### Introduction

The purpose of this document is to explain certain types of insurances and specific the risks involved.
To explain the risks mathematics is involved but I will try to keep it as simple as possible.
In order to do that the examples are also simple. The reality is much more complex but that is the subject of an advanced course.

What we are going to study are three types of insurance policies:

1. Type 1 is called a Funeral Insurance .
In a Funeral Insurance the policy holder, in this case his relatives, are always paid a fixed amount when the policy holder dies.
2. Type 2 is called a Death risk Insurance
In a Death risk Insurance the relatives (primarily his wife or her husband) are paid a fixed amount when the policy holder dies before a certain age.
3. Type 3 is called a Life Insurance
In a Life Insurance the policy holder is paid each year a certain amount after a certain age until the policy holder dies.
In all of those cases you can gain money and there is a risk involved that you loose money.
1. In a Funeral Insurance the relatives loose money the older the policy holder becomes.
2. In a Death risk Insurance the relatives are worst of the shortly the policy holder dies after the maturity age.
3. In a Life Insurance the policy holder is best of the older the policy holder becomes.
To explain this in more detail we assume that the dead rate follows what is called a normal distribution.
A normal distribution means that when the average age is 75 that the number of people that die at each age before 75 is identical as the number of people that dei after 75. For example this menas that the number of people that die at age 70 and 80 is the same. In reality the dead rate does not follow a normal distribution.

```                x
x     x
x         x
x           x
x<----------->x
x               x
x                 x
x                     x
xxx                           xxxx
2     1     3
Figure  1
```
• The point 1 identifies the highest dead rate and the average age of the population.
• The points 2 and 3 define the spread.

### Example 1 Funeral Insurance

Funeral insurance is an insurance where relatives are paid a certain amount after the holder has died to pay for the cost of the funeral
In principle Funeral Insurance places a life long obligation for the holder to pay the insurance company a certain amount.

Suppose the policy holder wants to receive 1000 Euro after his dead, starts at age 20 and that the average age of the population is 70. years. In that case the holder has to pay 1000/50 = 20 Euro each year. The issue is what are the benefits for the issurance company (IS).

1. When the holder dies at age 70 there is no loss nor profit for the IS. (except for interests received)
2. Suppose the holder dies at age 50. In that case the holder has paid 30*20 = 600 Euro and will receive 1000. That means the IS will loose 400 Euro on this policy.
3. Suppose the holder dies at age 90. In that case the holder has paid 70*20 = 1400 Euro and will receive 1000. That means the IS will gain 400 Euro on this policy.
If you consider those 3 cases in total the IS will not make any money on this type of issurance (assuming a normal distribution of the dead rate.)

The interesting case is case #3. In that case the holder will loose on the policy. If the holder stops payments before he dies than the IS company will pay nothing and will make a profit.

### Example 2 Death risk Insurance

Death risk insurance is an insurance where relatives are paid a certain amount when the holder has died before a certain age
The whole idea behind a Death risk Insurance is that the people who die at old age, pay for the people who die young. In the case of a normal distribution this decision point is the average age. That means 50% get something and 50% nothing.
In order to calculate the yearly amount to pay is more complicated and depends about the spread. See Figure 1.
• In the case when the spread is zero all people die very close to the age of 70. However 50% will die before 70 and they will get 1000 Euro and 50% after 70 and they will get nothing . That means on the average each person will receive 500 Euro. 500 divided by 50 years = 10. That means each person has to pay 10 Euro each year. (Assuming spread = zero)
• In reality there is a spread. That means certain percentage will die before 70. Those people will pay in less and the other people (those that will die after 70) have to pay more.
When the spread is 10 the Yearly pay is 10.86
When the spread is 20 the Yearly pay is 11.75
When the spread is 30 the Yearly pay is 12.4

