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Beta and Coskewness Pricing: Perspective from Probability
Weighting
YUN SHI, XIANGYU CUI, XUN YU ZHOU∗
ABSTRACT
The security market line is often flat. We hypothesize that probability weighting plays a role
and that one ought to differentiate periods in which agents overweight and underweight tails.
Overweighting inflates the probability of extremely bad events and demands greater compensation
for beta risk, whereas underweighting is the opposite. Overall, these two effects offset each other,
resulting in a nearly flat return-beta relationship. Similarly, overweighting the tails enhances the
negative relationship between return and coskewness, while underweighting reduces it. We offer
a psychology-based explanation regarding the switch between overweighting and underweighting,
and we support our theories empirically.
∗Shi is at Academy of Statistics and Interdisciplinary Sciences, Faculty of Economics and Management, EastChina Normal University, Shanghai, China. Email: [email protected]. Cui is at School of Statistics and Manage-ment, Shanghai Institute of International Finance and Economics, Shanghai University of Finance and Economics,Shanghai, China. Email: [email protected]. Zhou is at Department of Industrial Engineering andOperations Research and The Data Science Institute, Columbia University, New York, NY 10027, USA. Email:[email protected]. Shi acknowledges financial support from the National Natural Science Foundation of Chinaunder grants 71971083, 71601107, and from the State Key Program of National Natural Science Foundation of Chinaunder grant 71931004. Cui acknowledges financial support from the National Natural Science Foundation of Chinaunder grant 71601107. Zhou acknowledges financial support through a start-up grant at Columbia University andthrough the Nie Center for Intelligent Asset Management.
1 Introduction
The classical CAPM theory of Sharpe (1964), Lintner (1965), and Mossin (1966) asserts that
expected returns increase with beta, leading to an upward-sloping security market line. However,
empirical studies have shown that the return–beta slope is flat or even downward-sloping; see e.g.,
Black, Jensen, and Scholes (1972), Fama and French (1992), and Baker, Bradley, and Wurgler
(2011). Various explanations have been offered for why beta is not or is negatively priced, ranging
from misspecification of risk (Jagannathan and Wang, 1996), to inefficiency of market proxies (Roll
and Ross, 1994), to market frictions (Black et al., 1972; Frazzini and Pedersen, 2014), and to
aggregate disagreement (Hong and Sraer, 2016).
We approach the beta anomaly from a different perspective, one that involves probability weight-
ing. The central task of asset pricing is to characterize how expected returns are related to risk
and to investors’ perceptions of risk. Probability weighting affects the perception of risk, especially
in the two tails of the market returns; therefore, it is poised to play a role in beta pricing. On the
other hand, a positive/negative skewness measures the risk of large poitive/negative realizations
and can be viewed as a part of tail risk. As a result, probability weighting is also relevant to skew-
ness/coskewness pricing because substantially right-/left-skewed events, such as winning a lottery
(Barberis and Huang, 2008; Bordalo, Gennaioli, and Shleifer, 2012) or encountering a catastrophe
(Kelly and Jiang, 2014; Bollerslev, Todorov, and Xu, 2015; Kozhan, Neuberger, and Schneider,
2013) are more attractive/undesirable to an investor under probability weighting.
In this paper, we provide a theory for beta and coskewness pricing using rank-dependent utilities
(Quiggin, 1982; Schmeidler, 1989; Abdellaoui, 2002; Quiggin, 2012). The key difference between the
rank-dependent utility theory (RDUT) and the classical expected utility theory (EUT) is that in the
former the utility is weighted by a probability weighting function assigned to ranked outcomes.1 So
RDUT nests EUT but also captures risk attitude toward probabilities, especially those represented
in the two tails of the return distributions. We first derive an equilibrium asset pricing formula
1Another prominent theory that extends EUT to include probability weighting is the cumulative prospect theory(CPT; Tversky and Kahneman (1979) and Tversky and Kahneman (1992)). Based on CPT, Barberis, Mukherjee,and Wang (2016) construct a TK factor and find its predicting power derives mainly from probability weighting.Hence, we focus on RDUT in this paper.
2
for a representative RDUT economy, and we then examine the signs and magnitudes of the risk
premiums in covariance and coskewness. Our results show that the pricing kernel in this economy
is the product of the marginal utility and the derivative of probability weighting, which implies
that the shape of the weighting function matters for pricing.
The dominant view in the behavioral finance literature is that individuals overweight probabil-
ities of both extremely good and extremely bad events, rendering an inverse S-shaped probability
weighting function (Tversky and Kahneman, 1992; Wu and Gonzalez, 1996; Prelec, 1998; Hsu, Kra-
jbich, Zhao, and Camerer, 2009; Tanaka, Camerer, and Nguyen, 2010). Some authors, however,
have documented empirical evidence that individuals sometimes underweight tail events (Hertwig,
Barron, Weber, and Erev, 2004; Humphrey and Verschoor, 2004; Harrison, Humphrey, and Ver-
schoor, 2009; Henrich, Heine, and Norenzayan, 2010), leading to S-shaped weighting functions.
Accordingly, Barberis (2013) comments that the coexistence of overweighting and underweighting
poses a challenge in the study of extreme events. Employing the method proposed in Polkovnichenko
and Zhao (2013), we use S&P 500 index option data from January 4, 1996 to April 29, 2016 to back
out option-implied probability weighting functions, updated monthly. Among the total 244 months
tested in our study, about 89.3% of the time the implied weighting functions are either S-shaped
(about 37.7%) or inverse S-shaped (about 51.6%). This observation suggests that individuals tend
to overweight or underweight the two tails simultaneously (Tversky and Kahneman, 1992; Gonzalez
and Wu, 1999; Humphrey and Verschoor, 2004). We further use the implied weighting functions
to construct a monthly probability weighting index that reflects investors’ perceptions of and re-
sponses to tail events, based on which we divide the market into two regimes: overweighting and
underweighting. The ratio between the number of months during which the market is in the over-
weighting regime and during which it is in the underweighting one is around 57% to 43%. Given
this empirical finding, we can separate the overweighting and underweighting regimes and analyze
the corresponding beta and coskewness pricing separately and respectively.
Our main theoretical result is a closed-form expression of a three-moment CAPM, with risk
premiums of the covariance and coskewness of any given asset with respect to the market. These
premiums depend on the utility function, as well as on the probability weighting function evaluated
3
at the probability value of having a positive market return. During an overweighing month (in
which the weighting function is inverse S-shaped), the agent overweights both tails. Overweighting
the left tail marks the RDUT agent as more risk averse toward bad states (related to the convex
part of the weighting function) than her EUT counterpart, whereas overweighting the right tail
marks her as less risk averse (the concave part) toward good states. While these two effects seem
to offset each other, the empirical fact that the average probability of having a positive market
return is close to 1 for any reasonably long sample period indicates that only the convex part of
the weighting function is relevant. The intuition of this is that, when the probability of having
a positive market return is large enough, the chance of observing an extreme negative realization
is much smaller than that of observing the same size positive realization. Thus, the impact of
extremely bad states is much stronger than that of extremely good states, which causes the agent
to pay more attentions to the former. This one-sided effect results in an enhanced, positive risk
premium with respect to the covariance with the market. Symmetrically, during an underweighting
month in which the agent underweights both tails of the market return distribution, she is less risk
averse toward bad states and less risk seeking toward good ones. Once again, the left tail effect
dominates and only the former behavior is relevant because the probability of having a positive
market return now falls in the concave domain of the weighting function. In this case, the agent
demands less compensation for the covariance with the market compared to an EUT maximizer,
and she may even be willing to pay for the risk if and when she sufficiently underweights the bad
states. Finally, for a sufficiently long sample period covering comparable runs of overweighting and
underweighting months, the elevated beta during the former periods and the reduced beta during
the latter periods may cancel each other, leading to an overall flat or possibly downward-sloping
security market line (see Figure 7(a)).
Kraus and Litzenberger (1976) and Harvey and Siddique (2000) show that a typical EUT agent
is willing to pay for the coskewness, implying a negative premium in coskewness. Probability
weighting is likely to strengthen this skewness-inclined preference during an overweighting period.
This can be explained as follows. When the asset has positive coskewness with the market, the asset
return has a fat right tail with respect to the market portfolio. The hope of a larger gain relative to
4
the market due to overwheighting further increases the attraction of the risky asset and requires even
less compensation when compared to the case of no probability weighting. Symmetrically, when
the coskewness is negative, the fear of a larger loss, which is also amplified by the overweighting,
reduces the attraction of the risky asset and hence demands greater compensation. Combined,
overweighting the tails inflates the negative relationship between expected return and coskewness,
generating a significantly negative premium of the coskewness (see Figure 7(b)). When the agent
underweights both tails of the market return, the probability weighting introduces a skewness-
averse preference. Underweighting the tails deflates the negative relationship between expected
return and coskewness, resulting in an insignificant (or even potentially positive) premium.
Our empirical study confirms the above key theoretical predictions. To carry out this study,
the first important task is to identify the characteristics and level of probability weighting for
any given period. We develop several indices for this purpose, using S&P 500 option data for
the period January 4, 1996 – April 29, 2016. After deriving the non-parametric, option-implied
probability weighting functions as described earlier, we propose an “implied fear index” (IFI) and
an “implied hope index” (IHI) as the average values of the derivatives of the weighting function
around 1 and 0 respectively. The two indices have highly similar dynamics (see Figure 6(a)),
with a correlation coefficient of 0.8. Hence, we use the average of the two to define one of our
probability weighting indices, PWI1, to reflect the level of probability weighting. Moreover, to
cross examine how robust PWI1 is, we introduce a second probability weighting index, PWI2, by
fitting the parametric specification of Prelec’s weighting function.2 Moreover, PWI1 and PWI2
have a correlation coefficient of 0.949 (see also Figure 6(b) and Figure 6(c) for a comparison of
their dynamics). Therefore, we use PWI1 alone as the probability weighting index in our empirical
study, and we use its median to divide between the overweighting regime and the underweighting
one.
We then perform portfolio analysis to study the impact of probability weighting on the mean-
covariance and mean-coskewness relationships, respectively. We use all of the common stocks listed
on the NYSE, AMEX and NASDAQ for the period January 1991 – April 2016. The one-way
2Prelec’s weighting function generates inverse S-shaped and S-shaped functions, determined by a single parameter.
5
sort portfolio analysis (Table 2 and Figure 7) shows that the covariance spread portfolio has a
significantly positive return during overweighting months, a significantly negative return during
underweighting months, and a slightly positive (but insignificant) average return for the entire
period. The coskewness spread portfolio, on the other hand, has a significantly negative return
during overweighting months, an insignificantly negative return during underweighting months,
and a notably negative average return for the entire period. Moreover, the two-way sort portfolio
analysis (used to separate the effects of covariance and coskewness) yields similar results (Table 3).
Finally, a Fama–MacBeth cross-sectional regression analysis (Fama and MacBeth, 1973) further
confirms all of the results (Table 4).
The results themselves regarding conditional risk-return tradeoffs during periods when investors
overweight or underweight tail probabilities do not, however, answer the question as to why indi-
viduals switch between overweighting or underweighting. Dierkes (2013) estimates a probability
weighting function month by month from index options and finds that the former varies over time
in a way that is related to a sentiment index proposed by Baker and Wurgler (2006). Yet that
paper stops short of providing a psychological explanation as to why this variation takes place.
Baker and Wurgler (2006)’s index is constructed based on Initial Public Offering (IPO) numbers,
first-day return of IPO, close-end fund discount, turnover ratio, and share of equity/debt issues,
attributes to which individuals do not typically pay direct attention. So it is unlikely that the
changes in this sentiment index are responsible for the variation in probability weighting. In the
present paper, we propose a psychology-based hypothesis, involving both beliefs and preferences,
to explain how probability weighting responds directly to changes in the outside environment. We
use Tversky and Kahneman (1973)’s availability heuristic to understand how investors’ estimation
of tail probabilities is impacted by recently experienced rare yet significant events, such as extreme
price movements and massive volatilities, which are often portrayed sensationally by the news me-
dia. We apply the recent theories in the psychology and behavioral finance literatures (Herold and
Netzer, 2010; Woodford, 2012; Steiner and Stewart, 2016) that probability weighting, as a choice
of preferences, is a matter of responding optimally to various challenges associated with decision-
making under uncertainty, to deduce that probability weighting varies endogenously as the market
6
environment changes.
