EPV renewal expenses

[Neetu Verma]

In assignment X2 ,Question no 2.6.
In this question,I am confused to find the EPV of renewal expenses b'coz when we find the value of ä[40]:20 @ 4% upto age 55 and @ 5% thereafter(b'coz allowing for the extra force of mortality) but EPV of renewal expenses calculate @4% upto age 60 instead of @4% upto age 55 & @5% thereafter.
Anyone tell me why we take only 4%?

 

[rsmallela]

Extra mortality...

The expression (given in solution on page-9) for EPV of renewal expenses uses 4% of interest for entire term, but denotes the survival probabilities as p' after age 55 to represent extra mortality.

This is correct, because the question actually mentions there is extra mortality after age 55 and not extra interest. Given a constant force of mortality, we find out mathematically, how much this extra mortlity would increase interest - just for simplicity in evaluating assurance or annuity functions.

Now going further keeping an eye if this extra mortality after age 55 is still considered till the end...With compounding of expenses at 3%, we would want to calculate 'a' function with a revised interest of 1% through out -- still, with extra mortality after age 55.

This means, we need to evaluate a[40]:19 at interest 1% and at extra mortality after age 55.
This is exactly what you found out in part (i) of the question.

So, we are good on this part....let me know if this makes sense to you...

I posted another query on this question...a few days back...but, while reviewing your question, I got an answer for that too....

My query was about evaluating A[40]:20. A premium conversion relationship was used after calculating a[40]:20 which takes care of extra mortality. The 'd' in this premium conversion relationship used just 4% which is correct...because interest is always 4% if you had already taken care of extra mortality in your 'a' functions....

Thanks,
Raj

 

[Neetu Verma]

The expression (given in solution on page-9) for EPV of renewal expenses uses 4% of interest for entire term, but denotes the survival probabilities as p' after age 55 to represent extra mortality.

This is correct, because the question actually mentions there is extra mortality after age 55 and not extra interest. Given a constant force of mortality, we find out mathematically, how much this extra mortlity would increase interest - just for simplicity in evaluating assurance or annuity functions.
Raj

ok .........if u r right .
so why we change interest for evaluating assurance & annuity ( A[40]:20 &
ä[40]:20 ).
I am confuse. it's not clear.......
please explain again.
thanks
neetu

 

[rsmallela]

Extra mortality

Ok...for a moment, let's think of not meddling with interest rate - let it be 4% throught term. let's deal with only mortality - AM select up to age 55, and AM select with constant additional mortality of 0.00956945 from age 55 onwards. Note that at age 55, we have no longer select effect, we can just use ultimate values.

To evaluate a[40]:20 (with dots on a), you split into two terms - upto age 55 and 55 onwards to allow for extra mortality from age 55. You will need to calculate a55:5 at interest 4% and extra mortality as noted. let's evaluate it:

a55:5 = 1 + v*p55 + v^2*2p55+ v^3* 3p55 + v^4*4p55. where v= 1/1.04.
p55 = exp[-(mu55 + 0.00956945)] = 0.98630 on taking mu55 from AM92 tables.
p56 = exp[-(mu56 + 0.00956945)] = 0.985786
etc..calculate up to p58.

Write 2p55 as p55 * p56, 3p55 as p55*p56*p57, 4p55 as p55*p56*p57*p58.

On substituting these values, I got a55:5 = 4.505.

This way you don't need to change interest rate and still deal with it.

Now, coming to why we change interest: It's because
a) Question has given us value of a55:5 at 5%. So, we must provide solution using given input.
b) Secondly, in calculation of p55, p56 etc., we allowed for additional mortality by simply adding it to mu55, mu56 etc., Actually, it is not clear from question if mu55 is constant over a period of one year. our calculation assumes, even mu55 is constant over each year of age. Otherwise it should have been strictly:
p55 = exp [- (integral s=0 to 1 (mu55+s) ds + 0.00956945)]

By splitting over period of one year, our value came very very close - a55:5 at 5% given in question i.e. 4.503.

Probably, I confused you more???? I do not find an easy way to explain though....hope you are clear about how come this 4 % became 5% with given additional mortality...if not let me know...probably from that you may figure out a different perspective...

Thanks,
Raj