Forward transition rates in a multi-state model

Transcription

1 Forward transition rates in a multi-state model Marcus C. Christiansen, Andreas J. Niemeyer August 3, 2012 Institute of Insurance Science, University of Ulm, Germany

2 Page 2 Forward transition rates Actuarial Research Conference August 3, 2012 Agenda Introduction Definition of forward rates Forward rates for cycle-free multi-state models Conclusion

3 Page 3 Forward transition rates Actuarial Research Conference August 3, 2012 Introduction Definition of forward rates Forward rates for cycle-free multi-state models Conclusion

4 Page 4 Forward transition rates Actuarial Research Conference August 3, 2012 Introduction forward rates are a well-known concept in the interest rate world in the last decade: transfer to mortality rates Norberg (2010) makes the first attempt to define forward rates in a multi-state model Norberg (2010) shows the limits of forward rates our paper has two major objectives: 1. formulation of a sound definition of forward rates in a multi-state model 2. demonstration of a framework where forward rates are a helpful concept

5 Page 5 Forward transition rates Actuarial Research Conference August 3, 2012 Introduction Definition of forward rates Forward rates for cycle-free multi-state models Conclusion

6 Page 6 Forward transition rates Actuarial Research Conference August 3, 2012 Definition of forward rates Definition - first attempt: no arbitrage Let P t (T ) be the price at time t of a zero-coupon bond with maturity T which cannot default. We define the forward rate by ρ t (T ) := lim u T r t (T, u), where r t (T, u) is the compounded forward rate. definition only valid for bonds with deterministic payoff concept not appropriate for mortality and multi-state models

7 Page 7 Forward transition rates Actuarial Research Conference August 3, 2012 Definition of forward rates (2) Definition - second attempt: optimistic yield For a zero-coupon bond with price P t (T ), that is endangered to default, we define the forward rate ρ t (τ), t τ T, by P t (T ) = e T t ρ t(τ)dτ. for bonds with deterministic payoff both definitions are equivalent definition is not appropriate for e.g. credit default swap, term life assurance

8 Page 8 Forward transition rates Actuarial Research Conference August 3, 2012 Definition of forward rates (3) Definition - final version: substitution rule Let (m 1,..., m n ) be given hazard rates and let M be a set of integrable functions that give the time t value of different claims depending on (m 1,..., m n ). We call µ 1,..., µ n the forward rates of M and (m 1,..., m n ) if they are F t -measurable and ( f (µ 1,..., µ n ) = E Q f (m1,..., m n ) ) F t for all f M. in case the first two definitions are appropriate, all definitions are equivalent the definition is always valid, since the set M can be adjusted the forward rates can depend on the product

9 Page 9 Forward transition rates Actuarial Research Conference August 3, 2012 Introduction Definition of forward rates Forward rates for cycle-free multi-state models Conclusion

10 Page 10 Forward transition rates Actuarial Research Conference August 3, 2012 Forward rates for cycle-free multi-state models For given hazard rates the set M can be adjusted such that the forward rates are unique. What about the other way round? goal: conditions on hazard rates m such that the following functions are included in M: all common benefits (for staying in one state and for the transition into another state) standardized ( products, i.e. E Q e ) T t m ij (τ)dτ F t = e T t µ ij (t,τ)dτ

11 Page 11 Forward transition rates Actuarial Research Conference August 3, 2012 Forward rates for cycle-free multi-state models (2) Assumption: Dependency structure m i(i+1) (u) and m ij (u) F t -independent for all i, j S, i + 1 < j with m ij (u) := m ij (u) m (i+1)j (u) Example: simple disability insurance a i m ad (s) = m ad (s) + m id (s) d m ai (s), m id (s) and m ad (s) are F t -independent

12 Page 12 Forward transition rates Actuarial Research Conference August 3, 2012 Results Theorem 1: sufficient condition Under the dependency structure from above we can securitize the products. Theorem 2: necessary condition (work in progress) For all t [0, T ] let m i (t) be of the form t t m i (t) = m i (0) + α i (τ, m i (τ)) dτ + β i (τ, m i (τ)) dwτ i 0 0 where m i, α i and β i meet some week requirements. a Then the dependency structure is also necessary. a this includes all common models as e.g. Bauer (2010) and Dahl (2004)

13 Page 13 Forward transition rates Actuarial Research Conference August 3, 2012 Introduction Definition of forward rates Forward rates for cycle-free multi-state models Conclusion

14 Page 14 Forward transition rates Actuarial Research Conference August 3, 2012 Conclusion It is possible to define forward rates for multi-state models. However, they can depend on the product. Consequently, the drawbacks presented in Norberg (2010) are no contradiction to the definition. By considering a special dependency structure the forward rates do not depend on the product in a cycle-free multi-state model.

15 Page 15 Forward transition rates Actuarial Research Conference August 3, 2012 Contact Institute of Insurance Science University of Ulm Andreas Niemeyer Thank you very much for your attention!

16 Page 16 Forward transition rates Actuarial Research Conference August 3, 2012 Literature Bauer, D., Benth, F.E., and Kiesel, R. (2012). Modeling the forward surface of mortality. To appear in SIAM Journal of Financial Mathematics. Dahl, M. (2004). Stochastic Mortality in Life Insurance: Market Reserves and Mortality-Linked Insurance Contracts. Insurance: Mathematics and Economics, 35: Norberg, R. (2010). Forward mortality and other vital rates - are they the way forward? Insurance: Mathematics and Economics, 47(2):