Transcript MEASURING THE VALUE OF DYNAMIC CORRELATIONS

DYNAMIC CONDITIONAL
CORRELATION MODELS
OF TAIL DEPENDENCE
Robert Engle
NYU Stern
DEPENDENCE MODELING FOR CREDIT PORTFOLIOS
Venice 2003
Two Frontiers
• We are celebrating over 20 years of
research in Volatility Modeling
• The simple GARCH(1,1) has
transformed our risk measurement
• And there are many many
extensions
• Multivariate Volatility
• High Frequency or Real Time
Volatility
MULTIVARIATE
• Multivariate GARCH has never been
widely used
• Asset allocation and risk
management problems require large
covariance matrices
• Credit Risk now also requires big
correlation matrices to accurately
model loss or default correlations
WHERE DO WE USE
CORRELATIONS?
• TWO CANONICAL PROBLEMS
– Forecasting risk and Forming optimal
portfolios
– Pricing derivatives on multiple
underlyings
– Credit Risk uses both of these tools
Joint Density
P2,T
P1,T
APPLICATIONS
•
Portfolio Value at Risk
–
•
joint empirical distribution
Option payoff
–
•
joint risk neutral distribution
Payoff of option that both assets are
below strikes
Probability of one or more Defaults
•
–
•
joint empirical distribution of firm value
Pricing Credit Derivatives
–
joint risk neutral distribution of firm value
P2,T
P1,T
P1,T < P1,0 -VaR
Probability
that the
portfolio
looses more
than K
P2,T
K1
Put Option
on asset 1
Pays
P1,T
Both options
Payoff
Option on
asset 2
Pays
K2
OPTIONS
• Value of each option depends only
on the marginal risk neutral
distribution
• Correlation between the payoffs
depends on the joint distribution.
• Optimal portfolios including options
• Value an option that pays only when
both are in the money.
CREDIT RISK
• Credit Risk correlation is like this
problem where K’s are default
points and prices are firm values
• Credit Default Swaps (CDS) are like
options (written puts) in firm value.
• Credit Default Obligations (CDO) in
lower tranches are minima of sums
of firm values.
Symmetric Tail Dependence
P2,T
P1,T
Lower Tail Dependence
P2,T
P1,T
P2,T
K1
Put Option
on asset 1
Pays
P1,T
Both options
Payoff
Option on
asset 2
Pays
K2
Joint Distribution
• Under joint log normality, these
probabilities can be calculated
• Under other distributions,
simulations are required
• Copulas are a new way to formulate
such joint density functions
• How to parameterize a Copula to
match this distribution?
JOINT DISTRIBUTIONS
• Dependence properties are all
summarized by a joint distribution
• For a vector of kx1 random
variables Y with cumulative
distribution function F
F  y1,..., yk   P Y1  y1,...,Yk  yk 
• Assuming for simplicity that it is
continuously differentiable, then the
density function is:
k F  y 
f  y1,..., yk  
y1...yk
UNIVARIATE
PROPERTIES
• For any joint distribution function F,
there are univariate distributions Fi
and densities fi defined by:
Fi  yi   P Yi  yi   F  ,.., , yi , ,...,  
Fi
fi  yi  
yi
•
is a uniform random
variable on the interval (0,1)
• What is the joint distribution of
Ui  Fi Yi 
U  U1,...,Uk 
COPULA
• The joint distribution of these
uniform random variables is called a
copula;
– it only depends on ranks and
– is invariant to monotonic
transformations.
U  U1,...,Uk  ~ C u1,.., uk 
• Equivalently F  y   C  F  y  ,..., F  y  
1
1
k
k
C  u   F  F11  u1  ,..., Fk1  uk  
COPULA DENSITY
• Again assuming continuous
differentiability, the copula density
is
kC u 
c u  
u1 ,...uk
• From the chain rule or change of
variable rule, the joint density is the
product of the copula density and
the marginal densities
f  y   c u  f1  y1  f2  y2  ... fk  yk 
Tail Dependence
Upper and lower tail dependence:

lim Pr  r


lim Pr r1,T  K1, r2,T  K 2,  U
 1
 0
1,T
 K1, r2,T  K 2,
L
Pr  r1,T  K1,   Pr  r2,T  K 2,   
• For a joint normal, these are
both zero!
