QBank: 5.6

In the solution:
Deaths are recorded as as age x last bday ---> LIFE YEAR

At start of rate interval, a life would be age 48 exactly and no assumption is needed

How would this change if death was recorded as age x nearest bday, or age x next bday?

Correct my understanding if wrong, if the rate interval is calendar year, then all lives will change age label at that specific date, say Jan 1. If age is recorded as next bday, then there's average age at start of interval x-1/2, so the age at start of interval is not x. Why is there even an average age at the start of the interval is everyone's age is suppose to change simultaneously?

This is so confusing. Anyone here have more exercises on these which I could work on??

 

If age is defined as age x next birthday, the life year rate interval begins at the (x-1)th birthday and ends at the xth birthday.

For age x nearest birthday, the interval begins at age x-1/2 and ends at age x+1/2.

You need to keep in mind that there are two different definitions of age going on. The insurer will have a recorded age based on whatever perverse age definition they happen to use, but you will have your exact actual age as well (ie the current date less your date of birth).

As an example, say age is defined as "age last birthday at previous 1 Jan", so we have a calendar year rate interval starting at 1 Jan. I was born in 1979, so according to this definition, my age at the start of the rate interval (ie at 1 Jan 2009) is 29. It doesn't matter exactly when in 1979 I was born, my age will still be recorded as 29, as at 1 Jan 2009 that was my last birthday.

However, my exact age at the start of the rate interval will be different depending on when in the year I was born. If I was born on 30 Dec 1979, my exact age at the start of the interval would be just over 29. But if I was born on 2 Jan 1979, my exact age at the start of the interval would be just under 30. There is a range of possible exact ages from 29 to 30 that will give the same age according to the insurer's definition.

If we assume that birthdays are distributed uniformly throughout the calendar year, on average birthdays will occur at 1 July, so the average age at the start of the rate interval will be 29.5.

 

Thanks Michael

So when the text says "all lives changing age label at Jan 1", it is refering to the recorded ages by the insurer, right? The thing is, wouldn't insurers have the date of birth already, so why use this method of recording ages?

Let me try to understand this; so insurers have deaths recorded via some method (age last bday, next bday.. blablabla) so we're trying to match this death data to some exposure-to-risk measure which may be recorded in a different way, correct?

Do you think it matters if my death data is recorded as age next bday on Jan1 1982, but my exposed-to-risk is measured as at 31 Dec 1981? My point is, is the one day difference significant? Or is it dependant on the age recorded in exposed-to-risk?

 

(i) Yes, the lives are changing age label - the age value that the insurer has given them. It does seem odd that insurers wouldn't collect proper date of birth information, but apparently they don't always do so

You can imagine that they might just ask for your age at the time of taking out a policy - which would give rise to the definition "age last birthday at previous policy anniversary"

(ii) That's right. Age will be recorded at time of death using some definition of age. It could be age last birthday at time of death, or age last birthday at previous policy anniversary, or anything else they feel like... If the deaths are being recorded in a different computer system to the policies it's easy to imagine that they could be recorded inconsistently. We need to measure the exposed-to-risk using the same age definition as our deaths.

That's why we do all the messing around with Px(t) and P'x(t). We define P'x(t) so that it uses the same age definition as our deaths. We use this to calculate the exposed-to-risk. Then we try to write P'x(t) in terms of the policy data Px(t) we have actually collected.

(iii) Not exactly sure what you're asking here... I can't imagine that one day would make much difference for a life year. You could run into issues if this was a calendar year rate interval though, say if you had a age definition "age last birthday at prev 1 Jan" then the difference between measuring at 31 Dec and 1 Jan the next year would be significant!

 

Lemme rephrase:

Do you think it matters if my death data is recorded as age x next bday on Jan1 1982, but my exposed-to-risk is measured as age x last bday at 31 Dec 1981?

In this case, we want to use exposure as age x next bday at Jan1 1982 and age x next bday at Jan1 1983, so need to approx using age x-1 last bday at 31 Dec 1981 and age x last bday at 31 Dec 1982 ??