We then test the above hypothesis empirically. We choose different groups of variables that
capture, directly or indirectly, the market environment changes, including price return variables,
newspaper-based variables, investors survey variables, VIX, and Baker and Wurgler (2006)’s sen-
timent index. We find that there are high correlations between PWI1, the return variables, and
the media variables, confirming that a direct experience of extreme events and/or grandstanding
media coverage can prompt people to adjust their beliefs/preferences. There are considerable cor-
relations between PWI1 and the survey variables, showing (albeit perhaps circumstantial) evidence
that probability weighting is related to investors’ opinions regarding the market. Interestingly, our
study yields that PWI1 and Baker and Wurgler (2006)’s sentiment index are unrelated, suggesting
that, rather intuitively, factors such as IPOs and equity/debt issues are not sources of time variation
in probability weighting because individuals typically do not pay attention directly to these factors.
In an online appendix, we use these highly correlated variables as alternative probability weighting
indices to repeat the empirical analysis on the conditional return and covariance/coskewness rela-
tionship, and we find that the results remain. This additional empirical check demonstrates that
our main results are robust in that they are independent of the choice of a particular probability
weighting index.
The paper proceeds as follows. We present and solve the portfolio selection problem under
RDUT in Section 2. We derive the equilibrium asset pricing formula and the three-moment
CAPM model in Section 3. We present the empirical analysis of the conditional return and co-
variance/coskewness relationship in Section 4. We propose and empirically test a psychology-based
explanation of the time variation in probability weighting in Section 5. We conclude in Section 6.
All proofs are presented in the Appendix.
7
2 Portfolio Selection under Rank-dependent Utility
2.1 Basic Setting
Consider a one-period-two-date market. The set of possible states of nature at date 1 is Ω and
the set of events at date 1 is a σ-algebra F of subsets of Ω. There are n assets (one risk-free asset
and n − 1 risky assets), whose prices at time t are Pt1, Pt2, · · · , Ptn, t = 0, 1. These assets are
priced by
P0i = E[mP1i], i = 1, 2, · · · , n,
where m is the pricing kernel, which is an F-measurable random variable such that P(m > 0) = 1,
E[m] <∞ and E[mP1i] <∞ (i = 1, 2, · · · , n).
The representative agent in the market has an RDUT preference given by
U(X) =
∫u(x)w′(1− FX(x))dFX(x),
where u(·) is an (outcome) utility function, w(·) a probability weighting function, and FX(·) the
cumulative distribution function (CDF) of the random payoff X. Specific parametric classes of
probability weighting functions proposed in the literature include those by Tversky and Kahneman
(1992)
w(p) =pα
(pα + (1− p)α)1/α,
and by Prelec (1998)
w(p) = exp(−(− log(p))α),
where, in both cases, 0 < α < 1 corresponds to inverse S-shaped weighting functions (i.e., first
concave and then convex; hence overweighting both tails) and α > 1 corresponds to S-shaped
one (i.e., first convex and then concave; hence underweighting both tails). In this paper, we will
8
use Prelec’s weighting functions, which are plotted in Figure 1 with different α’s. The smaller
the α(< 1), the higher degree of overweighting, and the larger the α(> 1), the higher degree of
underweighting.
[Insert Figure 1 here]
We make the following assumptions on the market.
ASSUMPTION 1. (i) The probability space (Ω,F ,P) admits no atom.
(ii) u is strictly increasing, strictly concave, continuously differentiable on (0,∞), and satisfies the
Inada condition: u′(0+) = ∞, u′(∞) = 0. Moreover, without loss of generality, u(∞) > 0.
The asymptotic elasticity of u is strictly less than one: limx→∞xu′(x)u(x) < 1.
(iii) w is strictly increasing and continuously differentiable on [0, 1] and satisfies w(0) = 0, w(1) =
1.
2.2 Portfolio Selection
Use xi to denote the portfolio weight in asset i and let shorting be disallowed. Then, the
representative investor faces a one-period portfolio selection problem:
(P ) maxxi
∫u(x)w′(1− F
RP(x))dF
RP(x),
s.t. RP =∑n
i=1 xiRi,∑ni=1 xi = 1,
xi ≥ 0, i = 1, 2, · · · , n,
(1)
where RP is the total return of the portfolio and Ri is the total return of asset i.
Noting P0i = E[mP1i], we have E[mRi] = 1. Thus, problem (P ) is, equivalently,
(P ′) maxRP∈F
∫u(x)w′(1− F
RP(x))dF
RP(x),
s.t. E[mRP ] = 1,
P(RP ≥ 0) = 1.
(2)
9
This problem specializes to the one solved by Xia and Zhou (2016).3 Define a function N by
N(q) = −∫ 1−
w−1(q)Q−m(1− p)dp, q ∈ [0, 1],
where Q−m(p)∆= supx ∈ R | Fm(x) < p, p ∈ (0, 1], is the lower quantile function of m, and w is
the dual of w: w(p) = 1− w(1− p).
Applying Theorem 3.3 of Xia and Zhou (2016), we have the following result.
THEOREM 1. Assume m has a continuous CDF Fm, and
∫ 1−
0(u′)−1(µN ′(w(p)))Q−m(1− p)dp <∞
for all µ > 0, where N is the concave envelope of N and N ′ the right derivative of N . Then the
total return of the optimal portfolio is
R∗P = (u′)−1(λ∗N ′ (1− w(Fm(m)))
),
where the Lagrange multiplier λ∗ is determined by E[m(u′)−1
(λ∗N ′ (1− w(Fm(m)))
)]= 1.
3 Equilibrium Asset Pricing and Three-Moment CAPM
3.1 Equilibrium Asset Pricing Formula
Denote by
RM =
n∑i=1
xMi Ri
the total return of the market portfolio where xMi is the market capitalization weight of asset i
with∑n
i=1 xMi = 1, and by rM , ri and rf the return rates of the market portfolio, the asset i, and
3In Problem (2.2) of Xia and Zhou (2016), setting I = 1, u0i(·) = 0, u1i(·) = u(·), wi(·) = w(·), βi = 1, E[ρe1i] = 1,
e0i = 0, and c1i = RP , we recover our Problem (P ′).
10
the risk-free asset, respectively. Define a function m by
m(x) =w′(1− F
RM(x))u′(x)
(1 + rf )E[w′(1− F
RM(RM ))u′(RM )
] , x > 0. (3)
THEOREM 2. When the market is in equilibrium with RM having a continuous CDF FRM
, the
pricing kernel is given by
m = m(RM ) ≡w′(1− F
RM(RM ))u′(RM )
(1 + rf )E[w′(1− F
RM(RM ))u′(RM )
] . (4)
Moreover, if RM has a continuously differentiable density function fRM
, then, for any risky asset
i, we have the three-moment CAPM:
E[ri] = rf +ACov(ri, rM ) +1
2BCov(ri, r
2M ) + o(r2
M ), (5)
where the risk premiums A and B are given as
A =− (1 + rf )m′(1)
=− E−1[w′(1− FRM
(RM ))u′(RM )]
·[w′(1− F
RM(1))u′′(1)− w′′(1− F
RM(1))u′(1)f
RM(1)], (6)
B =− (1 + rf )m′′(1)
=− E−1[w′(1− FRM
(RM ))u′(RM )]
·[w′(1− F
RM(1))u′′′(1) + w′′′(1− F
RM(1))u′(1)f2
RM(1)
− w′′(1− FRM
(1))[2u′′(1)fRM
(1) + u′(1)f ′RM
(1)]]. (7)
The equilibrium pricing kernel m (indicated by expression (4)) is not only influenced by the
utility function u, but also by the probability weighting function w evaluated at 1−FRM
(RM ) and,
consequently, by the market return distribution. To demonstrate how all the three work together
in determining pricing kernel, we use Figure 2. There, we consider Prelec’s probability weighting
11
function, the CRRA utility function and two types of market scenarios:
• Market I: 1− FRM
(1) > e−1, i.e., 1− FRM
(1) lies in the convex domain (concave domain) of
w(·) when α < 1 (α > 1).
• Market II: 1−FRM
(1) < e−1, i.e., 1−FRM
(1) lies in the concave domain (convex domain) of
w(·) when α < 1 (α > 1).4
In Figure 2, both market scenarios are considered. Take rM to be skew-normal with mean
7.6%, standard deviation 15.8% and skewness -0.339, which can be easily verified to be a case
of Market I. For Market II we take rM to be skew-normal with mean -7.6%, standard deviation
15.8% and skewness -0.339.5 Figures 2(a) and 2(c) show the plots of the pricing kernel against the
market return. They show that, in both market scenarios, the RDUT agent overweighting both
tails pays higher prices in the extreme states, resulting in a U-shaped pricing kernel.6 The case of
underweighting both tails has the opposite characteristics, with a bell-shaped pricing kernel.
Figures 2(b) and 2(d) depict the implied relative risk aversion, −m′(RM ) · RM/m(RM ), at
RM = 1, which can be interpreted as the investor’s risk attitude at the initial wealth level, as a
function of α. The classical CRRA case (the black dot line) has a flat relative risk aversion, as
expected. The red and blue solid lines show how probability weighting can change risk attitudes
by influencing investors’ perceptions of tail risk. In Market I, the implied relative risk aversion is
a decreasing function of α; so a higher level of overweighting the tails leads to a larger relative
risk aversion and hence demands a higher risk compensation. In particular, the risk aversion is
negative when one sufficiently underweights, yielding a risk-seeking behavior. In contrast, the
implied relative risk aversion is increasing in α under Market II, which implies that the more (less)
overweighting (underweighting) the tails the less risk aversion is entailed.
It may seem puzzling why overweighting tails strengthens risk aversion in Market I, but weakens
it in Market II. The reason lies in the interaction between probability weighting and market return
4Note that, e−1 is the solution of the equation exp(−(− log(p))α) = p for any α; hence it is the reflection pointseparating the convex and concave domains of Prelec’s weighting function.
5Later we will show empirically that the US markets have historically been in Market I.6U-shaped pricing kernels have been presented in literature under various settings. For example, Shefrin (2008)
finds that the pricing kernel has a U-shape when agents have heterogeneous risk tolerance. Baele, Driessen, Ebert,Londono, and Spalt (2019) observe that pricing kernels under the cumulative prospect theory (CPT) are also U-shaped.
12
distribution. Under Market I in which the probability of a positive market return is sufficiently
higher, the chance of observing or experiencing an extreme negative realization is smaller than that
of the same size positive realization. Thus, the impact of an extremely bad event, once occurred,
is stronger than that of extremely good states, making the agent pay more attentions to the left
tail. This leads to the additional level of risk aversion arising from overweighting the left tail
dominating that from overweighting the right tail, strengthening the overall risk aversion of the
agent. In contrast, Market II underlines a smaller probability of a positive market return; hence a
symmetric reasoning yields that risk aversion is overall weakened.7
[Insert Figure 2 here]
3.2 Risk Premiums
In the three-moment CAPM model, the terms A and B are the risk premiums of the covariance
and the coskewness, respectively. The signs of the two parameters determine the risk preference with
respect to covariance and coskewness. As A and B depend on the utility function, the weighting
function, and the distribution of the market return in a complex way, it is difficult to analytically
study their signs in general. However, if we choose a specific weighting function - in this case,
Prelec’s - then an analytical examination of the signs becomes possible.
As noted earlier, the market return distribution also impacts pricing and risk premiums. We
have already introduced two scenarios, Market I and Market II, depending on whether 1−FRM
(1)
lies in the convex or concave domain of w. Notice that 1 − FRM
(1) = P(RM > 1), i.e., the
probability that the market has a positive return rate. So this quantity reflects the trend of the
overall economy/market. Let us first examine which type of Market is more plausible for the S&P
500. Following De Giorgi and Legg (2012), we assume that the S&P 500 annual return rate, rM ,
follows either a normal distribution, a skew-normal distribution, or a log-normal distribution. Based
on the annual historical data of the S&P 500 from January 1946 to January 2009, the annual return
7This also explains why, with overweighting of tail events, the left tail of the U-shaped pricing kernel is moreprominent in Market I and the right tail of the U-shaped pricing kernel is more prominent in Market II; see Figures2(a) and 2(c).
13
rate, rM , has a mean of 7.6% and a standard deviation of 15.8%. Moreover, in the case of skew-
normal, its skewness is estimated to be -0.339. Based on these values, assuming rM is respectively
normal, skew-normal, and log-normal, we can estimate the corresponding value of 1 − FRM
(1) to
be respectively 0.6847, 0.6980, and 0.6543, all greater than e−1. Next, we consider the S&P 500
monthly return. Following Polkovnichenko and Zhao (2013), we use the daily historical data of the
S&P 500 from January 4, 1996 to April 29, 2016 and estimate an EGARCH model for the S&P 500
with 1250 daily index return data up to the end of each month. Then, we simulate 100,000 samples
based on the EGARCH model and compute the nonparametric probability density function for the
monthly return of the S&P 500. We find that 1 − FRM
(1) > e−1 holds for all the months tested.
