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

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

3. Problem Legal  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. Problem Economical  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. Problem Include 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. Problem Exclude 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. Model Introduce economic rational decision  TR m,t = PU m,t. SP m,t BE +max(PP m,t + FV m,t ;0) TR m,t = technical reserves before decision is made PU m,t = paid up amount SP m,t = single premium for one unit insured amount PP m,t = direct value premium payment FV m,t = future value of right to make decision in a year  VP m,t = max(PP m,t + FV m,t ;0)  PP m,t = ΔPU m. SP m,t - P  FV m,t = 1 p x+m. E t Q [{exp  (t,t+1)r(s)ds}VP m+1,t+1 ]

8. Model Tree 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. Model Building a tree  Trinomial tree (up, middle, down)  Time between nodes free  Work backwards Last premium Normal Premium

10. Model Last premium node  In nodes where to decide to pay the last (n th ) premium V j,t =MAX (ΔPU n,t. SP j,t - P ; 0)  Premium at j+1 will be passed; others paid V j,t+1 V j+1,t+1 V j-1,t+1 Don’t pay: 0 Pay: >0

11. Model Normal node  Value the node looking forward  Number of normal nodes depends on stepsize  V j,t = Δt p x. e -rΔt. (p u V j+1,t+1 + p m V j,,t+1 +p d V j-1,,t+1 ) V j,t+1 V j+1,t+1 V j-1,t+1 V j,t pdpd pmpm pupu Normal or premium node Normal or premium node Normal or premium node Normal node

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

13. Model Premium node (except last premium)  Decide whether to pay the premium  V j,t = MAX (ΔPU m,t. SP j,t - P + Δt p x. e -rΔt. (p u V j+1,t+1 + p m V j,,t+1 +p d V j-1,,t+1 ) ; 0) V j,t+1 V j+1,t+1 V j-1,t+1 V j,t pdpd pmpm pupu Normal node Premium node

14. Results Initial 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. Results Value 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 (z n is zerorate until maturity, m is # premium)

16. Results Release 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. Results In 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. Applications Mortality (model)  Assume best estimate (BE) mortality differs from tariff:q x BE =  q x tariff  Standard mortality formulas n p x  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. Applications Mortality (EEB)  Search for Early Exercise Boundary: the line above which premium payment is irrational

20. Applications Paid 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 m th premium is the first not to be paid: . PU m,t. SP m,t  Decision: max(PP m,t + FV m,t ;- . PU m,t. SP m,t )  Value in force policy can be lower than paid up value

21. Applications Paid 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