CT5 Sept 2013 Q17 & Q20

[Laura]

Hi, I had some queries on the following questions:-
Q17) how do we obtain the exp(1.8) in the second integral? I've already ccounted for the survival and discounting for the prior 20 years in the exp(-1.6) so not sure what else I've missed.

Q20) I am confused on the method to obtain the independent prob from the 2 dependent decrements given - shouldn't we find the force of mortality and work out the independent prob from there? The method seems to assume then the dependent decrement takes place uniformly across the year hence it's halved : hence the halved dependent prob of the A decrement not taking place is multiplied by the independent prob of B decrement to obtain the dependent B decrement prob? I'm not sure if I've seen this in the core reading

Thanks!

 

[Joe Hook]

I presume you're looking at the examiners report here. If we assume that the life is alive after 20 years we can calculate the EPV of the benefits from this point as:

integral from 20 to infinity of (t * exp(-0.09)t) dt

If you calculate this then along the way you'll end up with an exp(-1.8) term as you evaluate that at the limit of 20. You will probably have set up an expression that allows for discounting and survival probability directly, in which case you won't see an exp(-1.8) e.g. the second integral can be written as:

integral from 20 to infinity of (0.04 * t * exp(-0.09t+0.2) dt

The method from question 20 is no longer examinable. We have reworded this question in our materials to read:

"It has been established that the forces of withdrawal are now only 50% of those assumed in the table above for the ages of 85 and 86. The underlying forces of mortality are unchanged.

Construct a revised decrement table to reflect this change."

With this question we'd need to find the underlying forces of mortality and withdrawal, reduce the force of withdrawal by 50% and then use these forces to reconstruct the table.

Hope this helps.
Joe

 

[Laura]

I presume you're looking at the examiners report here. If we assume that the life is alive after 20 years we can calculate the EPV of the benefits from this point as:

integral from 20 to infinity of (t * exp(-0.09)t) dt

If you calculate this then along the way you'll end up with an exp(-1.8) term as you evaluate that at the limit of 20. You will probably have set up an expression that allows for discounting and survival probability directly, in which case you won't see an exp(-1.8) e.g. the second integral can be written as:

integral from 20 to infinity of (0.04 * t * exp(-0.09t+0.2) dt

The method from question 20 is no longer examinable. We have reworded this question in our materials to read:

"It has been established that the forces of withdrawal are now only 50% of those assumed in the table above for the ages of 85 and 86. The underlying forces of mortality are unchanged.

Construct a revised decrement table to reflect this change."

With this question we'd need to find the underlying forces of mortality and withdrawal, reduce the force of withdrawal by 50% and then use these forces to reconstruct the table.

Hope this helps.
Joe

Thanks Joe, but I'm still confused as I have only gotten exp(-1.6) from surviving to age 40 and discounting to age 20. I have not managed to obtain exp(-1.8) nor exp(0.2)

 

[Joe Hook]

So allowing for the benefits from age 40 to infinity ie time 20 to infinity:

We construct this integral as integral from 20 to infinity of tp20 * mu_20+t * e^-delta*t * t

tp20 = 20p20 * (t-20)p40 = exp(-0.03*20)*exp(-(t-20)*0.04) = exp(-0.04t+0.2)

So the expression equals

= exp(-0.04t+0.2) * 0.04 * exp(-0.05t) * t = exp(-0.09t+0.2) * 0.04 * t dt

When we plug 20 into that we'll have an exp(-1.8) term and an exp(0.2) term.

 

[Laura]

Thanks very much for your help Joe!