Permanent Income Protection Policy

[tatos]

Trying to understand something:

Product - permanent income protection in the event of illness or accident, up to the age of retirement (age 65 in this case)

Is there a reason why someone who is aged 64 now and takes out a policy for this will receive a LOWER premium than someone with the same risk profile except for the fact that this person is aged 63 now?

When I use the UK life tables and example pension scheme to calculate a theoretical price from scratch I seem to be getting a premium that increases with age up to about age 58 and then starts decreasing. I can't seem to see why..

 

[Calum]

Think about what the sum at risk is as the insured life gets closer to age 65.

What would you charge someone who turned 65 tomorrow?

What would you charge someone who turned 65 in a month?

What would you charge someone who turned 65 in a year?

Your calculations sound like they are doing what you would expect them to, by the way.

 

[tatos]

Thanks Calum. I did sort of think along those lines but, to use your example to illustrate where I'm getting stuck:

Case 1:
If he's 64 and 11 months today then his greatest probability of getting ill (assumption) is between now and 65. That means the expected cost of paying this person their monthly benefit or income is greatest.

Case 2:
If he's 64 and 10 months today then he could get ill between 64.10-64.11 or he would have to survive from 64.10 to 64.11 and then get ill between 64.11-65.
Getting ill between 64.10 and 64.11 (I assume) has a lower probability than that probability in Case 1. So the expected cost for this month is lower.
Getting ill between 64.11 and 65 is the same probability as that in Case 1, with the only difference being that he first needs to survive till age 64.11. So that would surely decrease the overall probability of getting sick in this period. Hence that would also result in a lower expected cost than that in case 1.

So both of the expected costs in case 2 appear (to me) to be lower than the expected cost in case 1. Hence the MONTHLY premium to cover these expected costs will be lower? (as opposed to total premium which would clearly be larger because there are 2 months of cover as opposed to 1)

What am I failing to see or understand?

Also, you said that "Your calculations sound like they are doing what you would expect them to, by the way". Have you worked with such a product before? I ask because I tried to scrape some data off a site where an insurance company offers such a product and I was getting this very trend. I'm still in disbelief but hoping to see the light!

 

[Calum]

You're pretty close - but remember the sum at risk also changes depending on when the illness develops.

You're quite right in your thinking that the older you get, the more likely you are to become ill - your "force of illness" is always increasing.

But on the other hand, what does the insurer pay out (in total) for inception at 64y11m compared to 64y10m?

 

[tatos]

You're pretty close - but remember the sum at risk also changes depending on when the illness develops.

You're quite right in your thinking that the older you get, the more likely you are to become ill - your "force of illness" is always increasing.

But on the other hand, what does the insurer pay out (in total) for inception at 64y11m compared to 64y10m?

Oh! I think I get it now

So, for example,

If the insured is covered for 50,000 per month and he's 64:11 now then he can get

A: 50000 if he falls ill now

If he's 64:10 now then he can get

B: 100,000 (50,000 for 2 months till he is 65) if he falls ill before 64:11

or

C: 50,000 if he falls ill between 64:11 and 65 having survived till 64:11.

So, I factor in the probabilities of these events happening and then find the present values of these amounts.

Then, on the premium side of things:

If 64:10 today, then the insurer receives a premium (X) now, and again (X) at 64:11 with probability corresponding to survival AND not getting ill. This second X is discounted to today.

If 64:11 then we receive a premium (Y) now, equal to the present value of A above.

Now to test this, if I let (X) be equal to (Y), then the premium received now at age 64:10, i.e. (Y), will cover an amount not more than the present value of A.

And, the present value of C is less than the present value of A but it's probably quite close to the present value of A because the probability of surviving one month, even at age 64:10, is relatively high (at least according to the table I am using). Anyway, the PV of C seems to be covered by the first (Y) that we definitely receive.

However, the present value of B is obviously much greater than the present value of A though. So the second (Y) that we will probably receive and discounted to today would not cover that!

So (Y) is insufficient as a premium for someone aged 64:10 now, because it only just covers the expected cost due to C, but doesn't nearly cover the cost expected due to B.

So I think I'm seeing what's happening now...

And I seem to see that the turning point (where the premiums stop increasing with age and then start decreasing) is dependent on several assumptions made:

Interest rate used
Benefit amount chosen
Probabilities of survival and falling ill

Changing these can change the sensitivity of your results..

Am I reasoning this out correctly now? Or is there more to this puzzle?

Many thanks!

 

[Calum]

Bingo. The risk of becoming ill becomes higher over time, but as you get closer to 65 the sum the insurer has to pay starts to drop. There's a point where the two interact and hit a maximum.

 

[tatos]

Bingo. The risk of becoming ill becomes higher over time, but as you get closer to 65 the sum the insurer has to pay starts to drop. There's a point where the two interact and hit a maximum.

Fantastic... thanks Calum!