Predicting IBNYR Events and Delays II. Discrete Time

Abstract

An IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. A previous paper developed a continuous-time model of the IBNYR process in which both the Poisson rate at which events occur and the parameters of the delay distribution are unknown random quantities; a full-distributional Bayesian method was then developed to predict the number of unreported events. Using a numerical example, the success of this approach was shown to depend upon whether or not the occurrence dates were available in addition to the reporting dates. This paper considers the more usual practical situation in which only discretized epoch information is available; this leads to a loss of predictive accuracy, which is investigated by considering various levels of quantization for the same numerical example.

Understanding Reporting Delay in General Insurance

“…We extend the model considered in Jewell [1,2] to an inhomogeneous marked Poisson point process. We define Λ(t) ≥ 0 to be the instantaneous claims frequency and w(t) ≥ 0 to be the instantaneous exposure at time t ≥ 0.…”

“…From Jewell [1,2] and Norberg [5,6] we immediately obtain the following likelihood function for parameters Λ and Θ L N,(T ,S ) =1,...,N (Λ, Θ) = e…”

“…The lower limit was chosen to be u (1) = 7 days. For u (2) we test different reporting delays κ = 3, 6, 9 or 12 months. Note that κ months is not well-defined in terms of number of days.…”

“…Note that κ months is not well-defined in terms of number of days. We set u (2) equal to 89, 181, 273 or 365 which is the minimal number of days that κ consecutive calendar months can have. The accident periods (τ i−1 , τ i ] with i ≤ m − κ are then fully observed in the middle layer at time τ m , and we have…”

“…We study this reporting delay function in continuous time based on real individual claims data. The data is used to calibrate a non-stationary version of the statistical model considered in Jewell [1,2]. This calibration then provides an estimation of the number of incurred but not yet reported (IBNYR) claims.…”