Page 1 of 11
Archives of Business Research – Vol. 12, No. 7
Publication Date: July 25, 2024
DOI:10.14738/abr.127.17331.
Bouaddi, B. (2024). Market Extreme Moves and the Industries' Probability of Crash and Jump. Archives of Business Research, 12(7).
78-88.
Services for Science and Education – United Kingdom
Market Extreme Moves and the Industries' Probability of Crash
and Jump
Mohammed Bouaddi
American University in Cairo, Department of Economics,
AUC Avenue, P.O. Box 74, New Cairo 11835, Egypt
ABSTRACT
We investigate the dependence of stock returns in the extreme. We use a flexible
model for the probability of extreme moves in bad and good times. We then
estimate the conditional probability of an industry to crash when the market
crashed and the probability of the industry to jump when the market jumps. The
results show that the dependence of industries' returns to the market returns is
asymmetric and significant in the extreme. In addition, it shows that; the probability
of an industry to crash is at least seven times higher than the probability to jump.
The results also highlight that the probability of an industry the crash or jump is
higher in non-recession periods than in recession periods due to surprise effect.
That is, the probabilities of crash and jump are procyclical.
Keywords: Copula, Tail dependence, Probability of crash, Probability of jump, asymmetric
extreme dependence.
INTRODUCTION
Correlations are higher than normal correlations when both the equity selected portfolios and
the market are on the downside while these correlations cannot be distinguished from normal
correlations when both the equity portfolio and the market are on the upside [3]. Similar results
highlighting asymmetries in correlations can be found in [7,8,15,17,18,10, and 26] among
others.
While the existing literature focuses more on linear correlations that captures the average
linear dependence our approach focuses on the nonlinear dynamics of the conditional
probability of individual returns to crash when the market crushes and of the probability of
individual returns to jump when the market jumps. Our approach complements the existing
literature in two folds. First, in the existing literature the approach adopted was backward
looking and linear (linear correlation) while our approach is forward looking. That is, the linear
correlation is a backward-looking assessment of the relationship, the conditional probability is
forward looking by nature since is provide the likelihood of an event to happen. Second, given
that significant linear correlation does not imply tail dependence and vice-versa, analyzing the
dependence in the tails is an important improvement of the literature. Actually, most of
distributions capture an average dependence between the key variables nevertheless some of
them exhibit tail dependence and others don’t.1
1
See Joe (1997) for an extensive literature on the dependence concept.
Page 2 of 11
79
Bouaddi, B. (2024). Market Extreme Moves and the Industries' Probability of Crash and Jump. Archives of Business Research, 12(7). 78-88.
URL: http://doi.org/10.14738/abr.127.17331
We use asymmetric copula and asymmetric marginal distributions to model the probabilities
of jump and crash of a given industry. More precisely, we define and estimate these
probabilities when both the market and the industry returns are below a given threshold (crash
probability) and when both the market and the industry returns are above a given threshold
(jump probability).
We used a flexible model for the joint distribution of returns instead of assuming multivariate
normality. That is, our approach enables us to disentangle the marginal dynamics from
dependence by modeling separately the marginal distributions and using an asymmetric copula
to capture the non-linear dependence. Second, we estimate the probabilities in the tail of the
distribution.
Our results show that the probability of the industry to crash when the market crushed is higher
than the probability to jump when the market is booming. This finding in the extreme behavior
of returns generalize the results on linear correlation as shown by [3,6,9,10,13, and 32] among
others.
We find that the extreme tail dependence between stocks and the market returns are much
higher for extreme downside moves than for upside moves. In addition to the existing
literature, we found that the probability of extreme crash is higher in a non-crisis period than
during a crisis-period. We argue that this additional asymmetry is due to the surprise effect
because in good times investors don't expect the market to experience a strong crash and
overreact by exercising heavy short selling as a result of global panic. This is due to the fact that
the fire sales produce negative (extreme) returns across stocks in various industries [1, 2, 6, 10,
11, 16, 22,24, 28, 29, 31, 34 and 35).
The rest of the paper is organized as follows. In Section 2, we present the full specification of
our methodology. Section 3 is allocated to the presentation of results. Section 4 is dedicated to
the discussion of our empirical findings. Section 5 concludes.
SYSTEMATIC PROBABILITY OF CRASH AND JUMP
The probability of the crash of the industry, i.e. the probability of industry returns to be below
the market value at risk (VaR) given that the market is below its VaR, is given by
for 0<α<0.5. The probability of the jump of the return of the industry, i.e. the probability of
industry returns to be higher than the (1-α) level market VaR given that the market is higher
than its VaR is given by
Pi,t+1
Jump = Pr(ri,t+1 > VaRm,t
α
|rm,t+1 > VaRm,t
α
) =
Pr(ri,t+1>VaRm,t
α ,rm,t+1>VaRm,t
α )
Pr(rm,t+1>VaRm,t
α )
(2)
for 0.5<α <1.
