# Difference between assuming distribution of birthdays over calendar year/policy year/rate interval?

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Hi

I am very confused when it comes to the different assumptions that we have to make and when they apply. For instance, what is the difference between assuming a uniform distribution of birthdays over the calendar year compared to assuming a uniform distribution of birthdays over the rate interval? And if possible, can you please give me a quick example that highlights exactly when I would use one assumption over the other?

by (1.4k points)

Hi there!

The application of the assumption (eg uniform distribution of birthdays) will always apply over a specified period of time (1 year in length). If the start date and end date of this period of time are specified by a DATE in the (calendar) year (eg 31 December), and they are a year apart, then we say the assumption applies over a calendar year.

Here’s an example of this in action! If you are given the number of lives aged x last at 31 December 1999, then assuming birthdays are uniformly distributed over the calendar year (31 December 1998 to 31 December 1999, to be precise), half of the xth birthdays will occur from 31 Dec 1998 to 30 June 1999, while the other half of the xth birthdays will occur from 30 June 1999 to 31 Dec 1999.

Now, if you wish to apply the assumption (say UDB) between a period of time (1 year) that is between policy anniversaries, then we say that we assume UDB over the policy year (since this “year” is specified with reference to policy anniversaries). The time between two policy anniversaries is a policy year!

Here’s an example of this in action! If you are told that at a particular policy anniversary you have the number of lives aged x last birthday, then if we assume UDB over the policy year (ie from the last policy anniversary to this policy anniversary we are currently considering), half the lives turn age x between the last PA and the last PA plus half a year, while the other half of the lives turn age x between the last PA plus a half and the PA we are currently considering.

Finally, let’s consider the rate interval (RI). To be on the safe side, we only ever assume that deaths occur on average half way through the rate interval. This can imply that deaths occur uniformly over the RI - and this is useful in helping us to attach an exact age to our force of mortality. When dealing with the POPULATION/EXPOSURE data, we never really assume UDB or UDD over the rate interval since our rate interval cannot (at least for policy year and life year RIs) in most cases be “placed” in calendar time. You typically make such assumptions over a calendar or policy year, as in my examples above.

Hope this makes sense and that I haven’t made any typos.