Loss Reserving

Loss Reserving

Chain Ladder, Mack Standard Errors, Bornhuetter–Ferguson, Cape Cod and the Bootstrap

The run-off triangle

Claims from one accident (or underwriting, report) year are paid and reported over several years. Arranging the cumulative amounts C_i,k with the accident year i down the rows and the development age k across the columns gives a triangle whose lower right is still unknown: the oldest year has every age, the most recent only its first. The last cell of each row is the latest diagonal, the amount known at the valuation date. The reserving problem is to complete the triangle: the completed last column is the ultimate claims U_i of each year, and the difference U_i − C_i,latest is the unpaid (for a paid triangle) or the IBNR in the broad sense (for a reported triangle, unreported plus development on known claims). The tool takes the triangle cumulative, or incremental and sums each row; each line may start with the year as a label. Thousands separators inside a number are removed ("37,017,487"), so a comma-separated list whose values are all groups of three digits would be read as one number: separate such values with spaces or tabs.

Age-to-age factors and their averages

The development method (Friedland 2010, chapter 7) assumes that the ratio of consecutive cumulative amounts, F_i,k = C_i,k+1 / C_i,k, is the same for every accident year apart from noise. The individual factors are averaged down each column:

• Volume-weighted average over all years, f_k = Σ_i C_i,k+1 / Σ_i C_i,k: the classical chain ladder, each year weighted by its claims, and the estimator Mack's model is built on.
• Simple (arithmetic) average of the individual factors, which gives every year the same weight.
• The same two averages over the latest n years only, when the older years are no longer representative of current case reserving or settlement practice.
• Medial average: the latest n factors less one high and one low (Friedland's "latest 5 × 1"); with fewer than three factors it is the simple average.
• Geometric average of the latest n, the n-th root of their product.

Every average is tabulated and the analyst chooses one as the selected factor for all ages; a judgemental selection column by column is not offered. The selected factors are multiplied from the last age back to each earlier age to give the cumulative development factors CDF_k = f_k f_k+1 ⋯ f_K × tail, whose reciprocal is the percent developed (reported or paid) at age k.

Tail factor

When development has not finished at the last observed age, a tail factor carries the last column to ultimate. It may be entered (an industry benchmark, the reported-to-paid ratio at the last age, or a curve fitted elsewhere) or fitted here to the selected factors above 1.000 by one of the two curves Sherman (1984) and the CAS Tail Factors Working Party describe: exponential decay, ln(f_k − 1) = a + b k, and the inverse power curve, ln(f_k − 1) = a + b ln k. The tail is the product of the extrapolated factors over the next 100 development periods. An exponential fit whose slope is not negative, or an inverse power fit whose exponent is not below −1, does not converge and is refused with a note. The tail multiplies every year's ultimate and so has leverage over the whole reserve; it deserves the analyst's attention more than any single factor.

Mack's standard errors

Mack (1993) showed that the chain ladder is the estimate of a stochastic model with three assumptions: the expected next cumulative amount is C_i,k f_k given the row so far; its variance is C_i,k σ_k², proportional to the amount already observed; and accident years are independent. Under them the volume-weighted factors are unbiased and uncorrelated, the variance parameters are estimated by

σ̂_k² = Σ_i C_i,k (F_i,k − f_k)² / (I − k − 1),

and the last one, which has a single factor and no degrees of freedom, by the extrapolation σ̂_K² = min(σ̂_K−1⁴ / σ̂_K−2², min(σ̂_K−2², σ̂_K−1²)), which continues an exponentially decreasing series. The mean squared error of a year's reserve (Theorem 3 of the paper) is

mse(R_i) = U_i² Σ_{k ≥ I+1−i} (σ̂_k² / f_k²) (1 / C_i,k + 1 / Σ_{j ≤ I−k} C_j,k),

the first term in the bracket the process variance (the future is random), the second the estimation variance (the factors are estimated from a finite triangle). The total reserve's mean squared error adds the covariances between years, which share the same estimated factors (the Corollary); the tool reports it as the estimation column of the total row. The standard error is the root of the mse and the coefficient of variation its ratio to the reserve. Mack's own reading: a reserve from another method within one standard error of the chain ladder is not significantly different, and a young year with a small reserve carries a large coefficient of variation (80 % for accident year 2 of the Taylor–Ashe triangle) because the reserve itself is small, not because the data are poor. The standard errors belong to the volume-weighted chain ladder without a tail; when another average or a tail is selected the tool runs Mack's model separately and lists its reserves beside the selected ones (the tail extension of Mack 1999 is not implemented).

Expected claims, Bornhuetter–Ferguson and Cape Cod

The development method multiplies the latest diagonal by the CDF, so for an immature year a small random amount is leveraged by a large factor (14.4 for the first year of the Taylor–Ashe triangle). Premium-based methods (Friedland 2010, chapters 8–10) replace that leverage by an a priori expectation:

• Expected claims: U_i = P_i × ELR_i, the earned premium times an a priori expected loss ratio, independent of the claims reported so far; its reserve may be negative when the claims to date already exceed the expectation.
• Bornhuetter–Ferguson (1972): the expected claims are used only for the part not yet developed, R_i = P_i ELR_i (1 − 1/CDF_i) and U_i = C_i,latest + R_i. It is a credibility weighting of the development and expected-claims ultimates with the percent developed as the weight, and is the standard method for immature years.
• Cape Cod (Stanard 1985, Bühlmann): the same formula with the loss ratio estimated from the triangle itself, ELR = Σ C_i,latest / Σ P_i / CDF_i, the denominator the used-up (earned and developed) premium. With Gluck's (1997) decay factor d < 1 each year i gets its own ratio in which year j is weighted by d^|i−j|, so that a trend in the loss ratio is followed; d = 1 is one ratio for all years.

