Describing Copulas and Testing Fits

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Describing Copulas and Testing Fits
Gary Venter Guy Carpenter Instrat
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Architecture

• Review copulas
• Methods for describing copulas
• Comparing empirical and fitted copulas
• Goodness-of-fit examples
• Absolute and relative versions

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1. Review Copulas

• Convenient way to build joint distribution from
  individual distributions
• Insurance dependency often stronger in large
  events
• Can model this by copula methods
• Quantifying dependency
• Degree of dependency
• Parts of distribution most strongly related

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Modeling via Copulas

• Specify joint distribution of probabilities
• Copula of distribution of probabilities is joint
  distribution of losses
• Can specify where in the probability range
  relationship is strongest
• Conditional distribution easily expressed
• Simulation readily available

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Formal Rules Bivariate

• F(x,y) C(FX(x),FY(y))
• Joint distribution is copula evaluated at the
  marginal distributions
• C(u,v) F(FX-1(u),FY-1(v))
• u and v are unit uniforms, C maps I2 to I
• Any joint distribution expressed via a copula
• FYX(yx) C1(FX(x),FY(y))
• Conditional distribution is derivative of copula
• E.g., C(u,v) uv
• F(x,y) FX(x)FY(y), so independence copula
• C1(u,v) v Pr(VltvUu)

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Correlation

• Kendall tau and rank correlation depend only on
  copula, not marginals
• Not true for linear correlation
• Can calculate tau as 4EC(u,v) 1 .
  4??I2C(u,v)c(u,v)dudv ¼ and rho as
  12??I2C(u,v)dudv ¼
• Tau generalizes to multivariate ???cC
  2-n/½ 2-n. Indep ???2-n,max ½
• Looks like 3???C 2-n/½ 2-n for rho.

triples excess over ¼
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Rho and Tau Are Related(for continuous copulas,
from Nelsen)
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Example C(u,v) Functions

• Frank -a-1ln1 gugv/g1, with gz e-az 1
• t(a) 1 4/a 4/a2?0a t/(et-1)dt
• Gumbel exp- (- ln u)a (- ln v)a1/a, a ? 1
• t(a) 1 1/a
• HRT u v 1(1 u)-1/a (1 v)-1/a 1-a
• t(a) 1/(2a 1)
• Normal C(u,v) B(p(u),p(v)a) i.e., bivariate
  normal applied to normal percentiles of u and v,
  correlation a
• t(a) 2arcsin(a)/p

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Archimedean Copulas

• Defined by generator function f
• C(u,v) f-1f(u)f(v)
• Gumbel and Frank are examples
• A fairly easy way to create a copula
• Some calculations also easier

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2. Describing Copulas

• Correlation coefficients tell you more about
  parameter than about copula
• Tail coefficients tell more about copula
• For many copulas this is zero for all parameters
  e.g., normal copula
• For others it is always positive how positive
  depending on correlation
• Contours of joint distribution illuminating

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Copulas Differ in Tail EffectsLight Tailed
Copulas Joint Unit Lognormal
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Copulas Differ in Tail EffectsHeavy Tailed
Copulas Joint Unit Lognormal
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Descriptive Functions

• Correlation and tail coefficients are scalar
  descriptors
• Descriptive functions are univariate functions
  mapping 0,1 ? 0,1 that describe features of
  the copula
• Usually z?0,1 indicates a region of the unit
  square (or cube or hypercube, )
• E.g., ultz, or u,vltz or C(u,v)ltz
• Function(z) calculates a scalar on this region
  often an integral over region

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Copula on Diagonal I(z) C(z,z)
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Copula Distribution Function

• K(z) ??C(u,v)ltzc(u,v)dudv
• Compare to I(z) ??u,vltzc(u,v)dudv
• Genest and Rivest JASA 1993
• Can be interpreted as PrC(u,v) lt z
• Archimedean copulas K(z) z f(z)/f(z)
• f(z) exp?yz dt/t K(t) for y?(0,1)
• All copulas are Archimedian
• K looks similar for many copulas but the density
  k looks less so

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Two Copulas Fit to Same Data
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Tail Concentration FunctionsVenter PCAS 2002

• L(z) Pr(UltzVltz) Pr(Ultz Vltz)/z
• R(z) Pr(UgtzVgtz) Pr(Ugtz Vgtz)/(1 z)
• L(z) C(z,z)/z I(z)/z
• R(z) 1 2z C(z,z)/(1 z)
• L(1) 1 R(0)
• Action is in R(z) near 1 and L(z) near 0
• lim R(z), z-gt1 is R, and lim L(z), z-gt0 is L
• Generalizes L(z) Pr(Ultz Vltz Wltz)/z

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LR Function (L below ½, R above)
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Other Descriptive Functions

• Tau can be calculated by
  -14?01?01C(u,v)c(u,v)dvdu.
• Cumulative tau using u,vltz
  J-(z) z24?0z?0z
  C(u,v)c(u,v)dvdu/C(z,z).
• J(z) z24?z1?z1 C(u,v)c(u,v)dvdu/C(z,z)
• Expected value of V given Ultz M(z)
  E(VUltz) ?0z?01 vc(u,v)dvdu/z.
• Basially bivariate, but could condition on all
  other variables ltz