Therefore, we conclude that Market I is a more likely scenario for the S&P 500, and hence we focus
on this case in the following discussion.
Let us now examine first the sign of A. From its expression on the right hand side of (6), it
follows that the first term (including the factor in front of the brackets) is positive, representing
the part of the risk aversion arising from the outcome utility function u. The second term is
more complicated, representing the part of the risk aversion arising from the probability weighting,
whose sign depends on whether the quantity 1 − FRM
(1) lies in the convex or concave domain of
the weighting function w.
PROPOSITION 1. Assume that the market is in equilibrium with RM having a density function
fRM
, w(p) = exp(−(− log(p))α) and 1− FRM
(1) > e−1 (Market I). Then,
(i) A > 0 if 0 < α < 1 (overweighting case).
(ii) A < 0 if α > α∗ for some α∗ > 1 (underweighting case).
To interpret Proposition 1, consider first the case 0 < α < 1 when w is inverse S-shaped and
the agent overweights both extremely good and extremely bad states. Overweighting the left tail
makes the RDUT agent more risk averse toward bad states than her CRRA counterpart, and
overweighting the right tail makes her less risk averse toward good states. While these two effects
seem to offset each other, as explained intuitively earlier the condition 1− FRM
(1) > e−1, which is
consistent with the empirical observation for the S&P 500, dictates that only the former effect is
14
relevant.8 This results in a positive risk premium with respect to the covariance with the market;
see the solid red line in Figure 3.
There is more to it. Write
A =w′(1− F
RM(1))u′(1)
E[w′(1− FRM
(RM ))u′(RM )]
[−u′′(1)
u′(1)+w′′(1− F
RM(1))
w′(1− FRM
(1))fRM
(1)
]. (8)
The first term inside the brackets, −u′′(1)u′(1) > 0, measures the risk aversion arising from the outcome
utility function u in the classical EUT framework. The second term,w′′(1−F
RM(1))
w′(1−FRM
(1)) fRM (1), which
arises from the probability weighting independent of the outcome utility function, is also positive
under Market I and when 0 < α < 1. This second term elevates the level of risk aversion. This
may shed light on the equity premium puzzle (Mehra and Prescott, 1985): the reason the empirical
data of the S&P 500 implies an extremely implausibly high level of risk aversion within the classical
EUT framework is that the latter has not accounted for the part of the risk aversion emanating
from probability weighting.
When α > 1, w is S-shaped and the agent underweights the two tails of the market return
distribution. This entails less risk aversion toward bad states and less risk seeking toward good
states. However, the condition 1 − FRM
(1) > e−1 implies that 1 − FRM
(1) is now in the concave
domain of w and hence only the former behavior is relevant. In this case, the agent demands
less compensation for the beta risk compared with a classical utility maximizer, and, indeed, even
becomes willing to pay for the risk (i.e., A < 0) when she sufficiently underweights bad states (i.e.,
when α is sufficiently large). This case is illustrated by the dashed blue line in Figure 3.9
[Insert Figure 3 here]
Next we investigate the sign of B. First, when there is no probability weighting, we have
B = −E−1[u′(RM )]u′′′(1) < 0 assuming that u′′′ > 0, which is satisfied by most commonly used
utility functions.10 This implies that a typical EUT agent is willing to pay for the coskewness; see
8This observation is straightforward from the mathematical expression of A, (6), which depends on w only throughits value at 1 − FRM
(1), a point in the convex domain of w.9Figure 3 is completely consistent with Figure 2(b).
10Brockett and Golden (1987) refer to the class of increasing utility functions with derivatives that alternate in
15
also Harvey and Siddique (2000). In the presence of probability weighting, the first term of B (see
(7)) is negative if, again, u′′′ > 0. The second and third terms are more complicated, in that their
signs depend on the utility function, the weighting function, and the market return distribution,
all intertwined. However, as in the case of A, whether 1 − FRM
(1) lies in the convex or concave
domain of w is also critical for the sign of B. We provide several sufficient conditions for B to be
positive or negative.
PROPOSITION 2. Assume that the market is in equilibrium with RM having a continuously dif-
ferentiable density function fRM
, and that u(x) = 11−γx
1−γ where γ > 0, w(p) = exp(−(− log(p))α)
where α > 0, and 1− FRM
(1) > e−1. Then,
(i) B < 0 if 0 < α < 1 and γ ≥ γ∗ :=f ′RM
(1)
2fRM(1) .
(ii) B < 0 if α > 1 and γ > γ(α) where γ(α) is a threshold depending on α.
(iii) B > 0 if α > 1 and γ∗ ≤ γ < γ(α), where γ(α) is a threshold depending on α.
We have noted earlier that in the absence of probability weighting, B < 0 for “most commonly
used utility functions,” suggesting an inherent skewness-inclined preference in the classical EUT
framework. Now, when the agent overweights both tails of the market return (the case of 0 < α < 1),
the proof of Proposition 2 (Appendix C) shows that w′′′(1 − FRM
(1)) > 0. Thus probability
weighting is likely to enhance the skewness-inclined preference when overweighting occurs. This
enhancement can be explained intuitively as follows. When the asset has positive coskewness with
the market, the asset return has a fat right tail with respect to the market portfolio. The hope of
a larger gain relative to the market, which is strengthened by overweighting, further increases the
attraction of the risky asset and requires even less compensation when compared to the case of no
probability weighting. Symmetrically, when the coskewness is negative, the fear of a larger loss,
which is also amplified by the overweighting, reduces the attraction of the risky asset and hence
demands greater compensation. Combined, overweighting the tails inflates the negative relationship
between expected return and coskewness, generating a significantly negative B. This enhancement
sign as the class that contains “all commonly used utility functions.” In particular, u′′′ > 0 implies that the utilityfunction has nonincreasing absolute risk aversion, which is one of the essential properties of a risk-averse individual;see Harvey and Siddique (2000).
16
is confirmed by comparing the empirical results across Panels A and B in Table 1.11
When the agent underweights both tails of the market return (i.e., α > 1), w′′′(1 − FRM
(1))
becomes negative, which implies that the probability weighting introduces a skewness-averse pref-
erence. Underweighting the tails deflates the negative relationship between expected return and
coskewness. Whether B is eventually positive or negative depends on which one dominates: the
skewness-inclination inherent in u or the skewness-aversion in w. Proposition 2-(ii) and -(iii) pro-
vide sufficient conditions under which one of these is the case. These results are illustrated by the
dashed blue line in Figure 4 and by the values of B in Panel C of Table 1.12
[Insert Table 1 here]
[Insert Figure 4 here]
4 Empirical Analysis
Our empirical analysis tests our preceding theory regarding how probability weighting impacts
the risk premiums on covariance and coskewness. To this end, we first identify the characteristics
and level of probability weighting from option data. Then, we provide one-way sort and two-way
sort portfolio analyse to study the effect of probability weighting on the mean-covariance and mean-
coskewness relationships, respectively. Finally, we run a Fama-MacBeth regression to investigate
the impact of probability weighting on cross-sectional asset pricing.
4.1 Option-implied Probability Weighting Functions and Indices
Option data have been used to estimate empirical pricing kernels (see, e.g., Rosenberg and
Engle, 2002). As the pricing kernel under RDUT is the product of the marginal utility and the
derivative of probability weighting (Theorem 3.1), upon specifying a suitable utility function we
can obtain implied probability weighting functions from option price data.
11The additional condition γ > γ∗ in Proposition 2-(i) is purely technical, as it is used to control the product termu′′w′′ in the proof (Appendix C). Indeed, from that proof it is clear that this condition is sufficient, but by no meansnecessary. In Panel A of Table 1, all the B values are negative even if the condition is violated. Moreover, a numericalplot when α = 0.5 shows that B < 0 for any γ > 0; see Figure 4(a).
12Again, the condition γ ≥ γ∗ in Proposition 2-(iii) is only a technical condition used in the proof. In Panel C ofTable 1 we observe positive B when γ < γ∗.
17
We use S&P 500 index option data and S&P 500 index data to derive the option-implied
probability weighting functions and to construct the corresponding probability weighting indices
for any given period of time (monthly in this paper). These indices are then used to gauge the
overall probability weighting level in the market for the period concerned. The option data and
index data are obtained from the OptionMetrics daily files, with a time window from January 4,
1996 to April 29, 2016.13
More specifically, following the procedure of Polkovnichenko and Zhao (2013), with some vari-
ations, we derive implied probability weighting functions in three steps. First, at the end of each
month during the period January 4, 1996 to April 29, 2016, we use both the S&P 500 index and
index option data to construct the implied volatility curves for the options with the nearest matu-
rity and the second nearest maturity. We then compute the implied volatility curve for the options
with one-month maturity by spline interpolation. Next, compute the one-month maturity option
prices based on the implied volatility curve and extract the nonparametric risk-neutral probability
density function for the one-month return of the S&P 500 index, fQRM
, by taking the second-order
derivative of the one-month maturity option prices with respect to the strike prices (see Bliss and
Panigirtzoglou, 2002 and Kostakis, Panigirtzoglou, and Skiadopoulos, 2011 for details).14 Second,
at the end of each month, we estimate an EGARCH model for the S&P 500 index with 1250 daily
S&P 500 index return data up to the end of the month. Then, simulate 100,000 samples based on
the EGARCH model and compute the nonparametric real-world probability density function for
the one-month return of the S&P 500 index, fRM .15 Third, we estimate the probability weighting
13OptionMetrics is a provider of historical options data for empirical and econometric research. We access Option-Metrics via Wharton Research Data Services (WRDS).
14Different from Polkovnichenko and Zhao (2013), who directly select monthly option quotes closest to 28, 45, and56 days from expiration and compute the risk-neutral probability density function for the 28, 45, and 56-day returns,we focus on the last trading day of each month. As there are no traded options with one-month maturity at the endof each month in our experiments, we need to numerically compute the prices of one-month maturity options basedon the options with other maturities. Therefore, we apply the nonparametric method in Bliss and Panigirtzoglou(2002) and Kostakis et al. (2011) to derive the risk-neutral probability density function, which is more suitable forour analysis.
15We use the empirical S&P 500 index returns to estimate the physical probability density function. Investors’actual beliefs about the return distribution can differ from the empirical one. However, the probability weightingfunction is already the result of a combination of (largely erroneous) beliefs and (biased) preferences; see a detaileddiscussion in Section 5.1.
18
function according to the following equation:
w(1− p) =
∫ 1
pw′(1− y)dy =
∫ +∞
F−1
RM(p)w′(1− FRM (x))fRM (x)dx
=
∫ +∞
F−1
RM(p)
m(x)
(1 + rf )E[w′(1− F
RM(RM ))u′(RM )
]u′(x)
fRM (x)dx (9)
=
∫ +∞
0
fQRM
(x)
u′(x)dx
−1 ∫ +∞
F−1
RM(p)
fQRM
(x)
u′(x)dx,
where the function m is defined in (3). Based on this we derive w numerically by specifying the
utility function to be CRRA with γ = 2.16 The solid red lines in Figure 5 show two typical types of
implied probability weighting functions derived in our study, an inverse S-shaped one (overweighting
both tails) and an S-shaped one (underweighting both tails). Among the total 244 months tested
in our study, most of the time (about 89.3%) the implied weighting functions are either S-shaped or
inverse S-shaped, an observation that reconciles with the well-documented result that individuals
tend to underweight or overweight the two tails simultaneously.
[Insert Figure 5 here]
Probability weighting is captured by the derivative of the weighting function w, mainly around
the two ends of its domain, p = 1 and p = 0. We propose two indices, those of “implied fear” and
“implied hope”, capture the representative agent’s biases toward extremely good and bad events:
Implied fear index (IFI) =∑99
i=95 0.2w′(0.01i),
Implied hope index (IHI) =∑5
i=1 0.2w′(0.01i),
which are the average values of the derivatives of the probability weighting function around 1 and
0, respectively. When IFI > 1 (IHI > 1), the agent overweights extreme bad (good) events. Figure
6(a) shows the monthly time series of these indices. Evidently, the two indices have very similar
16The derived probability weighting function depends on the choice of the utility function. However, we have testedmany alternative settings, such as CRRA with γ = 1, 3, 4 and exponential utility u(x) = −e−ax with a = 1, 2, 3, 4,and found that our empirical results still hold, including the sort portfolio analysis and Fama-MacBeth regression.
