Model Risk of Correlation Products
PRESENTED TO: Risk Magazine's Training Course: Advanced Correlation Modelling & Analysis
BY: Martin Goldberg, [email protected] of Model ValidationRisk ArchitectureCitigroupNew York, New York
DATE: May 11, 2006
PLACE: New York City
Page 2 of 44Martin Goldberg, Citigroup April, 2006
Outline How does your model perform
Testing copula density and marginals separately
Estimating correlations and volatility - implied vs. historical
Stress Events, Extreme Value Theory, conditional dependencies, and VaR
Summary
Page 3 of 44Martin Goldberg, Citigroup April, 2006
Outline How does your model perform
Testing copula density and marginals separately
Estimating correlations and volatility - implied vs. historical
Stress Events, Extreme Value Theory, conditional dependencies, and VaR
Summary
Page 4 of 44Martin Goldberg, Citigroup April, 2006
Performance metrics
Most models can calibrate to the price of some standard instrument at
inception. What distinguishes good from bad is the performance over time or
over changing markets
Backtesting
Ex Ante Profit Attribution Analysis
Paper Trading
Hypothetical Paper Trades
Stress Tests
Page 5 of 44Martin Goldberg, Citigroup April, 2006
Sources of Error
There are many possible sources of model risk. Some that I will discuss today
are:
Calibration error - the model does not fit the market, either because
It is not flexible enough
It is too flexible and you are calibrating to noise
Input error - the data you are fitting your model with are wrong or come from
a different market
Regime shift - the model used to work well, but the market has changed and
the model can’t handle the changes.
Wild West error - there are not enough data to distinguish a good model from
a bad one
Page 6 of 44Martin Goldberg, Citigroup April, 2006
Not Flexible Enough - Simplistic models
No correlation skew as an analogy to B_S models without vol skew
CAPM one-factor models with no correlation between alphas
Multiplying VaR by and neglecting GARCH-type effects
Confusing correlation with contagion, or assuming constant correlation
t
Page 7 of 44Martin Goldberg, Citigroup April, 2006
Too Flexible - Careful Fit to Noise
In a variance-covariance Value-at-Risk calculation, or in a portfolio optimization, a key
component is a correlation matrix.
A matrix calculated from a year of daily returns has rank at most 252, no matter how
many timeseries you put in.
Random Matrix Theory [3], [4] describes the principal components you would get if
the timeseries were pure noise. How does it compare to the eigenvalues of, say, the
S&P 500 correlation matrix?
About 94% of the spectrum is random, leaving the meaningful signal a rank of about
30.
Moral: Whenever possible, use a control group of synthetic data of known
properties to separate out real effects from artefacts of the modeling process.
Page 8 of 44Martin Goldberg, Citigroup April, 2006
Calibrating to wrong data
First example
Hedge fund managers and private equity funds are required to post monthly
estimates of their portfolio value.
The estimates are noisy due to illiquidity
The managers, either unconsciously or deliberately, “smooth” the earnings
by shading their estimates.
Correlations due to similar smoothing, or from market moves?
Next example
Historical correlations for trading in liquid markets should probably use daily
timeseries
For buy-and-hold, probably less frequent, like monthly or quarterly.
Related issues in estimation of default correlation from equity data.
Spectral analysis, for example [5].
Page 9 of 44Martin Goldberg, Citigroup April, 2006
Regime Shifts
Abrupt jumps should be handled differently than gradual non-stationarity.
Key questions for jumps:
Is this really a jump and not just a fat tailed diffusion?
Is the jump a one-time regime shift invalidating all prior history, or is it a
feature of a jumpy market?
Is there a news story explaining the jump?
Was there a knock-on effect (possibly contagion?) in other markets?
Page 10 of 44Martin Goldberg, Citigroup April, 2006
Illiquid and New Markets and Cowboys
Here you need to use heuristics and trader intuition, since there isn’t anything
else.
A wide enough bid-ask spread covers all sins.
This is not exactly model risk since there is no model as such.
Anything worth doing is worth doing badly at first.
Page 11 of 44Martin Goldberg, Citigroup April, 2006
Outline How does your model perform
Testing copula density and marginals separately
Estimating correlations and volatility - implied vs. historical
Stress Events, Extreme Value Theory, conditional dependencies, and VaR
Summary
Page 12 of 44Martin Goldberg, Citigroup April, 2006
Quick Introduction to Copula Theory
Copula theory is a generalization of the concept of correlation.
