months is
obtained. The applied non-parametric bootstrap simulation techniques might differ
somehow, although I have no reason to believe it to be the case. Nonetheless, obtaining an evaluation period of six instead of twelve months suggests that prospective
utility, and MLA, is sensitive to the method used when simulating returns distributions.
A problem with GARCH approach occurs when the full Swedish data sample
is considered, and analysis using this approach is made on two subsamples of data
instead. The non-parametric approach produces a twelve-month evaluation period
over the full sample though, in line with BT, which suggests that MLA can explain
the EPP of Sweden. This result is new to the literature, since previous studies
have not applied MLA preferences to the equity premiums of non-U.S. economies.
Furthermore, overall longer evaluation periods are obtained under conditional heteroskedasticity when studying the two subsamples of data. For example, over the
period from July 1961 to December 2003, evaluation periods of ten and fifteen
months are obtained using the bootstrap and GARCH methods, respectively.
Throughout the analysis, longer evaluation periods are produced under conditional heteroskedasticity. Does this result fail BT’s explanation to the EPP? Since
BT’s twelve-month evaluation period is quite approximative, and the evaluation
periods obtained here are of similar magnitude, I would say the answer is no. However, the obtained difference between the two approaches further indicates that the
MLA model is sensitive to the method of simulating returns. This relates to the
returns’ distributional assumption, and thus to the shape of the simulated returns
6. Conclusions
67
distribution. Therefore, the two simulation techniques are analyzed with respect
to the how the first four unconditional moments evolve as the evaluation period
lengthens. Plausibly, the skewness of the returns distribution is important for the
loss-averse investor, which is intuitive since loss aversion induces an asymmetric
preference over gains and losses. Further research on prospective utility in relation
to the distributional skewness is suggested.
The sensitivity of the paper’s results are analyzed with respect to various modifications. One interesting finding is that the probability weighting function seems
to be important for MLA. When altering the weighting function parameter so that
probabilities are treated linearly, the two approaches to simulating returns produce
equal evaluation periods irrespective of the value function’s parameter-values and
whether U.S. or Swedish data is considered. This indicates that MLA investors
become less sensitive to the distributional assumption made on returns when probabilities are objective. Further research investigating this issue is suggested.
68
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
References
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Engle, R. (1982), "Autoregressive conditional heteroskedasticity with estimates of
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prices", Journal of Political Economy 96, 246-73.
Frennberg, P. and B. Hansson (1992), "Computation of a monthly index for Swedish
stock returns: 1919-1989", Scandinavian Economic History Review 1, 3-27.
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Gneezy, U., A. Kapteyn, and J. Potters (2003), "Evaluation periods and asset prices
in a market experiment", Journal of Finance 58, 821-38.
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Haigh, M and J. List (2006), "Do professional traders exhibit myopic loss aversion? An experimental analysis", Journal of Finance 60, 1.
Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton,
New Jersey.
Kahneman, D. and A. Tversky (1979), "Prospect theory: An analysis of decision
under risk", Econometrica 47, 263-91.
Kocherlakota, N. (1996), "The equity premium: It’s still a puzzle", Journal of Economic Literature 34, 42-71.
Lamoureux, C. and G. Zhou (1996), "Temporary components of stock returns: What
do the data tell us?", Review of Financial Studies 9, 4, 1033-59.
Mehra, R. and E. Prescott (1985), "The equity premium puzzle", Journal of Monetary Economics 15, 145-61.
Samuelson, P. (1963), "Risk and uncertainty: A fallacy of large numbers", Scientia 98, 108-13.
Siegel, J. and R. Thaler (1997), "Anomalies: The equity premium puzzle", Journal
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Thaler, R. (1985), "Mental accounting and consumer choice", Marketing Science
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myopia and loss aversion on risk taking: An experimental test", Quarterly Journal
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70
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
Appendix
Figure A.1: Prospective Utility as a Function of the Evaluation Period Using U.S.
Data in Real Terms
Panel A: 1926:1 − 2003:12
prospective utility
0.06
stocks using GARCH
stocks using bootstrap
bonds using bootstrap
0.04
0.02
0
−0.02
2
4
6
8
10
12
evaluation period (months)
14
16
18
16
18
prospective utility
Panel B: 1926:1 − 1990:12
stocks using GARCH
stocks using bootstrap
bonds using bootstrap
0.04
0.02
0
−0.02
−0.04
2
4
6
8
10
12
evaluation period (months)
14
The figure shows prospective utility as a function of the investor’s portfolio evaluation period using
U.S. data in real terms. The portfolio either consists of one hundred percent stocks or one hundred
percent long-term bonds. Two approaches to simulating returns distributions are considered; the
non-parametric bootstrap approach, and the parametric approach using a GARCH model. Panels
A and B consider different data sample periods.
Appendix
71
Figure A.2: Prospective Utility as a Function of the Evaluation Period Using
Swedish Data in Real Terms
Panel A: 1919:1 − 2003:12
prospective utility
0.08
stocks using bootstrap
bonds using bootstrap
0.06
0.04
0.02
0
−0.02
2
4
6
8
10
12
14
16
evaluation period (months)
18
20
22
24
20
22
24
20
22
24
prospective utility
Panel B: 1919:1 − 1961:6
0.06
stocks using GARCH
stocks using bootstrap
bonds using bootstrap
0.04
0.02
0
−0.02
2
4
6
8
10
12
14
16
evaluation period (months)
18
prospective utility
Panel C: 1961:7 − 2003:12
0.1
stocks using GARCH
stocks using bootstrap
bonds using bootstrap
0.05
0
2
4
6
8
10
12
14
16
evaluation period (months)
18
The figure shows prospective utility as a function of the investor’s portfolio evaluation period
using Swedish data in real terms. The portfolio either consists of one hundred percent stocks or
one hundred percent long-term bonds. Two approaches to simulating returns distributions are
considered; the non-parametric bootstrap approach, and the parametric approach using a GARCH
model. Panels A-C consider different data sample periods.
72
Essay 2. Myopic Loss Aversion, the Equity Premium Puzzle, and GARCH
Figure A.3: Prospective Utility as a Function of the Evaluation Period Using U.S.
Data with Bills as the Alternative Investment
prospective utility
Panel A: 1926:1 − 2003:12
stocks using GARCH
stocks using bootstrap
bills using bootstrap
0.1
0.05
0
2
4
6
8
10
12
evaluation period (months)
14
16
18
16
18
prospective utility
Panel B: 1926:1 − 1990:12
0.1
stocks using GARCH
stocks using bootstrap
bills using bootstrap
0.05
0
2
4
6
8
10
12
evaluation period (months)
14
The figure shows prospective utility as a function of the investor’s portfolio evaluation period using
U.S. data. The portfolio either consists of one hundred percent stocks or one hundred percent
short-term Treasury bills. Two approaches to simulating returns distributions are considered; the
non-parametric bootstrap approach, and the parametric approach using a GARCH model. Panels
A and B consider different data sample periods.
Appendix
73
Figure A.4: Prospective Utility as a Function of the Evaluation Period Using
Swedish data with Bills as the Alternative Investment
Panel A: 1919:1 − 2003:12
prospective utility
0.3
stocks using bootstrap
bills using bootstrap
0.2
0.1
0
5
10
15
20
25
evaluation period (months)
30
prospective utility
Panel B: 1919:1 − 1961:6
0.15
stocks using GARCH
stocks using bootstrap
bills using bootstrap
0.1
0.05
0
5
10
15
20
25
evaluation period (months)
30
prospective utility
Panel C: 1961:7 − 2003:12
0.4
stocks using GARCH
stocks using bootstrap
bills using bootstrap
0.3
0.2
0.1
0
5
10
15
20
25
evaluation period (months)
30
The figure shows prospective utility as a function of the investor’s portfolio evaluation period using
Swedish data. The portfolio either consists of one hundred percent stocks or one hundred percent
short-term Treasury bills. Two approaches to simulating returns distributions are considered; the
non-parametric bootstrap approach, and the parametric approach using a GARCH model. Panels
A-C consider different data sample periods.