19
dynamics and move together most of the time (their correlation coefficient is 0.8). This reiterates
the aforementioned simultaneous underweighting/overweighting of the two tails.
Given the highly correlated dynamics of IFI and IHI, we use their average as an option-implied
probability weighting index to reflect the level of probability weighting:
PWI1 =IFI + IHI
2. (10)
Figure 6(b) depicts the movement of PWI1 during the testing period.
To cross examine how robustly PWI1 captures the degree of probability weighting in the market
over time, we introduce a second probability weighting index by fitting the parametric specification
of Prelec’s weighting function. Specifically, we calibrate Prelec’s parameter α by minimizing the
Euclidean distance between Prelec’s weighting function and the obtained nonparametric weighting
function:
α = arg minα
99∑i=1
|w(0.01i)− exp(−(− log(0.01i))α)|2,
which we call the implied α. Note that for Prelec’s weighting function, 0 < α < 1 corresponds to
inverse S-shaped (overweighting) and α > 1 to S-shaped (underweighting), with α = 1 dividing
the two. The greater (smaller) α, the greater the degree of overweighting (underweighting). This
suggests that the following index, PWI2, also reflects the level of probability weighting:
PWI2 =1
α. (11)
The percentages of time when these indices indicate the occurrence of overwheighting are re-
markably close to each other: investors overweight extreme events 56%-59% of the time and un-
derweight extreme events 41%-44% of the time. Moreover, Figure 6(b) and (c) show that PWI1
and PWI2 have very similar dynamics. Indeed, they have a correlation coefficient 0.949. Thus,
in the following analysis, we focus on PWI1, and we use it to separate the whole sample period
(from January 1996 to April 2016) into two categories: overweighting periods and underweighting
20
periods. Specifically, we calculate the median level of PWI1 over the entire period. When the
PWI1 value at the end of month t is above its median level, the next month t+ 1 is labelled as an
overweighting period; otherwise, t+ 1 is an underweighting period.17
[Insert Figure 6 here]
4.2 One-way Sort Portfolio Analysis
We now conduct one-way sort portfolio empirical analysis to test the mean-covariance and
mean-coskewness relationships during different probability weighting periods. In our study, we
use all the common stocks listed on the NYSE, AMEX, and NASDAQ to perform cross-sectional
analysis. Data on prices, returns, and shares outstanding are obtained from the Centre for Research
in Security Prices (CRSP) monthly files, which range from January 1991 to April 2016.18
At the end of each month t from January 1991 to April 2016, we use the immediate prior
60-month sample covariance and sample coskewness to estimate the pre-ranking covariance and
coskewness in month t,
covari,t =1
59
59∑j=0
(ri,t−j − ri)(rM,t−j − rM ),
coskewi,t =1
59
59∑j=0
(ri,t−j − ri)(r2M,t−j − r2
M ),
where ri,t−j is the monthly return rate of stock i in month t− j, rM,t−j is the monthly return rate
of the value-weighted market portfolio in month t− j, and ri, rM and r2M are the sample means of
ri,t, rM,t, and r2M,t, respectively.
At each t, we classify stocks into 10 covariance-sorted groups based on their pre-ranking co-
variance. Group 1 (10) contains stocks with the lowest (highest) covariance. For each group, we
build an equal-weighted portfolio. Then, we construct a spread portfolio (10-1) that longs Port-
17Given that the percentages of overweighting periods and underweighting periods are 56%-59% and 41%-44%respectively, it is reasonable to use the median as the separating point. We have also used the value 1 to separatethese periods, and we found the main results to be the same.
18CRSP is a provider of historical options data for use in empirical research and econometric studies. We accessCRSP via Wharton Research Data Services (WRDS) at http://www.whartonwrds.com.
21
folio 10 and shorts Portfolio 1. We then calculate the next month (t + 1)’s post-ranking portfolio
returns for these portfolios and repeat the procedure for the next month. Similarly, we construct
the coskewness-sorted portfolios.
Table 2 reports the average next month’s returns of covariance-sorted portfolios and coskewness-
sorted portfolios for the entire period, overweighting periods and underweighting periods, respec-
tively, where the overweighting periods and underweighting periods are determined by the index
PWI1, as described earlier. We observe that the spread covariance-sorted portfolio (10-1) has a
significantly positive monthly return during the overweighting periods and a significantly negative
monthly return during the underweighting periods. Meanwhile, the spread coskewness-sorted port-
folio (10-1) has a significantly negative monthly return during the overweighting periods and an
insignificantly negative monthly return during the underweighting periods. Moreover, the negative
return of this spread portfolio nearly doubles its size from an average of -1.01% monthly over the
entire period to an average of -1.81% monthly during the overweighting periods.
[Insert Table 2 here]
Figures 7 (a) and (b) plot the regression lines of the average next month’s returns on the
average covariance and on the average coskewness, respectively. The return–covariance line has a
steep upward (downward) slope during the overweighting (underweighting) periods. The return–
coskewness line has a steep downward slope during the overweighting periods and a nearly flat slope
during the underweighting periods. These findings are consistent with our theoretical predictions
in Section 3.2. Finally, for the entire sample period, the return–covariance line is almost flat,
reconciling with the beta anomaly, and the return–coskewness line is downward-sloping, conforming
to the skewness-inclined preference that is well documented in the literature.
[Insert Figure 7 here]
4.3 Two-way Sort Portfolio Analysis
Next, we conduct two-way sort portfolio analysis to separate the effects of covariance and
coskewness. For each month t from January 1996 to April 2016, we classify all the stocks into five
22
coskewness-sorted groups based on their pre-ranking coskewness. Then, in each coskewness-sorted
group, we further group the stocks into five covariance-sorted groups based on their pre-ranking
covariance. Thus, there are now a total of 25 groups. For each group, we construct the equal-
weighted portfolio, calculate next month (t + 1)’s post-ranking portfolio returns, and repeat the
procedure from the next month onwards. As the stocks have similar values of coskewness within
the same coskewness-sorted group, we can thereby control the influence of the coskewness and
compare the performance of the five covariance-sorted portfolios. Similarly, we also analyze the
25 covariance-coskewness-sorted portfolios based first on their pre-ranking covariance and then on
their pre-ranking coskewness.
Table 3 reports the average next month’s returns for the entire period, overweighting periods
and underweighting periods, respectively, where the overweighting periods and underweighting
periods are determined by PWI1. Panel A shows that with the coskewness controlled, the spread
portfolios (5-1) still have significantly positive monthly returns during the overweighting periods
and significantly negative monthly returns during the underweighting periods. Panel B, on the other
hand, yields that with covariance controlled the spread portfolios (5-1) have significantly negative
monthly returns during the overweighting periods and insignificantly monthly returns during the
underweighting periods.
[Insert Table 3 here]
Our sort portfolio analysis suggests that one can significantly improve the performance of the
long-short covariance-sorted and coskewness-sorted strategies by making use of the additional infor-
mation from the probability weighting indices proposed in this paper. It confirms that probability
weighting is an important factor for pricing securities and building profitable portfolios.
4.4 Fama-MacBeth Cross-Sectional Regression
We conduct a Fama-MacBeth regression to investigate the impact of probability weighting
on cross-sectional asset pricing. Following Bali, Engle, and Murray (2016), we first compute the
covariance and coskewness of risky stock i for each month t from January 1996 to April 2016, by
23
means of a 60-month-window regression:
ri,τ = ki,t + covariancei,trM,τ + coskewnessi,tr2M,τ + εi,τ , τ = t− 59, · · · , t,
where ki,t is the intercept and εi,τ is the residual. In contrast to computing the covariance and
coskewness from the samples directly, as in Subsection 4.2 and 4.3, the above regression rules
out interactions between covariance and coskewness. Next, for each t, we consider the following
cross-sectional regression:
ri,t+1 = kt +Atcovariancei,t +Btcoskewnessi,t + εi,t+1, i = 1, · · · , n.
The significance of the risk premiums A and B is determined by a t-test on the time series At
and Bt.19
Panel A of Table 4 reports the Fama-MacBeth regression results, in which the market over-
weighting and underweighting periods are divided by the index PWI1. If we do not separate the
time periods based on probability weighting, the overall risk premium of covariance, A, is not sig-
nificant and the risk premium of coskewness, B, is modestly negative. These are consistent with the
empirical findings in the literature. When we do separate the whole period into two regimes (over-
weighting and underweighting), however, A becomes significantly positive during overweighting
periods and significantly negative during underweighting periods, and B almost doubles its overall
value during overweighting periods yet becomes insignificant during underweighting periods. Again,
these results reaffirm our theoretical and prior empirical findings.
[Insert Table 4 here]
19Besides Bali et al. (2016)’s approach, Harvey and Siddique (2000) propose to compute covariancei,t andcoskewnessi,t by the following:
ri,τ = ki,t + covariancei,trM,τ + εi,τ , τ = t− 59, · · · , t,
coskewnessi,t =159
∑59j=0(εi,t−j)(r
2M,t−j − r2M )√
160
∑59j=0 ε
2i,t−j · 1
59
∑59j=0(rM,t−j − rM )2
,
where rM and r2M are the sample means of rM,τ and r2M,τ , respectively. We obtain the same results based on thismethod (omitted here due to space).
24
Finally, we note that during overweighting periods, the mean-covariance tradeoff and mean-
coskewness tradeoff are also economically significant. Indeed, the standard deviation of covariance
is 0.8586 and that of coskewness is 10.8123. Thus, during an overweighting period, a one-standard-
deviation increase in covariance is associated with a roughly 0.76% (the std of covariance times
A, i.e., 0.8586× 8.888× 10−3) increase in expected monthly return, and a one-standard-deviation
decrease in coskewness is associated with a roughly 0.89% (the std of coskewness times B, i.e.,
(−10.8123)× (−0.823)× 10−3) increase in expected monthly return. This finding strengthens the
notion that probability weighting has important implications in pricing and investment.
4.5 Robustness with Control Variables
Finally, we conduct a Fama-MacBeth regression to test the robustness of our main findings
after controlling six other variables. These control variables are Size (Size), Book-to-market ratio
(BM), Momentum (Mom), Short-term reversal (Rev), Idiosyncratic volatility (IV ol), and Idiosyn-
cratic skewness (ISkew). Among them, the first four are well-studied factors long known to be
relevant in asset pricing, and the last two are relatively new in the literature. Liu, Stambaugh, and
Yuan (2018) argue that the beta anomaly arises from beta’s positive correlation with idiosyncratic
volatility (IV ol). Barberis and Huang (2008) find that CPT investors are willing to pay more for
lottery-like stocks. An, Wang, Wang, and Yu (2020) observes that lottery-like stocks significantly
underperform their non-lottery-like counterparts, especially among stocks where investors have lost
money. This type of stocks has high idiosyncratic skewness, which could be largely independent of
the co-skewness with the market. If probability weighting has an impact on the latter, as we have
demonstrated, then, very likely, it has an impact on the former as well.
25
Following Bali et al. (2016), these control variables are computed as follows
Sizei,t = log
(PRCi,t · SHROUTi,t
1000
),
BMi,t = log
(1000 ·BEi,y
PRCi,y · SHROUTi,y
),
Momi,t = 100
11∏j=1
(ri,t−j + 1)− 1
,Revi,t = 100 · ri,t,
IV oli,t = 100 ·√
1
56
∑59j=0 ε
2i,t−j ·
√12,
ISkewi,t =160
∑59j=0 ε
3i,t−j(
160
∑59j=0 ε
2i,t−j
)3/2,
where y is the end of a fiscal year, t is the end of a month between June of year y + 1, and May of
year y + 2, PRCi,t is the price of stock i at t, SHROUTi,t is the number of shares outstanding of
stock i at t, BEi,y is the book value of common equity of stock i at y, PRCi,y is the price of stock
i at y, SHROUTi,y is the number of shares outstanding of stock i at y, ri,t−j is the return rate of
stock i during month t− j, and εi,t−j , j = 0, · · · , 59, are the residuals of the following regression:
ri,τ = ki,t + δ1i,tMKTτ + δ2
i,tSMBτ + δ3i,tHMLτ + εi,τ , τ = t− 59, · · · , t,
where MKTτ is the return of the market factor during month τ and SMBτ and HMLτ are the
returns of the size and value factor mimicking portfolios, respectively, during month τ .20
The Fama-MacBeth regression results with control variables are reported in Panel B of Table
4. They show that even with the commonly used variables controlled in a cross-sectional analy-
sis, our main empirical finding that the risk premium of covariance (coskewness) is significantly
positive (negative) during overweighting periods and significantly negative (insignificant) during
underweighting periods still stands.
