# Violation of Uniform distribution assumption - Will this affect what age mu and q applies to ?

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Suppose we are dealing with census data and the rate interval beginning for lives classified age $$x$$, starting at the policy anniversary where they were aged $$x$$ nearest birthday.

It is clear that estimates for $$q_x$$ applies to age $$x$$ because that is the average age in the beginning of the rate interval by assuming birthdays are uniformly distributed across the policy year.

However, now suppose it is discovered that half of the policies were taken out on the day before a birthday and the rest are distributed evenly over the rest of the year. How would this affect the age that $$q$$ applies to ? Based on the new info, the average age in the beginning of the rate interval is $$x-\frac{3}{4}$$ instead of $$x$$ as before. Does this now mean that $$q$$ applies to $$x-\frac{3}{4}$$ now ? The memo I am dealing with says it does not change and this partly makes sense because we are not changing any assumptions about deaths. But it is also confusing because the average age in the beginning is not $$x$$ anymore.

Also how would this affect the census data and calculation of $$E^c_x$$?

answered Apr 8 by (1,170 points)
edited Apr 8

Thank you for your question. I assume you are referring to tutorial 4, question 4 of section 5. I will write everything in detail here because this question is fairly delicate in nature.

You are correct in saying that $$q_x$$ applies to age x on the basis of the assumption stated.

Let me consider your question on part (ii) first. Now, if 1/2 the policies were effected a day before the birthday, this does NOT impact the age at the beginning of the rate interval. Why? Because the average age at the beginning of the rate is determined WITH REFERENCE to the nearest birthday, and we can still safely assume that birthdays are uniformly distributed over the policy year. In other words, assuming that policies are taken out a day before the birthday has NO impact on when the birthdays themselves happen.

Now, were you able to find the corresponding exposures for part (i)? If not, we can proceed as follows, and the reasoning below is important for part (ii) (note that PA = policy anniversary). In order to find your exposure, define $$P_x(t)$$  as the number of policies (or pensioners) in force aged x nearest on 31 December of year t.  We need $$P_x(t) \ast$$, which is the number of policies  in force at 31 December year t, age x nearest on the previous PA.

• Now, if PAs are uniformly distributed over the calendar year, then the average policy anniversary date is 30 June.
• Of those lives aged x nearest on 31 December year t (i.e. your $$P_x(t)$$) , assuming birthdays are uniformly distributed over the calendar year (1 July to 30 June, to be precise), 1/2 will turn age x between 30 June and 31 December of year t (i.e. they will be between ages (x-1/2,x) on 30 June), while the other half will turn age x between 1 Jan and 1 July of year t+1. Hence, can you see that 1/2 $$P_x(t)$$ are aged x nearest on the average PA date of 30 June?
• Similarly, consider the those lives aged x+1 nearest on 31 December of year t. Argue, using the same logic as in the point above, that 1/2 $$P_{x+1}(t)$$ are aged x nearest on the average PA date of 30 June.
• So, $$P_x(t) \ast$$ = 1/2 $$P_x(t)$$) + 1/2 $$P_{x+1}(t)$$.
• You are now in a position to calculate the exposure via the usual method in the slides.

Let's return to part (ii). Even though I did not want you to consider it in the question, what is the impact of the additional information on the exposure? Does it affect our chain of reasoning in the bullet points above? Let's check.

• We know that of the $$P_x(t)$$'s, 1/2$$P_x(t)$$ have a PA one day before a birthday, but for the other half the PAs can still be assumed to be distributed uniformly over the calendar year. The latter can be treated as we did in part (i).
• There is also NO reason stopping us from assuming that birthdays are uniformly distributed over the policy year.
• But tread carefully. At 31 December year t, if birthdays are uniformly distributed over the calendar year, then 1/4$$P_x(t)$$  have their xth birthday 6 months before, and the other 1/4$$P_x(t)$$ have their xth birthday 6 months after.
• Clearly, the (first) 1/4$$P_x(t)$$  will be aged x nearest birthday on the policy anniversary PRECEEDING 31 December year t, since their birthday occurs one day after the PA! The other 1/4$$P_x(t)$$ will be age x nearest birthday on the PA AFTER 31 December year t (they are, in fact, not needed).
• You can now do a similar argument with the $$P_{x+1}(t)$$'s that have birthdays one day before the PAs, and show that 1/4$$P_{x+1}(t)$$ are indeed aged x nearest on the policy anniversary preceding 31 December year t.
• You can also show that the overall impact will NOT change the answer for the exposure you found in part (i).

I hope this helps. I would also be interested to see your arguments pertaining to the $$P_x(t)$$'s for part (i) of the question. Let me know if you think anything is wrong with my answer.

commented Apr 8 by (430 points)

Thanks for the detailed response Mario! I did manage to get the exposures in part (i) but was just struggling with the reasoning for part (ii)

commented Apr 8 by (430 points)

Would I be correct in saying that it would affect the value that q applies to if we were dealing with a life year rate interval instead of a policy year rate interval as in this question ?

commented Apr 9 by (1,170 points)

Good question, but I don't think that a life year rate interval would ever apply to a situation where policy anniversaries are involved (well maybe in certain perverse (and exotic) cases it may, but they will most certainly not be asked in BUS3018F). Maybe you can convince me otherwise?

Secondly, if your attempt at finding the exposure in part (i) differed from my workings above, do you mind please posting it here (even briefly)? It may be good for the others to see an alternative methodology to approach the exposure calculation.