Transcript Presentation

The Value of non
enforceable Future
Premiums in Life Insurance
Pieter Bouwknegt
AFIR 2003 Maastricht
Outline
Problem
Model
Results
Applications
Conclusions
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
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
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
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
Model
Introduce 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]
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
Model
Building a tree
Trinomial tree (up, middle, down)
Time between nodes free
Work backwards
Normal
Normal
Premium Normal
Normal
Last
premium
Model
Last premium node
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
Premium at j+1 will be passed; others paid
Model
Normal 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
Vj,t+1
Normal or premium
node
Vj-1,t+1
Normal or premium
node
pu
Normal
node
Vj,t
pm
pd
Model
Premium node (example values)
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)
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
Premium node
Vj,t
pm
pd
Vj,t+1
Normal node
Vj-1,t+1
Normal node
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
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 (zn is
zerorate until maturity, m is # premium)
m
1
2
3
4
5
zn
4,68%
6,80%
7,46%
6,39%
8,05%
VP
780,48
0
0
0
0
zn
4,68%
3,09%
3,14%
3,67%
3,29%
VP
780,48
2 675,49
1 602,43
578,66
267,61
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
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
Applications
Mortality (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
Applications
Mortality (EEB)
Search for Early Exercise Boundary: the line above
which premium payment is irrational
Early exercise boundary and mortality
6.00%
5.00%
50%
4.00%
100%
3.00%
200%
2.00%
400%
1.00%
0.00%
1
2
3
premium number
4
5
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 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
Applications
Paid up penalty (EEB)
Study different values for  and early exercise
boundary
Early exercise boundary
12,00%
10,00%
0%
8,00%
2.50%
6,00%
5%
100%
4,00%
-10%
2,00%
0,00%
1
2
3
4
5
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