Essay 3
Prospect Theory and Higher
Moments
1
Introduction
Behavioral finance has emerged as an alternative approach to financial economics
largely because of the difficulties of the traditional theory. The most acclaimed
behavioral model of individual decision-making under risk is prospect theory. Kahneman and Tversky (1979) demonstrate a number of individual violations of neoclassical expected utility based on experimental evidence, and in spirit of these
violations they propose prospect theory as a more realistic model. Although successful in many applications, the original version has its drawbacks. For one, utility
can be derived from gambles of only two outcomes, and, for the other, the attractive property of first-order stochastic dominance does not hold. As a resolution,
Tversky and Kahneman (1992) introduce cumulative prospect theory (CPT), where
utility is derived from gambles of any number of outcomes, and first-order stochastic
dominance holds.
CPT is perhaps the most complete summary of the experimental evidence on
attitude to risk. Under CPT, investors derive utility by using a specific value function, and by weighting probabilities subjectively. The latter feature transforms the
outcome distribution so that small probabilities are over-weighted, which magnifies
the tails of the distribution, and moderate to large ones are under-weighted. The
value function differs from standard concave utility functions, e.g. power utility, in
three main respects. First, utility is derived from changes in wealth relative to a
reference point, as opposed to final levels of wealth. Second, the value function is
concave over gains, implying risk aversion, but convex over losses, reflecting a riskseeking behavior in that domain. Third, losses loom larger than gains do, causing
76
Essay 3. Prospect Theory and Higher Moments
for a kink in the value function at the reference point. This last property, referred to
as loss aversion, implies a high sensitivity for small changes in wealth. In contrast,
standard concave utility functions display local risk-neutrality.1
There exists a number of applications of prospect theory and its modified version
CPT in financial economics research. Shefrin and Statman (1985) apply prospect
theory to help explain the disposition effect, which concerns the disposition of individual investors to sell wining stocks too early, and hold on to losers for too long.2
A plausible clarification of the ambiguous endowment effect, which refers to the
individual tendency to value something more heavily once owned, is put forward
by Kahneman, Knetsch, and Thaler (1990). The most celebrated finance application of CPT, and in particular loss aversion, is presented by Benartzi and Thaler
(1995) however. Stocks are perceived as more risky among loss-averse investors if
evaluated frequently, since losses occur with greater probability over shorter time
horizons. Benartzi and Thaler (1995) show that "myopic loss aversion" can explain
the historical magnitude of equity premium over bonds if evaluated yearly. Consequently, they propose a behavioral explanation to the infamous equity premium
puzzle of Mehra and Prescott (1985).3 Barberis, Huang, and Santos (2001) generalize Benartzi and Thaler’s (1995) single-period model in a multi-period general
equilibrium context. They argue that loss aversion alone does not produce a large
enough equity premium, and incorporate an investor sensitivity for prior outcomes
as a resolution, causing for a time-varying loss aversion.
The current paper relates to the literature on CPT portfolio choice. Levy, De
Giorgi, and Hens (2003) are first to show that CPT efficient portfolios are, in fact,
also mean-variance efficient, provided that returns are normally distributed. They,
also, prove that the standard two-period capital asset pricing model (CAPM) is
consistent with CPT. The mean-variance optimization algorithm can thus be employed when constructing CPT efficient portfolios, something that is quite remarkable since CPT stands in such sharp contrast to the assumptions of mean-variance
analysis, namely expected utility maximization and global risk aversion. Levy and
Levy (2004) and Barberis and Huang (2005) present analogous proofs to solidify the
result. However, the normality assumption weakens the general understanding of
CPT in portfolio choice issues. Financial returns distributions are often skewed and
fat-tailed, which are characteristics that the normal distribution cannot model since
1
Loss aversion is related to the concept of first-order risk aversion. See Epstein and Zin (1990).
Recently, however, Barberis and Xiong (2006) argue that prospect theory predicts the opposite
of the disposition effect.
3
The puzzle concerns the inability to explain the historical magnitude of U.S. equity premium
within a standard consumption-based general equilibrium model at reasonable parameter values.
2
1. Introduction
77
it is fully determined by the mean and the variance. A natural question that comes
to mind is how CPT utility is related to the higher-order moments, e.g., skewness
and kurtosis. Since loss aversion implies an asymmetric preference over gains and
losses, and probability weighting magnifies the tails of the returns distribution, the
question is relevant. Furthermore, are CPT portfolios mean-variance efficient under
more general distributional assumptions than normality? These issues are addressed
in the current paper.
Few previous studies within the literature of optimal asset allocation consider
higher-order moments. Kraus and Litzenberger (1976) present an unconditional
three-moment CAPM, and find that investors with standard concave utility functions like skewness. This result is in line with Arditti (1967), who shows that most
standard concave utility functions, e.g., logarithmic and power utility imply a preference for skewness, since they fulfill the condition of non-increasing absolute risk
aversion. Harvey and Siddique (2000) expand the conditional CAPM to include
coskewness with the market, which helps to explain the cross-section of equity returns. Furthermore, Ågren (2006) presents a technical extension to the work of Benartzi and Thaler (1995) by incorporating conditional heteroskedasticity in returns,
in contrast to the original temporal independence assumption. The results show
that overall longer evaluation periods are needed under conditional heteroskedasticity when considering both U.S. and Swedish data. Consequently, Ågren (2006)
argues that prospect theory utility is sensitive to the distributional assumption of
returns, especially concerning the skewness.
Assuming returns are normal inverse Gaussian (NIG) distributed, this paper
addresses the implications of higher moments for CPT portfolio choice. The NIG
distribution, presented by Barndorff-Nielsen (1997), is a four parameter distribution
with the desirable property of parameter-dependent higher-order moments. Eriksson, Forsberg, and Ghysels (2005) present a useful transformation of the NIG distribution’s parameters so that its probability density can be expressed as a function of
the first four cumulants. Cumulants are a set of distributional descriptive constants
just like moments are. The first and second cumulants equal the respective first
and second central moments, i.e. the mean and the variance, while skewness and
kurtosis are simple normalizations of the third and forth cumulants, respectively.
The transformed alternative parameterization makes a straightforward link between
utility and cumulants possible so that the effects of a change in one specific distributional characteristic, such as skewness, can be analyzed in isolation, i.e., without
78
Essay 3. Prospect Theory and Higher Moments
affecting the other ones.4 This makes the NIG distribution highly suitable for the
current investigation.
The paper considers a risky portfolio with NIG distributed return in a singleperiod framework. There are two main objectives. First, an analysis of portfolio
utility in relation to the return’s distributional characteristics is conducted, where
three kinds of investor preferences are considered: CPT, CPT without investor
probability weighting, which is referred to as expected loss aversion (ELA), and
expected power (EP) utility.5 In this way, the implications of probability weighting
and loss aversion can be separated, and compared with traditional utility theory.
Second, the CPT portfolio choice is examined by optimizing the allocation to a risky
and a relatively risk-free asset. Both analyses involve model-parameter calibrations
to empirical estimates.
I show that investor utility is positively related to the portfolio’s mean, and
negatively related to its variance, irrespective of the preference scheme. Intuitively,
the result for the variance is somewhat surprising considering that CPT investors are
risk-seeking over losses. Loss aversion dominates however, implying a preference for
low-variance portfolios. Furthermore, the relation between utility and skewness is
negative when ELA preferences are considered, but turns positive when probability
weighting is introduced, i.e., when CPT preferences are embraced in full. This
shows that CPT investors display a preference for skewness through the probability
weighting function. Essentially, loss aversion makes the ELA investor sensitive to
the probability of small losses, while CPT investors, over-weighting the probability
of extreme outcomes, care more about the probability of large losses. While CPT
investors prefer lottery-type gambles with positively skewed outcomes as they might
receive a large gain, the ELA investor is averse to such gambles since they incur a
small but almost sure loss.
Utility and kurtosis are positively related under ELA, but inverse hump-shape
related under CPT. The relation is difficult to explain, and is quite sensitive to the
level of loss aversion and degree of probability weighting. The extent to which the
investor suffers from loss aversion in relation to her degree of probability weighting
determines the relation between CPT utility and kurtosis.
What implications do these results have for the optimal asset allocation? To answer this question, the CPT portfolio choice problem is analyzed under the NIG distributional assumption. Related research includes Aït-Sahalia and Brandt (2001),
who study the optimal set of predictive variables for portfolio choice over differ4
Throughout the paper, I will time and again use the collective term: distributional characteristics, when referring to the mean, the variance, the skewness, and the kurtosis as a whole.