It is interesting to note that, in Panel B of Table 4, IVol exhibits a significantly negative rela-
20The IV oli,t and ISkewi,t are estimated base on monthly data in a rolling window fashion. We have also estimatedthe IV oli,t and ISkewi,t based on daily data in each month t, and the main results still hold.
26
tionship with expected return only during underweighting periods. In other words, the mispricing
(the negative risk premiums for beta and idiosyncratic volatility) only appears when investors un-
derweight the tail risk, which echoes the finding in Liu et al. (2018). On the other hand, similar to
coskewness, ISkew shows a significantly negative relation with expected return only during over-
weighting periods. This implies that overweighting both tails leads to RDUT investors developing
a strong taste for both coskewness and idiosyncratic skewness.
5 Dynamism of Probability Weighting
We have uncovered a conditional relationship, both theoretically and empirically, between the
covariance/coskewness and the average return, where the conditioning variable is probability weight-
ing. However, we have left an important question unanswered: Why do individuals sometimes
overweight tail events and sometimes underweight them? In this section we offer a psychological
explanation of the switch between overweighting and underweighting, and then test our theory
against empirical evidence.
5.1 A Hypothesis on Time Variation in Probability Weighting
Fox and Tversky (1998) propose a two-step model of the psychology of tail events. The first step
is about beliefs: an individual, based on her beliefs, estimates the probabilities of tail events. This
step does not yet involve any decisions. The second step is about preferences: with these estimated
probabilities at hand, the individual assigns weights to the tail events to assist her decision. So
let us say that the true probability of a tail event is p, which is unknown to the individual; she
will first assess it to be w1(p) – a subjective estimation of p – and then further weight it, relevant
to her decision, to be w2(w1(p)). The final probability weighting function w – such as the one
we have been dealing with in this paper – is indeed a composition of the two distortion functions:
w = w2 w1.
Research in psychology distinguishes between decisions from description and those from expe-
rience (see e.g., Barberis, 2013). A decision from description occurs when the true probabilities
are known to the individual, in which case the first step in the Fox and Tversky (1998) model can
27
be skipped. A decision from experience occurs when the individual is not privy to the objective
probabilities and hence has to learn them by experiencing them over time. In this case, she will
undergo both steps. It is widely held in the psychology literature that, when making decisions from
description, there is consistent overweighting of tails; that is, the aforementioned w2 function, and
hence the w function (w ≡ w2 w1 = w2), are both more likely to be inverse S-shaped. However,
Hertwig et al. (2004) argue via lab data that when making decisions from experience people may
underweight tails.21 This observation reconciles with the S-shape of the weighting functions during
certain months inferred from option data in the present paper. Mathematically, it is also consistent
with – or, at least not contradictory to – the formula w = w2 w1: while w2 is robustly inverse
S-shaped, the presence of w1 may render the final w a different shape, including S-shaped.
The above theory may explain the presence of overweighting or underweighting of tail distri-
butions in stock investing, but not the switch between overweighting in some months and under-
weighting in others. To understand this dynamism of probability weighting we need to understand
the underlying psychological mechanism that triggers the switch. For this we propose a coherent
hypothesis, involving both beliefs and preferences and built on Fox and Tversky (1998)’s two-step
model, to explain how probability weighting reacts to changes in the outside environment, and
hence to account for the switch.
Tversky and Kahneman (1973) put forth the so-called availability heuristic, which refers to an
apparent tendency to give more weight to easily recalled events in estimating the probability of
an event happening. Several studies have examined the availability heuristic in finance, ranging
from asymmetric reactions of stock prices to consumer sentiment reports (Akhtar, Faff, Oliver,
and Subrahmanyam, 2012) to investors’ tendency to repurchase previously held stocks (Nofsinger
and Varma, 2013). Goetzmann, Kim, and Shiller (2017) test the availability heuristic directly on
investors’ probability estimates of a major stock crash – a left tail extreme event. They observe
that recently experienced exogenous rare events significantly – although often incorrectly – impact
an investor’s estimating of crash probabilities. The authors present evidence that articles in the
financial press change investor crash beliefs: those with negative sentiment lead to higher crash
21The robustness and interpretation of this finding, however, is still being debated; see, e.g., pp. 613–614 of Barberis(2013).
28
probability assessments following market downturns, but those with positive sentiment have al-
most no effect. Finally, they find that subjective crash probabilities are highly dynamic: they vary
significantly over time and are correlated to measures of jump risk such as the VIX and to the
occurrence of extreme negative stock returns. The availability heuristic thereby provides a psy-
chological ground as to why people’s probability beliefs react closely to recent exogenous shocks
and, as a consequence, change over time. Moreover, Barberis (2013) argues that the availability
heuristic can explain both overestimation and underestimation of tails for stock investing: during or
immediately after a major stock crash (such as the most recent one due to COVID-19), events like
market circuit-breaks and spectacular rebounds “receive disproportionate media coverage, making
it easier for us to recall them, and tricking us into judging them more probable than they actually
are”. On the other hand, in the years leading up to a crash (such as the big bull run before the
pandemic), massive one-day stock market ups and downs (both of, say, more than 10%) are rare
at best, for which reason we hardly recall tail events and hence assign lower probabilities to them.
While beliefs change dynamically, so do preferences. Many studies argue that probability
weighting, while seemingly biased and irrational, actually serves a purpose. It is employed –
most likely subconsciously or unconsciously – to respond optimally to various challenges associated
with decision-making under uncertainty, such as limited information processing capacity (including
limited attention), information loss, or over-optimism. Herold and Netzer (2010) show that inverse
S-shaped probability weighting is an optimal response to S-shaped valuation of rewards. Woodford
(2012) argues that the choices of probability weighting - termed perception strategies - are them-
selves optimal in minimizing the mean-square error of the subjective evaluation of probabilities.
Steiner and Stewart (2016) demonstrate that overweighting of small probability events optimally
mitigates errors due to randomness. Now, if probability weighting is endogenous and is an optimal
response to the external environment, then the former will be forced to change over time because,
inevitably, the latter will do so. Finally, like underestimation, if investors have not seen tail events
for a long time, then they may underweight or even simply ignore them so as to optimally divert
their ability and attention to what they think are more important scenarios.
Given the support from the psychology and behavioral finance literatures for the feedback nature
29
of probability weighting on tail events – that is, that environmental changes are fed back to our
perceptions on the tails in both beliefs and preferences – we hypothesize that the probability
weighting is time varying. Most of the time, the change is slow and insignificant, especially when
no extreme events are occurring, during which time a prior overweighting may gradually change to
underweighting. However, a sudden and significant event, aka a “black swan”, such as a pandemic,
a stock crash or even an unexpected news report, may trigger a major, characteristic change in the
probability weighting, including that from underweighting to overwheighting.22
5.2 Empirical Tests
We now test empirically our hypothesis that switches between overwheighting and underweight-
ing are associated with (sometimes dramatic) changes in the market environment. We choose vari-
ables in different categories that capture these changes. The first category consists of proxies of
extreme price movement. Let Maxi and Mini be, respectively, the largest (most positive) and small-
est (most negative) daily returns of asset i in a month, and Maxave and Minave be the respective
average values across all stocks. Moreover, the monthly returns of all stocks form a cross-sectional
distribution, whose 1% and 99% quantiles are denoted by Q01 and Q99, respectively. We also take
the largest and smallest daily returns of the S&P 500 in each month, denoted by MaxSP , MinSP ,
respectively, and the standard derivation of the daily excess returns of the market portfolio, denoted
by σMKT .
The next natural choice is the VIX, the Chicago Board Options Exchange (CBOE) Market
Volatility Index, a well-known measure of the stock market’s expectation of volatility based on
S&P 500 index options. This index has been widely accepted as the best available barometer of
investors’ fear since its introduction in the 1990s.
We also consider two media-based variables. The first is the Equity Market Volatility (EMV)
tracker, which is based on eleven major U.S. newspapers.23 The EMV index counts the newspaper
22We reiterate that our theory here is both belief- and preference-based. Beliefs and preferences are always en-tangled, and it is both challenging and interesting to disentangle them even for classical EUT (Ross, 2015). It iseven thornier to decouple them in the context of probability weighting (Barberis, 2013, Goetzmann et al., 2017).Luckily, for our purpose in this paper we need to understand only the combined effect of the dynamics of beliefs andpreferences on probability weighting.
23The eleven newspapers are Boston Globe, Chicago Tribune, Dallas Morning News, Houston Chronicle, Los Ange-
30
articles that contain at least one term in each of E: economic, economy, financial, M: stock
market, equity, equities, Standard and Poors (and variants), V: volatility, volatile, uncertain,
uncertainty, risk, risky. The second variable we consider is the Financial Stress Indicator (FSI),
which is based on five U.S. newspapers.24 This index also counts articles related to financial
distress. Specifically, four financial sentiment dictionaries are used to first measure the sentiment
of each article’s title, and then the FSI is the total number of articles in a month with net negative
connotations.
In addition to these news-based sentiment indices, we also consider a more general and widely
used sentiment index proposed by Baker and Wurgler (2006). This index is constructed by the first
principal component of five different sentiment proxies, including IPO numbers, first-day return of
IPO, close-end fund discount, turnover ratio and share of equity/debt issues.
The final category is the survey variables. We consider Shiller’s U.S. Crash Confidence Index,
which is the percentage of respondents who think that the probability of a US market crash occur-
ring in the next six months is strictly less than 10%.25 We also take the Bull and Bell Indices based
on the Investor Sentiment Survey of the American Association of Individual Investors (AAII). The
Investor Sentiment Survey, which has been administered weekly to members of the AAII since 1987,
measures the percentage of individual investors who are bullish, neutral, or bearish on the stock
market for the next six months.
We present all of the above variables, together with the PWI1 index, and their summary
statistics in Table 5. Figure 8 graphs the PWI1 index together with variables selected from each
group in Table 5.26 It is evident from Figure 8 that all of the indices, except the sentiment index,
move along with PWI1.
les Times, Miami Herald, New York Times, San Francisco Chronicle, USA Today, Wall Street Journal, and Washing-ton Post. The monthly EMV data can be downloaded from http://www.policyuncertainty.com/EMV monthly.html.
24The five U.S. newspapers are Boston Globe, Chicago Tribune, Los Angeles Times, Wall Street Journal and Wash-ington Post. The monthly FSI data can be downloaded from http://www.policyuncertainty.com/financial stress.html.
25The monthly data of this index is available at https://som.yale.edu/faculty-research-centers/centers-initiatives/international-center-for-finance/data/stock-market-confidence-indices/united-states-stock-market-confidence-indices.
26There are a total of 15 variables in Table 5. It would be visually incoherent were we to plot them all in the samefigure. Therefore, we choose a representative from each group to display in Figure 8. For example, we select MaxSPand MinSP from the return variables to represent the extreme market movements. If we instead had used Maxaveand Minave as the representative, they would have displayed patterns similar to the green and red bars shown inFigure 8.
31
[Insert Table 5 here]
[Insert Figure 8 here]
Panel A of Table 6 reports the correlations between PWI1 and the other variables. VIX has
the highest correlation with PWI1 (0.815). The return variables representing extreme events are
all highly correlated with PWI1, the smallest one being Q99, which still has 0.338. The media
variables also have high correlations, as does the Crash index in the survey group. The Bull/Bear
survey indices are correlated with PWI1, if not as significantly. Finally, the sentiment index is
uncorrelated with PWI1. Panel B of Table 6 reports the number and proportion of months for
which an alternative variable and PWI1 give different labels (overweighting/underweighting). We
can see that the variables highly correlated with PWI1, such as VIX, Minave, Maxave, have small
numbers of discrepancy.
[Insert Table 6 here]
Figure 9 further plots the switching patterns of the overweighting periods and the underweight-
ing periods under five indices. These are remarkably similar.
[Insert Figure 9 here]
The findings summarized by Table 6, Figure 8, and Figure 9 strongly support our theory that
the dynamic of PWI1 is driven by market movements. The high correlation between PWI1, the re-
turn variables (and their proxies such as VIX), and the media variables confirms our conjecture that
a direct experience of extreme events and/or sensational media coverage quickly causes investors to
pay more attention to tail risk and to adjust their beliefs/preferences accordingly. The considerable
correlation between PWI1 and the survey variables underlines a synchronization between proba-
bility weighting and investors’ sentiments regarding the market. Intriguingly, the fact that PWI1
and Baker and Wurgler (2006)’s sentiment index are largely unrelated proves our hypothesis from
a different angle: the constituents of this particular index (IPO numbers, first-day return of IPO,
close-end fund discount, turnover ratio, and share of equity/debt issues) are obscure and opaque,
32
not directly observable and easily comprehensible by investors; hence, they do not respond to them,
at least not immediately or dramatically.
