CM1 chapter 27 q3

in the solution to this question it calculates the probability of Amit dieing in the 18th month as: p60*q61*1/12

It states that deaths are uniformly distributed, however my confusion is the above reads to me that amit is dieing in the 13th month. Why are we not evaluating the probably that amit is Alive after 18 months multiplied by q61*1/12. I hope that makes sense

Thanks for your help

 

Consider any month from age 61 to age 62 exact. Let x = {1,2,...,12} represent months past age 61 then P(dying in month x) = P(survived to start of month x)*P(dying in month x | survived to start of month x). We are interested specifically in x = 6 but we can do the general calculation.

Using our assumption of uniform deaths we can calculate P(survived to start of month x) as \( l_{61+([x-1] /12)} / l_{60} \). We can use interpolation to calculate the numerator. Similarly P(dying in month x | survived to start of month x) = \( (d_{61} / 12) / l_{61+([x-1] /12)} \). Noting the numerator here doesn't depend on x as we assume the same number of deaths occur in each of the 12 months in that year period under the uniform assumption. Also note the use of x-1 as we are interested in survival to start of the month.

Multiplying these together to get the probability we want gives us \( l_{61+([x-1] /12)} / l_{60} * (d_{61} / 12) / l_{61+([x-1] /12)} = (d_{61} / 12) / l_{60} \). This doesn't depend on x as the relevant terms cancel, thus the probability is the same for any one month in the period from age 61 to age 62 exact.

This is the same as the probability represented in the notes as we can see in the definition of \(p_{60} * (1/12)* q_{61} \) we get \( l_{61} / l_{60} * (1/12) * d_{61} / l_{61} \) and cancelling gives us the above. We can use this version in the case of the uniform assumption.

Another way to think about it is to consider lots of people taking out this exact policy at age 60 exact and asking how many policies are going to pay out in month 18. If we take the AM92 tables and say, for sake of argument that \(l_{60} \) people take out this policy, then we have 9,287.2164 policies in force at the start. How many of these pay out in month 18? Well if we assume uniform deaths over the period age 61 to age 62 exact, then we would expect \( d_{61} / 12 \) policies out of this starting population to pay out. Note that this is the same for any month in that year, i.e. the same for month 13 through to 24! As a proportion of the starting population (which we take as probability of any one policy paying out in this month) this gives us \( [d_{61} / 12] / l_{60} \). The key thing to remember here is that we want everything relevant to the starting population so we divide by \( l_{60} \).

I hope this helps.