[Paper Review] Modelling Annuity Portfolios and Longevity Risk with Extended CreditRisk

Jonas Hirz, Uwe Schmock+1 more|arXiv (Cornell University)|May 18, 2015

Insurance, Mortality, Demography, Risk Management3 citations

TL;DR

This paper extends CreditRisk+ to model annuity portfolios and longevity risk by introducing stochastic life tables driven by common risk factors representing causes of death. It enables exact, numerically stable calculation of one-period loss distributions and derives regulatory risk measures, with applications to stress testing and demographic forecasting using Australian data.

ABSTRACT

Using an extended version of the credit risk model CreditRisk$^+$, we develop a flexible framework to estimate stochastic life tables and to model credit, life insurance and annuity portfolios, including actuarial reserves. Deaths are driven by common stochastic risk factors which may be interpreted as death causes like neoplasms, circulatory diseases or idiosyncratic components. Our approach provides an efficient, numerically stable algorithm for an exact calculation of the one-period loss distribution where various sources of risk are considered. As required by many regulators, we can then derive risk measures for the one-period loss distribution such as value at risk and expected shortfall. Using publicly available data, we provide estimation procedures for model parameters including classical approaches, as well as Markov chain Monte Carlo methods. We conclude with a real world example using Australian death data. In particular, our model allows stress testing and, therefore, offers insight into how certain health scenarios influence annuity payments of an insurer. Such scenarios may include outbreaks of epidemics, improvement in health treatment, or development of better medication. Further applications of our model include modelling of stochastic life tables with corresponding forecasts of death probabilities and demographic changes.

Motivation & Objective

• To develop a flexible, computationally efficient framework for modeling annuity and life insurance portfolios including actuarial reserves.
• To incorporate stochastic mortality by modeling deaths through common risk factors representing disease categories and idiosyncratic components.
• To enable exact calculation of one-period loss distributions under multiple risk sources, supporting regulatory risk measures like VaR and expected shortfall.
• To provide estimation procedures for model parameters using classical and Markov chain Monte Carlo methods.
• To demonstrate real-world applicability through stress testing scenarios such as epidemics or medical advancements using Australian death data.

Proposed method

• Extends the CreditRisk+ model to include stochastic mortality by introducing common risk factors for death causes such as neoplasms and circulatory diseases.
• Models deaths as a Poisson mixture where default intensities are driven by stochastic risk factors, enabling exact computation of the one-period loss distribution.
• Uses a factor-based structure to capture dependence across portfolios and demographic groups, ensuring numerical stability.
• Applies classical estimation and Markov chain Monte Carlo (MCMC) methods to calibrate model parameters from publicly available death data.
• Derives regulatory risk measures (e.g., value at risk, expected shortfall) directly from the exact loss distribution.
• Enables stress testing by perturbing risk factor intensities to simulate health scenarios like disease outbreaks or medical improvements.

Experimental results

Research questions

• RQ1 How can CreditRisk+ be extended to model stochastic life tables and longevity risk in annuity and life insurance portfolios?
• RQ2 What is the impact of different death causes—such as neoplasms or circulatory diseases—on annuity payment risk when modeled as stochastic risk factors?
• RQ3 Can an exact, numerically stable algorithm compute the one-period loss distribution when multiple risk sources are present?
• RQ4 How do stress testing scenarios, such as epidemics or medical advancements, affect the distribution of annuity losses?
• RQ5 What estimation techniques, including MCMC, are effective for calibrating model parameters using real demographic data?

Key findings

• The extended CreditRisk+ framework enables exact and numerically stable computation of the one-period loss distribution for annuity and life insurance portfolios.
• The model successfully captures dependence in mortality risk through common stochastic factors representing major causes of death.
• Regulatory risk measures such as value at risk and expected shortfall can be derived directly from the exact loss distribution.
• Stress testing reveals how changes in health scenarios—like improved treatments or disease outbreaks—affect annuity payment risk.
• The model supports the construction of stochastic life tables with forecasts of death probabilities and demographic trends using real Australian death data.
• Parameter estimation is feasible using both classical methods and MCMC, enhancing model flexibility and robustness.