### Example 3 Life Insurance or Pension Plan

A Pension Plan is the reverse of a Death risk Insurance. The holder get nothing if you die before a certain age and you get something each year if you die after a certain age.
• In the case when the spread is zero all people die very close to the age of 70. However 50% will die just before 70 and they will get nothing and 50% after 70 and they will get 1000 Euro for one year in this case. That means on the average each person will receive 500 Euro. 500/50 = 10. That means each person has to pay 10 Euro. (Assuming spread = zero)
• In reality there is a spread. That means:
• more people will die younger and and they will pay less into the total pension fund.
• more people will die older that means over more years pension has to be paid.
The result will be that the larger the spread the less will be paid each year.
When the spread is 0 the Yearly pay out is 1000
When the spread is 10 the Yearly pay out is 108,36
When the spread is 20 the Yearly pay out is 52,87
When the spread is 30 the Yearly pay out is 39,07
 Spread Type 1 Type 2 Type 3 0 20 10 1000 10 20 10,86 108,36 20 20 11,75 52,87 30 20 12,40 39,07

### Asymetric Distribution

In the above examples the dead rate was a normal distribution. In reality this is not the case.
The following table shows the results when the spread after the average age is 50% smaller as before
 Spread Type 1 Type 2 Type 3 0 20 10 1000 10 21,73 14,91 296,84 20 23,56 16,68 139,24 30 24,07 17,25 84,01
• Example 1 Funeral Insurance
In the case of a symetrical normal distribution of the dead rate the influence of the spread is zero because on average the same number of people die before and after the average age and because in all cases the Isurance Company paids.
In the case of an asymetrical distribution the dead rate increases more after the average age as compared with the symetrical normal distribution. That means that the total number of contributions for all the policy holders decreases. As a consequence all the contributors, assuming that the payout stays the same, have to pay more.
They have to pay more the larger the spread.
• Example 2 Death risk Insurance
in the case Example 2 Death risk Insurance the same results apply as in the case of the Funeral Insurance . That means in the case that the people after reaching the average age die at a higher rate the more the policy holders have to pay.
• Example 3 Life Insurance or Pension Plan
the same opposite results apply as in the first two cases. That means in the case that the people after reaching the average age die quicker the more the policy holders should recieve each year, other wise the Insurance Company makes a profit.

### Lessons to be learned

In the first three examples (normal distribution) the rules of the games (simulations) are simple and rather static. There is no inflation, there is no interest considered, the average age of the population is constant and the dead rate follows strict rules. The reality is more complex.
What are the risks involved :
1. Increase in average age of the population
Increase in average you can observe in the above examples when you go from asymetrical to more symetrical distribution of the dead rate.
In the following discussion it is assumed that people become older and the average age increases but not the maturity date.
• For a Funeral Insurance this means that the policy holders have to pay longer. That is benificiary for the IC.
• For a Death risk Insurance this means that less people will die before maturity date. The result is that the IC has to pay less. This is beneficiary for the IC.
• For a Life Insurance or Pension Plan this means if the contributions stay the same the yearly benefits will decrease. If the yearly benefits should stay the same, than the yearly contributions should increase.

An increase in contributions is always difficult to manage. This is the case

• For a Funeral Insurance when the average age decreases assuming that the payout stays the same
• For a Death risk Insurance this is the case when the average age before maturity decreases, because less contributions will be made.
• For a Life Insurance or Pension Plan this is the case when the average population becomes older. In that case the payouts will increase while the income increases less.
In all those cases (in general) the Insurance Companies will loose money because when and IC increases the contributions, based on the current dead rate statistics, than this will only effect the future while (in general) the participants should have paid those same amounts in the past. This is what is called a time delay effect.

Specific for a Life Insurance or Pension Plan it is interesting to see that if the IC keeps both the contributions and the yearly benefits the same that the IC looses money. On the other hand this loss will be offset by a gain in the Death risk Insurance business.

2. The biggest risk for an Insurance Company is to calculate a competitive price which includes future conditions.
For example: if the average age increases the yearly contributions for a Life Insurance or Pension Plan should increase assuming the benefits stay the same. This means that the policy holder each year has to pay a higher price. It is even worse, the policy holder should also get an adjustment bill for the previous years.
This risk becomes clear if the average age of the people increases after maturity date and the policy holder starts to recieve his yearly benefits. In that case the policy holder will recieve more than original estimated and paid for and the IC will loose money. The most obvious (painly) solution is to cut the yearly benefits.
3. The general risk for an IC is that for each insurance type you need experts.
The general risk for a policy holder that he needs different advisers for each insurance type.

Created: 17 September 2012