A copula is expressed as a quantile of the distribution in N dimensions.
A copula in one dimension is a tautology - x% of the data are at or below the x
% quantile.
The one-dimensional copula density, also called a marginal, is a uniform
distribution from 0 to 1, with the copula
density of each point being its quantile in the data series.
The copula is related to the copula density by
),(
),Pr(),(),(
)0,0( VUc
vVuUvuCvu
Page 13 of 44Martin Goldberg, Citigroup April, 2006
Example
In two dimensions, the copula density of changes in USD Libor and JPY Libor
looks like
Note that it is not actually continuous, since some days are unchanged.
3m Libor Weekly Changes Copula Density Rank Correlation 10.6%
0.0%
20.0%
40.0%
60.0%
80.0%
100.0%
0.0% 20.0% 40.0% 60.0% 80.0% 100.0%
JPY Libor
US
D L
ibor
Page 14 of 44Martin Goldberg, Citigroup April, 2006
Copulas again
The Copula for this data is at each point (x,y) the fraction of the data where X<x and Y<y.
For a good introduction to copulas see [9].
10
%
20
%
30
%
40
%
50
%
60
%
70
%
80
%
90
%
10
0% 10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
JPY Libor
USD Libor
Cumulative Copula of weekly Changes in 3M Libor
80%-100%
60%-80%
40%-60%
20%-40%
0%-20%
Page 15 of 44Martin Goldberg, Citigroup April, 2006
Marginal Probability Distributions
The first step in constructing a copula is to transform all the marginals to uniform
densities.
Not everything is Gaussian.
Alternative distributions, which can be fitted to data and give better results
usually than a Gaussian or lognormal, are Student-t, CEV, Madan’s VG, and my
favorite for exploratory data analysis, the Tukey gXh distribution [15],[16]
Where g controls skew and h controls smile. This nests Gaussian, lognormal,
Student-t, and even Cauchy distributions, and is more tractable than CEV or VG.
Later I will show what can go wrong if the marginals are mis-specified.
Art form to trade-off between accuracy/flexibility and tractability/speed.
2
2
1)(
hZgZ
gXh eg
eBAZX
Page 16 of 44Martin Goldberg, Citigroup April, 2006
Fitting the Tukey gXh distribution to single-B bond spreads
-5 -4 -3 -2 -1 0 1 2 3 4 50
500
1000
1500
2000
2500
3000
3500
Absolute change in yield (%)
Normal
Rescaled
gXh
Comparison of normal, rescaled normal and (g X h) distribution fits to 10 day changes in idiosyncratic spread for single-B bonds using EJV data. Rescaled Cumulative Normal fits at 99th percentile.
Page 17 of 44Martin Goldberg, Citigroup April, 2006
Families of Copulas
There are many “families” of copulas described in the literature [7].
Note that most of the literature stops at 2 dimensions; more than that is
much harder to work out the math, and there are constraints.
Some popular families of copulas are:
#
Parameters
in 2-D
Name
0 Independence
1 Gaussian, Plackett, Frank, Gumbel, Morgenstern
2 Student-t, but most others have no good names -
see [7] page 149 and following
As many as
needed
Bernstein (see [8]) which is the copula equivalent
of a Taylor series
Page 18 of 44Martin Goldberg, Citigroup April, 2006
Properties of Copulas
Any multivariate density can be expressed as a copula connecting marginal densities.
The copula and the marginals are completely separate - any marginal pdf’s can be
connected by any valid copula.
For distributions with continuous marginals the copula is unique.
The average of two copulas may not be a copula, but the average of two copula
densities is a copula density.
A Gaussian copula connecting two Gaussian marginals is a Pearson (ordinary)
correlation.
Pearson Correlation is not a good measure if the copula is not Gaussian or any marginal
is not Gaussian.
The rank correlation is a non-parametric copula equivalent, for any distributions, of
Pearson correlations for multivariate normals. Spearman rho is easier than Kendall Tau.
Spearman rank correlation in 2-D is the Pearson correlation between ranks of the
entries of the 2 data series.
Easy to compute in Excel using the RANK() function.
