The Value of non enforceable Future Premiums in

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

Outline z Problem z Model z Results z Applications z Conclusions

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

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

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

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

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

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

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

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

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

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

Model Premium node (except last premium) z Decide whether to pay the premium z 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 Premium node Vj, t pm pd Vj, t+1 Normal node Vj-1, t+1 Normal node

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

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

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

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

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

Applications Mortality (EEB) z Search for Early Exercise Boundary: the line above which premium payment is irrational

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

Applications Paid up penalty (EEB) z Study different values for and early exercise boundary

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