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Micro correlations and tail dependence

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									Micro correlations and
   tail dependence
            Roger Cooke
      Resources for the Future &
             Dept. Math
               TU Delft
        Vine-Copula workshop
           Dec. 16-17, 2008
Subsequently, insurers were required by September 30, 2007 to make “true up” filings based on their actual
    reinsurance costs and pass on to the insureds the actual savings which resulted from the expanded FHCF
    coverage. …State officials are perplexed over the failure of the 2007 insurance reforms to bring about
    meaningful rate reduction. In October 2007, the OIR served Allstate with broad subpoenas, demanding an
    explanation of the criteria Allstate used when it began dropping 300,000 homeowners policies starting in
    2005 and justification for its rate filings.
FLORIDA INSURANCE REFORM: One year after a massive legislative overhaul, is Florida any closer to finding a solution to its
    property insurance problems? By Fred E. Karlinsky, Shareholder, and Richard J. Fidei, Partner
   Colodny, Fass, Talenfeld, Karlinsky & Abate, P.A.
           Background:
Catastraphe insurance market failure
… insurers were required by September 30, 2007 to make “true up” filings based on their actual reinsurance costs
and pass on to the insureds the actual savings which resulted from the expanded FHCF coverage. …State
officials are perplexed over the failure of the 2007 insurance reforms to bring about meaningful rate reduction. In
October 2007, the OIR served Allstate with broad subpoenas, demanding an explanation of the criteria Allstate
used when it began dropping 300,000 homeowners policies starting in 2005 and justification for its rate filings.
FLORIDA INSURANCE REFORM: One year after a massive legislative overhaul, is Florida any closer to finding a solution to its p roperty insurance problems? By Fred E. Karlinsky, Shareholder,
and Richard J. Fidei, Partner Colodny, Fass, Talenfeld, Karlinsky & Abate, P.A.
“Here in Lee County, property insurance premiums are proving to be the straw that broke
the camel's back in so many foreclosure proceedings. A mortgage is in default if insurance
premiums are not paid. These high premiums, coupled with escalating adjustable
mortgages, are the main reasons there were more than 1,000 foreclosures in July with no
end in sight.” http://www.taylor.house.gov
             Micro Correlations
Let X1,...X2N be identically distributed random
variables with variance σ2 and COV(Xi,Xj) = c. Then


ρ(  i=1..NXi ,  i=N+1..2NXi) =

   N2 c
                 → 1 (N → )
Nσ2 + N(N-1) c
       Eg σ2=1, c=0.01, N = 1000


ρ(  i=1..NXi ,  i=N+1..2NXi) = 0.91
 Lp processes may be the ‘natural’ way to get
     tail dependent, fat tail distributions
 (X1,..) have a L1 symmetric distribution with gamma
  prior if, for any n, the n-dimensional marginal
  density is given by
p(x1,…xn) =  e-λΣxi -1a e-a (1/()) d.

Setting n = 1 and integrating over λ, the univariate
  density =
p(x) = a / (a+x)+1
 The sum of n L1 vbls is a (gamma gransform
              of) an Ln process
The density of the sum of the n variables is

p(Σi=1..n Xi = r) =  (1/(n))nrn-1e-r -1ae-a(1/())d

= rn-1a(+n) / [()(a+r)+n].
            Tail Dependence
For variables X,Y with the same distribution, the
  upper tail dependence of X and Y is

UTD(X,Y) = limR P(X>R  Y>R) / P(X>R).
The upper tail dependence of sums of N L1 variables is



        k,j=0..N-1 (1/2)j+k (+k+j) / [ ()k!j!]
 (1/2) 
         k=0..N-1 (+k) / [k!()]
UTD, correlation depend on shape and
                on n
Lower tail dependence
 (no closed formula)
Lower quadrant conditional
       correlations
Idem, log scale
                   idea
• Fix claims type (eg coastal)
• Fix time window
• Model claims aggregation as Lp process
• Find aggregation level and indemnity level
  R st correlation Sum < R is small.
• Private insurance markets can work under
  claim level R.
• Above R, own risk or govt risk
       illustration




              N/2     N
Policy packages

								
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