PaidIncurredChain
Description
The Paidincurred Chain model (Merz, Wuthrich (2010)) combines claims payments and incurred losses information to get a unified ultimate loss prediction.
Usage
1  PaidIncurredChain(triangleP, triangleI)

Arguments
triangleP 
Cumulative claims payments triangle 
triangleI 
Incurred losses triangle. 
Details
The method uses some basic properties of multivariate Gaussian distributions to obtain a mathematically rigorous and consistent model for the combination of the two information channels.
We assume as usual that I=J. The model assumptions for the LogNormal PIC Model are the following:
Conditionally, given Θ = (Φ_0,...,Φ_I, Ψ_0,...,Ψ_{I1},σ_0,...,σ_{I1},τ_0,...,τ_{I1}) we have
the random vector (ξ_{0,0},...,ξ_{I,I}, ζ_{0,0},...,ζ_{I,I1}) has multivariate Gaussian distribution with uncorrelated components given by
ξ_{i,j} \sim N(Φ_j,σ^2_j),
ζ_{k,l} \sim N(Ψ_l,τ^2_l);
cumulative payments are given by the recursion
P_{i,j} = P_{i,j1} \exp(ξ_{i,j}),
with initial value P_{i,0} = \exp (ξ_{i,0});
incurred losses I_{i,j} are given by the backwards recursion
I_{i,j1} = I_{i,j} \exp(ζ_{i,j1}),
with initial value I_{i,I}=P_{i,I}.
The components of Θ are indipendent and σ_j,τ_j > 0 for all j.
Parameters Θ in the model are in general not known and need to be estimated from observations. They are estimated in a Bayesian framework. In the Bayesian PIC model they assume that the previous assumptions hold true with deterministic σ_0,...,σ_J and τ_0,...,τ_{J1} and
Φ_m \sim N(φ_m,s^2_m),
Ψ_n \sim N(ψ_n,t^2_n).
This is not a full Bayesian approach but has the advantage to give analytical expressions for the posterior distributions and the prediction uncertainty.
Value
The function returns:

Ult.Loss.Origin Ultimate losses for different origin years.

Ult.Loss Total ultimate loss.

Res.Origin Claims reserves for different origin years.

Res.Tot Total reserve.

s.e. Square root of mean square error of prediction for the total ultimate loss.
Note
The model is implemented in the special case of noninformative priors.
Author(s)
Fabio Concina, fabio.concina@gmail.com
References
Merz, M., Wuthrich, M. (2010). Paidincurred chain claims reserving method. Insurance: Mathematics and Economics, 46(3), 568579.
See Also
MackChainLadder
,MunichChainLadder
Examples
1  PaidIncurredChain(USAApaid, USAAincurred)

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