Tail cmvt slides

Category - Based Tail Comovement
Christophe Villa
AUDENCIA - Nantes and CREST
February, 25 2010
Christophe Villa () EM Lyon February, 25 2010 1 / 22

Co-authors
Arthur Charpentier
University of Rennes and Ecole Polytechnique
Emilios Galariotis
Audencia Nantes School of Management

Comovement
there are numerous patterns of comovement in the data

Comovement
common factors in the returns of certain groups of assets

Comovement
stocks within the same industry

Comovement
small stocks

Comovement
small stocks
value stocks

Comovement
small stocks
value stocks
closed-end funds

Comovement
small stocks
value stocks
closed-end funds
what is the source of this comovement ?

Comovement
small stocks
value stocks
closed-end funds
what is the source of this comovement ?
why do certain assets comove while others do not ?

Traditional view : Fundamentals - based comovement
derived from economies without frictions and with rational investors

assets comove because their "fundamental values" comove

fundamental value = rational forecast of future cash ‡ows discounted
at rate appropriate for risk

in this view, assets comove because of :

correlated news about their cash ‡ows

correlated changes in their discount rates

changes in interest rates

changes in risk aversion

correlated changes in rational perception of risk

correlated changes in rational perception of risk
useful framework for understanding many types of comovement

Evidence on comovement
Twin stocks (Froot/Dabora [1999])

claims to same cash-‡ow stream, but traded in di¤erent locations (eg
Royal Dutch / Shell)

Royal Dutch, traded primarily in New York, is a claim to 60% of the
cash ‡ow

cash ‡ow
Shell, traded primarily in London, is a claim to the remaining 40%

cash ‡ow
under traditional view of comovement, expect them to move in lockstep

cash ‡ow
under traditional view of comovement, expect them to move in lockstep
in fact, Royal Dutch comoves more with the U.S. stock market, Shell
with the U.K. market
rRD,t rSH,t = α + 0.207 rS&P,t 0.428 rFTSE ,t + εt

Evidence on comovement ctd.
Closed-end country funds (Hardouvelis et al. [1994], Bodurtha et al.
[1995])

[1995])
funds traded in one location, fund assets in another

[1995])
under traditional view of comovement, expect fund returns and NAV
returns to move together closely

[1995])
in fact, fund returns comove as much with market where fund is traded
as with market where assets are traded

[1995])
Domestic closed-end funds (Lee et al. [1991])

[1995])
Domestic closed-end funds (Lee et al. [1991])
closed-end funds invested in large stocks often comove with small
stocks

Small stocks and value stocks (Fama/French [1995])

there is a strong common factor in returns of small stocks and in
returns of value stocks (Fama/French [1993])

Fama/French [1995] test whether these factors are due to cash-‡ow
news

news
do …nd cash-‡ow factors but they line up poorly with the return factors

news
Commodities (Pindyck/Rotemberg [1990])

news
…nd strong comovement in price changes of seven commodities

news
…nd strong comovement in price changes of seven commodities
hard to explain comovement through news about aggregate demand

Category-based comovement
Barberis/Shleifer [2003]

many investors allocate funds at the level of an asset category

simpli…es the portfolio problem

makes it easier to evaluate money managers

categories are based on a salient common characteristic

impressive past performance often spurs category formation

categories can be identi…ed by looking at labels on money managers’
products

categories can be identi…ed by looking at labels on money managers’
products
e.g. small-cap, large-cap, growth, index

Category-based comovement ctd.
suppose categories are adopted by noise traders with correlated
sentiment

sentiment
if the noise traders a¤ect prices
) assets will comove simply because they are classi…ed into the same
category

sentiment
if the noise traders a¤ect prices
) assets will comove simply because they are classi…ed into the same
category
even if fundamentals are uncorrelated

Consider an simple economy with a riskless asset and 2n risky assets

risky asset i is a claim to a single liquidating dividend Di,T to be paid
at some later time T:
Di,T = Di,0 + ei,1 + ... + ei,T
where
et = (e1,t , ..., e2n,t )0
~ N (0, ΣD ) , i.i.d. over time

Di,T = Di,0 + ei,1 + ... + ei,T
where
et = (e1,t , ..., e2n,t )0
Denote the price of asset i at time t as Pi,t and the return as
∆Pi,t = Pi,t Pi,t 1

Di,T = Di,0 + ei,1 + ... + ei,T
where
et = (e1,t , ..., e2n,t )0
Some investors group the risky assets into two categories, X and Y

