This code is provided by Peter England on behalf of EMC Actuarial and Analytics Ltd as an educational resource.
This repository includes stochastic reserving functions in R (with R Markdown) and Python (with Jupyter) to reproduce the examples in:
England & Verrall (2006). Predictive distributions of outstanding liabilities in general insurance. Annals of Actuarial Science, 1, II, 221-270. https://doi.org/10.1017/S1748499500000142
and
England, Verrall & Wüthrich (2018/2019). On the lifetime and one-year views of reserve risk, with application to IFRS 17 and Solvency II risk margins. Insurance: Mathematics and Economics (2019) https://doi.org/10.1016/j.insmatheco.2018.12.002. A preprint is also available at SSRN (2018) https://ssrn.com/abstract=3141239
England & Verrall (2006) shows how predictive distributions of outstanding liabilities in general insurance can be obtained using bootstrap or 'Bayesian' MCMC techniques for clearly defined statistical models.
England, Verrall & Wüthrich (2018/2019) brings together analytic and simulation-based approaches to reserve risk, applied to the traditional actuarial view of risk over the lifetime of the liabilities and to the one-year view of Solvency II. It also connects the lifetime and one-year views of risk. Predictive distributions are used to estimate risk margins under Solvency II and risk adjustments under IFRS 17.
Also included is an example modus operandi for stochastic reserving when faced with a new claims triangle. The example shows the steps to go through and the exhibits to explore to understand what is driving variability.
The functions and examples include the following:
- Methods for the lifetime and one-year (and beyond) views of risk
- Analytic (closed form), bootstrap, and 'Bayesian' MCMC approaches
- Mack's model, the overdispersed Poisson chain ladder model (with constant and non-constant scale parameters), and the overdispersed negative binomial chain ladder model (with constant and non-constant scale parameters)
- Cost-of-Capital risk margins for Solvency II
- Risk adjustments for IFRS 17 using risk measures applied to the distribution of discounted outstanding liabilities
- Sensitivity analysis to identify influential data points
Also included are the following useful features:
- Parametric bootstrapping options, in addition to the traditional non-parametric approach
- User-defined scale (variance) parameters at the forecasting stage for bootstrap and MCMC methods
- Scaling of simulated reserves to target estimates of ultimate claims using multiplicative or additive scaling
Specific details for Python and R can be found in the README files in their respective folders.
The code is provided as-is and may contain errors. Questions and reporting of issues to peter@emc-actuarial.com are welcome but there is no active support and a response is not guaranteed.
Incremental claims triangle:
Incremental claims graph:
Cumulative claims graph:
Link ratios triangle, with volume-weighted chain ladder development factors:
Link ratios graph:
Analytic results:
Bootstrap results for the lifetime and one-year views of risk:
Sequence of one-year views of risk, connecting the one-year view with the lifetime view:
Example reserve development (fan) chart for a single origin year:
Solvency II cost-of-capital risk margin:
IFRS 17 risk adjustments using risk measures applied to the distribution of discounted outstanding liabilities:
IFRS 17 risk adjustments equivalent to a cost-of-capital risk margin:
Cost-of-capital future capital profiles under different bases:
Maximum likelihood vs MCMC parameter estimates:
MCMC vs Bootstrap results summary:
Residuals by development period, with Mack's sigma (variance) parameters