 

I'm a little confused by those age definitions. Is your age definition "x last birthday", measured at time 31 Dec 1981? Or "age x last birthday at the previous/next/closest 31 Dec"?

I assume you're asking about the latter, let me know if I've got the wrong end of the stick.

Say you had
dx = deaths aged x next birthday at prev or coincident 1 Jan
and
Px,t = lives aged x last birthday at prev or coincident 31 Dec, measured at time t after 1 Jan 1982

Say we were interested in the period 1 Jan 1982 to 31 Dec 1982

For exposure to risk, we need to integrate
P'x(t) = lives aged x next birthday at prev or coincident 1 Jan
= lives aged x-1 last or coincident birthday at prev or coincident 1 Jan
from t = 0 to 1
and would approximate this integral with
1/2 * (P'x(0) + P'x+1(1))

P'x(0) = lives aged x next birthday at 1 Jan 1982
ie lives whose birthdays are between 2 Jan 1982 and 1 Jan 1983 inclusive

This is almost but not quite the same as Px-1(0)
Px-1(0) = lives aged x-1 last birthday at 31 Dec 1981
= lives aged x next or coincident birthday at 31 Dec 1981
ie lives whose birthdays are between 1 Jan 1982 and 31 Dec 1982 inclusive.

I'd imagine this is close enough, but if you really wanted to you could approximate P'x(t) by taking
364/365 * Px-1(t) (the people whose birthdays are between 2 Jan 1982 and 31 Dec 1982)
+ 1/365 * Px(t) (the people whose birthdays are on 1 Jan 1982)

I think. I may have got the details wrong in here, this stuff gets confusing!

At any rate, I think there's going to be a trivial difference from just taking P'x(t) = Px-1(t), and this depends on the assumption that birthdays are distributed uniformly over the calendar year.

Does this help or are you even more confused now? I think I've just managed to confuse myself...

 

No, your explaination is quite clear. But a few questions I have to ask:

This equation here, does come to you automatically once you have determined that this is a calendar year interval?

I feel silly asking this but I don't understand why 1/365*Px(t) is used to approximate those ppl whose bdays are on Jan 1 82, but at the same time, I cannot think of better replacement for Px(t).

Maybe it's the date Jan 1 83 that's bothering me.

Sorry if I made u more confused. But thanks for taking the time to explain this.

 

1/2 * (P'x(0) + P'x+1(1))

This equation here, does come to you automatically once you have determined that this is a calendar year interval?

Yes, you have to be careful with calendar year intervals to change the x to x+1 as well as increasing the time in the second term.


I feel silly asking this but I don't understand why 1/365*Px(t) is used to approximate those ppl whose bdays are on Jan 1 82, but at the same time, I cannot think of better replacement for Px(t).


Hmm, I think I got that stuff wrong... Told you I'd confused myself!

Px-1(0) = lives aged x-1 last birthday at 31 Dec 1981
= lives aged x next or coincident birthday at 31 Dec 1981
ie lives whose xth birthdays are between 31 Dec 1981 and 30 Dec 1982 inclusive.

Px-2(0) = lives aged x-2 last birthday at 31 Dec 1981
= lives aged x-1 next or coincident birthday at 31 Dec 1981
ie lives whose xth birthdays are between 31 Dec 1982 and 30 Dec 1983

So we'd take 363/365 * Px-1(0) to get lives whose birthdays are between 2 Jan 1982 and 30 Dec 1982
and 2/365 * Px-2(0) to get lives whose birthdays are between 31 Dec 1982 and 1 Jan 1983.

I wouldn't worry about this though, I haven't seen any exam questions like this.

 

Thanks! We're getting bit too carried away with that aren't we?

My big challenge now is to identify what type of rate interval the question wants; looking at recent questions, the way the question states this is getting less obvious and more often I need to crack my head to figure out what they're talking about! For instance, that Q10iii from April 2007.