Page 19 of 44Martin Goldberg, Citigroup April, 2006
Tail Dependence
The Upper Tail Dependence is defined for any copula density c(U1,U2) as
By flipping the < to > and replacing the u with 1-u, we can get a Lower Tail Dependence
More generally, in N dimensions, the hypercube has 2N corners, so we can define 2N Tail
Dependences
Tail Dependence is my preferred definition of contagion.
A Gaussian Copula has zero tail dependence - a very extreme move in one dimension is
never simultaneous with a similarly big move in any other dimension.
)(u
)1/(),Pr(1
),()( 21 uuUuU
uuuC
u
Page 20 of 44Martin Goldberg, Citigroup April, 2006
Fiendish Copula Density
Upper and lower tail dependence of 1;
middle “local dependence” -1
The rank correlation is constructed to be
exactly zero.
It is more pathological than what you will
ever actually find, but it is a good stress
test.
You can find funnel-shaped and galaxy-
shaped copula densities in real data, but
in a less exaggerated form than below.
Extreme Funnel Extreme Galaxy
Fiendish Copula Density
0%
20%
40%
60%
80%
100%
0% 20% 40% 60% 80% 100%
Gaussian Copula Density
Page 21 of 44Martin Goldberg, Citigroup April, 2006
Fiendish Copula
100
200
300
400
500
600
700
800
900
1000
X
Y
Fiendish Cumulative Copula It is not immediately obvious why this is
so fiendish.
Although all the theory is done using the
cumulative distributions, the copula
densities are more informative and make
prettier pictures.
Caveat: Copula theory is not nearly as
well developed for more than 2
dimensions. The standard cheat scheme
used in Credit Derivatives is to assume
every underlying looks like all the others,
which means all 2-dimensional slices
look alike.
Page 22 of 44Martin Goldberg, Citigroup April, 2006
Distinguishing Correlation from Contagion
Non-Linear CorrelationContagion can be defined as a significant difference in the association between large moves (tail events) relative to the association between smaller moves (ordinary days). This is Tail Dependence. As an example study, here is a test of contagious increases in Pearson correlation between Brent oil and kerosene, using 215 pairs of weekly historical spot data. Since you always should use a control, I have also used 215 pairs of random numbers with the same correlation of 63%.
Random fake normals correlated 62.82%
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
Brent vs Kerosene 62.82% CorrelationAll weekly data
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Brent % Moves
Ker
o %
Move
s
Page 23 of 44Martin Goldberg, Citigroup April, 2006
Tonsured densityTonsured Density
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Null hypothesis is an elliptical distribution, so eliminate center of distribution where
Xba
b
i
a
i 2
2
2
2
Page 24 of 44Martin Goldberg, Citigroup April, 2006
Nonlinearity? Tonsured correlation
Tonsured Correlation
0.6
0.7
0.8
0.9
1
Tonsured Fake data
Tonsured real data
Page 25 of 44Martin Goldberg, Citigroup April, 2006
Tail Dependence for Brent and Kerosene
Upper Tail Dependence - Brent and Kero
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
U
Lam
bda
Lower Tail dependence - Brent and Kero
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
00.10.20.30.40.50.6
u
Lam
bda
�Upper tail dependence, but not lower, is found in the US Treasury curve[14].
�Barry Schachter’s gloriamundi.org website has many other such tail dependence papers.
Page 26 of 44Martin Goldberg, Citigroup April, 2006
Another Parametric Correlation Curve
Bjerve and Doksum[1] have a correlation curve, which does not generalize
well to n>2 data series.
Regress Y against X, then the local variance
The local regression slope
And the global standard deviation of the X variable
And use these to get a local correlation
Note this is not symmetric in X and Y
)|()(2 xXYVarx )(x
)()()(
)(222 xx
xx
Page 27 of 44Martin Goldberg, Citigroup April, 2006
Causes of Non-constant Correlation
Use of a standard Pearson correlation assumes multivariate normal distributions.
The changes in correlation could be due to contagion, or just to skewed or fat-tailed
underlyings.
Kurtosis alone still has elliptical distribution and does not have much effect on
correlation.
Copula theory - not well developed for n>2. For example, a Student-t copula has the
same degrees of freedom for every pair of variables.
If two Gaussian marginals are associated by a Gaussian copula, the rank-correlation is
constant, and equal to the Pearson correlation.