5
For EP utility, the constant relative risk aversion power utility function is employed.
2. Cumulative Prospect Theory
79
ent preference schemes, among them CPT, and Berkelaar, Kouwenberg, and Post
(2004), who analyze the optimal investment strategy of CPT investors when assuming general Ito processes for asset prices. The two studies do not consider probability
weighting however, but analyze what I refer to as ELA preferences. Neither do they
consider the effects of skewness and kurtosis on the portfolio choice.
Consistent with Aït-Sahalia and Brandt (2001) and Berkelaar et al. (2004), I
find strong horizon effects in the investor’s asset allocation. The portion of stocks
progresses heavily as the horizon increases. Moreover, the results suggest that CPT
optimal portfolios are not mean-variance efficient under the NIG assumption, with
the investor typically placing a relatively larger weight on stocks when higher moments are taken into account.
The rest of the paper is outlined as follows: Section 2 introduces CPT, and
explains how to derive CPT utility under a distributional assumption. Section 3
presents the NIG distribution in general, as well as in a more useful alternative form.
Section 4 analyzes investor utility as a function of the portfolio’s mean, variance,
skewness, and kurtosis. Section 5 turns to the optimal portfolio choice of CPT
investors. Section 6 concludes.
2
Cumulative Prospect Theory
Tversky and Kahneman (1992) present two cornerstone functions for CPT utility:
a value function over outcomes, v(·), and a weighting function over cumulative
probabilities, w(·). The CPT utility of a gamble G with stochastic return X is
defined as
(1)
U(G) ≡ E w [v(X)] ,
where E w [·] is the unconditional expectations operator under subjective probability
weighting, indicated by w, and v(X) is the value function.
2.1
Value Function
The value function derives utility from gains and losses, and not from final wealth as
traditional utility functions do. Tversky and Kahneman (1992) suggest the following
functional form:
(
if x ≥ x̄
(x − x̄)γ
,
(2)
v(x) =
γ
−λ(x̄ − x) if x < x̄
80
Essay 3. Prospect Theory and Higher Moments
Figure 1: The Value Function
10
5
value
0
−5
−10
λ=1, γ=1
λ=2.25, γ=0.6
λ=2.25, γ=0.88
−15
−10
−5
0
return
5
10
The figure illustrates the cumulative prospect theory value function over returns (%) for a few
parameter-value combinations, and with a zero reference return. Tversky and Kahneman (1992)
suggest λ̂ = 2.25 and γ̂ = 0.88.
where outcomes x are separated into gains and losses with respect to a reference
point x̄, which is thought of as a sure alternative to the risky gamble.6
The value function in (2) exhibits loss aversion when λ > 1, which is motivated
by the experimental finding that individual investors are more sensitive to losses
than to gains. Although its expected value is positive, a fifty-fifty bet of wining
$200 or losing $100 is generally rejected, since a loss of $100 is perceived as more
painful than a gain of $200 is enjoyable. Moreover, (2) allows for risk aversion over
gains but risk-seeking over losses when γ < 1. Consider a gamble with a fifty percent
chance of wining $100 or nothing to the alternative of receiving $50 for sure. Most
individuals would prefer the sure gain to the risky gamble since they are risk-averse
over gains. They prefer the expected value to the gamble. In comparison, consider
a gamble with a fifty percent probability of losing $100 or nothing. When choosing
between this gamble and the alternative of giving up $50 for sure, experimental
evidence shows that individuals generally prefer to take on the gamble. They are
risk-seeking over losses, and, hence, favor the gamble to its expected value.
Figure 1 illustrates the value function for a few parameter-value combinations
and with a zero reference return. Loss aversion causes the value function to be kinked
at the reference point, reflecting a dramatic change in marginal utility. With γ < 1,
the value function becomes concave over gains and convex over losses. Tversky and
6
When considering the gamble of investing in a portfolio of risky assets, a common assumption
is to let the average return on a risk-free asset represent the investor’s reference return.
2. Cumulative Prospect Theory
81
Kahneman (1992) conduct individual experiments, and estimate the value function’s
parameters to λ̂ = 2.25 and γ̂ = 0.88.
Figure 2: The Weighting Function
1
weighted probability
0.8
0.6
0.4
τ = 1.00
τ = 0.80
τ = 0.65
0.2
0
0
0.2
0.4
0.6
probability
0.8
1
The figure illustrates the cumulative prospect theory weighting function for a few parameter values.
Tversky and Kahneman (1992) suggest τ̂ = 0.65.
2.2
Probability Weighting
The probability weighting function w(·) applies to cumulative probabilities. Essentially, it over-weights small probabilities so that the tails of the distribution are
magnified. This feature of CPT stems from experimental evidence showing that
individuals perceive extreme events as more likely to occur than they really are.7
Furthermore, moderate to large probabilities are under-weighted, which reflects the
pessimism individuals might feel toward a relatively sure outcome. Tversky and
Kahneman (1992) propose the following function:
w(P ) =
(P τ
Pτ
,
+ (1 − P )τ )1/τ
(3)
where P is the objective cumulative probability, and τ ∈ (0, 1] is a function parameter.8 In (3), cumulative probabilities are weighted non-linearly to the extent
determined by τ . Since it is cumulative probabilities that are weighted and not
the actual ones, CPT is consistent with first-order stochastic dominance.9 More7
For instance, why do people buy lottery tickets?
Other functional forms of probability weighting have been proposed. See, e.g., Prelec (1998).
9
The original version of prospect theory weights actual probabilities, and, therefore, lacks the
property of first-order stochastic dominance.
8
82
Essay 3. Prospect Theory and Higher Moments
over, probability weighting should not be associated with a change of probability
measure, since the weighted probabilities, in fact, need not sum up to one.10
Figure 2 illustrates w(P ) for a few values of τ . When τ = 1, the function
collapses so that w(P ) = P , and the CPT investor treats probabilities linearly. A
value of τ < 1 introduces probability weighting, and the lower the value the more
prominent the weighting becomes. Tversky and Kahneman (1992) suggest τ̂ = 0.65
by way of individual experiments.11
2.3
Incorporating a Distributional Assumption
Consider a risky portfolio, G, with a stochastic return, X, that is continuously
distributed. CPT utility, defined in (1), is then derived as
U(G; θ) ≡ U(θ)
Z x̄
Z ∞
v(x)dw(1 − F (x)) +
v(x)dw(F (x))
= −
x̄
−∞
Z x̄
Z ∞
v(x)w0 (1 − F (x))f (x)dx +
v(x)w0 (F (x))f (x)dx,
=
x̄
(4)
−∞
where f (·) is the probability density function of X, F (·) is the corresponding cumulative distribution function, v(·) is the value function in (2), w(·) is the weighting
function in (3), and θ is a vector of parameters. Tversky and Kahneman (1992) consider gambles with discrete outcomes for which CPT utility is expressed differently.
Similar to Barberis and Huang (2005), the expression presented here is adjusted to
allow for continuous probability distributions.
Notice, in (4), that the weighting function applies differently in the domain of
gains and in the domain of losses. Moreover, utility is expressed using both the
Riemann-Stieltjes integral as well as the Riemann integral. Although the former
expression is, perhaps, easier to relate to Tversky and Kahneman’s (1992) discrete
representation, the latter one is attractive for computational reasons.
Expression (4) holds under any continuous distributional assumption for X. In
this paper, I assume X is NIG distributed.
10
For this reason, Kahneman and Tversky (1979) refer to the weighted probabilities as decision
weights.
11
Actually, Tversky and Kahneman (1992) estimate τ to 0.61 in the gains domain, and 0.69
in the loss domain. For simplicity, I approximate the value of τ with the average of these two
estimates.
3. Normal Inverse Gaussian Distribution
3
83
Normal Inverse Gaussian Distribution
The NIG distribution is introduced by Barndorff-Nielsen (1997) in an application
to stochastic volatility modeling. It is a mixture of the normal distribution and the
inverse Gaussian (IG) distribution.12 Formally, if a normally distributed variable X
has its variance drawn from the IG distribution, i.e.,
X|[Z = z] ∼ N(μ, z),
where
¶
µ q
Z ∼ IG δ, α2 − β 2 ,
then X is NIG distributed with parameters α, β, μ, and δ. Since I apply a result of
Eriksson et al. (2005), their standard parametrization is used: ᾱ = δα and β̄ = δβ.
