Modelling Dependence with Copulas - IFOR

9 Modelling Extremal Events in Practice
9.1 Pricing Risky Insurance Contracts
Consider a portfolio consisting of n risks X 1 ,... ,X n which are positive. Let the
risks represent potential losses in dependent lines of business for an insurance company.
Suppose that losses not greater than k 1 ,... ,k n can be accepted, and that the
insurance company wants to buy protection against the situation where l or more
of the losses simultaneously exceed their respective thresholds k 1 ,... ,k n . Suppose
that a reinsurance company offers to sell a contract which makes a payout only in
thecasethatatleastl of the n risks exceeds their thresholds. To begin with we will
look at the probability that a payout is made. Later we will look at the expected
size of the payout under the assumption that losses which exceed their thresholds
are paid in full.
Assume historical data are available allowing estimation of
Let
• marginal distributions;
• pairwise rank correlations.
N = | { i ∈{1,...,n}|X i >k i
}
|
be the number of losses exceeding their thresholds and let
L l = 1 {N≥l}
n ∑
i=1
(
Xi 1 {Xi>k i})
be the loss to the reinsurer.
Suppose that the contract the insurance company is interested in is the one
for which the losses are paid in full when all losses exceed their thresholds. If the
insurance company holds this contract it has full protection against simultaneous
big losses in all lines of business no matter how big the total loss is. The probability
of payout is then given by
P[N = n] =H(k 1 ,... ,k n ).
Unfortunately a good estimation of the joint distribution, H, is not realistic due to
lack of data and theoretical difficulties. However, data from the previous year(s) enable
estimation of the Kendall’s tau rank correlation matrix via the sample version.
Furthermore, estimation of univariate marginal distributions is well understood.
Hence we can assume that rank correlations and margins are given. Since
H(x 1 ,... ,x n )=Ĉ(F 1(x 1 ),... ,F n (x n )),
the payout probability can not be calculated without choosing a suitable copula
representing the dependence structure among the n risks. The standard approach
might be to choose a Gaussian copula, given by
Cρ Ga
l
(u) =Φ n ρ l
(Φ −1 (u 1 ),... ,Φ −1 (u n )),
where Φ n ρ l
denotes the joint distribution function of the n-variate standard normal
distribution function with linear correlation matrix ρ l and Φ −1 denotes the inverse
of the distribution function of the univariate standard normal distribution. To
parametrizise the Gaussian copula from the estimated Kendall’s tau rank correlation
matrix the relation
ρ l = sin(πτ/2)
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