Copulas can be made flexible enough to represent arbitrarily screwy associations.
Mathematical theories use the copula, which is the cumulative function; easier to
visualize copula density.
Page 28 of 44Martin Goldberg, Citigroup April, 2006
Problems with Pearson Correlation
High leverage data points
Page 29 of 44Martin Goldberg, Citigroup April, 2006
Picking the right copula
Fitting the data exactly with a copula density of delta functions at each data point
is valid, but not very useful. Trade-off between smooth function and accuracy [2].
Hurd et al[10] use an expansion in Bernstein copulas for the copula between
EUR/GBP and USD/GBP implied by the volly smile surfaces of traded options. FX
is special because triangular arbitrage gives an implied correlation. On some days,
the copula is bimodal and they need 11 terms to get the extra hump qualitatively
right. Is this noise or a feature?
Malvergne and Sornette[11] find that most of the time, a Gaussian copula works
for most currency pairs and most pairs of stocks, but not for pairs of commodities.
However, if your portfolio is actively traded or has optionality, this may not be
the case. Boyson et al[12] find contagion between hedge fund styles, so a fund-of-
funds is not well-modeled by a Gaussian copula.
Some other caveats are found in [13], where they show how hard it is to find the
best parametric copula to fit to real data.
Page 30 of 44Martin Goldberg, Citigroup April, 2006
Is stock market contagion an urban legend?
On the previous slide I mentioned[11] that a Gaussian copula works well for equities.
Does this contradict the received wisdom that correlations between stocks goes to one
in a crash?
One example of the studies of correlation in a crash is [17]. Note that they always use
ordinary Pearson correlations.
In contrast, [18] notes that a constant association with a fat-tailed distribution leads to
the appearance of changing correlation.
This is an example of invoking higher-order effects to explain illusory phenomena
caused by not fully capturing the lower order effects. In quantum chemistry, this is
called lack of basis set saturation.
A similar example is the simpler versions of jump-diffusion models, where they pretend
the diffusion part is lognormal or Gaussian, pretend that the drift term is linear, and then
invoke jumps to explain the rest. Jumps in financial time series are real, but they are
always accompanied by headlines in the Wall Street Journal or Financial Times. If your
model has more jumps than headlines, fix the diffusion part.
Page 31 of 44Martin Goldberg, Citigroup April, 2006
Outline How does your model perform
Testing copula density and marginals separately
Estimating correlations and volatility - implied vs. historical
Stress Events, Extreme Value Theory, conditional dependencies, and VaR
Summary
Page 32 of 44Martin Goldberg, Citigroup April, 2006
Nonstationary time series
Historical estimate is always a convolution with a filter.Square wave - equally weighted data more recent than -RiskMetrics exponential decay
Whatever other filter you like, or better, one suggested by the data
Greg Sullivan[6] describes an optimal estimator for the length of the square
wave filter, which could be easily modified to find, for example, the optimal
exponent for the RiskMetrics method.
For a stationary time series, the error in the correlation estimate is
In the absence of any obvious regime shift, the non-stationarity can be
modeled as a linear drift
t )1( 2
1
2||
)(
2
b
btat
Page 33 of 44Martin Goldberg, Citigroup April, 2006
Nonstationary time series 2
Then the optimum observation period, dictated by the data itself, is
This represents the trade-off between longer periods reducing statistical
noise, and shorter periods being closer to stationary. The paper[6] has detailed
instructions for finding this optimum point.
Of course there are data series where there is a genuine regime shift, like the
start of a new equity listing due to IPO, and there the cutoff is obvious.