The premium box takes one earned premium per accident year and, optionally, that year's a priori loss ratio; a common ratio can be entered instead. Premiums should be on the same basis as the claims (on-level and trended where the book has changed); the tool makes no adjustment.

Bootstrap of the over-dispersed Poisson chain ladder

Renshaw and Verrall (1998) showed that a generalized linear model for the incremental claims X_i,k with an over-dispersed Poisson distribution, E[X_i,k] = m_i,k, Var[X_i,k] = φ m_i,k, log link and a row and a column parameter gives exactly the chain-ladder forecasts. England and Verrall (1999, 2002) use it to obtain the predictive distribution of the reserve by a two-stage bootstrap:

1. Fit: the chain-ladder fitted incrementals are obtained by running the latest diagonal backwards with the volume-weighted factors and differencing; the Pearson residuals r = (X − m)/√m are scaled by √(n/(n − p)), n the number of cells and p = 2I − 1 parameters, to allow for the degrees of freedom; the dispersion is φ = Σ r² / (n − p). The two corner cells (first row, last column and last row, first column) have zero residuals by construction and are left out of the resampling set.
2. Resample: a pseudo-triangle is built from X* = m + r* √m with residuals drawn with replacement, cumulated, and projected by the chain ladder; this gives the estimation error. Each future increment is then replaced by a draw from a gamma (or a scaled Poisson) distribution with mean equal to the forecast and variance φ × mean; this adds the process error. The sum of the draws is one bootstrap reserve.

Repeated R times, the bootstrap reserves form the predictive distribution: their mean (slightly above the chain-ladder reserve, the reserve being a convex function of the data), their standard deviation (the prediction error, comparable with the analytic root-mse of the ODP model and with Mack's standard error), and their percentiles. The 75th and 95th percentiles are the usual reporting points; the 99.5th is the quantile of the ultimate reserve risk that capital standards refer to. Mack's and the bootstrap's errors differ because the variance assumptions differ (proportional to the cumulative amount in Mack, to the increment in the ODP model); on the Taylor–Ashe triangle Mack gives a 13 % coefficient of variation and the bootstrap about 16 %. The seed makes a run reproducible; 1,000 resamples take a fraction of a second, 20,000 under a second for a ten-year triangle.

Reading the results

• Figure 1, the development pattern: parallel curves support the common-pattern assumption; a year that crosses the others, or a kink along a calendar diagonal, points to a change in case reserving, settlement speed or mix that a latest-n average or a Berquist–Sherman adjustment should address before the factors are trusted.
• Figure 2, the factor plot: the scatter at each age is what σ_k measures; a drift of the recent years' factors away from the older ones at one age is the reason to use a latest-n average.
• The comparison table: the development method responds fully to the latest diagonal, the expected-claims method not at all, Bornhuetter–Ferguson in between, and Cape Cod differs from Bornhuetter–Ferguson only where the a priori ratio differs from the triangle's own experience. Differences within one Mack standard error are not significant.
• The worked lines on the page are for the most recent accident year, the one the methods disagree about most.

Worked examples on the page

The default triangle is the Taylor and Ashe (1983) paid-claims triangle as printed in Mack (1993, Table 1), also the GenIns data set of the R package ChainLadder: factors 3.49, 1.75, 1.46, 1.174, 1.104, 1.086, 1.054, 1.077, 1.018; σ_k²/1000 of 160, 37.7, 42.0, 15.2, 13.7, 8.19, 0.447, 1.15, 0.447 (the paper prints 0.477 for the last, a misprint of its own rule); total reserve 18,680,856 with standard error 2,447,095 (13 %); the ODP dispersion 52,601. The example selector loads Friedland's U.S. Industry Auto reported and paid triangles with their premiums and a priori ratios (chapters 7 to 10: selected factors by the simple average of the latest three years; the exhibits print the factors and CDFs rounded to three decimals and multiply with the rounded values, giving a reported ultimate of 569,172,456, expected claims 569,281,839, Bornhuetter–Ferguson 569,091,348 and Cape Cod 570,800,677 with the 69.5 % ratio; the page works with the unrounded averages and lands within 0.1 % of them, the expected claims exactly), the RAA triangle of the ChainLadder vignette (total IBNR 52,135, Mack s.e. 26,909) and the England and Verrall (2001) incremental triangle with a negative cell. The test script checks the library against all of them and against a numpy bootstrap of 200,000 resamples.

Limits of the tool

• One selected average for every age: factors cannot be chosen column by column, and no adjustment of the triangle (Berquist–Sherman, on-level premiums, large-claim capping) is made.
• Mack's standard errors are for the volume-weighted chain ladder without a tail; the Mack (1999) tail extension and the Munich chain ladder (paid and incurred together) are not offered.
• Annual development is assumed for the tail curves; quarterly triangles fit but the extrapolation horizon of 100 periods then covers 25 years.
• The bootstrap is the ODP residual bootstrap only; the one-year (solvency) view of Merz and Wüthrich, the gamma GLM and the lognormal models of England and Verrall are not included.
• Negative incremental cells are accepted by the chain ladder and the residuals, but a fitted increment that is negative gets no process error.