Di,T = Di,0 + ei,1 + ... + ei,T
where
et = (e1,t , ..., e2n,t )0
Some investors group the risky assets into two categories, X and Y
suppose assets 1 through n in category X, assets n + 1 through 2n in Y

assume the following cash-‡ow structure:

for an asset i 2 X
ei,t = ψM fM,t + ψS fX ,t +
q
1 ψ2
S ψ2
M εi,t

for an asset i 2 X
q
1 ψ2
S ψ2
M εi,t
for an asset j 2 Y
ej,t = ψM fM,t + ψS fY ,t +
q
1 ψ2
S ψ2
M εj,t

for an asset i 2 X
q
1 ψ2
S ψ2
M εi,t
for an asset j 2 Y
q
1 ψ2
S ψ2
M εj,t
fM,t is the market-wide shock

for an asset i 2 X
q
1 ψ2
S ψ2
M εi,t
for an asset j 2 Y
q
1 ψ2
S ψ2
M εj,t
fX ,t and fY ,t are the group-speci…c shocks

for an asset i 2 X
q
1 ψ2
S ψ2
M εi,t
for an asset j 2 Y
q
1 ψ2
S ψ2
M εj,t
εi,t and εj,t are idiosyncratic shocks

for an asset i 2 X
q
1 ψ2
S ψ2
M εi,t
for an asset j 2 Y
q
1 ψ2
S ψ2
M εj,t
ψM and ψS are constants that control the relative importance of the
three components

for an asset i 2 X
q
1 ψ2
S ψ2
M εi,t
for an asset j 2 Y
q
1 ψ2
S ψ2
M εj,t
ψM and ψS are constants that control the relative importance of the
three components
Each shock has unit variance and is orthogonal to the other shocks

Suppose that

Suppose that
category-based investors are noise traders, who channel funds in and
out of the categories depending on their sentiment

Suppose that
the economy also contains "fundamental traders" who invest at the
level of individual assets and act as arbitrageurs

Suppose that
A simple representation for asset returns is then
∆Pi,t = ei,t + ∆uX ,t , for i = 1, ..., n
∆Pi,t = ei,t + ∆uY ,t , for i = n + 1, ..., 2n
where
uX ,t
uY ,t
N
0
0
, σ2
u
1 ρu
ρu 1
i.i.d. over time

Suppose that
A simple representation for asset returns is then
∆Pi,t = ei,t + ∆uX ,t , for i = 1, ..., n
∆Pi,t = ei,t + ∆uY ,t , for i = n + 1, ..., 2n
where
uX ,t
uY ,t
N
0
0
, σ2
u
1 ρu
ρu 1
i.i.d. over time
uX ,t (uY ,t ) can be thought of as time t noise trader sentiment about
the securities in category X (Y )

so long as arbitrage is limited in some way, two assets in same
category comove not only because of correlated cash-‡ow news, but
because of a correlated sentiment shock

assets in the same category comove too much

assets in di¤erent categories comove too little

and reclassifying an asset into a new style raises its correlation with
that category

that category
category-based view suggests that there is excess comovement in
stock prices, with stocks in the same category moving together more
than their intrinsic values say that they should.

that category
category-based view suggests that there is excess comovement in
stock prices, with stocks in the same category moving together more
than their intrinsic values say that they should.
comovement = correlation

Evidence on extreme comovement
Mr. A. Greenspan: "work that characterizes the distribution of
extreme events would be useful as well."

Mr. A. Greenspan: "work that characterizes the distribution of
extreme events would be useful as well."
During the unfolding …nancial crisis stocks have experienced both
extreme movement and extreme comovement in returns.
) Category-based tail comovement

No simple analogue of the correlation coe¢ cient for extreme
dependencies ) various statistical measures.

Poon, Rockinger and Tawn (2004) derive a general multivariate
framework with two types of extreme dependence structures that
allow for both asymptotic dependence and independence.

Poon, Rockinger and Tawn (2004) derive a general multivariate
framework with two types of extreme dependence structures that
allow for both asymptotic dependence and independence.
four types of dependence: perfect dependent, independent,
asymptotically dependent and asymptotically independent

Quantile dependence
Quantile dependence is a measure of the dependence in the tails of
the distribution and describes the limiting proportion that

Quantile dependence
one margin exceeds a certain threshold

Quantile dependence
given that the other margin has already exceeded that threshold.

Quantile dependence
given that the other margin has already exceeded that threshold.
If Z1 and Z2 are random variables with distribution functions F1 and
F2,
) there is quantile dependence in the lower tail at threshold α,
whenever
P Z2 F 1
2 (α) j Z1 F 1
1 (α) 6= 0
) there is quantile dependence in the upper tail at threshold α,
whenever
P Z2 F 1
2 (α) j Z1 F 1
1 (α) 6= 0

Asymptotic dependence
Upper tail dependence index between Z1 and Z2 is de…ned as
λU = lim
α!1
P(Z1 F 1
1 (α) j Z2 F 1
2 (α))
= lim
α!1
P(Z2 F 1
2 (α) j Z1 F 1
1 (α)),

λU = lim
α!1
P(Z1 F 1
1 (α) j Z2 F 1
2 (α))
= lim
α!1
P(Z2 F 1
2 (α) j Z1 F 1
1 (α)),
Lower tail dependence index is
λL = lim
α!0
P(Z1 F 1
1 (α) j Z2 F 1
2 (α))
= lim
α!0
P(Z2 F 1
2 (α) j Z1 F 1
1 (α)).

