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)



Cumulative claims payments triangle


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 Θ = (Φ_0,...,Φ_I, Ψ_0,...,Ψ_{I-1},σ_0,...,σ_{I-1},τ_0,...,τ_{I-1}) we have

    • the random vector (ξ_{0,0},...,ξ_{I,I}, ζ_{0,0},...,ζ_{I,I-1}) 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,j-1} \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,j-1} = I_{i,j} \exp(-ζ_{i,j-1}),

      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_{J-1} 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.


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,


Merz, M., Wuthrich, M. (2010). Paid-incurred chain claims reserving method. Insurance: Mathematics and Economics, 46(3), 568-579.

See Also



PaidIncurredChain(USAApaid, USAAincurred)

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