The NIG(ᾱ, β̄, μ, δ) probability density function is given by
ᾱ
exp
fNIG (x; ᾱ, β̄, μ, δ) =
πδ
h q
i
µq
¶ K ᾱ 1 + ( x−μ )2
µ ¶
1
δ
β̄μ
β̄
2
q
x ,
ᾱ2 − β̄ −
exp
δ
δ
1 + ( x−μ )2
δ
(5)
where x ∈ R, ᾱ > 0, 0 < β̄ < ᾱ, δ > 0, μ ∈ R, and K1 is the modified Bessel
function of third order with index 1. The mean, the variance, the skewness, and the
kurtosis of X ∼ NIG(ᾱ, β̄, μ, δ) are given by
β̄δ
,
E[X] = μ + q
2
ᾱ2 − β̄
V [X] =
S[X] =
and
δ 2 ᾱ2
2
(ᾱ2 − β̄ )3/2
3β̄
,
2
ᾱ(ᾱ2 − β̄ )1/4
,
(6)
(7)
(8)
2
K[X] =
12β̄ + 3ᾱ2
q
,
2
ᾱ2 ᾱ2 − β̄
(9)
respectively.
While the normal distribution has zero skewness and a kurtosis equal to three,
equations (8) and (9) show that a NIG distributed variable has parameter-dependent
12
The IG distribution is defined over the interval [0, ∞). The name stems from the fact that the
cumulant generating function of an IG distributed variable is the inverse of the cumulant generating
function of a normally (Gaussian) distributed variable.
84
Essay 3. Prospect Theory and Higher Moments
skewness and kurtosis. Explicitly, the parameters of the NIG density can be interpreted as follows: ᾱ and β̄ are shape parameters with β̄ expressing the skewness of
the distribution, and, when β̄ = 0, ᾱ determining the amount of excess kurtosis.13
The parameter μ is a location parameter, and δ is a scale parameter.14
To illustrate the NIG distribution’s ability to capture the characteristics of financial returns distributions, consider the real six-month returns of the S&P 500
composite index from Ibbotson Associates. Table 1 reports on summary statistics.
Over the sample period of January 1926 to December 2003, the returns average
at 5.56 percent, with a standard deviation of 15.13 percent. The skewness and
kurtosis equal 1.07 and 9.21, respectively, indicating that the data is non-normally
distributed. Indeed, the Jarque-Bera test of normality is highly significant. Figure
3 illustrates both the empirical stock returns distribution (panel A), as well as two
approximations, where both NIG and normality are assumed (panel B). Notice that
the NIG distribution captures both the skewness and the kurtosis of the empirical
distribution.
Figure 3: Empirical and Approximate Distributions
Panel A: Empirical Distribution
0.1
0.05
0
−100
−50
0
50
100
Panel B: Approximate Distributions
normal
NIG
0.03
0.02
0.01
−100
−50
0
50
100
The figure illustrates the empirical distribution of S&P 500 real six-month returns (%) from January
1926 to December 2003 (panel A), together with two approximate distributions, namely the normal
and normal inverse Gaussian distributions (panel B).
13
Excess kurtosis refers to the amount of kurtosis that exceeds that of the normal distribution.
To read more on the NIG distribution and its use in stochastic volatility modeling, see, e.g.,
Andersson (2001) and Forsberg (2002).
14
U.S. 30-day bill, real returns
Horizon (months)
1
6
12
0.06
0.37
0.77
2.37
8.36
13.54
-5.39
-13.10
-16.22
0.52
2.25
4.06
-1.68
-0.83
-0.38
18.59
8.56
6.08
0.12
0.16
0.19
9 868
1 300
385
(<0.001)
(<0.001)
(<0.001)
936
931
925
S&P 500, real returns
Horizon (months)
1
6
12
0.90
5.56
11.50
53.64
113.31
258.99
-25.48
-44.03
-59.00
5.85
15.13
24.29
1.62
1.07
2.17
19.80
9.21
20.71
0.15
0.37
0.47
11 357
1 663
12 747
(<0.001)
(<0.001)
(<0.001)
936
931
925
The table reports on summary statistics for continuously compounded returns on the S&P 500 composite index and a U.S. 30-day Treasury bill, provided
by Ibbotson Associates. The time period stretches from January 1926 to December 2003. One-month, six-month and twelve-month horizons are considered.
Jarque-Bera is a test over skewness and kurtosis under the null of normality, where skewness equals zero and kurtosis is equal to three. p-values are in
parentheses.
No. obs.
Jarque-Bera
Mean (%)
Max. (%)
Min. (%)
Std. dev. (%)
Skewness
Kurtosis
Sharpe ratio
S&P500,
nominal returns
One-month
horizon
0.99
42.56
-29.73
5.62
0.39
12.45
0.18
3 488
(<0.001)
936
Table 1: Summary Statistics for Financial Returns
3. Normal Inverse Gaussian Distribution
85
86
Essay 3. Prospect Theory and Higher Moments
3.1
An Alternative Parameterization
Analyzing the relationship between CPT utility and the distributional characteristics of the portfolio’s return is complicated given its standard parameterization.
Although the mean (6) and the variance (7) are quite easily altered by varying μ and
δ, respectively, it seems difficult to change, e.g., the distributional skewness without
affecting another central moment. It would be preferable to parameterize the NIG
distribution as a function of its mean, variance, skewness and kurtosis directly, instead of indirectly via the standard parameters α, β , μ, and δ. Such an alternative
parameterization would imply that an individual moment’s influence on utility can
be analyzed in isolation, i.e., without affecting the other moments.
Eriksson et al. (2005) show that if the first four cumulants of X exist, and fulfill
a regularity condition, the NIG density can be expressed as a function of these first
four cumulants. Cumulants are a set of descriptive constants of a distribution just
like moments are, and, in some instances, they are more useful than moments.15
The result of Eriksson et al. (2005) is very useful since the first and the second
cumulants equal the mean and the variance, respectively, and the skewness and the
kurtosis are simple normalizations of the third and forth cumulants.
Specifically, if we let κ1, κ2 , κ3 , and κ4 denote the first four cumulants of the
probability distribution of a stochastic variable X, the mean, the variance, the
skewness, and the kurtosis of X are given by
E[X] = κ1 ,
(10)
V [X] = κ2 ,
κ3
S[X] = 3/2 ,
κ2
(11)
(12)
κ4
+ 3,
κ22
(13)
and
K[X] =
respectively. Using (6)-(9) and (10)-(13), Eriksson et al. (2005) show that the NIG
parameters ᾱ, β̄, μ, and δ can be expressed as functions of the first four cumulants
15
To read more on moments and cumulants, see chapter 3 of Kendall and Stuart (1963).
4. Utility in Relation to Distributional Characteristics
87
κ1, κ2 , κ3 , and κ4 . The following parameter transformations are presented:
4/ρ + 1 κ22
,
ᾱ = 3 p
1 − ρ−1 κ4
signum(κ3 ) 4/ρ + 1 κ22
p
,
β̄ = 3
√
ρ
1 − ρ−1 κ4
s
signum(κ3 )
κ3
μ = κ1 −
(12/ρ + 3) 2 ,
√
ρ
κ4
and
δ=
s
3(4/ρ + 1)(1 − ρ−1 )
κ32
,
κ4
(14)
(15)
(16)
(17)
16
where ρ = 3κ4 κ2 κ−2
The transformation is valid under the regularity condition
3 −4.
ρ > 1.
Equations (5) and (14)-(17) imply an alternative parametrization of the NIG
density, denoted f¯NIG , which is a direct function of the first four cumulants, i.e.,
f¯N IG = f¯NIG (x; {κi }4i=1 ). Using this alternative NIG density, one can approximate
an empirical distribution by estimating its first four cumulants, {κi }4i=1 , instead of
estimating the standard NIG parameters ᾱ, β̄, μ and δ. More importantly, a single
distributional characteristic can be altered without affecting the other ones, making
the study of CPT utility in relation to a specific moment possible.