λU = lim
α!1
P(Z1 F 1
1 (α) j Z2 F 1
2 (α))
= lim
α!1
P(Z2 F 1
2 (α) j Z1 F 1
1 (α)),
Lower tail dependence index is
λL = lim
α!0
P(Z1 F 1
1 (α) j Z2 F 1
2 (α))
= lim
α!0
P(Z2 F 1
2 (α) j Z1 F 1
1 (α)).
0 λ 1

asymptotically dependent if λ > 0
) using multivariate EVT implicitly assume asymptotic dependence

asymptotically independent if λ = 0
) a range of extremal dependence models describe dependence but
have λ = 0

asymptotically independent if λ = 0
) a range of extremal dependence models describe dependence but
have λ = 0
perfectly dependent if λ = 1

Asymptotic independence
Although the random variables are asymptotically independent,
di¤erent degrees of dependence are attainable at …nite levels of α.

To this end, PRT introduce another (weak) tail dependence index

upper tail dependence index between Z1 and Z2 is de…ned as
ηU = lim
α!1
log(1 α)
log P(Z1 > F 1
1 (α), Z2 > F 1
2 (α))
,

ηU = lim
α!1
log(1 α)
log P(Z1 > F 1
1 (α), Z2 > F 1
2 (α))
,
lower tail dependence index is
ηL = lim
α!0
log(α)
log P(Z1 F 1
1 (α), Z2 F 1
2 (α))
.

ηU = lim
α!1
log(1 α)
log P(Z1 > F 1
1 (α), Z2 > F 1
2 (α))
,
ηL = lim
α!0
log(α)
log P(Z1 F 1
1 (α), Z2 F 1
2 (α))
.
η > 0, η = 1
2 and η < 1 loosely correspond respectively to when Z1
and Z2 are positively associated in the extremes, exactly independent,
and negatively associated.

ηU = lim
α!1
log(1 α)
log P(Z1 > F 1
1 (α), Z2 > F 1
2 (α))
,
ηL = lim
α!0
log(α)
log P(Z1 F 1
1 (α), Z2 F 1
2 (α))
.
η > 0, η = 1
2 and η < 1 loosely correspond respectively to when Z1
and Z2 are positively associated in the extremes, exactly independent,
and negatively associated.
λ = 0 and η 2 (0, 1) signi…es asymptotic independence, in which case
the value of η determines the strength of dependence within this class
(also known as dependence in independence).

Results
The Category-based comovement is also a Category-based
(weak) tail comovement

Results
Under the category-based comovement model, for two assets i and j
we have λ ∆Pi , ∆Pj = λ ei , ej = 0

Results
if two assets i and j, i 6= j, belong to the same category, i, j 2 X or Y ,
then
η ∆Pi,t , ∆Pj,t > η ei,t , ej,t

Results
if two assets i and j, i 6= j, belong to the same category, i, j 2 X or Y ,
then
η ∆Pi,t , ∆Pj,t > η ei,t , ej,t
suppose that asset j, previously a member of category Y , is reclassi…ed
as belonging to X. Then η ∆Pj,t , ∆PX ,t increases after j is added to
category X where ∆PX ,t = 1
n ∑
l2X
∆Pl,t .

Results ctd.
we assume that uX has heavy tails

Results ctd.
if two assets i and j, i 6= j, belong to the same category,
i, j 2 X or Y , then

Results ctd.
if uX has right-heavy tails
1 = λU ∆Pi,t , ∆Pj,t > λU ei,t , ej,t = 0

Results ctd.
if uX has right-heavy tails
1 = λU ∆Pi,t , ∆Pj,t > λU ei,t , ej,t = 0
if uX has left-heavy tails
1 = λL ∆Pi,t , ∆Pj,t > λL ei,t , ej,t = 0
We thus observe that even if cash-‡ow shocks are asympotically
independent, ∆Pi and ∆Pj are perfectly dependent.

Results ctd.

Results ctd.
Suppose that asset j, previously a member of category Y , is
reclassi…ed as belonging to X. Then

Results ctd.
if uX has right-heavy tails, λU ∆Pj,t , ∆PX ,t increases after j is
added to category X where ∆PX ,t = 1
n ∑
l2X
∆Pl,t .

Results ctd.
if uX has right-heavy tails, λU ∆Pj,t , ∆PX ,t increases after j is
added to category X where ∆PX ,t = 1
n ∑
l2X
∆Pl,t .
if uX has left-heavy tails, λL ∆Pj,t , ∆PX ,t increases after j is added
to category X where ∆PX ,t = 1
n ∑
l2X
∆Pl,t .

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