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More

• Cumulative tau could use just ultz
• Q correlation given u,v lt z or ultz really rank
  correlation since done on probabilities
• Q-(z) E(UVU,Vltz) E(UU,Vltz)E(VU,Vltz)
  /Var(UU,Vltz)Var(VU,Vltz)½
• Similar to Malevergne Sornette 2002 Review of
  Financial Studies and Charpentier 2003 ASTIN C
• Generalize to multi Q-(z) E(UVWU,V,Wltz)
  E(UU,V,Wltz)E(VU,V,Wltz)E(WU,V,Wltz)/
  Var(UU,V,Wltz)Var(VU,V,Wltz)Var(VU,V,Wltz)½

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3. Compare Empirical and Fitted Copulas

• Compare descriptive functions for empirical and
  fitted copulas
• Empirical copula at each point counts number of
  points it in each element
• Eyeball tests of relative fit
• Look for functions that best show fit
• Clearly distinguish bad fit from good fit
• Show difference between good fit and pretty good
  fit

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Can Also Do PP-Plots

• Fitted copulas at each sample point vs. empirical
  
• Also compute fitted conditional distributions at
  each sample point, sort, and compare to uniform
• PP-plots go from lt0,0gt to lt1,1gt and stay near
  diagonal so sometimes hard to identify good fit
• Subtracting YX makes them horizontal and then
  scale can be magnified
• PP-error plots

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4. Goodness-of-Fit Examples

• Motor and property losses in French windstorms
• Monthly changes in exchange rate against the US
  dollar for Yen, Swedish Krona, and Canadian
  dollar

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Auto and Property Claims in French Windstorms
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MLE Estimates of Copulas
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F/X Data

• Japan, Sweden and Canada currencies monthly
  change in U rate from 1971
• Correlations
• SJ 50
• SC 25
• JC 10
• Fit t-copula
• Takes correlation matrix
• Dof parameter n expresses tail strength
• This is gt in pairs with higher correlation

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F/X Fits

• MLE dof n 19.7
• Take n 1 as a bad fit
• Fit already close to normal copula but use n500
  to represent pretty good fit
• Reduces ln L by about 1
• Extra parameter for t vs. normal worth it by
  Akaike but not by other information criteria
• Can look pairwise and all together

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PP, I, K Functions

• Constrained to go from 0,0 to 1,1
• Makes it hard to see differences
• Subtracting empirical flattens out and shows
  differences better

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Frank clearly worse fit HRT best on left HRT vs.
Gumbel hard to tell on right
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Differences show up more clearly
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K differences are hard to read
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K errors show better but still hard to choose
normal vs. n 20
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PP Plot AP Empirical vs Fits
Cant tell much
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Frank shows up as clearly worse and HRT generally
best
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Bad fitting n1 shows but cant tell normal from
n20
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Bad fitting n1 shows but cant tell normal from
n20
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Harder to tell anything in regular PP-plot, but
n1 still looks off
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Hard to tell much from I(z) but differ-ences show
up well
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Again readily shows bad fit but hard to tell good
from pretty good PP, K and I work better as
errors and can tell HRT from Gumbel but not n500
from n20. All three give pretty much the same
story
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Tail Functions

• Important in insurance because usually sum of
  variables is taken
• If large losses are highly correlated this can
  impact sum
• Though different story is told, resolution is
  similar to that of PP, I, K
• Empirical L R noisy in tails

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Frank clearly worse fit HRT clearly best L(z) HRT
probably best R(z)
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R and L functions also distinguish n1 but not
normal from n20
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M (conditional mean) has two empiricals
conditioning on U or V lt z Even bad fit does not
always show up as bad here
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Frank clearly worse fit HRT best on left HRT vs.
Gumbel hard to tell on right
Two empirical conditional means
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Conditional and Partial Correlations

• J is build up of tau
• Q is conditional rank correlation
• Could try the other way also
• Both appear more sensitive to how good the fit is

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Quite noisy on left HRT better on right but not
great
Conditional rank correlation U,V lt z
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Not quite as noisy on left HRT still better on
right but still not great
Conditional rank correlation U lt z
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HRT generally better than others
Cumulative Tau U,V lt z
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Conditional rank correlation (here on Ultz) does
show some difference between normal and t with n
20
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Similar for conditioning on U, V lt z
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For cumulative tau (here conditioned on U, V lt
z), normal vs n20 can be discerned
For SJ a somewhat lower value of n may have
worked better compromise of only one tail
parameter
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5. Absolute Tests of Fit

• Compare empirical descriptive functions to
  confidence intervals based on best fitting copula
• Get confidence intervals by drawing from copula
  samples of the same size as original sample
• Look at percentiles of the descriptive functions
  at each z

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L and R shown with 50 and 80 confidence bands
from HRT generally good fit
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K errors also generally reasonable but more
outside 80
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Partial correlations show potential problems with
fit
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(No Transcript)
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SJ and JC ok but less than ideal. Suggests need
for more parameter copulas
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Conclusions

• For I, K and PP plots, error plots show fit
  quality better than standard plots
• Cumulative and conditional correlations appear to
  show fit differences best
• L and R are important for insurance applications
  but empiricals are noisy making judging fits
  harder
• M (conditional mean) differentiates less