In addition, we use these highly correlated variables as alternative probability weighting indices
to repeat the empirical analysis on the conditional return and covariance/coskewness relationship,
and we find that the results remain consistent. The results are presented in an online appendix.
This additional empirical check demonstrates that our main results are robust in that they are
independent of the choice of a specific probability weighting index.
6 Conclusions
We show that probability weighting is a key driver of beta and coskewness pricing. Individuals
tend to overweight or underweight the two tails simultaneously. Whether they overweight or under-
weight changes dynamically over time, to which we offer a psychology based explanation, supported
by empirical evidence that we present. We find that the total length of overweighting periods is
comparable to that of underweighting periods. If we analyze a sufficiently long sample period, the
aggregate effects of overweighting and underweighting may cancel out, potentially deceiving us into
overlooking the significance of probability weighting. It is, therefore, important to separate the
overweighting and underweighting periods and to study them individually and respectively. In do-
ing so, as demonstrated in this paper, we are able to understand the significant role that probability
weighting plays in pricing and in building profitable portfolios.
We carry out the theoretical analysis using an RDUT model and confirm its implications with
empirical tests. We believe that our model offers a partial explanation of the beta anomaly and
hints at a viable way in which to potentially resolve other puzzles in asset pricing.
Appendix
A: Proof of Theorem 2
When the market is in equilibrium, the total return of the optimal portfolio of the representative
investor equals the total return of the market portfolio, i.e., R∗P = RM . Applying Theorem 5.2 of
33
Xia and Zhou (2016), we obtain
m = (λ∗)−1w′(1− FRM
(RM ))u′(RM ).
However, the pricing formula of the risk-free asset yields
1 = E[m(1 + rf )] = E[m](1 + rf ),
which in turn implies
λ∗ = (1 + rf )E[w′(1− F
RM(RM ))u′(RM )
]. (12)
This establishes (4).
Next we rewrite
m = (λ∗)−1w′(1− FRM
(1 + rM ))u′(1 + rM ).
Applying the Taylor expansion up to the second order of rM , we can express the pricing kernel as
follows:27
m =(λ∗)−1w′(1− FRM
(1))u′(1)
+ (λ∗)−1[w′(1− F
RM(1))u′′(1)− w′′(1− F
RM(1))u′(1)f
RM(1)]rM
+1
2(λ∗)−1
w′(1− F
RM(1))u′′′(1)− w′′(1− F
RM(1))[2u′′(1)f
RM(1) + u′(1)f ′
RM(1)]
+ w′′′(1− FRM
(1))u′(1)f2RM
(1)r2M + o(r2
M ). (13)
Applying the pricing formula to the risky assets, we have
E[m(1 + ri)] = 1, i = 2, 3, · · · , n.
27Here, we Taylor expand FRM(1 + rM ) around 1, then expand w′(1 − FRM
(1 + rM )) around 1 − FRM(1), and
expand u′(1 + rM ) around 1.
34
Hence,
Cov(m, (1 + ri)) + E[1 + ri]E[m] = 1,
or
E[ri] =1− Cov(m, ri)
E[m]− 1 = rf +ACov(ri, rM ) +
1
2BCov(ri, r
2M ) + o(r2
M ),
thanks to (12) and (13).
B: Proof of Proposition 1
i) When α < 1, w is inverse S-shaped. Hence, 1 − FRM
(1) lies within the convex domain of w
under the assumptions of the proposition. Therefore, the two terms inside the brackets of (6) are
both negative. This implies A > 0.28
ii) When α > 1, w is S-shaped, and 1 − FRM
(1) lies within the concave domain of w. With
w(p) = exp(−(− log(p))α), we have
A ≡−w′(1− F
RM(1))
E[w′(1− FRM
(RM ))u′(RM )]
[u′′(1)−
w′′(1− FRM
(1))
w′(1− FRM
(1))u′(1)f
RM(1)
]
=−w′(1− F
RM(1))
E[w′(1− FRM
(RM ))u′(RM )]
[u′′(1)− αyα0 − α+ 1− y0
(1− FRM
(1))y0u′(1)f
RM(1)
],
where y0 = − log(1−FRM
(1)). It follows from the assumptions of the proposition that 0 ≤ y0 < 1,
which implies limα→+∞ αyα0 − α = −∞. Hence,
limα→+∞
−αyα0 − α+ 1− y0
(1− FRM
(1))y0u′(1)f
RM(1) = +∞.
Thus, there must exist α∗ > 1 such that when α > α∗, − αyα0−α+1−y0(1−F
RM(1))y0
u′(1)fRM
(1) > −u′′(1),
implying A < 0.
28This part of the proof does not depend on the specific form of w; it only requires 1−FRM(1) to be in the convex
domain of w.
35
C: Proof of Proposition 2
With w(p) = exp(−(− log(p))α), we have
B =−w′(1− F
RM(1))u′(1)
E[w′(1− FRM
(RM ))u′(RM )]
[γ(γ + 1)− αyα0 − α+ 1− y0
(1− FRM
(1))y0(−2γf
RM(1) + f ′
RM(1))
+(αyα0 − α+ 1− y0)2 + (1− α) + (1− α− y0)(αyα0 − y0)
(1− FRM
(1))2y20
f2RM
(1)
]
:=−w′(1− F
RM(1))u′(1)
E[w′(1− FRM
(RM ))u′(RM )][B1 +B2 +B3],
where B1, B2, B3 denote the three terms in the brackets, and y0 = − log(1 − FRM
(1)). Clearly,
B1 > 0.
i) When 0 < α < 1 and γ ≥f ′RM
(1)
2fRM(1) , we have B2 ≥ 0. We now examine B3. It follows from
1 − FRM
(1) > e−1 that 0 ≤ y0 < 1. On the other hand, the first-order condition of the function
(αyα0 − y0) is
∂
∂y0(αyα0 − y0) = α2yα−1
0 − 1 = 0,
whose solution is y∗0 = α2
1−α ∈ (0, 1). Then, the extreme values of this function on y0 ∈ [0, 1] are
maxy0∈[0,1]
(αyα0 − y0) = max α0α − 0, α1α − 1, α1+α1−α − α
21−α = α
1+α1−α − α
21−α > 0,
miny0∈[0,1]
(αyα0 − y0) = min α0α − 0, α1α − 1, α1+α1−α − α
21−α = α− 1 < 0.
When 1− α− y0 > 0, we have
(1− α) + (1− α− y0)(αyα0 − y0) ≥ (1− α) + (1− α− 0)(α− 1) = (1− α)α > 0.
When 1− α− y0 < 0, we have
(1− α) + (1− α− y0)(αyα0 − y0) ≥ (1− α) + (1− α− 1)(α1+α1−α − α
21−α ) = (1− α
21−α )(1− α) > 0.
36
When 1− α− y0 = 0, we have
(1− α) + (1− α− y0)(αyα0 − y0) = (1− α) > 0.
This establishes that B3 > 0, leading to B < 0. (Here we have actually proved that, in this case,
w′′′(1− FRM
(1)) > 0.)
ii) When α > 1, we consider B1 +B2 +B3 as a function of γ and α:
B1 +B2 +B3 = γ2 + (1 + 2g1(α)fRM
(1))γ − g1(α)f ′RM
(1) + g2(α)f2RM
(1),
where g1(α) and g2(α) are continuous functions of α:
g1(α) :=αyα0 − α+ 1− y0
(1− FRM
(1))y0,
g2(α) :=(αyα0 − α+ 1− y0)2 + (1− α) + (1− α− y0)(αyα0 − y0)
(1− FRM
(1))2y20
.
It is easy to show that, for any given finite α > 1, there exists a threshold
γ(α) :=
−(1+2g1(α)fRM
(1))+√
(1+2g1(α)fRM
(1))2−4(−g1(α)f ′RM
(1)+g2(α)f2RM
(1))
2 ,
if (1 + 2g1(α)fRM
(1))2 − 4(−g1(α)f ′RM
(1) + g2(α)f2RM
(1)) ≥ 0,
0, if (1 + 2g1(α)fRM
(1))2 − 4(−g1(α)f ′RM
(1) + g2(α)f2RM
(1)) < 0,
such that B1 +B2 +B3 > 0 (and, hence, B < 0) whenever γ > γ(α).
iii) As 0 ≤ y0 < 1 and 1− FRM
(1) lies within the concave domain of w, we have
αyα0 − α+ 1− y0 ≤ 0.
Thus, B2 ≤ 0 when γ ≥ γ∗. Now,
B1 +B3 = γ(γ + 1) + g2(α)f2RM
(1).
37
It is easy to prove that, for any given finite α > 1, there must exist a threshold
γ(α) :=
√
1−g2(α)f2RM
(1)−1
2 , if g2(α) ≤ 0,
0, if g2(α) > 0,
such that B1 +B3 < 0 when 0 ≤ γ < γ(α). This completes the proof.
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42
Table
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wit
hva
riou
sva
lues
ofre
lati
ve
risk
aver
sion
par
amet
erγ
.T
he
dis
trib
uti
onof
the
mark
etp
ort
foli
o’s
tota
lre
turn
isn
orm
al(µ
=1.
076,σ
=0.
158)
,sk
ew-n
orm
al(µ
=1.
076,σ
=0.
158,skewness
=−
0.3
39),
orlo
g-n
orm
al
(µ=
1.076,σ
=0.
158).
Th
ep
aram
eter
sγ∗ ,γ
andγ
are
defi
ned
inP
rop
osit
ion
2an
dit
sp
roof
(Ap
pen
dix
C).
Panel
A:
Over
wei
ghti
ng
case
(α=
0.5
)
Dis
trib
uti
on
ofRM
Para
met
ers
γ=
1γ
=2
γ=
3γ
=4
γ=
5
Norm
al
γ∗
=1.5
222
-34.6
634
-39.9
503
-43.4
616
-44.6
584
-43.4
897
Skew
-norm
al
γ∗
=2.3
799
-30.2
979
-36.4
819
-41.0
653
-43.2
547
-42.8
122
Log-n
orm
al
γ∗
=0.7
816
-32.5
639
-34.8
329
-35.4
924
-16.2
108
-12.7
188
Panel
B:
no
pro
babilit
yw
eighti
ng
case
(α=
1)
Dis
trib
uti
on
ofRM
Para
met
ers
γ=
1γ
=2
γ=
3γ
=4
γ=
5
Norm
al
γ∗
=1.5
222
-2.1
102
-6.5
066
-13.0
431
-21.2
156
-30.1
970
Skew
-norm
al
γ∗
=2.3
799
-2.0
897
-6.3
884
-12.7
085
-20.5
003
-28.8
738
Log-n
orm
al
γ∗
=0.7
816
-2.1
385
-6.3
540
-12.2
682
-19.2
142
-26.2
404
Panel
C:
Under
wei
ghti
ng
case
(α=
1.5
)
Dis
trib
uti
on
ofRM
Para
met
ers
γ=
1γ
=2
γ=
3γ
=9
γ=
10
Norm
al
γ∗
=1.5
222,γ
(α)
=1.4
765,γ
(α)
=8.2
577
8.9
976
15.8
929
20.7
699
-8.4
330
-20.4
957
Skew
-norm
al
γ∗
=2.3
799,γ
(α)
=1.4
129,γ
(α)
=7.1
000
-1.3
199
4.8
151
9.0
720
-16.7
676
-26.2
754
Log-n
orm
al
γ∗
=0.7
816,γ
(α)
=1.6
202,γ
(α)
=8.8
817
21.4
745
28.4
943
32.8
405
-1.3
287
-12.4
790
43
Table
2O
ne-
way
sort
resu
lts
bas
edon
PWI 1
.T
he
over
wei
ghti
ng
per
iod
san
du
nd
erw
eigh
tin
gp
erio
ds
are
sep
ara
ted
by
the
ind
exPWI 1
.A
llp
erio
ds
(All
),ov
erw
eigh
tin
gp
erio
ds
(Ove
r),
and
un
der
wei
ghti
ng
per
iod
s(U
nd
er)
incl
ud
e24
4m
onth
s,122
month
s,an
d122
month
s,re
spec
tive
ly.
Th
esp
read
por
tfol
io(1
0-1)
isco
nst
ruct
edby
lon
gin
gth
e10
thp
ortf
olio
and
shor
tin
gth
e1st
port
foli
o.
