PaidIncurredChain {ChainLadder} | R Documentation |
The Paid-incurred Chain model (Merz, Wuthrich (2010)) combines claims payments and incurred losses information to get a unified ultimate loss prediction.
PaidIncurredChain(triangleP, triangleI)
triangleP |
Cumulative claims payments triangle |
triangleI |
Incurred losses triangle. |
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 Log-Normal PIC Model are the following:
Conditionally, given \Theta = (\Phi_0,...,\Phi_I,
\Psi_0,...,\Psi_{I-1},\sigma_0,...,\sigma_{I-1},\tau_0,...,\tau_{I-1})
we have
the random vector (\xi_{0,0},...,\xi_{I,I},
\zeta_{0,0},...,\zeta_{I,I-1})
has multivariate Gaussian distribution
with uncorrelated components given by
\xi_{i,j} \sim N(\Phi_j,\sigma^2_j),
\zeta_{k,l} \sim N(\Psi_l,\tau^2_l);
cumulative payments are given by the recursion
P_{i,j} = P_{i,j-1} \exp(\xi_{i,j}),
with initial value P_{i,0} = \exp (\xi_{i,0})
;
incurred losses I_{i,j}
are given by the backwards
recursion
I_{i,j-1} = I_{i,j} \exp(-\zeta_{i,j-1}),
with initial value I_{i,I}=P_{i,I}
.
The components of \Theta
are independent and
\sigma_j,\tau_j > 0
for all j.
Parameters \Theta
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 \sigma_0,...,\sigma_J
and
\tau_0,...,\tau_{J-1}
and
\Phi_m \sim N(\phi_m,s^2_m),
\Psi_n \sim N(\psi_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.
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.
The model is implemented in the special case of non-informative priors.
Fabio Concina, fabio.concina@gmail.com
Merz, M., Wuthrich, M. (2010). Paid-incurred chain claims reserving method. Insurance: Mathematics and Economics, 46(3), 568-579.
MackChainLadder
,MunichChainLadder
PaidIncurredChain(USAApaid, USAAincurred)