DEFAULT CORRELATIONS
• Let Ii be the event that firm i
defaults,
• Then the default correlation is the
correlation between I1 and I2 which
can be computed conditional on
today’s information set.
• If the probability of default for each
firm is  , then:
Pr  I1 * I 2  1   2
1,2   
 1   
Default Correlations and
Tail Dependence
• When defaults are unlikely, these
are related to the tail dependence
measure
• Take the limit as  becomes small
L  
lim 1,2   
 0
1   
– Under normality or independence, the
limiting default correlation is zero
– Under lower tail dependence it is
positive.
Asset Allocation
• Optimal portfolios will be affected
by such asymmetries.
• The diversification is not as great in
a down market as it is in an up
market. Thus the risk is greater
than implied by an elliptical
distribution with the same
correlation.
• The optimal portfolio here would
probably hold more of the riskless
asset.
DYNAMIC
CORRELATIONS
• A joint distribution can be defined for any
horizon. Long horizon distributions can
be built up from short horizons
• Multivariate GARCH gives many possible
models for daily correlations. The implied
multi-period distribution will generally
show symmetric tail dependence
• Special asymmetric multivariate GARCH
models give greater lower tail
dependence.
TWO PERIOD RETURNS
• Two period return
is the sum of two
one period
continuously
compounded
returns
• Look at binomial
tree version
• Asymmetry gives
negative skewness
Low
variance
High
variance
Two period Joint Returns
• If returns are both
negative in the
first period, then
correlations are
higher.
• This leads to lower
tail dependence
Up Market
Down Market
Dynamic Conditional Correlation
• DCC model is a new type of multivariate
GARCH model that is particularly
convenient for big systems. See
Engle(2002) or Engle(2004)
• Motivation: the conditional correlation of
two returns with mean zero is:
t 
Et 1  r1,t r2,t 
Et 1  r
2
1,t
 E r 
t 1
2
2,t
DCC
• Then defining the conditional
variance and standardized residual
as
hi ,t  Et 1  ri 2,t  , and  i ,t  ri ,t / hi ,t
• All the volatilities cancel, giving
t 
Et 1 1,t 2,t 
Et 1 1,2t  Et 1  2,2 t 
 Et 1 1,t 2,t 
DCC
• The DCC method first estimates
volatilities for each asset and
computes the standardized
residuals.
• It then estimates the covariances
between these using a maximum
likelihood criterion and one of
several models for the correlations
• The correlation matrix is
guaranteed to be positive definite
GENERAL SPECIFICATION
h
•
rt past ~ N  0, Dt Rt Dt  , Dt  diag
•
Rt  diag  Qt 
•
Qt ~ linear vector time series process
•
L  .5 log Dt Rt Dt  rt '  Dt Rt Dt  rt 


1/ 2
Qt diag  Qt 
i ,t
1/ 2
1
DCC Example
• Let epsilon be an nx1 vector of
standardized residuals
• Then let
1
R    t t '
T
Qt  R 1        t 1 t 1 '  Qt 1
• And
Rt  diag Qt 
1/ 2
Qt diag Qt 
1/ 2
• The criterion to be maximized is:
L  .5 log Rt   t ' Rt1 t 
DCC Details
• This is a two step estimator
because the volatilities are
estimated rather than known
• It uses correlation targeting to
estimate the intercept.
• There are only two correlation
parameters to estimate by MLE no
matter how big the system.
• On average the correlations will be
the same as in the data.
Intuition and Asymmetry
• More specifically:
q1,2,t  1,2   1,t 1 2,t 1  1,2     q1,2,t 1  1,2 
• So that correlations rise when
returns move together and fall when
they move opposite.