Th
et-
stati
stic
sof
the
retu
rns
ofth
esp
read
por
tfol
ios
are
rep
orte
d.
***,
**,
and
*den
ote
sign
ifica
nce
atth
e1%
,5%
,and
10%
leve
ls,
resp
ecti
vely
.
Panel
A:
cova
riance
-sort
edp
ort
folios
12
34
56
78
910
(10-1
)t-
stati
stic
of
(10-1
)
All
1.0
5%
1.1
6%
1.1
8%
1.2
1%
1.3
4%
1.2
7%
1.2
1%
1.3
3%
1.1
8%
1.1
8%
0.1
3%
(0.2
3)
Over
1.0
9%
1.1
6%
1.2
1%
1.2
5%
1.4
7%
1.6
3%
1.6
3%
2.0
8%
2.2
4%
2.7
5%
1.65%*
(1.7
7)
Under
1.0
1%
1.1
6%
1.1
6%
1.1
6%
1.2
2%
0.9
1%
0.8
0%
0.5
8%
0.1
2%
-0.4
0%
-1.40%**
(-2.4
8)
Panel
B:
cosk
ewnes
s-so
rted
port
folios
12
34
56
78
910
(10-1
)t-
stati
stic
of
(10-1
)
All
1.7
3%
1.4
6%
1.4
2%
1.4
0%
1.2
0%
1.1
4%
1.2
0%
0.9
3%
0.9
1%
0.7
2%
-1.01%***
(-3.0
4)
Over
2.8
5%
2.2
4%
1.9
7%
1.7
9%
1.4
7%
1.4
9%
1.5
3%
1.0
7%
1.0
5%
1.0
4%
-1.81%***
(-3.0
1)
Under
0.6
1%
0.6
7%
0.8
7%
1.0
1%
0.9
3%
0.7
9%
0.8
7%
0.8
0%
0.7
8%
0.3
9%
-0.2
1%
(-0.7
9)
44
Table 3 Two-way sort results based on PWI1. The overweighting periods (Over) and under-weighting periods (Under) are separated by PWI1. The spread portfolio (5-1) is constructed bylonging the 5th portfolio and shorting the 1st portfolio. The t-statistics of the returns of thespread portfolios are reported. ***, **, and * denote significance at the 1%, 5%, and 10% levels,respectively.
Panel A: coskewness-covariance-sorted portfolios
Periods coskewness
covariance
1 2 3 4 5 (5-1) t-statistic of (5-1)
All
1 1.40% 1.36% 1.74% 1.55% 1.07% -0.32% (-0.83)2 1.15% 1.29% 1.34% 1.08% 1.03% -0.13% (-0.29)3 1.11% 1.23% 1.30% 1.20% 1.23% 0.13% (0.31)4 0.92% 1.02% 1.18% 1.23% 1.23% 0.31% (0.70)5 0.86% 1.27% 1.04% 1.09% 1.37% 0.51% (0.87)
Over
1 1.79% 1.69% 2.25% 2.51% 2.05% 0.26% (0.39)2 1.13% 1.23% 1.56% 1.49% 2.37% 1.25%* (1.68)3 1.08% 1.19% 1.37% 1.53% 2.37% 1.29%* (1.90)4 0.83% 1.00% 1.24% 1.55% 2.39% 1.56%** (2.18)5 0.67% 1.43% 1.46% 2.08% 3.01% 2.33%** (2.28)
Under
1 1.01% 1.04% 1.22% 0.59% 0.09% -0.91%** (-2.30)2 1.18% 1.34% 1.12% 0.68% -0.32% -1.50%*** (-3.36)3 1.13% 1.27% 1.23% 0.86% 0.09% -1.04%** (-2.36)4 1.02% 1.04% 1.12% 0.90% 0.08% -0.94%* (-1.91)5 1.04% 1.10% 0.61% 0.10% -0.26% -1.30%** (-2.32)
Panel B: covariance-coskewness-sorted portfolios
Periods covariance
coskewness
1 2 3 4 5 (5-1) t-statistic of (5-1)
All
1 1.70% 1.24% 1.08% 1.09% 0.76% -0.94%*** (-2.71)2 1.59% 1.29% 1.25% 1.05% 0.80% -0.79%*** (-3.30)3 1.61% 1.32% 1.23% 1.13% 1.05% -0.55%** (-2.30)4 1.72% 1.19% 1.24% 1.13% 1.07% -0.66%** (-2.47)5 1.12% 1.04% 1.36% 1.19% 1.03% -0.09% (-0.33)
Over
1 2.32% 1.33% 1.00% 1.00% 0.49% -1.83%*** (-2.88)2 2.02% 1.32% 1.21% 0.98% 0.72% -1.30%*** (-3.11)3 2.11% 1.59% 1.24% 1.26% 1.12% -0.99%** (-2.45)4 2.79% 1.93% 1.70% 1.42% 1.43% -1.35%*** (-2.99)5 2.42% 2.32% 2.63% 2.59% 2.29% -0.13% (-0.26)
Under
1 1.08% 1.16% 1.15% 1.18% 1.03% -0.05% (-0.19)2 1.16% 1.26% 1.28% 1.11% 0.87% -0.29% (-1.24)3 1.10% 1.04% 1.23% 1.01% 0.99% -0.12% (-0.46)4 0.66% 0.46% 0.78% 0.84% 0.70% 0.04% (0.14)5 -0.18% -0.25% 0.09% -0.21% -0.23% -0.06% (-0.21)
45
Table
4F
ama-
Mac
Bet
hre
gres
sion
resu
lts.
“All
per
iod
”is
from
Jan
uar
y19
96to
Ap
ril
2016
,w
hil
e“U
nder
wei
ghti
ng
per
iod
s”an
d“O
verw
eigh
tin
gp
erio
ds”
are
det
erm
ined
byPWI 1
ind
ex.
Pan
elA
rep
orts
resu
lts
wit
hou
tco
ntr
olva
riab
les.
Pan
elB
rep
ort
sre
sult
sw
ith
two,
fou
r,an
dsi
xco
ntr
olva
riab
les.
Th
eav
erag
eva
lues
ofth
eco
effici
ents
,w
hic
har
em
ult
ipli
edby
1000,
are
rep
ort
edab
ove
an
dth
eco
rres
pon
din
gt-
stat
isti
csar
ere
por
ted
bel
ow,
inp
aren
thes
es.
***,
**,
and
*d
enot
esi
gnifi
cance
at
the
1%
,5%
,an
d10%
leve
ls,
resp
ecti
vely
.T
he
row
lab
elle
d“A
dj.R
2”
rep
orts
the
aver
age
adju
sted
r-sq
uar
eof
each
regr
essi
on.
Th
ero
wla
bel
led
“n”
rep
ort
sth
eav
erag
enu
mb
erof
stock
sin
each
cros
s-se
ctio
nal
regr
essi
on.
Panel
A:
Fam
a-M
acB
eth
regre
ssio
nre
sult
sw
ithout
contr
ol
vari
able
s
All
per
iods
Over
wei
ghti
ng
per
iods
Under
wei
ghti
ng
per
iods
A1.5
18.89**
-5.87***
(0.6
5)
(2.2
0)
(-2.7
4)
B-0.44***
-0.82***
-0.0
5(-
2.8
7)
(-2.8
9)
(-0.5
3)
Adj.R
20.0
20.0
20.0
2n
2695
2819
2571
Panel
B:
Fam
a-M
acB
eth
regre
ssio
nre
sult
sw
ith
contr
ol
vari
able
s
All
per
iods
Over
wei
ghti
ng
per
iods
Under
wei
ghti
ng
per
iods
A1.2
42.3
01.7
27.47**
8.74***
7.23**
-4.99***
-4.14**
-3.79**
(0.6
5)
(1.2
0)
(1.0
3)
(2.2
8)
(2.6
5)
(2.5
9)
(-2.7
3)
(-2.3
5)
(-2.2
5)
B-0.38***
-0.33***
-0.24**
-0.66***
-0.60**
-0.43**
-0.0
9-0
.06
-0.0
5(-
2.9
4)
(-2.6
3)
(-2.4
7)
(-2.7
8)
(-2.5
6)
(-2.5
1)
(-1.0
5)
(-0.7
2)
(-0.5
3)
IVol
0.0
1-0
.06
-0.0
50.1
00.0
10.0
1-0.09*
-0.12***
-0.12***
(0.1
5)
(-1.3
4)
(-1.3
5)
(1.0
2)
(0.0
7)
(0.1
3)
(-1.6
9)
(-2.7
6)
(-2.6
9)
ISkew
-0.7
2-1.03*
-0.7
4-2.46**
-2.60**
-1.92**
1.0
10.5
40.4
4(-
1.1
7)
(-1.6
8)
(-1.3
9)
(-2.3
2)
(-2.4
8)
(-2.1
8)
(1.6
4)
(0.8
9)
(0.7
4)
Size
-1.66***
-1.43***
-2.23**
-1.66**
-1.09**
-1.21***
(-3.2
7)
(-3.2
3)
(-2.4
5)
(-2.1
3)
(-2.4
3)
(-2.7
9)
BM
1.68**
1.93***
1.3
01.6
42.06**
2.22***
(2.3
5)
(2.7
1)
(1.1
1)
(1.3
9)
(2.5
1)
(2.7
6)
Mom
0.0
1-0
.02
0.04**
(0.4
5)
(-0.5
9)
(2.2
5)
Rev
-0.34***
-0.54***
-0.15***
(-6.3
8)
(-5.9
4)
(-2.8
1)
Adj.R
20.0
30.0
40.0
50.0
30.0
40.0
60.0
20.0
30.0
4n
2693
2691
2689
2817
2815
2813
2569
2567
2565
46
Table 5 The variables used to test for the dynamism of probability weighting, and their summarystatistics. The data are collected from the OptionMetrics (OM), the Center for Research on SecurityPrices (CRSP), Kenneth R. French’s data library (FF), the Economic Policy Uncertainty website(EPU), the survey data from Robert Shiller’s Investor Confidence Surveys (ICS), Investor SentimentSurvey of the American Association of Individual Investors (AAII), or Jeffrey Wurgler’s website(BW). The starting months when the data are available are different for these indices, which arereported under the “Start Time” column, and the ending month is set to be April 2016 uniformly.For example, the period of VIX we use to compute summary statistics is from January 1990 toApril 2016.
Panel A: Variable Descriptions
Index Name Description Start Time Source
Option-implied variables:
PWI1 The probability weighting index at the end of a month, derivedfrom the S&P500 index options.
Jan. 1996 OM
VIX The 30-day expected volatility of the U.S. stock market at theend of a month, derived from the S&P500 index options.
Jan. 1990 CBOE
Return variables:Maxave, Minave For each stock, the largest and smallest daily returns during
a month are obtained. Maxave and Minave are the averagevalues of the largest and smallest daily returns across all stocks,respectively.
Jun. 1968 CRSP
Q01, Q99 In each month, the monthly returns of all stocks form a cross-sectional distribution. Q01 (Q99) is the 1 (99) percentage quan-tile of such cross-sectional distribution.
Jun. 1968 CRSP
MaxSP , MinSP MaxSP and MinSP are the largest and smallest daily returnsof S&P500 in each month.
Jun. 1968 CRSP
σMKT The standard derivation of the daily excess returns of the mar-ket portfolio during a month.
Jun. 1968 FF
Media variables:EMV The newspaper-based and Policy-Related Equity Market
Volatility (EMV) tracker, which is based on eleven majorU.S. newspapers: the Boston Globe, Chicago Tribune, DallasMorning News, Houston Chronicle, Los Angeles Times, MiamiHerald, New York Times, San Francisco Chronicle, USA To-day, Wall Street Journal, and Washington Post, and updatedmonthly.
Jan. 1985 EPU
FSI The newspaper-based Financial Stress Indicator of the U.S.market, which is constructed by considering five U.S. news-papers: the Boston Globe, Chicago Tribune, Los AngelesTimes, Wall Street Journal and Washington Post and updatedmonthly.
Jun. 1968 EPU
Survey variables:
Crash The average crash probability reported by individual investors,which is 1 minus the crash confidence level. The Crash Con-fidence Index is the percentage of individual respondents whothink that the probability of market crash for the next sixmonth is strictly less than 10%.
Jul. 2001 ICS
Bull, Bear The percentage of individual investors who are bullish andbearish on the stock market for the next six month.
Sep. 1987 AAII
Sentiment variable:Sentiment Sentiment index in Baker and Wurgler (2006), based on the
first principal component of FIVE (standardized) sentimentproxies.
