Projection to midpoint of usuage

[kze]

The notes say that assumptions used for pricing must stay valid for an average of

expected shelf-life of propose premium rates + expected duration of policy to termination or to next review date.

And then there's an example:
Suppose claim incidence rates have been estimated that apply on average, on 1/1/2019 for lives aged x. These rates are defined as i_x(0).
A product is to be launched on 1/1/2020 and premium rates will not be changed for 3 years.

Why do we assume lives start their policies on average, on 1/1/2021?
Why is the premium rate calculated assuming claim incidence rates of

(i_x(2)+i_x(3))/2 in the first policy year
(i_x(3)+i_x(4))/2 in the second policy year
.. and so on?

How does the expected duration of policy to termination or to next review data apply to the formula above?

 

[Anna Walklate]

Hi kze,

Yes, this is tricky. I find it always helps to draw a timeline for this sort of thing.

I think you have misread when we assume lives start their policies on average - the notes say "on average 1 July 2021", so this is the midpoint of the three-year period (1 Jan 2020 to 31 Dec 2022).

Now consider, on:
1/1/19 the claim incidence rates are ix(0)
1/1/20 the claim incidence rates are ix(1)
1/1/21 the claim incidence rates are ix(2)
1/1/22 the claim incidence rates are ix(3)

So in the first policy year, policies (which are assumed to start on average on 1/7/21) should experience an average of the 1/1/21 and 1/1/22 claim rates - which is what the 0.5(ix(2)+ix(3)) relates to.

In the second policy year, policyholders are now a year older (so x+1) and claims will occur a year later on average, so 0.5(ix+1(3)+ix+1(4)).

We will have to continue to calculate these claim incidence rates, bearing in mind how long the policy duration is.

I hope this helps!

Anna

 

[kze]

Hi kze,

Yes, this is tricky. I find it always helps to draw a timeline for this sort of thing.

I think you have misread when we assume lives start their policies on average - the notes say "on average 1 July 2021", so this is the midpoint of the three-year period (1 Jan 2020 to 31 Dec 2022).

Now consider, on:
1/1/19 the claim incidence rates are ix(0)
1/1/20 the claim incidence rates are ix(1)
1/1/21 the claim incidence rates are ix(2)
1/1/22 the claim incidence rates are ix(3)

So in the first policy year, policies (which are assumed to start on average on 1/7/21) should experience an average of the 1/1/21 and 1/1/22 claim rates - which is what the 0.5(ix(2)+ix(3)) relates to.

In the second policy year, policyholders are now a year older (so x+1) and claims will occur a year later on average, so 0.5(ix+1(3)+ix+1(4)).

We will have to continue to calculate these claim incidence rates, bearing in mind how long the policy duration is.

I hope this helps!

Anna

Hi Anna,

Sorry that was a typo. Should have been 1/7/2021 instead of 1/1/2021.
The example above is a lot clearer to me now.
Thanks!

 

[Meher]

Hi,
I still don't get this example.
1.where did Dec 2022 come into the picture?
2.as also July 2021?
3. How does the formula "expected shelf-life of propose premium rates + expected duration of policy to termination or to next review date" translate in the example?

Thanks

 

[Mark Willder]

Hi Meher

Do you have the 2021 set of the notes? If you have old notes then you will see different dates, as we move the dates on by one year each year.

1. December 2022 is the end of the time when the contract is sold. The question tells us it starts to be sold on 1/1/20 and is sold for 3 years, so the last day it is sold for is 31/12/22.

2. 1 July 2021 is then the midpoint between 1/1/20 and 31/12/22.

3. You need to be careful applying the formula you quote. The notes say below it that "The adjustment method will be different for different assumptions" so you can't just add the shelf-life and duration together.

The question says we have incidence rates for 1/1/19, so they are already 1 year out of date. The product will be sold for 3 years, so on average this makes the the incidence rates another 1.5 years out of date. So allowing for the shelf life we need to adjust for 2.5 years.

The above is ok for the first year of the policy, so we've averaged the year 2 and year 3 rate for age x to allow for the shelf life adjustment of 2.5 years.

We now need to allow for the duration of the policy. Considering the second policy year, this will start on average at 1/7/22, so we need to look at the average of years 3 and 4 for age x+1.

Similarly for the third policy year, this will start on average at 1/7/23, so we need to look at the average of years 4 and 5 for age x+2, and so on until we've covered the full duration of the policy.

Best wishes

Mark

 

[Vatsal Gupta]

Hi Mark,

I am a little bit confused by this example.
If the premium rates are not changing for the next 3 years then how does this mean that we need to adjust the rates for another 1.5 years?
Why we are taking the midpoint of 1/1/20 - 31/12/22?

Thanks

 

[Mark Willder]

Hi Mark,

I am a little bit confused by this example.
If the premium rates are not changing for the next 3 years then how does this mean that we need to adjust the rates for another 1.5 years?
Why we are taking the midpoint of 1/1/20 - 31/12/22?

Thanks

Hi Vatsal

Ideally we would change the premium rates every year. That way we make sure that we charge the correct amount.

However, for practical reasons we may decide to keep the premiums the same for 3 years. This means that for contracts bought in some years we will be charging too much and for contracts bought in other years we will be charging too little.

The forum discussion above looks at the case where claim rates are changing over time (expenses are likely to be changing too). For example, there may be an increasing trend. So by pricing at the midpoint we ensure that on average the premiums are correct, but we will be charging too much for contracts sold in the first year and too little for contracts sold in the third year (again assuming an increasing trend).

If instead we charged the premium that would be correct now, then we would be charging too little for contracts sold in the second year, and charging much too little for contracts sold in the third year (again assuming an increasing trend).

I hope this helps to explain the purpose of this adjustment.

Best wishes

Mark