CDRboot — Tweedie GLM Bootstrap Reserve Risk Model

Overview

CDRboot estimates the distribution of the one-year technical result (Claims Development Result) for non-life insurance reserve risk, as required under Solvency 2.

The capital requirement for reserve risk is defined as:

VaR(0.995) of L   where   L = C1 + R1 - R0

(equivalently L = -CDR with CDR = R0 - C1 - R1)

• R0 — opening reserves (IBNR reserves at the beginning of the year)
• C1 — claims payments during the year
• R1 — closing reserves at the end of the year (re-reserving)

The model addresses the three main challenges of Solvency 2 reserve risk modelling:

1. Process variance — inherent variability in future claim payments, captured via simulation
2. Parameter variance — uncertainty in GLM parameter estimation, captured via bootstrap resampling of Pearson residuals
3. Re-reserving — the dependency between next-year payments and closing reserves, handled by re-fitting a GLM on the extended triangle

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Methodology

The model follows the double bootstrap re-reserving methodology introduced by Diers (2007) and Ohlsson & Lauzenings (2008), combined with Tweedie GLMs from the exponential dispersion family.

Step 1 — Opening reserves

A Tweedie GLM is fitted on the observed incremental claims triangle using accident year and development year as the only explanatory variables (Chain Ladder structure):

log(E[Cij]) = ci + bj

Opening reserves are the sum of fitted values in the future triangle.

Step 2 — Bootstrap & next-year payment simulation

For each bootstrap replication:

• Adjusted Pearson residuals are resampled with replacement
• A pseudo-triangle is constructed and a new GLM is fitted
• Next-year diagonal payments are simulated from the fitted pseudo-triangle using the selected Tweedie distribution (Poisson, Gamma, Tweedie, or Inverse Gaussian)

Step 3 — Closing reserves (re-reserving)

For each simulation:

• Simulated next-year payments are added to the initial triangle
• A new GLM is fitted on the extended triangle (unobserved future cells remain NA and are excluded from the fit)
• Closing reserves are predicted from this updated fit

Output

The distribution of L = C1 + R1 - R0 is obtained from nBoot × nsim replications. The VaR at 99.5% of this distribution is the Solvency 2 capital requirement for reserve risk.

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Installation

# Install required packages (run once)
install.packages(c("tweedie", "statmod"))

# Load the function
source("CDRboot.R")

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Data

Sample Motor cumulative triangle (10×10, Belgian market illustration):

File	Description
triangle_auto.csv	Cumulative triangle (NA = unobserved future)
triangle_auto.RData	Same triangle as object triangle_auto

# From CSV
y <- as.matrix(read.csv("triangle_auto.csv", header = TRUE, na.strings = ""))

# From RData
load("triangle_auto.RData")
y <- as.matrix(triangle_auto)

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Usage

source("CDRboot.R")
load("triangle_auto.RData")
y <- as.matrix(triangle_auto)

# Smoke test (fast)
set.seed(42)
CDRboot(y = y, g = 1.28, nBoot = 2, nsim = 3)

# Full run (slow)
results <- CDRboot(y = y, g = 1.28, nBoot = 999, nsim = 1000)

Arguments

Argument	Type	Description
y	matrix (n × n)	Cumulative claims triangle
g	numeric	Tweedie power parameter
nBoot	integer	Number of bootstrap replications
nsim	integer	Number of simulations per bootstrap
plot	logical	Draw P&L density plot (default TRUE)

Tweedie power parameter g

Value	Distribution
1	Poisson
1 < g < 2	Tweedie (compound Poisson-Gamma)
2	Gamma
3	Inverse Gaussian

The optimal power can be selected by maximising the quasi log-likelihood over the observed triangle.

Importing your own triangle from a CSV

y <- as.matrix(read.csv("your_triangle.csv", header = TRUE, na.strings = ""))
CDRboot(y = y, g = 1.28, nBoot = 999, nsim = 1000)

Storing results for further analysis

results <- CDRboot(y = y, g = 1.28, nBoot = 999, nsim = 1000)

results$open_reserves   # Opening reserves R0
results$next_yr_tot     # Distribution of next-year payments C1
results$close_res       # Distribution of closing reserves R1
results$pnl             # Distribution of loss L = C1 + R1 - R0

# Capital requirement at 99.5%
quantile(results$pnl, 0.995)

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Output

The function prints:

• Quantiles of next-year payments (C1) at levels 50%, 75%, 90%, 95%, 99%, 99.5%, 99.95%
• Quantiles of closing reserves (R1) at the same levels
• Quantiles of the technical result / loss (P&L) at the same levels
• Mean and standard deviation of the P&L
• Opening reserves R0
• Pearson and Spearman correlations between C1 and R1
• A density plot of the P&L with empirical and normal fit overlay

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Empirical results (from thesis)

Results obtained on Belgian insurance market data (National Bank of Belgium, 10 accident years 2001–2010):

Line of Business	Settlement period	Best GLM	Opening Reserves	Capital Req.	% of Reserves
Motor	Short	Tweedie (1.28)	204,738 k€	23,002 k€	11.2%
Casualty	Medium	Poisson	212,594 k€	21,623 k€	10.2%
Motor TPL	Medium	Gamma	2,038,056 k€	138,000 k€	6.8%
Legal Protection	Long	Poisson	462,871 k€	35,464 k€	7.7%

Key findings:

• Re-reserving reduces capital requirements by 10–20% for lines with medium and long settlement periods
• Ignoring parameter variance (bootstrap) leads to serious underestimation of risk, especially for long-tail lines
• The Tweedie GLM outperforms Poisson in motor insurance, confirming that all members of the exponential dispersion family should be considered

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Dependencies

Package	Role
tweedie	rtweedie() simulation
statmod	Tweedie GLM family and rinvgauss() for Inverse Gaussian (g = 3)

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References

• Diers, D. (2008). Stochastic re-reserving in multi-year internal models. ASTIN Colloquium, Helsinki.
• England, P. & Verrall, R.J. (2002). Stochastic claims reserving in general insurance. British Actuaries Journal, 8(3).
• Merz, M. & Wüthrich, M.V. (2008). Modelling the claims development result for solvency purposes. CAS E-Forum.
• Ohlsson, E. & Lauzenings, J. (2008). The one-year non-life insurance risk. ASTIN Colloquium, Manchester.
• Renshaw, A.E. & Verrall, R.J. (1998). A stochastic model underlying the chain-ladder technique. B.A.J. 4.
• Wüthrich, M.V. (2003). Claims reserving using Tweedie's compound Poisson model. ASTIN Bulletin.

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Related projects

Part of a non-life quantitative risk series:

Project	Focus	Repo
Claim frequency	GLM vs ML for P(claim)	EDA-GLM-RF-XGB
Aggregate loss (this repo)	Frequency–severity Monte Carlo, VaR/TVaR	mtpl-loss-model
Reserve risk (Solvency II)	Tweedie GLM + double bootstrap CDR	CDRboot

Full portfolio overview: philippehardydata

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Author

Philippe le Hardÿ — Actuarial & Quantitative Risk Consultant
GitHub