Jun.1968 BW
47
Panel B: Summary Statistics
N Mean StDev Min P25 Median P75 Max
Option-implied variables:
PWI1 244 1.51 1.21 0.00 0.40 1.31 2.44 5.10VIX 316 19.85 7.52 10.42 14.02 18.24 23.73 59.89
Return variables:Maxave 577 7.04 2.17 3.69 5.37 6.57 8.25 18.77Minave 577 -5.74 1.75 -17.22 -6.75 -5.42 -4.43 -3.01Q99 577 23.68 11.66 -3.45 16.57 21.93 29.01 94.24Q01 577 -18.53 7.25 -53.59 -21.95 -17.29 -14.04 -2.82MaxSP 577 1.80 1.11 0.34 1.10 1.53 2.18 11.58MinSP 577 -1.73 1.39 -20.47 -2.09 -1.49 -1.00 -0.01σMKT 577 14.01 8.16 4.41 9.24 11.73 15.84 78.92
Media variables:EMV 377 19.68 8.11 8.03 14.76 17.59 22.30 69.84FSI 577 100.87 0.84 98.76 100.30 100.80 101.32 105.89
Survey variables:
Crash 179 66.68 7.38 51.12 61.08 67.48 71.16 85.43Bull 346 38.53 8.76 18.00 32.06 38.75 43.69 64.40Bear 346 30.31 8.21 14.00 24.25 29.13 35.50 59.00
Sentiment variable:Sentiment 577 0.08 1.00 -2.42 -0.37 0.09 0.58 3.20
48
Table
6PanelA
rep
orts
corr
elat
ion
sam
ong
15va
riab
les.
Th
eco
rrel
atio
ns
wit
hC
rash
are
com
pu
ted
base
don
the
month
lyd
ata
from
Jan
uar
y20
01to
Ap
ril
2016
.T
he
oth
erco
rrel
atio
ns
are
com
pu
ted
bas
edon
the
mon
thly
dat
afr
omJan
uary
1996
toA
pri
l2016.Panel
Bre
por
tsth
enu
mb
erof
mon
ths
for
wh
ich
anal
tern
ativ
eva
riab
lean
dPWI 1
give
diff
eren
tla
bel
sof
over
wei
ghti
ng/u
nd
erw
eighti
ng.
For
Q01,
Minave
and
MinSP
,an
over
wei
ghti
ng
per
iod
ison
ein
wh
ich
the
valu
eis
smal
ler
than
the
med
ian
of
all
per
iod
s.F
or
the
oth
erin
dic
es,
anov
erw
eigh
tin
gp
erio
dis
one
inw
hic
hth
eva
lue
isla
rger
than
the
med
ian
ofal
lp
erio
ds.
Panel
A:
Corr
elati
ons
am
ong
vari
able
s
PWI 1
VIX
Minave
Maxave
Q01
Q99
MinSP
MaxSP
σMKT
EM
VF
SI
Cra
shB
ear
Bull
VIX
0.8
15
Minave
-0.7
15
-0.7
08
Maxave
0.7
29
0.7
26
-0.9
27
Q01
-0.4
02
-0.4
17
0.7
92
-0.5
61
Q99
0.3
38
0.3
75
-0.2
53
0.5
54
0.2
60
MinSP
-0.5
29
-0.6
01
0.7
52
-0.5
90
0.6
46
0.0
07
MaxSP
0.6
09
0.7
15
-0.7
50
0.7
15
-0.5
36
0.1
60
-0.7
38
σMKT
0.5
81
0.7
30
-0.8
18
0.7
01
-0.6
86
0.0
56
-0.8
93
0.9
04
EM
V0.5
89
0.6
11
-0.7
89
0.6
76
-0.6
52
0.0
33
-0.7
67
0.7
71
0.8
29
FSI
0.6
35
0.6
14
-0.7
56
0.7
05
-0.5
62
0.1
94
-0.6
60
0.6
34
0.6
79
0.7
04
Cra
sh0.5
16
0.5
86
-0.3
85
0.3
63
-0.2
64
0.1
43
-0.4
54
0.4
58
0.4
87
0.3
92
0.1
67
Bea
r0.2
17
0.2
83
-0.1
78
0.0
35
-0.2
85
-0.2
41
-0.4
12
0.3
82
0.4
73
0.3
49
0.2
12
0.4
28
Bull
-0.1
60
-0.1
27
0.0
94
0.0
76
0.2
75
0.3
61
0.2
45
-0.1
83
-0.2
44
-0.2
05
-0.0
30
-0.3
51
-0.6
37
Sen
tim
ent
0.0
39
-0.1
26
-0.2
54
0.2
26
-0.2
70
0.0
63
0.0
35
-0.0
31
-0.0
38
0.1
01
0.1
07
-0.2
10
-0.1
89
0.0
87
Panel
B:
Num
ber
of
diff
eren
tly
lab
eled
month
s
VIX
Minave
Maxave
Q01
Q99
MinSP
MaxSP
σMKT
EM
VF
SI
Cra
shB
ear
Bull
Sen
tim
ent
Num
ber
38
34
34
79
90
72
68
76
50
58
59
113
132
115
Pro
port
ion
15.6
%13.9
%13.9
%32.4
%36.9
%29.5
%27.9
%31.1
%20.5
%23.8
%24.2
%46.3
%54.1
%47.1
%
49
Figure 1. Prelec’s probability weighting functions with different α’s. When α < 1, it has an inverseS-shape (i.e., overweighting both tails). When α > 1, it has an S-shape (i.e., underweighting bothtails). The point p∗ = e−1 separates the convex and concave domains of the probability weightingfunctions.
p
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
w(p;a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Overweighting (α = 0.5)No probability weighting (α = 1)Underweighting (α = 1.5)
50
Figure 2. The pricing kernels and implied relative risk aversions for CRRA and RDUT, underboth market scenarios. The CRRA has a relative risk aversion γ = 2 and the RDUT has Prelec’sprobability weighting functions. The dot black lines are those of the CRRA model, the solid redlines represent the RDUT model with overweighting the tails, and the dashed blue lines show theRDUT model with underweighting the tails. For Market I, the market return follows a skew-normaldistribution with mean 7.6%, standard deviation 15.8%, and skewness -0.339. For Market II, themarket return follows a skew-normal distribution with mean -7.6%, standard deviation 15.8%, andskewness -0.339. In (a), (b), α = 0.5 for overweighting and α = 1.5 for underweighting.
RM
0.7 0.8 0.9 1 1.1 1.2 1.3
Pricingkernel
m
0
1
2
3
4
5
6
7
8
Overweighting (α=0.5)
CRRA (α=1)
Underweighting (α=1.5)
(a) Pricing kernel (Market I)
α
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Implied
risk
aversionatinitialwealth
-8
-6
-4
-2
0
2
4
6
8
Overweighting
Underweighting
CRRA
(b) Implied risk aversion at RM = 1 (Market I)
RM
0.6 0.7 0.8 0.9 1 1.1
Pricingkernel
0
1
2
3
4
5
6
7
Overweighting (α = 0.5)CRRA (α = 1)Underweighting (α = 1.5)
(c) Pricing kernel (Market II)
α
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Implied
risk
aversionatinitialwealth
-5
0
5
10
15
20
25
Overweighting
Underweighting
CRRA
2
(d) Implied risk aversion at RM = 1 (Market II)
51
Figure 3. Relationship between A and α. The probability weighting function is that of Prelec withparameter α and the utility is CRRA with relative risk aversion γ = 2. The market return followsa skew-normal distribution with mean 7.6%, standard deviation 15.8%, and skewness -0.339.
α
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
A
-4
-3
-2
-1
0
1
2
3
4
Overweighting
Underweighting
α∗
52
Figure 4. Relationship between B and γ. The probability weighting function is that of Prelecwith α = 0.5 or α = 1.5 and the utility is CRRA with relative risk aversion γ. The market returnfollows a skew-normal distribution with mean 7.6%, standard deviation 15.8%, and skewness -0.339.
γ
0 5 10 15 20 25 30
B
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
(a) Overweighting Case α = 0.5
γ
0 5 10 15 20 25 30 35 40 45
B
-50
-40
-30
-20
-10
0
10
20
(b) Underweighting Case α = 1.5
53
Figure 5. Two typical option-implied probability weighting functions from our empirical study.The weighting function on September 30, 2008 is inverse S-shaped, implying overweighting of bothtails. Among the 244 months tested, there are 126 months (about 51.6%) in which the impliedweighting functions are inverse S-shaped. The weighting function on December 30, 2006 is S-shaped,yielding underweighting of both tails. Among the 244 months, there are 92 months (about 37.7%)in which the implied weighting functions are S-shaped.
probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Dis
tort
ed
pro
ba
bili
ty
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a) 2008-09-30
probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Dis
tort
ed p
robabili
ty
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) 2006-12-30
54
Figure 6. Time series of implied fear index(IFI), implied hope index(IHI), probability weightingindices PWI1 and PWI2. The dashed black line in Panel A separates overweighting and under-weighting. The dashed blue lines in Panel B and Panel C are the medians of the indices, whichalso separate overweighting and underweighting.
time1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016
Imp
lied
fe
ar
(Im
plie
d h
op
e)
-1
0
1
2
3
4
5
6
Implied hope
Implied fear
(a) Time series of IHI and IFI
time
1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016
PW
1
-1
0
1
2
3
4
5
6
(b) Time series of PWI1
time
1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016
PW
2
0
0.5
1
1.5
2
2.5
3
(c) Time series of PWI2
55
Figure 7. The relationship between the average next month’s returns and the average covariance(coskewness) of the ten covariance-sorted portfolios (ten coskewness-sorted portfolios).
(a) Covariance-sorted portfolios
(b) Coskewness-sorted portfolios
56
Figure
8.
Plo
tsof
pro
bab
ilit
yw
eigh
tin
gin
dex
(PWI 1
),V
IXin
dex
,E
qu
ity
Mar
ket
Vol
atil
ity
(EM
V)
ind
ex,
an
nu
ali
zed
vola
tili
tyof
Fam
a-F
ren
chm
arket
fact
or,
max
imu
md
aily
retu
rnof
S&
P50
0,m
inim
um
dai
lyre
turn
ofS
&P
500,
crash
bel
ief
index
,an
dse
nti
men
tin
dex
from
1996
to20
16.
Th
eva
lues
are
com
pu
ted
sem
ian
nu
ally
from
the
orig
inal
mon
thly
dat
a.P
rob
ab
ilit
yw
eighti
ng,
crash
bel
ief
an
dse
nti
men
tin
dic
esar
ep
rop
erly
scal
edor
adju
sted
inor
der
tofi
tin
toth
esa
me
figu
re.
Mor
esp
ecifi
call
y,p
rob
ab
ilit
yw
eighti
ng
ind
exis
scal
edto
hav
eth
esa
me
stan
dar
dd
evia
tion
asV
IX,
senti
men
tin
dex
issc
aled
toh
ave
hal
fst
and
ard
dev
iati
on
of
VIX
,an
dcr
ash
bel
ief
ind
exis
adju
sted
tob
eth
eor
igin
alcr
ash
bel
ief
ind
exm
inu
s55
.
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
-100
10
20
30
40
50
60
an
nu
al vo
latilit
y o
f F
F m
ark
et
facto
rm
axim
um
da
ily r
etu
rn o
f S
&P
50
0m
inim
um
da
ily r
etu
rn o
f S
&P
50
0
pro
ba
bili
ty w
eig
htin
g in
de
xcra
sh
be
lief
se
ntim
en
tV
IXe
qu
ity m
ark
et
vo
latilli
ty (
EM
V)
57
Figure
9.
Ove
rwei
ghti
ng
per
iod
san
du
nd
erw
eigh
tin
gp
erio
ds
gen
erat
edby
pro
bab
ilit
yw
eigh
tin
gin
dex
(PWI 1
),V
IXin
dex
(VIX
),E
qu
ity
Mar
ket
Vol
atil
ity
ind
ex(E
MV
),av
erag
em
axim
um
dai
lyre
turn
ofal
lst
ock
s(M
axave),
cras
hb
elie
fin
dex
(Cra
sh)
from
1996
to20
16.
For
each
ind
ex,
the
hig
her
par
tof
the
lin
eco
rres
pon
ds
toov
erw
eigh
ting
per
iod
san
dth
elo
wer
part
of
the
lin
eco
rres
pon
ds
tou
nd
erw
eigh
tin
gp
erio
ds.
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
EM
VV
IXP
WI 1
Max
ave
Cra
sh
58