4
Utility in Relation to Distributional Characteristics
This section presents an analysis of single-period portfolio utility in relation to the
portfolio return’s distributional characteristics. Three kinds of investor preferences
are considered, namely CPT, CPT without probability weighting, i.e. ELA, and
EP utility preferences. The first two cases are considered in order to separate the
effects of the value and weighting functions. EP utility preferences are considered
to compare CPT with traditional utility theory.
16
The function signum(x) equals the sign of x.
88
4.1
Essay 3. Prospect Theory and Higher Moments
Investor Utility with NIG Distributed Returns
Consider a single-period portfolio with NIG distributed stochastic return. Following
(4), CPT utility is derived as
U (θ) =
Z
∞
(x − x̄)γ w0 (1 − F̄NIG (x; {κi }4i=1 ))f¯NIG (x; {κi }4i=1 )dx
Z x̄
−λ
(x̄ − x)γ w0 (F̄NIG (x; {κi }4i=1 ))f¯NIG (x; {κi }4i=1 )dx,
x̄
(18)
−∞
where f¯NIG is the alternative NIG density function, F̄NIG is the corresponding cumulative distribution function, and w(·) is the probability weighting function (3).17
Utility parameters are gathered in θ = (γ, λ, τ , x̄, {κi }4i=1 )0 , where γ reflects risk
aversion over gains and risk-seeking over losses, λ measures loss aversion, τ determines the degree of probability weighting, x̄ is the reference return that separates
gains from losses, and {κi }4i=1 are the first four cumulants of the portfolio’s returns
distribution.
Consider the case when τ = 1 in (18). The weighting function in (3) then
collapses so that objective probabilities are considered, and utility becomes
U(θ)|τ =1 =
Z
∞
(x − x̄)γ f¯N IG (x; {κi }4i=1 )dx
Z x̄
−λ
(x̄ − x)γ f¯NIG (x; {κi }4i=1 )dx,
x̄
(19)
−∞
which is referred to as ELA utility.
EP utility under a NIG assumption is derived similarly to ELA utility, however
using a different utility function. Replacing the value function in (19) by the constant
1−η
x
relative risk aversion (CRRA) power utility function v(w) = w1−η , where w = 1+ 100
(x in percent) is final wealth, EP utility is formalized as
V (ψ) =
Z
∞
−∞
¡
¢
x 1−η
1 + 100
f¯N IG (x; {κi }4i=1 )dx,
1−η
(20)
where η is the parameter of constant relative risk aversion, and ψ = (η, {κi }4i=1 )0 is
a parameter vector.
17
To my knowledge, there is actually no closed
R x form expression of the NIG c.d.f. It is, however,
easily derived numerically using F̄N IG (x) = −∞ f¯N IG (t)dt.
4. Utility in Relation to Distributional Characteristics
4.2
89
Analysis Procedure
Utility is analyzed in relation to the portfolio return’s distributional characteristics
through the following procedure:
1. Consider one of the utility functions (18), (19) and (20), and calibrate its
parameters using experimental or empirical estimates.
2. Vary a cumulant value of choice and register the variation in derived utility.
Recall that a change in κi affects either the mean, the variance, the skewness,
or the kurtosis, according to (10)-(13).18
3. Illustrate utility as a function of the analyzed distributional characteristic
graphically.
4. Carry out steps 2 and 3 for the other cumulants.
5. Carry out steps 2-4 for the other utility functions.
The procedure involves a calibration of the parameters in its first step. I use the
Tversky and Kahneman (1992) estimates of λ̂ = 2.25, γ̂ = 0.88, and τ̂ = 0.65, for
CPT utility. The weighting function parameter is set to one when ELA utility is
considered, implying objective probabilities. The parameter of relative risk aversion
of EP utility is set to η = 3, which is reasonable.19 The first four cumulants,
{κi }4i=1 , are estimated using the historical monthly nominal returns of the S&P 500
composite index. Table 1 presents summary statistics. The mean and the standard
deviation equal 0.99 and 5.62 percent, respectively, while the skewness is 0.39 and
the kurtosis equals to 12.45. Moreover, the investor’s reference return for CPT
and ELA utility, x̄, is set to the risk-free nominal interest rate measured by the
average return on a U.S. 30-day Treasury bill, which equals 0.31 percent.20 Hence,
the gamble of investing in a single-period stock portfolio is considered, with the
reference investment being a risk-free bill.
The derivation of utility in the second step involves numerical integration or
quadrature. The Matlab programming function quad is applied.
18
Changing κ2 alters the variance of the distribution, as (11) shows, but the measures of the
skewness in (12) and the kurtosis in (13) are also affected. The latter changes are only matters of
normalization however, and are not of concern. Specifically, the actual distributional skewness is
not affected by κ2 , only its normalized measure.
19
See, e.g., Mehra and Prescott (1985).
20
The average T-bills return is from Ibbotson Associates.
90
4.3
Essay 3. Prospect Theory and Higher Moments
Results
Figures 4-7 illustrate ELA and EP utility as functions of the mean, the variance,
the skewness, and the kurtosis, respectively, in panels A. Analogous functions for
CPT utility are presented graphically in panels A of (8)-(11). To help clarify the
distributional variations, panel B of each figure displays the two outermost distributions of analysis. For example, since changes in the mean vary within the interval
of 0.5 percent to 1.4 percent, as panel A of figure 4 shows, panel B gives plots of two
distributions with respective means equal to 0.5 percent and 1.4 percent, all other
things equal.
Figure 4: Expected Utility in Relation to Mean
Panel A: Utility as a Function of Mean (%)
−0.5
−1
−0.485
ELA utility
EP utility
−0.49
−1.5
−2
0.4
−0.495
0.6
0.8
1
1.2
−0.5
1.4
Panel B: The Two Outermost Distributions
smallest mean
largest mean
ref. return
0.1
0.05
0
−20
−10
0
10
20
The figure plots expected loss-averse (ELA) and expected power (EP) utility as functions of the
distributional mean of a single-period risky investment with the other distributional characteristics
held constant (panel A), and the two outermost analyzed distributions (panel B). In panel A, ELA
(EP) utility is measured on the left (right) axis.
ELA and EP Utility
Figures 4 and 5 (panels A) show that ELA and EP utility are both positively related to the mean of the underlying returns distribution, and negatively related to
its variance. The intuition for ELA preferences is that a higher mean decreases the
probability of a loss, increasing utility, while a higher variance spreads the distribution and, hence, increases the probability of a loss, which decreases utility.
Illustrations of ELA and EP utility as functions of the skewness and the kur-
4. Utility in Relation to Distributional Characteristics
91
Figure 5: Expected Utility in Relation to Variance
Panel A: Utility as a Function of Variance (%)
0
−0.49
ELA utility
EP utility
−0.5
−0.492
−1
−0.494
−1.5
−0.496
−2
0.2
0.25
0.3
0.35
0.4
0.45
−0.498
0.5
Panel B: The Two Outermost Distributions
0.3
smallest variance
largest variance
ref. return
0.2
0.1
0
−20
−10
0
10
20
The figure plots expected loss-averse (ELA) and expected power (EP) utility as functions of the
distributional variance of a single-period risky investment with the other distributional characteristics held constant (panel A), and the two outermost analyzed distributions (panel B). In panel
A, ELA (EP) utility is measured on the left (right) axis.
tosis are presented in figures 6 and 7, respectively. The figures also show the two
outermost distributions, where the skewness equals either -2 or 2 (figure 6), and
the kurtosis is either 3 or 20 (figure 7). A slightly hump-shaped relation between
ELA utility and the skewness is shown. At reasonable levels of the skewness for
stock returns, say greater than -1, utility falls as the skewness rises.21 Intuitively,
when the skewness increases, the left tail of the distribution attenuates while the
right tail fattens, but the center mass moves in the opposite direction to preserve
the mean. Although the effect on the tails of the distribution increases ELA utility,
since the probability of large losses is reduced, the adjustment of the center mass
has a negative effect, since small losses become more probable. ELA utility falls
when the distributional skewness increases since loss aversion induces an investor
sensitivity to small losses.
Figure 7 presents ELA and EP utility plotted against a kurtosis between 3 and
20. The graph for ELA utility in panel A is clearly positively sloped, meaning
that ELA utility increases with kurtosis. A plot of the two outermost examined
distributions, found in panel B, helps in understanding this result. When kurtosis
increases, the distributional tail masses thicken but the center mass becomes more
21
In table 1, the stock market returns skewness is greater than -1 overall.