Page 3 of 11
80
Archives of Business Research (ABR) Vol. 12, Issue 7, July-2024
Services for Science and Education – United Kingdom
It is easy to show that, in terms of copula, we have
Pi,t+1
Crash =
C(Fi,t+1(VaRm,t)
α ,α)
α
for 0 < α < 0.5 (3)
And
Pi,t+1
Jump =
C(Fi,t+1(VaRm,t)
α ,α)
α
for 0.5 < α < 1 (4)
where Fi,t(.) is the marginal distribution of industry's i return.
In the two subsections below, we will specify the dynamics of the dependence model and its
marginals.
Dependence Modelling
To model the dependence between the industry and the market returns, we use the BB7 copula
[5]
C(u, v) = 1 − (1 − [(1 − u̅
θ
)
−δ + (1 − v̅
θ
)
−δ − 1]
−
1
δ)
1
θ
(5)
where u̅ = 1 − u, v̅= 1 − v, θ ≥ 1 and δ > 0. This copula nests other copulas when we restrict
its parameters. Actually, when θ = 1 we get copula family B4 Joe (1997), as δ → 0 we get family
B5 [5] and, as δ → +∞ and θ → +∞ we obtain independence between u and v. Also, family B7
and family B8 [5] obtain as extreme value limits from lower and upper tails.2 The lower tail
dependence of BB7 copula is λL = 2
−
1
δ independent of θ and the upper tail dependence is λL =
2 − 2
1
θ is independent of δ. Therefore, this copula is very flexible to capture different tail
dependence behavior (extreme dependence) which is of interest in our case.
Marginal Modelling
We use skewed t-distribution to model the marginal distribution of returns of the industries
and the market [14]. The probability distribution function has the following form:
f(rt+1) =
{
bc
σt
(1 +
1
λ−2
(
b(rt+1−μt
)+aσt
(1−τ)σt
)
2
)
−
λ+1
2
if rt+1 < μt −
a
b
σt
bc
σt
(1 +
1
λ−2
(
b(rt+1−μt
)+aσt
(1+τ)σt
)
2
)
−
λ+1
2
Otherwise
(6)
Where r is the return of the industry, λ is the shape parameter 2 < λ < ∞, the parameter τ
governs the asymmetry of the distribution (1 < τ < 1), μt and σt are the location and scale
2
For more details about these families of copula see Joe (1997) on page 153.
Page 5 of 11
82
Archives of Business Research (ABR) Vol. 12, Issue 7, July-2024
Services for Science and Education – United Kingdom
Where α, α, η and β capture the news effect, the leverage effect and the volatility memory
(persistence) respectively. If η < 0, the volatility reacts more to bad news relative to good
news. We estimate the model using Inference Functions for Margins (hereafter IFM). [23] show
that asymptotic normality of IFM estimator hold even when both margins and copula are mis- specified.5
RESULTS
In this section, we analyze the impact of the market conditions on the performance of
industries' returns. The data used in this section are collected from Kenneth R. French Data
Library.6 The data cover the period from July 1963 to December 2022. It consists of Fama- French five industries’ portfolios and market returns. The industries' returns and market
returns are constructed using value-weighted returns of all CRSP US firms listed on the NYSE,
AMEX, and NASDAQ. The five industries are Consumer, Manufacturing, HiTec, Healthcare and
Others (Mines, Construction, building material, Transportation, Hotels, Bus Services,
Entertainment, Finance).7 We use the NBER based recession indicators for the United States to
determine the recession dates.
Table 1: Descriptive statistics
Market Consumer Manufacturing HiTec Healthcare Other
Mean 0.9068 0.9816 0.9241 0.9459 1.0558 0.9328
Median 1.265 1.13 1.205 1.28 1.115 1.345
Maximum 16.61 21.77 17.2 19.89 29.52 20.24
Minimum -22.64 -24.99 -20.91 -22.58 -20.45 -23.6
Std. Dev. 4.4867 4.5901 4.409 5.4414 4.8037 5.3214
Skewness -0.4862 -0.2724 -0.4833 -0.3701 0.0267 -0.4728
Kurtosis 4.7488 5.412 5.3314 4.5834 5.3074 4.8179
Note: Descriptive statistics for the market index and the five industries.