• By adding another term we can
allow them to rise more when both
returns are falling than when they
are both rising.
DCC and the Copula
• A symmetric DCC model gives
higher tail dependence for both
upper and lower tails of the multiperiod joint density.
• An asymmetric DCC gives higher
tail dependence in the lower tail of
the multi-period density.
• q1,2,t  1,2   1,t 1 2,t 1  1,2     q1,2,t 1  1,2  
 1,t 1 2,t 1d1,2,t 1  1,2 
REFERENCES
• Engle, 2002, Dynamic Conditional Correlation-A
Simple Class of Multivariate GARCH Models,
Journal of Business and Economic Statistics
– Bivariate examples and Monte Carlo
• Engle and Sheppard, 2002 “Theoretical and
Empirical Properties of Dynamic Conditional
Correlation Multivariate GARCH”, NBER
Discussion Paper, and UCSD DP.
– Models of 30 Dow Stocks and 100 S&P Sectors
• Cappiello, Engle and Sheppard, 2002,
“Asymmetric Dynamics in the Correlations of
International Equity and Bond Returns”, UCSD
Discussion Paper
– Correlations between 34 International equity and bond
indices
TERM STRUCTURE OF
DEFAULT CORRELATIONS
• SIMULATE FIRM VALUES, AND
CALCULATE DEFAULTS
• VARIOUS ASSUMPTIONS CAN BE MADE.
– FOR EXAMPLE, ONCE DEFAULT OCCURS,
FIRM REMAINS IN DEFAULT
– OR, FIRMS CAN EMERGE FROM DEFAULT
FOLLOWING THE SAME PROCESS
– OR, CALCULATE THE HAZARD RATE –
PROBABLILITY OF DEFAULT GIVEN NO
DEFAULT YET.
• LOSS COULD BE STATE DEPENDENT OR
DEPEND ON HOW FAR BELOW
THRESHOLD THE VALUE GOES
UPDATING
• As each day passes, the remaining
time before maturity of a credit
derivative will be shorter and the
joint distribution of the outcome will
have changed.
• How do you update this distribution?
• How do you hedge your position?
Two period Joint Returns
CONCLUSIONS
• Dynamic Correlation models give a
flexible strategy for modeling nonnormal joint density functions or
copulas.
• Updating can be used to re-price or
re-hedge positions
• The model can be of equity values
or firm values and risk neutral or
empirical measures, depending on
the application.
Data
• Weekly $ returns Jan 1987 to Feb
2002 (785 observations)
• 21 Country Equity Series from FTSE
All-World Index
• 13 Datastream Benchmark Bond
Indices with 5 years average
maturity
Europe
BELGIUM*
DENMARK*
FRANCE*
GERMANY*
IRELAND*
AUSTRIA*
ITALY
THE
NETHERLANDS*
SPAIN
SWEDEN*
SWITZERLAND*
NORWAY
UNITED KINGDOM*
Australasia
AUSTRALIA
HONG KONG
JAPAN*
NEW ZEALAND
SINGAPORE
Americas
CANADA*
MEXICO
UNITED STATES*
GARCH Models
(asymmetric in orange)
•
•
•
•
•
•
•
•
•
GARCH
AVGARCH
NGARCH
EGARCH
ZGARCH
GJR-GARCH
APARCH
AGARCH
NAGARCH
•
•
•
•
•
•
•
•
•
3EQ,8BOND
0
1BOND
6EQ,1BOND
8EQ,1BOND
3EQ,1BOND
0
1EQ,1BOND
0
Parameters of DCC
Asymmetry in red (gamma)
and Symmetry in blue (alpha)
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
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RESULTS
• Asymmetric Correlations –
correlations rise in down markets
• Shift in level of correlations with
formation of Euro
• Equity Correlations are rising not
just within EMU-Globalization?
• EMU Bond correlations are
especially high-others are also
rising