32
22* )1(2
bt
Page 34 of 44Martin Goldberg, Citigroup April, 2006
Implied and Historical Correlations
I n t h e s t a n d a r d r i s k - n e u t r a l p r i c i n g o f d e r i v a t i v e s w i t h m u l t i p l e u n d e r l y i n g s s 1 a n d s 2 , a c o r r e l a t i o n e n t e r s t h e e q u a t i o n s a s :
MdwdwE
ddwtsdttds
ddwtsdttds
)(
,...),(,...)(
,...),(,...)(
21
2222222
1111111
W h e r e t h e d r i f t s a r e t h e r i s k - n e u t r a l d r i f t s , a n d t h e d i f f u s i o n s w a r e c o n t i n u o u s m a r t i n g a l e s w i t h n o d r i f t , d i s t r i b u t e d a s G a u s s i a n s u n d e r s o m e f o r w a r d m e a s u r e . T h e j u m p i n t e n s i t i e s w i t h p r o b a b i l i t i e s a r e n o t p r e s e n t i n p u r e d i f f u s i o n m o d e l s . T h e r e c a n b e c o r r e l a t i o n s b e t w e e n w 1 a n d w 2 , b e t w e e n
1 a n d 2 , a n d / o r b e t w e e n t h e a n d t h e w ,
o r e v e n m o r e i f t h e a r e s t o c h a s t i c : a s f e w a s o n e o r a s m a n y a s s i x o r m o r e p o s s i b l e m o d e l c o r r e l a t i o n s . W h a t w e o b s e r v e i n t h e r e a l w o r l d a r e h i s t o r i c a l t i m e s e r i e s S 1 a n d S 2 , w i t h h i s t o r i c a l c o r r e l a t i o n H m e a s u r e d a s t h e u s u a l P e a r s o n c o r r e l a t i o n e i t h e r b e t w e e n a r i t h m e t i c c h a n g e s [ S 1 ( t ) - S 1 ( t - 1 ) ] a n d [ S 2 ( t ) - S 2 ( t - 1 ) ] o r b e t w e e n l o g r a t i o s l o g ( S 1 ( t ) / S 1 ( t - 1 ) ) a n d l o g ( S 2 ( t ) / S 2 ( t - 1 ) ) . T h e q u e s t i o n i s w h a t r e l a t i o n s h i p i s t h e r e b e t w e e n t h e m o d e l M a n d t h e h i s t o r i c a l H .
Page 35 of 44Martin Goldberg, Citigroup April, 2006
Changing the skewness – fake data with correlation .5
6.2 5.0 3.7 2.3 1.1 0.1 -0.9 -1.9 -3.0 -4.2-7.5
-3.6
-0.7
2.1
5.9
0
0.1
0.2
0.3
0.4
0.5
0.6
Skew1
Skew2
Linear Correlation
Page 36 of 44Martin Goldberg, Citigroup April, 2006
Changing the Skew with Added Jumps – fake data with correlation .5
13.8 12.3 9.9 6.1 2.5 0.4 -0.9 -2.1 -3.5 -5.1-7.7
-3.4
0.3
11.2
0
0.1
0.2
0.3
0.4
0.5
0.6
Skew1
Skew2
Linear Correlation
Page 37 of 44Martin Goldberg, Citigroup April, 2006
Rank Correlations
In the absence of jumps, any amount of skewness or kurtosis leaves the rank
correlation between time series unchanged.
This is one of the best reasons to use copula theory.
Rank correlations are very little extra effort compared to Pearson
correlations, and do not assume Gaussian marginals.
Enhancing your model of the univariate marginals does not require redoing
the copula / rank correlation matrix.
Non-Gaussian copulas can capture tail dependence and not confuse it with the
tail shapes.
Page 38 of 44Martin Goldberg, Citigroup April, 2006
Outline How does your model perform
Testing copula density and marginals separately
Estimating correlations and volatility - implied vs. historical
Stress Events, Extreme Value Theory, conditional dependencies, and VaR
Summary
Page 39 of 44Martin Goldberg, Citigroup April, 2006
Extreme Value Theory
EVT says that, for quantiles outside the observation data set, the marginal pdf
can take only 3 different shapes:
Gaussian decay
Abrupt cutoff (Gumbel)
Power-law decay (Frechet)
Most financial time series have Frechet tails. Estimate the exponent, and fit it
with the bulk of the distribution, and you are done. You can use this to predict
the 99%ile (VaR), the 99.9%ile (Basel 2), or the 99.97%ile (economic capital for
a AA firm.) The result will differ from scaling up using the assumption of a
multivariate normal density.
What copula to use? A quick series of tests include tail dependences, stability
over the sample period, and the X test.
Page 40 of 44Martin Goldberg, Citigroup April, 2006
X test for copula skew
By construction, half the copula density is on the left half (0-.5), and half is on the
lower half (0-.5).
You can test for quadrant dependence by seeing how far off the density is from ¼ in
each quadrant, but that may not be very informative – this is just a crude measure of
rank correlation.
