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It is well known that when (re)insurance coverages involve a deductible, the impact of inflation of loss amounts is distorted, and the changes in claims paid by the (re)insurer cannot be assumed to reflect the rate of inflation. A particularly interesting phenomenon occurs when losses follow a Pareto distribution. In this case, the observed loss amounts (those that exceed the deductible) are identically distributed from year to year even in the presence of inflation. Nevertheless, in this paper we succeed in estimating the inflation rate from the observations. We develop appropriate statistical inferential methods to quantify the inflation rate and illustrate them using simulated data. Our solution hinges on the recognition that the distribution of the number of observed losses changes from year to year depending on the inflation rate.

A number of challenges arise when an insurance policy covers only loss amounts that exceed a threshold known as the deductible. The insurer typically does not know about losses that are less than this amount, making appropriate characterization of the loss distribution impossible. This can even give rise to misleading and/or paradoxical observations about the distribution.

An interesting example of this has been observed in actuarial practice. A reinsurer desired to understand the impact of inflation on loss amounts. However, upon exploring the losses that were reported to the reinsurer, it was found that no inflation was present. The losses reported to the reinsurer were only those that exceeded a fixed deductible, which did not change over time as is typically the case. The losses reported in different years had near identical distributions. Specifically, the reinsurer found that the distribution of reported losses in each year could be accurately described by the same Pareto distribution. Moreover, attempts to model inflation by employing various macroeconomic indexes (e.g., consumer price index) also failed to yield satisfactory results as the reinsurance data was industry specific. The details of this problem were obtained through personal communications with reinsurance industry practitioners.

The Pareto distribution arises quite often in modelling insurance losses. This distribution uniquely possesses a property that gives rise to the reinsurer's observation regarding the inflation of loss amounts.

To examine this phenomenon statistically, we simulated losses corresponding to 10 successive years. The numbers of losses in these years is assumed to be independent Poisson random variables with mean 1000, and all loss amounts are independent. These are common assumptions in insurance loss modelling. The losses occurring during the

Box-and-whisker plots of all loss amounts (a) and observed loss amounts (b).

All loss amounts

Observed loss amounts

The left-hand graph shows box-and-whisker plots of loss amounts in each year. Each box extends from the first quartile to the third quartile, with the median indicated by the line inside the box. The whiskers extend to the most extreme observations that are not more than 1.5 times the interquartile range outside the box. We see very clearly from the left-hand graph the impact that inflation has on the loss distribution. The right-hand graph in Figure

Table

Summary of simulated loss data.

Year | Number of losses | Average loss | Number of observed losses | Average of observed losses | Sum of observed losses |

1 | 1004 | 1.9813 | 37 | 365.3071 | |

2 | 971 | 2.1358 | 43 | 10.6640 | 458.5501 |

3 | 1029 | 2.1206 | 44 | 415.3972 | |

4 | 1063 | 2.3359 | 56 | 554.3648 | |

5 | 1026 | 2.3554 | 62 | 509.0030 | |

6 | 1030 | 2.5579 | 78 | 718.5715 | |

7 | 1003 | 2.7498 | 75 | 10.7216 | 804.1190 |

8 | 955 | 2.7866 | 71 | 10.0545 | 713.8679 |

9 | 982 | 3.1771 | 89 | 12.1130 | 1078.0582 |

10 | 1029 | 3.0533 | 92 | 906.5962 |

The rest of the paper is organized as follows. In Section

If we were observing every loss, then we would have a realization of the following array of random variables:

Suppose that the annual inflation rates for the observation period are represented by

We first present a simple and intuitively appealing approach to estimating the inflation rate when we assume that it is the same in each year. We can also view this as a method of estimating the average inflation rate during the observation period. That is, the inflation rate

We assume that

Now since the number of losses

Notice that the right-hand side of (

This approach allows us to estimate

A more general approach involves estimating the parameters

Note that we have an identifiability problem because

By cancelling multiplicative constants in the likelihood function and taking logs, we have

The latter approach does not assume any structure between

In practice, rather than simply assuming that all

In this section we provide numerical illustrations of the methods presented in Section

Applying the first method, we can estimate

In practice, we do not know that the loss inflation rate is the same each year, and our full maximum likelihood approach allows us to estimate the individual inflation rates. The estimates reported in Table

Maximum likelihood estimates of

2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|

0.0786 | 0.0116 | 0.1291 | 0.0526 | 0.1226 | −0.0196 | −0.0272 | 0.1205 | 0.0168 |

If we then impose the restriction that the inflation rate is the same each year, we can obtain the maximum likelihood estimate of

Point estimates and approximate 95% confidence intervals of

Full likelihood approach | First approach | |||

Parameter | Estimate | Asymptotic CI | Estimate | Bootstrap CI |

0.0503 | 0.0526 | |||

1.9858 | 1.9858 |

Having maximized the log-likelihood with and without the restriction that the inflation rate in each year is the same, we can perform a likelihood ratio test of the hypothesis that the inflation rates are the same. Using the LRT statistic in (

The authors sincerely thank Editor Tomasz J. Kozubowski and two anonymous referees for queries and suggestions that have guided them in revising the paper. The first author gratefully acknowledges the stimulating scientific atmosphere at the 38th ASTIN Colloquium in Manchester, United Kingdom, and Michael Fackler in particular for posing the problem whose solution makes the contents of the present paper. The second and third authors are grateful to the University of Wisconsin-Milwaukee for the most productive and pleasant stay during which results of the present paper had evolved to fruition.