The Value of non enforceable Future Premiums in Life Insurance

1. The Value of non enforceable Future Premiums in Life Insurance Pieter Bouwknegt AFIR 2003 Maastricht

2. Outline • Problem • Model • Results • Applications • Conclusions

3. ProblemLegal • The policyholder can not be forced to pay the premium for his life policy • Insurer is obliged to accept future premiums as long as the previous premium is paid • Insurer is obliged to increase the paid up value using the original tariff rates • Asymmetric relation between policyholder and insurer

4. ProblemEconomical • The value of a premium can be split in two parts • The value of the increase in paid up insured amount minus the premium • The value to make the same choice a year later • Valuation first part is like a single premium policy • Valuation second part is difficult, as you need to value all the future premiums in different scenarios

5. ProblemInclude all future premiums? • One can value all future premiums if it were certain payments: use the term structure of interest • With a profitable tariff this leads to a large profit at issue for a policy • However: can a policy be an asset to the insurer? • If for a profitable policy the premiums stop, a loss remains for the insurer • Reservation method can be overoptimistic and is not prudent

6. ProblemExclude all future premiums? • Reserve for the paid up value, treat each premium as a separate single premium • No profit at issue (or only the profit related with first premium) • A loss making tariff leads to an additional loss with every additional premium • A loss making tariff is not recognized at once • Reservation method can be overoptimistic and is not prudent

7. ModelIntroduce economic rational decision • TRm,t = PUm,t. SPm,tBE +max(PPm,t + FVm,t;0) • TRm,t = technical reserves before decision is made • PUm,t = paid up amount • SPm,t= single premium for one unit insured amount • PPm,t = direct value premium payment • FVm,t = future value of right to make decision in a year • VPm,t = max(PPm,t + FVm,t;0) • PPm,t = ΔPUm . SPm,t - P • FVm,t = 1px+m . EtQ[{exp(t,t+1)r(s)ds}VPm+1,t+1]

8. ModelTree problem • The problem looks like the valuation of an American put option • Use an interest rate tree consistent with today’s term structure of interest (arbitrage free) • Start the calculation with the last premium payment for all possible scenario’s • Work back (using risk neutral probabilities) to today • Three types of nodes

9. ModelBuilding a tree • Trinomial tree (up, middle, down) • Time between nodes free • Work backwards Last premium Normal Normal Premium Normal Normal

10. ModelLast premium node • In nodes where to decide to pay the last (nth) premium Vj,t=MAX (ΔPUn,t . SPj,t - P ; 0) • Premium at j+1 will be passed; others paid Vj+1,t+1 Don’t pay: 0 Vj,t+1 Pay: >0 Vj-1,t+1 Pay: >0

11. ModelNormal node • Value the node looking forward • Number of normal nodes depends on stepsize • Vj,t = Δtpx. e-rΔt . (pu Vj+1,t+1 + pm Vj,,t+1 +pd Vj-1,,t+1) Vj+1,t+1 Normal or premium node pu Normal or premium node Vj,t+1 Normal node Vj,t pm pd Vj-1,t+1 Normal or premium node

12. ModelPremium node (example values) • Value premium Current Future Node -4 1 0 Market>tariff Do not pay -1 2 1 Market>tariff Pay premium 2 3 5 Market<tariff Pay premium

13. ModelPremium node (except last premium) • Decide whether to pay the premium • Vj,t = MAX (ΔPUm,t . SPj,t - P + Δtpx. e-rΔt . (pu Vj+1,t+1 + pm Vj,,t+1 +pd Vj-1,,t+1) ; 0) Vj+1,t+1 Normal node pu Vj,t+1 Vj,t Premium node Normal node pm pd Vj-1,t+1 Normal node

14. ResultsInitial policy • Policy is a pure endowment, payable after five years if the insured is still alive • Insured amount € 100 000 • Annual mortality rate of 1% • Tariff interest rate at 5% • Five equal premiums of € 16 705,72

15. ResultsValue of premium payments • If the value of a premium VP is nil then do not pay • If it is positive then one should pay • A high and low interest scenario in table (zn is zerorate until maturity, m is # premium)

16. ResultsRelease of profit • When tariffrate<market rate: no release at issue • When tariffrate>market rate: full loss at issue • If interest rates drop below tariff rate a loss arises due to the given guarantee on future premiums • If a premium is paid and the model did not expect so, a profit will arise, attributable to “irrational behavior” • The behavior of the policyholder can not become more negative then expected

17. ResultsIn or out of the money • A simple model is to consider the value of all future premiums together and the insured amounts connected to them • If the future premiums are out of the money (value premiums exceeds the value of the insured amount) then exclude all premiums from calculations • If the future premiums are in the money (value premiums lower then the value of the insured amount) then include all premiums in calculations • This model gives essentially the same results

18. ApplicationsMortality (model) • Assume best estimate (BE) mortality differs from tariff: qxBE =  qxtariff • Standard mortality formulas npx • When  is small: healthy person • Policy (pure endowment) is more valuable to the policyholder, because he “outperforms” the tariff mortality • When  is large: sick person • Policy (pure endowment) is less valuable to the policyholder, he must be compensated with higher profit on interest

19. ApplicationsMortality (EEB) • Search for Early Exercise Boundary: the line above which premium payment is irrational

20. ApplicationsPaid up penalty (model) • Assume that the paid up value of the policy is reduced with a factor  when the premium is not paid • Value reduction when the mth premium is the first not to be paid:  . PUm,t . SPm,t • Decision: max(PPm,t + FVm,t;-  . PUm,t . SPm,t) • Value in force policy can be lower than paid up value

21. ApplicationsPaid up penalty (EEB) • Study different values for  and early exercise boundary

22. Conclusions • Valuation of future premiums should be considered • Economic rationality introduces prudent reservation • Important influence on the release of profit • Use of trees is complicated and time consuming • In/out of the money model gives roughly same results • Possible to study behavior of policyholder using economic rationality concept