92
Essay 3. Prospect Theory and Higher Moments
Figure 6: Expected Utility in Relation to Skewness
Panel A: Utility as a Function of Skewness
−0.8
−0.493
ELA utility
EP utility
−1
−0.494
−1.2
−0.495
−1.4
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−0.496
Panel B: The Two Outermost Distributions
smallest skewness
largest skewness
ref. return
0.1
0.05
0
−20
−10
0
10
20
The figure plots expected loss-averse (ELA) and expected power (EP) utility as functions of the
distributional skewness of a single-period risky investment with the other distributional characteristics held constant (panel A), and the two outermost analyzed distributions (panel B). In panel
A, ELA (EP) utility is measured on the left (right) axis.
Figure 7: Expected Utility in Relation to Kurtosis
Panel A: Utility as a Function of Kurtosis
−0.5
−0.4945
−1
−0.495
ELA utility
EP utility
−1.5
2
4
6
8
10
12
14
16
18
−0.4955
20
Panel B: The Two Outermost Distributions
0.15
smallest kurtosis
largest kurtosis
ref. return
0.1
0.05
0
−20
−10
0
10
20
The figure plots expected loss-averse (ELA) and expected power (EP) utility as functions of the
distributional kurtosis of a single-period risky investment with the other distributional characteristics held constant (panel A), and the two outermost analyzed distributions (panel B). In panel
A, ELA (EP) utility is measured on the left (right) axis.
4. Utility in Relation to Distributional Characteristics
93
peaked and concentrated around the mean. Although extreme negative returns
become more likely, the effect is not large enough to offset the implications of a fall
in the probability of small losses. Again, it is the effect on the probability of small
losses that is decisive for the outcome. ELA utility rises since the probability of
small losses decreases.
The results for ELA utility contrast to EP utility, which rises with the skewness,
and falls when the kurtosis increases. The former result is expected following Arditti
(1967), who shows that most standard concave utility functions, e.g., logarithmic
and power utility imply a preference for skewness, since they fulfill the condition
of non-increasing absolute risk aversion. The latter result, however, is new to the
literature as far as the author is aware of. Intuitively, EP utility maximizers are most
sensitive to the probability of larger outcomes since they do not exhibit first-order
risk aversion. Thus, EP utility falls as the kurtosis increases.
Figure 8: CPT Utility in Relation to Mean
Panel A: Utility as a Function of Mean (%)
−2
−2.2
CPT utility
−2.4
−2.6
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
Panel B: The Two Outermost Distributions
smallest mean
largest mean
ref. return
0.1
0.05
0
−20
−10
0
10
20
The figure plots cumulative prospect theory (CPT) utility a function of the distributional mean of
a single-period risky investment with the other distributional characteristics held constant (panel
A), and the two outermost analyzed distributions (panel B).
CPT Utility
Let us now turn to CPT preferences, and include probability weighting in the analysis. Figures 8 and 9 illustrate CPT utility as respective functions of the mean and
the variance. The graphs are similar to the ones for ELA utility; high-mean and
94
Essay 3. Prospect Theory and Higher Moments
Figure 9: CPT Utility in Relation to Variance
Panel A: Utility as a Function of Variance (%)
CPT utility
−1.5
−2
−2.5
0.2
0.25
0.3
0.35
0.4
0.45
Panel B: The Two Outermost Distributions
0.3
smallest variance
largest variance
ref. return
0.2
0.1
0
−20
−10
0
10
20
The figure plots cumulative prospect theory (CPT) utility a function of the distributional variance
of a single-period risky investment with the other distributional characteristics held constant (panel
A), and the two outermost analyzed distributions (panel B).
low-variance portfolios are preferred by CPT investors too. However, the results
for the skewness and the kurtosis change dramatically. Compared with ELA preferences, figures 10 and 11 show that utility now rises with the skewness, and is inverse
hump-shape related to the kurtosis. Probability weighting causes small (cumulative) probabilities to be over-weighted so that the tails of the returns distribution
are magnified. Hence, with a change in the skewness or the kurtosis, the effects on
the probability tail-masses, i.e. the probability of extreme outcomes, is of greater
importance. CPT utility rises since an increasing skewness attenuates the left tail.
Of course, the probability of small losses still increases with a larger skewness, but
the over-weighting of small probabilities dominates this effect.
The relation to the kurtosis is more complicated to explain. The inverse humpshaped function in figure 11 makes it unclear which aspect of the distributional
change, following an increase in the kurtosis, that is most important for CPT utility. A larger kurtosis accentuates the tails, which raises the probability of large
losses, while making the distribution more pointy, decreasing the probability of
small losses. Since the first effect has bad implications for utility, and the second
has good ones, the inverse hump-shaped function is likely the result of a balance
between the two effects at the specified preference-parameter values, i.e., the ones
provided by Tversky and Kahneman (1992).
4. Utility in Relation to Distributional Characteristics
95
Figure 10: CPT Utility in Relation to Skewness
Panel A: Utility as a Function of Skewness
−1.5
CPT utility
−2
−2.5
−3
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Panel B: The Two Outermost Distributions
smallest skewness
largest skewness
ref. return
0.1
0.05
0
−20
−10
0
10
20
The figure plots cumulative prospect theory (CPT) utility a function of the distributional variance
of a single-period risky investment with the other distributional characteristics held constant (panel
A), and the two outermost analyzed distributions (panel B).
Figure 11: CPT Utility in Relation to Kurtosis
Panel A: Utility as a Function of Kurtosis
−2.1
CPT utility
−2.15
−2.2
4
6
8
10
12
14
16
18
20
Panel B: The Two Outermost Distributions
0.15
smallest kurtosis
largest kurtosis
ref. return
0.1
0.05
0
−20
−10
0
10
20
The figure plots cumulative prospect theory (CPT) utility a function of the distributional kurtosis
of a single-period risky investment with the other distributional characteristics held constant (panel
A), and the two outermost analyzed distributions (panel B).
96
4.4
Essay 3. Prospect Theory and Higher Moments
Sensitivity Analysis
So far, the analysis has assumed Tversky and Kahneman’s (1992) estimates of the
value and weighting functions’ parameters, i.e., (λ, γ, τ ) = (2.25, 0.88, 0.65), but
with τ = 1 for ELA utility. Are the obtained results sensitive to changes in these
estimates? The question is analyzed by fixing the distributional parameters, i.e., the
first four cumulants at their empirical estimates, and by varying the CPT preference
parameters.
Parameter-value variations do not have any drastic effects on the results for
the mean or the variance. Utility is negatively related to the variance so long as
the investor is loss-averse, i.e., λ > 1. This is the case despite a heavy degree
of investor risk-seeking over losses, measured by γ. A preference for high-variance
portfolios appears when λ = 1 however. Indeed, if the investor is risk-neutral with
(λ, γ) = (1, 1), she only has concern for a large return, irrespective of the level of
risk, and the probability of large returns increases with a higher variance.
Not so surprising, the results for the skewness and the kurtosis turn out to be
quite parameter sensitive, especially to the weighting function parameter τ . Recall
that the investor’s preference for skewness and kurtosis changes quite dramatically
when introducing probability weighting. Figure 6 shows a negative relation between
ELA utility and the skewness when the skewness is greater than -1, while in figure
10, where probability weighting is considered, a clear positive relation is presented.
What degree of probability weighting is sufficient to achieve this positive relation?
Experimenting with different values, a τ of 0.90 turns out to be adequate. In fact,
the CPT investor has a preference for skewness so long as τ ≤ 0.90, regardless of the
level of loss aversion or degree of risk aversion/risk-seeking. Probability weighting
is clearly the driving source of the CPT preference for skewed portfolios.
