Expected Claims with Random Frequency and Reporting Delays

Summary

The document formulates an insurance claims problem in which policyholder heterogeneity is represented by a mixed Poisson process: the Poisson intensity is scaled by an independent exponential random variable. It then distinguishes accident or claim arrival times from later reporting times, with independent uniform delays, and defines the count of claims reported by a given horizon.

The goal is to find the expected aggregate amount of reported claims and compare it with the expected total amount based on the underlying claim count. The setup highlights that delayed reporting changes which claims are included by a fixed date, even when claim sizes are independent and identically distributed. However, the document poses the problem without deriving the expectation or supplying numerical or empirical evidence. Its definitions and assumptions are the useful content; conclusions require carefully handling the random intensity, delay window, and the claim size mean.

Key ideas

• A mixed Poisson process models claim counts with a random intensity multiplier.
• Claims may be counted at their reporting times rather than their occurrence times.
• The reported count by a horizon depends on both arrival timing and reporting delay.
• Expected aggregate claims require assumptions about claim sizes and their relation to the count process.
• The document states a mathematical problem but does not provide its solution.

Tags

# Poisson modelling of non-life insurance claims with reporting delay


# Poisson modelling of non-life insurance claims with reporting delay












I am considering a portfolio of car insurance policies. In order to capture the individual history (driving skills, age, etc.) of policyholders, it is assumed that the claim numbers $N(t)$ are modeled by a mixed Poisson process, that is: $$N(t) := \hat{N}(\theta t), \quad t>0$$ where $\theta\sim\mathcal{E}(1)$ is the exponentially-distributed mixing variable, which is independent of the Poisson process $\hat{N}(t)$ and with intensity $1$.

However, in my case, general claims in an insurance portfolio are not reported at the arrival times $T_i$ but at times $T_i + V_i$ with a delay $V_i > 0$. I assume that $V_i\sim U(0,1)$ is uniformly-distributed.

An example for such delays is a policyholder who is injured in a car accident and does not have the opportunity to call his agent, immediately.

So I derive the number of claims reported up to time $t$, given by: $$\bar{N}(t):=\sum_{j=1}^{N(t)}1_{[0,t]}(T_j+V_j)$$

I assume that $V_j$ is independent of $T_j$ for each $j$.

I assume that the claim sizes $X_i$ are I.I.D and I want to determine the expected total claim amount $\bar{S}(t)$, that is $\mathbb{E}[\bar{S}(t)]$, where, $$\bar{S}(t):=\sum_{j=1}^{\bar{N}(t)}X_j.$$ Furthermore, I also want to compare it with the expected value of $S(t)$ (with respect to $N(t)$).

Thanks in advance.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.