Table 2: Marginal models’ estimation
Market Consumer Manufacturing HiTec Healthcare Other
γ0 1.6971*** 1.6959*** 1.5870*** 1.8234*** 1.9395*** 1.8532***
γ1 -0.7487* -0.7503*** -0.5411 -0.9598*** -0.7277*** -0.7952***
γ2 0.7671* 0.7998*** 0.4337 0.9966*** 0.7292*** 0.8633***
σ 0.1357 0.1237 0.2313* -0.0125 2.1502*** 0.2209*
α 0.2247*** 0.2483*** 0.2294*** 0.2881*** 0.0443 0.1739**
η -0.1800*** -0.1189*** -0.1922*** -0.1057*** -0.3240*** -0.1940***
β 0.8905*** 0.8935*** 0.8555*** 0.9327*** 0.2851* 0.8863***
λ 9.2187*** 7.3743*** 6.7706 14.5132*** 9.9652*** 10.1422***
τ -0.2099*** -0.1121 -0.1937** -0.1349* -0.1004 -0.1638*
5 We refer the reader to the following papers dealing with the properties of IFM estimator: Joe and Xu (1996), Shih
and Louis (1995), Louzada and Ferreira (2016) and Ko and Hjort (2019), Baillien et. al. (2022) among others.
6 The data is publicly available from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
7 The industries description as well as their four-digit SIC codes can be found in Appendix A and in more details at the
Kenneth R. French - Data Library.
Page 10 of 11
87
Bouaddi, B. (2024). Market Extreme Moves and the Industries' Probability of Crash and Jump. Archives of Business Research, 12(7). 78-88.
URL: http://doi.org/10.14738/abr.127.17331
[10] Coval, J., and E. Stafford, 2007. Asset fire sales (and purchases) in equity markets. Journal of Financial
Economics 86, 479-512.
[11] Diamond, D. and R. Rajan, 2010. Fear of Fire Sales and the Credit Freeze. Bank for International
Settlements, Working Papers, No 305.
[12] Fama, E. F., 1965. The behavior of stock-market prices. Journal of Business 38, 34-105.
[13] French, K. R., G.W. Schwert, and R. F. Stambaugh, 1987. Expected stock returns and volatility, Journal of
Financial Economics 19, 3-29.
[14] Hansen, B.E., 1994. Autoregressive conditional density estimation. International Economic Revue 35, 705-
730.
[15] Hong, Y., J. Tu, and G. Zhou, 2007. Asymmetries in stock returns: Statistical tests and economic evaluation.
Review of Financial Studies 20, 1547-81.
[16] Ivashina, V., and D. Scharfstein, 2010. Bank Lending during the Financial Crisis of 2008. Journal of
Financial Economics 97, 319-338.
[17] Jiang, L., W. Ke, and G. Zhou, 2018a. Asymmetry in stock comovements: An Entropy Approach. Journal of
Financial and Quantitative Analysis 53, 1479-507.
[18] Jiang, Lei, Esfandiar Maasoumi, Jiening Pan, and K. Wu, 2018b. A test of general asymmetric dependence.
Journal of Applied Econometrics 33, 1026-43.
[19] Jondeau, E, 2016. Asymmetry in tail dependence in equity portfolios. Computational Statistics and Data
Analysis 100, 351-68.
[20] Joe, H., 1997, Multivariate Models and Dependence Concepts. Monograph in Statistics and Probability 73,
London: Chapman and Hall.
[21] Joe, H., and J. J. Xu, 1996. The estimation method of inference functions for margins for multivariate
models. Technical Report 166, Department of Statistics, University of British Columbia. doi:
http://dx.doi.org/10.14288/1.0225985
[22] Kang, W., A. Hameed, and S. Viswanathan, 2010. Stock Market Declines and Liquidity. Journal of Finance
65, 257-293.
[23] Ko, V., and N.L. Hjort, 2019. Model robust inference with two-stage maximum likelihood estimation for
copulas. Journal of Multivariate Analysis 171, 362-381.
[24] Krishnamurthy, A. 2009. Amplification Mechanisms in Liquidity Crises. American Economic Journal 2,1-33.
[25] Kyle, A. S., and W. Xiong, 2001. Contagion as a wealth effect. Journal of Finance 56, 1401-1440.
[26] Longin, F., and B. Solnik, 2001. Extreme correlation of international equity markets. Journal of Finance 56,
649-676.
[27] Louzada, F., and P. H. Ferreira, 2016. Modified inference function for margins for the bivariate clayton
copula-based SUN tobit model. Journal of Applied Statistics 43, 2956-2976.
[28] Mirza, H., D. Moccero, S. Palligkinis, and C. Pancaro, 2020. Fire sales by euro area banks and funds: What is
their asset price impact? Economic Modelling 93, 430-444.