The Value of non enforceable Future Premiums in Life Insurance

Pieter Bouwknegt

AFIR 2003 Maastricht

Outline

Problem

Model

Results

Applications

Conclusions

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

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

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

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

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]

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

ModelBuilding a tree

Trinomial tree (up, middle, down)

Time between nodes free

Work backwards

Last premium

Normal NormalNormal NormalPremium

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,t+1

Vj+1,t+1

Vj-1,t+1

Don’t pay: 0

Pay: >0

Pay: >0

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,t+1

Vj+1,t+1

Vj-1,t+1

Vj,t

pd

pm

pu

Normal or premiumnode

Normal or premiumnode

Normal or premiumnode

Normal node

ModelPremium node (example values)

Value premium

2

1

3Market<tariff

Market>tariff

Market>tariff -4

-1

2

CurrentFuture

0

1

5

Do not pay

Pay premium

Pay premium

Node

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,t+1

Vj+1,t+1

Vj-1,t+1

Vj,t

pd

pm

pu

Normal node

Normal node

Normal node

Premium node

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

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)m zn VP zn VP1 4,68% 780,48 4,68% 780,482 6,80% 0 3,09% 2 675,493 7,46% 0 3,14% 1 602,434 6,39% 0 3,67% 578,665 8,05% 0 3,29% 267,61

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

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

ApplicationsMortality (model)

Assume best estimate (BE) mortality differs from tariff: qx

BE = 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

ApplicationsMortality (EEB)

Search for Early Exercise Boundary: the line above which premium payment is irrational

Early exercise boundary and mortality

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

1 2 3 4 5

premium number

50%

100%

200%

400%

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

ApplicationsPaid up penalty (EEB)

Study different values for and early exercise boundary

Early exercise boundary

0,00%

2,00%

4,00%

6,00%

8,00%

10,00%

12,00%

1 2 3 4 5

0%

2.50%

5%

100%

-10%

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