The positive relation between ELA utility and the kurtosis, previously explained
to be driven by loss aversion, is presented in figure 7. When probability weighting
is introduced, figure 11 presents an inverse hump-shaped relation however. Varying
the parameter values, it is quite obvious that the level of loss aversion and the
degree of probability weighting have counteracting effects on utility. When λ > 1
and (γ, τ ) = (1, 1), i.e. the investor suffers from "pure" loss aversion and weights
probabilities linearly, utility is positively related to the kurtosis. The loss-averse
investor’s sensitivity to the probability of small losses causes this result. On the
contrary, when τ < 1, γ = 1, and λ > 1 but close to one, i.e. the investor is mildly
loss-averse and distorts probabilities, utility is negatively related to the kurtosis,
which concerns the probability of large losses and the investor’s probability overweighting of such. In the general case of λ > 1 and τ < 1, the interplay between the
5. Optimal Portfolio Choice with NIG Distributed Returns
97
level of loss aversion and the degree of probability weighting implies an inverse humpshaped relation, where the relation is first negative at low values of the kurtosis, but
turns positive at larger ones. With (λ, γ) = (2.25, 0.88), the relation to the kurtosis
is positive and monotonic when 0.75 < τ ≤ 1, but inverse hump-shaped related
when τ ≤ 0.75.
5
Optimal Portfolio Choice with NIG Distributed
Returns
This section turns to the single-period portfolio choice of CPT investors. Aït-Sahalia
and Brandt (2001) and Berkelaar, Kouwenberg, and Post (2004) conduct similar
studies, however without investigating the effects of higher-order moments on optimal asset allocation. Neither do the two studies consider probability weighting,
but focus on loss aversion and the ELA investor’s behavior. Having found that
probability weighting is a crucial ingredient of CPT when returns are non-normally
distributed, a complete study of CPT portfolio choice includes this property.
The optimal allocation to a risky asset and a relatively risk-free asset is examined
under the assumption of a NIG distributed portfolio return. To examine the effects
of skewness and kurtosis on the portfolio choice, the normality assumption is also
considered in comparison to the NIG. I study the investment strategies of both the
ELA investor, who applies objective probabilities, and the complete CPT investor,
who weights probabilities subjectively.
5.1
Data Set
The risky and the relatively risk-free assets are represented by continuously compounded real returns of the S&P 500 composite index and a U.S. 30-day Treasury
bill, respectively. Real and not nominal returns are used in the analysis, since real
returns are more kind to NIG approximations; the regularity condition does not hold
for nominal returns, while real returns cause no problem.22 Investment horizons of
one, six, and twelve months are considered, where a moving window is used when
calculating the lower frequency data.
Summary statistics of the data across all frequencies are reported on in table
1. Over the sample period of January 1926 to December 2003, the monthly real
aggregate stock return has averaged at 0.90 percent, compared with the real bill
22
Nominal Treasury bills have empirical returns distributions that are far from "bell-shaped",
resulting in cumulant estimates that do not fulfill the NIG regularity condition.
98
Essay 3. Prospect Theory and Higher Moments
return of 0.06 percent. The empirical monthly standard deviations of the two assets
are 5.85 and 0.52 percent. The mean returns increase at longer horizons, but so do
the standard deviations, naturally. Yearly returns average at 11.50 and 0.77 percent
and have standard deviations of 24.29 and 4.06 percent for the stock and bill assets,
respectively.
Over the one-, six-, and twelve-month horizons the skewness of the empirical
stock returns distributions are 1.62, 1.07, and 2.17, respectively, and the respective
kurtosis are 19.80, 9.21, and 20.71. Hence, neither the skewness nor the kurtosis
is monotonically increasing or decreasing as the horizon increases. All data series,
including the ones for real bill returns, deviate from normality to such an extent
that the Jarque-Bera test statistics are significant throughout.
5.2
Portfolio Choice Problem
Formally, the portfolio choice problem is stated as
max E w [v(X)] =
qs ,qtb
Z
∞
(x − x̄)γ w0 (1 − F (x; ξ))f (x; ξ)dx
Z x̄
−λ
(x̄ − x)γ w0 (F (x; ξ))f (x; ξ)dx,
x̄
(21)
−∞
subject to
X = qs Xs + qtb Xtb ,
(22)
and
qs + qtb = 1,
qs , qtb ∈ [0, 1],
(23)
where qs (qtb ) denotes the weight of stocks (bills), X is the composed portfolio’s
stochastic return, f (x; ξ) is the probability density function of X, F (x; ξ) is the
corresponding cumulative distribution function, ξ is a vector of distributional parameters, and Xs (Xb ) is the stochastic return on the stock (bill) asset. The portfolio’s
return is assumed either NIG or normally distributed. The constraints (23) imply
that short selling is not allowed.23
23
The optimization problem (21) is solved by using the Matlab constrained minimization routine
fmincon.
5. Optimal Portfolio Choice with NIG Distributed Returns
99
Table 2: Single-Period Portfolio Choice of ELA Investors
λ=1
λ = 2.25
λ=3
λ=1
λ = 2.25
λ=3
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
Panel A: NIG Assumption
One-Month Horizon
Six-Month Horizon
qs
qtb
S
qs
qtb
S
1
0
0.154
1
0
0.368
1
0
0.154
1
0
0.368
1
0
0.154
1
0
0.368
0.042 0.958 0.166
1
0
0.368
0.052 0.948 0.173
1
0
0.368
0.061 0.939 0.178
1
0
0.368
0.035 0.965 0.160 0.213 0.787 0.387
0.037 0.963 0.162 0.333 0.667 0.390
0.038 0.962 0.163 0.445 0.555 0.385
Panel B: Normality Assumption
One-Month Horizon
Six-Month Horizon
qs
qtb
S
qs
qtb
S
1
0
0.154
1
0
0.368
1
0
0.154
1
0
0.368
1
0
0.154
1
0
0.368
0.064 0.938 0.178 0.553 0.447 0.381
0.051 0.949 0.172
1
0
0.368
0.047 0.953 0.169
1
0
0.368
0.048 0.952 0.170 0.232 0.768 0.389
0.037 0.963 0.162 0.209 0.791 0.386
0.034 0.966 0.159 0.205 0.795 0.386
Twelve-Month
qs
qtb
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Horizon
S
0.473
0.473
0.473
0.473
0.473
0.473
0.473
0.473
0.473
Twelve-Month
qs
qtb
1
0
1
0
1
0
1
0
1
0
1
0
0.502 0.498
1
0
1
0
Horizon
S
0.473
0.473
0.473
0.473
0.473
0.473
0.488
0.473
0.473
The table shows optimal portfolio weights of stocks (qs ) and Treasury bills (qtb ) of an expected
loss-averse investor with single-period objective:
max E[v(X)],
qs ,qtb
where E[·] is the expectations operator,
½
v(x) =
(x − x̄)γ
−λ(x̄ − x)γ
if x ≥ 0
,
if x < 0
and x̄ is the average return on Treasury bills. The portfolio return X is assumed either NIG
distributed (panel A) or normally distributed (panel B). The investor horizon is either one, six, or
twelve months. S is the Sharpe ratio. Restrictions qs , qtb ∈ [0, 1] and qs + qtb = 1 are imposed in
the optimization.
100
5.3
Essay 3. Prospect Theory and Higher Moments
Results
Table 2 reports on the optimal portfolio weights of stocks and bills of an ELA investor
with loss aversion parameter λ equal to 1, 2.25, or 3, and risk aversion/risk-seeking
parameter γ equal to 0.6, 0.88, or 1. Panel A presents the results under the NIG
assumption, and panel B under normality. The sharpe ratio, i.e. the mean divided
by the standard deviation of the optimal portfolio composition, is also provided.
The results show that an investor who does not value losses any more than she
does gains, i.e. λ = 1, allocates one hundred percent to stocks over all horizons,
irrespective of the degree of risk aversion/risk-seeking and whether NIG or normality
is assumed. Loss aversion is the investor’s main source of aversion to risk, and
without it she is practically risk-neutral.
Consistent with previous studies such as Aït-Sahalia and Brandt (2001), the
investor’s portfolio choice displays large horizon effects. Larger weights are placed
on stocks as the horizon increases. Under the NIG assumption, an ELA investor
with (λ, γ) = (2.25, 0.88) increases her allocation to stocks from 5.2 to one hundred
percent when the investment horizon rises from one to six months. This is quite a
dramatic increase.24 With a higher loss aversion of λ = 3, the allocations to risky
stocks over the horizons are also very progressive; 3.7 percent at the one-month, 33
percent at the six-months, and one hundred percent at the yearly horizon. Benartzi
and Thaler (1995) explain that loss-averse investors perceive stocks as less risky
at longer horizons, since losses occur with smaller probability.25 On the contrary,
Merton (1969) and Samuelson (1969) show that the portfolio choice under traditional
expected utility preferences are horizon independent, so long as returns are i.i.d.26
The ELA investor allocates to a fairly similar portfolio under normality as she
does under the NIG assumption, as panel B shows. Previously, it was found that
the ELA investor cares about the probability mass surrounding the reference return,
particularly the probability of small losses. Similar weights are obtained under the
NIG and normality assumptions since higher moments primarily affect the distributional tails.
