The Run-Off Triangle
A run-off triangle (development triangle) tabulates cumulative paid claims $C_{i,j}$ where:
- $i = 1, \ldots, n$ is the accident year (or underwriting year)
- $j = 1, \ldots, n$ is the development year
- Only cells with $i + j - 1 \le n$ are observed (the “upper triangle”)
The lower triangle ${C_{i,j} : i + j - 1 > n}$ must be projected to estimate the outstanding claims reserve.
Example triangle (cumulative paid losses, £000s):
| Acc.\Dev. | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| 2020 | 1,001 | 1,855 | 2,423 | 2,988 | 3,410 |
| 2021 | 1,113 | 2,103 | 2,774 | 3,422 | ? |
| 2022 | 1,265 | 2,433 | 3,233 | ? | ? |
| 2023 | 1,490 | 2,873 | ? | ? | ? |
| 2024 | 1,725 | ? | ? | ? | ? |
IBNR (Incurred But Not Reported) reserves cover claims that have occurred but not yet been reported. IBNER (Incurred But Not Enough Reserved) covers reported claims still developing. The total Outstanding Claims Provision includes both.
Chain-Ladder Method
The chain-ladder (development method) assumes that cumulative claims develop by age-to-age factors that are stable across accident years.
The development factor from column $j$ to $j+1$ is:
\[\hat{f}_j = \frac{\sum_{i=1}^{n-j} C_{i,j+1}}{\sum_{i=1}^{n-j} C_{i,j}}, \quad j = 1, \ldots, n-1\]This is a volume-weighted average: larger triangles in column $j$ receive proportionally more weight.
Projection algorithm:
- Compute all $\hat{f}_j$ from the observed upper triangle.
- Project: $\hat{C}{i,j+1} = \hat{C}{i,j} \cdot \hat{f}_j$ for each unseen cell, row by row.
- The reserve for accident year $i$ is: $\hat{R}i = \hat{C}{i,n} - C_{i,n+1-i}$ (projected ultimate minus current diagonal).
- Total reserve: $\hat{R} = \sum_i \hat{R}_i$.
The chain-ladder is equivalent to the weighted least squares regression of $C_{i,j+1}$ on $C_{i,j}$ with weights $1/C_{i,j}$, providing a least-squares interpretation.
Tail factor: Beyond the last observed development year, a tail $\hat{f}{\infty}$ is applied. Common methods: curve-fitting (exponential, inverse power), benchmark from industry data, or $\hat{f}{\infty} = 1.000$ (assuming full development).
Bornhuetter-Ferguson Method
The Bornhuetter-Ferguson (BF) method blends the chain-ladder projection with an a priori (external) loss estimate $\mu_i$ (e.g., from pricing):
\[\hat{C}_{i,n}^{\text{BF}} = C_{i,j_i} + \mu_i \cdot (1 - 1/\hat{f}_i^{\text{ult}})\]where $j_i = n + 1 - i$ is the current development year for accident year $i$, and $\hat{f}i^{\text{ult}} = \hat{f}{j_i} \cdot \hat{f}{j_i+1} \cdots \hat{f}{n-1}$ is the cumulative development factor to ultimate.
The factor $(1 - 1/\hat{f}_i^{\text{ult}})$ is the expected unreported percentage, so BF adds the expected unreported IBNR to the actual paid claims.
Comparison with chain-ladder:
| Method | Ultimate = | Weight on data | Weight on prior |
|---|---|---|---|
| Chain-ladder | $C_{i,j_i} \times \hat{f}_i^{\text{ult}}$ | 100% | 0% |
| BF | $C_{i,j_i} + \mu_i(1-1/\hat{f}_i^{\text{ult}})$ | $1/\hat{f}_i^{\text{ult}}$ | $1 - 1/\hat{f}_i^{\text{ult}}$ |
| Bornhuetter (credibility) | $Z_i\bar{X}_i + (1-Z_i)\mu_i$ | $Z_i$ | $1-Z_i$ |
BF is preferred for immature accident years (small $1/\hat{f}^{\text{ult}}$) where few claims have been paid and the chain-ladder amplifies noise.
Cape Cod Method
The Cape Cod (Bornhuetter-Ferguson with endogenous a priori) avoids specifying $\mu_i$ externally. Instead, the expected loss ratio is estimated from the data:
\[\hat{\text{ELR}} = \frac{\sum_i C_{i,j_i}}{\sum_i P_i / \hat{f}_i^{\text{ult}}}\]where $P_i$ is the premium for accident year $i$ and $P_i / \hat{f}_i^{\text{ult}}$ is the used-up premium (the portion of premium that covers already-developed claims). Then:
\[\mu_i = \hat{\text{ELR}} \times P_i\]Cape Cod is particularly useful when the a priori loss ratio varies across accident years and no external benchmark is available.
Mack’s Model and Uncertainty
Mack’s model (1993) provides a distribution-free framework for quantifying the prediction error of the chain-ladder reserve, without assuming a parametric distribution.
Assumptions:
- $E[C_{i,j+1} \mid C_{i,1}, \ldots, C_{i,j}] = C_{i,j} \cdot f_j$ (development factors explain all structure)
- $\text{Var}(C_{i,j+1} \mid C_{i,j}) = C_{i,j} \cdot \sigma_j^2$ (variance proportional to cumulative)
- Accident years are independent
Under these assumptions, Mack derived:
\[\widehat{\text{MSE}}(\hat{R}_i) = \hat{C}_{i,n}^2 \sum_{j=j_i}^{n-1} \frac{\hat{\sigma}_j^2 / \hat{f}_j^2}{\sum_{k=1}^{n-j} C_{k,j}}\]with $\hat{\sigma}j^2 = \frac{1}{n-j-1}\sum{i=1}^{n-j} C_{i,j}!\left(\frac{C_{i,j+1}}{C_{i,j}} - \hat{f}_j\right)^2$.
The total reserve standard error is:
\[\text{SE}(\hat{R}) = \sqrt{\sum_i \widehat{\text{MSE}}(\hat{R}_i) + 2\sum_{i<k} \text{Cov}(\hat{R}_i, \hat{R}_k)}\]Stochastic Reserving: ODP and Bootstrap
The Over-Dispersed Poisson (ODP) model assumes incremental payments $c_{i,j} = C_{i,j} - C_{i,j-1}$ satisfy:
\[E[c_{i,j}] = x_i\,y_j, \quad \text{Var}(c_{i,j}) = \phi\,x_i\,y_j\]where $x_i$ are accident year parameters, $y_j$ are development year parameters, and $\phi > 0$ is the dispersion parameter ($\phi = 1$ is standard Poisson). MLE of $x_i, y_j$ exactly reproduces the chain-ladder development factors.
The bootstrap ODP method:
- Fit the ODP model to obtain Pearson residuals $r_{i,j} = (c_{i,j} - \hat{c}{i,j})/\sqrt{\hat{c}{i,j}}$.
- Resample residuals with replacement; reconstruct a pseudo-triangle.
- Re-fit the chain-ladder to each pseudo-triangle and project.
- Repeat 10,000 times to obtain the empirical distribution of the reserve.
This distribution captures process variance (random claim payments) and parameter uncertainty (estimation error in the factors).
IFRS 17 implications: Under IFRS 17, reserves must be a contractual service margin (CSM) plus risk adjustment (RA). The RA quantifies non-financial risk using a confidence interval (TVaR at 75% is common), making stochastic reserving methods an accounting requirement rather than merely an internal actuarial tool.