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

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

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

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

Agenda  Actuarial vs financial pricing  Monte Carlo simulations  Cash flow model  Open questions

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

Single pure premiums Actuarial pricing : Financial pricing : with

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

Probability distribution functions 0 0,2 0,4 0,6 0, Discounted future costs Actuarial Financial

Sensitivity analysis

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

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

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

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

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

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

Simulation of provisions and capital –Tree simulations  Two possibilities: –Independent trajectories

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

Tree simulations P 1 (t) K 1 (t) P N (t) K N (t) t = 1

Comparison with non-life reinsurance business  Number of claims : Poisson( )  Severity of claim : Pareto(A,  )  Let  vary  Fix so that we obtain the same pure premium  Compare premium with both models  For usual values of , results not significantly different

Cost of capital  CAPM :  What is the  for this contract? –Same  for the whole company? –Specific  for this line of business?  How to estimate it?

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