Table 3 reports on the optimal asset allocation to stocks and bills of a CPT
investor with probability weighting parameter τ = 0.65, and varying value function
parameters. The investor weights probabilities so that the portfolio’s distribution is
subjectively transformed, magnifying its tails. Panel A presents the results under
24
The weight on stocks is one hundred percent at the yearly horizon as well.
The stock return’s probability mass moves further away from the reference return as the
horizon increases.
26
Barberis (2000) shows that this result breaks down if returns are somehow predictable, e.g.,
mean-reverting.
25
5. Optimal Portfolio Choice with NIG Distributed Returns
101
Table 3: Single-Period Portfolio Choice of CPT Investors
λ=1
λ = 2.25
λ=3
λ=1
λ = 2.25
λ=3
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
γ
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
= 0.6
= 0.88
=1
Panel A: NIG Assumption
One-Month Horizon
Six-Month Horizon
qs
qtb
S
qs
qtb
S
1
0
0.154
1
0
0.368
1
0
0.154
1
0
0.368
1
0
0.154
1
0
0.368
0.048 0.952 0.170 0.325 0.675 0.390
0.066 0.934 0.179 0.449 0.551 0.385
0.075 0.925 0.181 0.550 0.450 0.381
0.033 0.967 0.158 0.197 0.803 0.384
0.032 0.968 0.157 0.240 0.760 0.389
0.031 0.969 0.156 0.263 0.737 0.390
Panel B: Normality Assumption
One-Month Horizon
Six-Month Horizon
qs
qtb
S
qs
qtb
S
1
0
0.154
1
0
0.368
1
0
0.154
1
0
0.368
1
0
0.154
1
0
0.368
0.041 0.959 0.165 0.193 0.807 0.384
0.032 0.968 0.158 0.172 0.828 0.378
0.030 0.970 0.155 0.167 0.833 0.377
0.032 0.968 0.158 0.136 0.864 0.364
0.024 0.976 0.149 0.114 0.886 0.350
0.023 0.977 0.147 0.108 0.892 0.345
Twelve-Month
qs
qtb
1
0
1
0
1
0
1
0
1
0
1
0
0.421 0.579
0.457 0.543
0.502 0.498
Horizon
S
0.473
0.473
0.473
0.473
0.473
0.473
0.490
0.489
0.488
Twelve-Month
qs
qtb
1
0
1
0
1
0
0.351 0.649
0.391 0.609
0.460 0.540
0.209 0.791
0.182 0.818
0.176 0.824
Horizon
S
0.473
0.473
0.473
0.490
0.490
0.489
0.474
0.464
0.461
The table shows optimal portfolio weights of stocks (qs ) and Treasury bills (qtb ) of a cumulative
prospect theory investor with single-period objective:
max E w [v(X)],
qs ,qtb
where E w [·] is the expectations operator under probability weighting,
½
(x − x̄)γ
if x ≥ 0
v(x) =
,
−λ(x̄ − x)γ if x < 0
and x̄ is the average return on Treasury bills. The probability weighting parameter is set to
τ = 0.65. The portfolio return X is assumed either NIG distributed (panel A) or normally distributed (panel B). The investor horizon is either one, six, or twelve months. S is the Sharpe ratio.
Restrictions qs , qtb ∈ [0, 1] and qs + qtb = 1 are imposed.
102
Essay 3. Prospect Theory and Higher Moments
the NIG distributional return assumption, and panel B under normality. First,
compared with the results of table 2, the horizon effects are still present, which
does not come as a surprise. Second, the results at the monthly horizon resemble
the corresponding ones obtained without probability weighting, where only a minor
portion of stocks is chosen. Whether a NIG or a normality assumption is applied
does not seem to matter here either. Essentially, the stock returns’ variance is too
dominating at the monthly horizon for them to be attractive.
There are quite striking differences between tables 2 and 3 at the longer horizons
however. Consider the optimal weights under the NIG assumption in panel A,
with an investment horizon of six months. Instead of investing fully in stocks, the
CPT investor with Tversky and Kahneman (1992) estimates of the value function
parameters places 45 percent in stocks and 55 percent in bills. The intuition is that
the probability weighting investor perceives stocks as more risky, since the left tail
is magnified. Although stocks are positively skewed at the six-month horizon, which
is a positive for CPT utility, they are not skewed enough to offset the fear of a large
loss, which is enhanced by the large stock distributional kurtosis. Thus it seems
that kurtosis has a negative effect on CPT utility in this case.
Under the normal distribution, Levy et al. (2003), among others, show that CPT
is consistent with mean-variance efficiency. Hence, the optimal portfolios presented
in panel B of table 3 are mean-variance efficient. Are the CPT portfolios obtained
under the NIG assumption (panel A) mean-variance efficient too? Considering the
large differences in optimal weights shown in panels A and B, this does not seem to be
the case. For instance, at the yearly horizon, the CPT investor with (λ, γ) = (3, 0.88)
chooses to allocate 45.7 percent in stocks under the NIG assumption, but only 18.2
percent under normality. Such a disparity between optimal allocations indicates that
the there are other aspects of the distribution besides the mean and the variance
that are important to the CPT investor. Plausibly, the positive skewness of the
yearly stock returns distribution makes the CPT investor want to deviate from the
mean-variance portfolio, and choose a portfolio composition with a larger weight of
stocks.
Consider the Sharpe ratios of table 3. Since the optimal portfolios of panel B are
obtained under the normal distribution, which is fully characterized by the mean
and the standard deviation, it is fair to believe that these portfolios have the largest
attainable Sharpe ratio. However, the fact that the portfolios are mean-variance
efficient undermines this reasoning. Mean-to-variance efficiency sets and mean-to
standard deviation efficiency sets are not equivalent. This, plausibly, explains why
some optimal portfolios of panel A, obtained under the NIG distribution, have larger
6. Conclusions
103
Sharpe ratios than the corresponding portfolios obtained under normality. For instance, at the monthly horizon with (λ, γ) = (2.25, 1) , the Sharpe ratio is 0.155
under normality, but 0.181 under the NIG assumption.
6
Conclusions
The paper examines the CPT utility of a NIG distributed portfolio return in a
single-period context. The NIG assumption allows for a straightforward approach
to analyzing utility in relation to the return’s distributional characteristics mean,
variance, skewness, and kurtosis. Moreover, the optimal portfolio choice is analyzed, paying special interest to the implications of higher moments and probability
weighting, which have received little attention in the previous literature. The main
findings can be summarized as follows: First, CPT investors prefer high-mean and
low-variance portfolios, since such portfolios imply smaller loss-probabilities. Second, skewness typically has a negative impact on utility when probability weighting
is not considered. Once probabilities are subjectively transformed however, a clear
preference for skewness appears. This shows that CPT investors display a preference
for skewness through the probability weighting function. Third, utility is positively
related to kurtosis when the investor treats probabilities objectively, but inverse
hump-shape related when introducing probability weighting. The latter result is
quite sensitive to the level of loss aversion in relation to the degree of probability
weighting.
What implications do these results have for the portfolio choice? To answer
this question, the CPT optimal asset allocation is analyzed under the NIG distributional assumption. Consistent with the previous literature, CPT investors are
progressive in their allocation to stocks over the investment horizon. While the
optimal portfolio might only consist of a small portion of stocks at the monthly
horizon, the CPT investor with Tversky and Kahneman (1992) parameter estimates
will prefer an all-stocks portfolio at the yearly horizon. Furthermore, the optimal
portfolio composition may differ quite dramatically when higher-order moments are
accounted for. Specifically, CPT portfolios do not seem to be mean-variance efficient under the NIG assumption, and they typically consist of a relatively larger
portion of stocks. Probability weighting causes this result. Since higher moments
are important to the CPT investor, the main priority is not mean-variance efficiency
but a more complicated preference-scheme including all first four moments.
104
Essay 3. Prospect Theory and Higher Moments
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