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C. Frantz, X. Chenut and J.F. Walhin Secura Belgian Re

Pricing and capital allocation for unit-linked life insurance contracts with minimum death guarantee. C. Frantz, X. Chenut and J.F. Walhin Secura Belgian Re. Sum at risk. Insurer’s liability for a death at time t:. Financial index S t. Time t. The problem. How to price it ?

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C. Frantz, X. Chenut and J.F. Walhin Secura Belgian Re

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  1. Pricing and capital allocation for unit-linked life insurance contracts with minimum death guarantee C. Frantz, X. Chenut and J.F. Walhin Secura Belgian Re

  2. Sum at risk Insurer’s liability for a death at time t: Financial index St Time t The problem • How to price it ? • Capital allocation ?

  3. The actuary: it is an insurance contract • Solution: equivalence principle Expected value of future losses Two approaches … • The financer: it is a contingent claim • Solution: hedging on the financial market Black-Scholes put pricing formula

  4. … and two risk managements • Financial approach : hedging on financial markets • Actuarial approach : reserving and raising capital

  5. Agenda • Actuarial vs financial pricing • Monte Carlo simulations • Cash flow model • Open questions

  6. First question:actuarial or financial pricing? • Hypotheses : • Complete and arbitrage-free financial market • Constant risk-free interest rate • Financial index follows a GBM: Simple expressions for the single pure premium in both approaches

  7. Single pure premiums Actuarial pricing : Financial pricing : with

  8. Monte Carlo simulations • Goal : distribution of the future costs • 3 processes to simulate : • Financial index • Death process • Hedging strategy (financial approach only)

  9. Probability distribution functions 1 0,8 0,6 0,4 0,2 Actuarial Financial 0 0 10 20 30 40 50 60 Discounted future costs

  10. Sensitivity analysis

  11. Sensitivity analysis

  12. Conclusion • Financial approach is better • BUT only makes sense if the hedging strategy is applied ! • Difficult to put into practice (especially for the reinsurer) • Conclusion : actuarial approach has to be used

  13. Second question :How to fix the price ? • Base : single pure premium • + Loading for « risk » • Answer : cash flow model

  14. Cash flow model • Insurance contract = investment by the shareholders • Investment decision: cash flow model • Price P fixed according to the NPV criterion

  15. Open questions • How much capital to allocate? • How to release it through time? • What is the cost of capital?

  16. Risk measures and capital allocation • Coherent risk measures (Artzner et al.) • Conditional tail expectation (CTE): where • Capital to be allocated at time t:

  17. One-period vs multiperiodic risk measures • Problem: intermediate actions during development of risk • Addressed recently in by Artzner et al. • Capital at time t : • to cover all the discounted future losses? • to pay the losses for x years and set up provisions at the end of the period? • We applied the one-period risk measure to the distribution of future losses at each time t

  18. Two possibilities: • Independent trajectories • Tree simulations Simulation of provisions and capital

  19. P(t) K(t) t = 1 Independent trajectories

  20. P1(t) K1(t) PN(t) KN(t) t = 1 Tree simulations

  21. Comparison with non-life reinsurance business • Number of claims : Poisson(l) • Severity of claim : Pareto(A,a) • Let a vary • Fix l so that we obtain the same pure premium • Compare premium with both models • For usual values of a (1,5-2,5), results not significantly different

  22. Cost of capital • CAPM : • What is the b for this contract? • Same b for the whole company? • Specific b for this line of business? • How to estimate it?

  23. Conclusions • Actuarial approach • Pricing and capital allocation using simulations • Other questions: • Asset model: GBM, regime switching models, (G)ARCH, …? • Risk measure? Threshold a? • Capital allocation and release through time?

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