Far more informative a test is (in a 2D copula density) drawing an X from upper left
to lower right and upper right to lower left. If the density is elliptical, all 4 triangles
will be mirrored images of each other.
Test comparing each of the 4 to the average of all 4.
Extend to quartets of diamond-shaped sub-regions?
This X-test is not in the literature, as far as I know, but it is referred to in [10].
The extension of the X test to N dimensions involves 2N diagonals.
Page 41 of 44Martin Goldberg, Citigroup April, 2006
Summary
Not everything is multivariate (log-)Gaussian.
Rank correlations are a better and more flexible measure than correlations.
Copula theory is useful.
Tradeoffs between accuracy and speed, complexity vs ease of use.
Liquidity of market important in selecting model.
Get the simpler features right before invoking higher-order effects.
Keep up with the literature – there are lots of good ideas out there.
Disclaimer: This talk represents my personal views and is not
intended to be in agreement with anything Citigroup says or does.
Page 42 of 44Martin Goldberg, Citigroup April, 2006
References
1. Bjerve, S. and Doksum, K.A. (1993). Correlation curves: measures of association
as functions of covariates. Ann. of Statist., 21, 890-902.
2. http://xxx.lanl.gov/abs/physics?papernum=0406023 Maximum Entropy
Multivariate Density Estimation: An exact goodness-of-fit approach
3. http://arxiv.org/abs/cond-mat?papernum=0111503 Noisy Covariance Matrices
4. http://arxiv.org/abs/cond-mat?papernum=0205119 Noisy Covariance Matrices II
5. http://www.ofce.sciences-po.fr/pdf/dtravail/wp2003-07.pdf and
http://ideas.repec.org/p/fce/doctra/0307.html, Iacobucci, Spectral Analysis for
Economic Time Series
6. Greg Sullivan, Risk 8(8), August 1995, page 36, Correlation Counts
7. H. Joe, “Multivariate Models and Dependence Concepts” Chapman&Hall, 1997
8. http://greywww.kub.nl:2080/greyfiles/center/2003/doc/122.pdf Multivariate
Option Pricing Using Dynamic Copula Models by R.W.J. van den Goorbergh, C.
Genest, B.J.M. Werker
Page 43 of 44Martin Goldberg, Citigroup April, 2006
References
9. http://gro.creditlyonnais.fr/content/wp/copula-survey.pdf Copulas for Finance - A
Reading Guide and Some Applications by Bouyé, Durrleman, Nikeghbali, Riboulet,
Roncalli
10.http://www2.warwick.ac.uk/fac/soc/wbs/research/wfri/rsrchcentres/ferc/
wrkingpaprseries/wp05-20.pdf Using Copulas to Construct Bivariate Foreign
Exchange Distributions with an Application to the Sterling Exchange Rate Index by
Matthew Hurd, Mark Salmon, Christoph Schleicher
11.http://arxiv.org/abs/cond-mat?papernum=0111310 Testing the Gaussian Copula
Hypothesis for Financial Assets Dependences Authors: Y. Malevergne , D. Sornette
12.http://www.nber.org/papers/w12090 Is There Hedge fund contagion? Authors: N.
Boyson, C. Stahel, R. Stulz
13.http://www.crest.fr/pageperso/fermanian/pitfalls_copula.pdf Some Statistical pitfalls
in Copula Modeling for Financial Applications Authors J Fermanian, O Scaillet
14.http://gloriamundi.org/picsresources/mjasnw.pdf Nonlinear Term Structure
Dependence by M Junker, A Szimayer, N Wagner
Page 44 of 44Martin Goldberg, Citigroup April, 2006
References
15.http://fic.wharton.upenn.edu/fic/papers/02/0225.pdf
16.http://fic.wharton.upenn.edu/fic/papers/02/0226.pdf
17.http://arxiv.org/abs/cond-mat?papernum=0302546 Dynamics of market
correlations: Taxonomy and portfolio analysis Authors: J.-P. Onnela, A.
Chakraborti, K. Kaski, J. Kertesz, A. Kanto
18.http://ideas.repec.org/p/cam/camdae/0319.html. "Changing Correlation and
Portfolio Diversification Failure in the Presence of Large Market Losses by
Sancetta, Satchell
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