A reserve is an estimate, and every estimate carries uncertainty. This mini-course shows how to measure it with the two most widely used stochastic models for triangles, what their results mean and what they leave out.
By the end of this mini-course you will be able to:
The best estimate is a mean. For the risk margin, capital, the IFRS 17 risk adjustment, or simply to know whether a deviation is normal, you need a measure of dispersion around that mean.
Prediction error has two parts: process error, because even with the exact model future claims are random, and estimation error, because the factors were estimated from limited data. In small triangles the second usually weighs more than people expect.
Mack (1993) showed that the chain ladder with weighted factors is unbiased under three assumptions: the mean of the next value is proportional to the current one, so is its variance (with parameter \(\sigma_j^2\)), and origin years are independent. Without assuming any distribution, he derived the standard error:
The first term in the parentheses is process error; the second is estimation error. The error of the total is not the sum of the errors by year, because every year shares the same estimated factors. The original version excludes the tail; Mack published an extension in 1999 that includes it.
Renshaw and Verrall (1998) showed that an over-dispersed Poisson (ODP) generalized linear model reproduces the chain ladder. England and Verrall (2002) describe how to simulate its distribution: (1) fit the model and compute expected increments; (2) obtain Pearson residuals and adjust them for degrees of freedom; (3) resample the residuals to build thousands of synthetic triangles; (4) in each one, re-estimate the factors and project the future adding process noise.
The result is a full distribution of the reserve, from which you read the mean, standard deviation and percentiles. It requires positive column sums of increments and residuals without patterns; if they show patterns, the model needs adjusting before use.
Bootstrap percentiles feed the IFRS 17 risk adjustment, economic capital and stress tests. Always review the coefficient of variation by year: mature years can be volatile in relative terms even when the amount is small.
No triangle model captures a regime change that is not in the data: a legal reform, inflation different from the past, or a change in mix. Those risks are added with judgment, scenarios or explicit margins. Estimated variability is a floor, not a ceiling.
Using the series triangle, we measure chain ladder variability without a tail, because neither model in its basic form includes one. That is why base IBNR is lower than in mini-course 2: the difference is exactly the effect of the 1.01 tail. Amounts in millions of pesos.
| Year | IBNR, no tail | Standard error | Coefficient of variation |
|---|---|---|---|
| 2021 | 0.0 | 0.0 | |
| 2022 | 77.9 | 7.0 | 9.0% |
| 2023 | 271.9 | 19.0 | 7.0% |
| 2024 | 768.8 | 47.2 | 6.1% |
| 2025 | 2,073.2 | 85.5 | 4.1% |
| Total | 3,191.7 | 110.9 | 3.5% |
| Statistic | Value |
|---|---|
| Mean | 3,192.9 |
| Standard deviation | 129.2 |
| 50th percentile | 3,193.0 |
| 75th percentile | 3,280.3 |
| 90th percentile | 3,361.1 |
| 99.5th percentile | 3,519.7 |
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Primary sources and reference texts, all publicly accessible at the time of publication.
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