Technical summary
This page is a technical summary of the Internship Project carried out at Barrie & Hibbert in conjunction with Heriot-Watt University in 2008. The project looked at improvements in stochastic mortality modelling.

Under the forthcoming Solvency II risk management framework for the insurance industry, the preferred method for estimating risks is the 1-year value-at-risk (VaR) method, instead of the run-off method. The 1-year VaR method also assesses the risk-based capital, but with the difference that it only considers the variability within the next year. The aim of this internship project was therefore to extend Barrie & Hibbert’s current mortality model by incorporating a stochastic trend factor, in a way that will enable it to capture a significant risk factor under the new framework.

The approach

The mathematical expression of the enhanced model represents the observed probability of a life of a specified age dying in a certain year. The expected estimate at varying time of the probability of a life with varying age dying in varying years, is shown in Figure 1 [2].

Figure 1: Mortality table PMA92 and the projection tables.

The enhanced model has an observable stochastic process which models the year-by-year random transient change, and is related to another two non-observable stochastic processes which model the long-term true trend in mortality [1][5]. The stochastic drivers for all three of the stochastic processes follow independent distributions. Finally, there is an exponential term representing the trend factor which changes the tilting of the expected mortality [3].

Using parameters estimated from observed data by maximizing a likelihood function, Monte Carlo simulation was used to assess the risk-based capital. The distribution of the observed mortality probability is shown in Figure 2.

Figure 2: The distribution of the observed mortality probability.

The corresponding distribution of the annuity price is shown in Figure 3, and some resulting figures for the risk-based capital in Table 1.

Figure 3: The distribution of the annuity prices. Annuity prices are quoted as a multiple of the required annual payment

References

[1] Cairns, A.J.G., Blake, D., and Dowd, K., (2006) Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk, ASTIN Bulletin, 36: 79-120.
[2] C.M.I.B. (2007) The “Library” of Mortality Projections, Continuous Mortality Investigation, Working Paper, Number 27.
[3] Turnbull C., and McCulloch C. (2004) A Stochastic Model for Mortality, Barrie & Hibbert Internal Paper.
[4] Yang, S. (2000) A note on the Investigation of a Stochastic Mortality Model, Private communication between S. Yang and Barrie & Hibbert.
[5] Yang, S. (2001) Reserving, Pricing and Hedging for Guaranteed Annuity Options, Ph.D Thesis, Heriot-Watt University.

Click on the link below to view the full mathematics of the case study.

Download 'BarrieAndHibbertCaseStudy.pdf' (295 Kb).