Micro correlations and tail dependence

Roger CookeResources for the Future &

Dept. MathTU Delft

Vine-Copula workshopDec. 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 property 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 CorrelationsLet X1,...X2N be identically distributed random

variables with variance σ2 and COV(Xi,Xj) = c. Then

ρ( i=1..NXi , i=N+1..2NXi) =

N2c → 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 -1ae-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

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

5.00E-01

6.00E-01

7.00E-01

8.00E-01

9.00E-01

0 10 20 30 40 50 60

correlation(1,2)

n=1

n=3

n=5

n=15

n=50

shape factor

UTD

Lower tail dependence(no closed formula)

PROB(Z1,Z2 <R) / Prob(Z1<R) Zi = sum of n; a=3, v=3

R= mean * fraction: n= 0.5 0.25 0.1 0.01 1 0.557984 0.36059 0.173613 0.019704031 4 0.569028 0.273852 0.047931 2.65813E-05 8 0.628236 0.305849 0.036901 3.31393E-06

Lower quadrant conditional correlations

Idem, log scaleconditional correlations of sums of N L1

variables, given each summand < fraction x mean log scale

0.0001

0.001

0.01

0.1

10 10 20 30

N

cond

ition

al

prob

abili

ty 110100unconditional

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

NN/2Policy packages