X Assignment X3.3

Discussion in 'CM1' started by Han, Feb 12, 2022.

  1. Han

    Han Keen member

    Hello,

    I have a couple of queries regarding question 3.3 of the X assigments.

    1) For the survival benefit, the solution uses AM92 Select mortality for the D/D factor and PMA92C20 for the annuity factor. However, the numerator of the D/D factor is D_65, so shouldn't this be found under the basis of PMA02C20? Another example would be: If we are given the same basis in the question such that we have AM92 Select before age 65 PMA92C20 after age 65, and we need to calculate l_75/l_[40], would we be getting l_75 from the PMA92C20 basis?
    Note: I am aware that there is no D_x data in PMA92C20 but my question relates to how we should be finding mortality rates given different basis.

    2) When calculating the death benefit, the solution shows the increasing term assurance for 25 years because this is the deferment period. However, I'd like to know why we aren't over-calculating the EPV when we have a term assurance of 25 years. The 25th term of a term assurance includes
    v^25* 25p[40] *q65. Here, q65 is the probability that a life aged exactly 65 dies in the next year. However, wouldn't the survival benefit already kick in once the life reaches aged 65? So wouldn't we be counting both the survival benefit and death benefit at age 65 in this case?

    Thanks in advance and please let me know if anything above is unclear.
     
  2. Joe Hook

    Joe Hook ActEd Tutor Staff Member

    Hi, answers below:

    1)

    Dx+n/Dx is defined as v^x+n*lx+n / v^x * lx = v^n * lx+n/lx or v^n * npx. You could calculate this directly of course but if you have AM92 mortality and 4% you can use the D/D factor. In this question D65/D[40] represents v^25 * 25p[40] or in other words the 25 year discount factor multiplied by the 25 year survival probability. As the mortality over this period from age 40 to 65 is AM92 (select) D65/D[40] works.

    If the basis changed at say 65 and the annuity was payable at age 75 then we would need to split the discount factor * survival probability into two parts.

    Either:

    D[65]/D[40] * v^10 * 10p65 (PMA92C20) or
    v^35 * 25p[40] (AM92 select) * 10p65 (PMA92C20)

    Basically, you can't mix and match your numerator/denominator in l/l expressions. They must both come from the same table.

    2)

    You may want to have a look at level/increasing term assurances again. The final term in a level/increasing term assurance of 25 years would have v^25 * 24p[40] * q64 ie 25 year discount factor multiplied by the probability that the life dies in the 25th year. To die in the 25th year the live must survive 24 years and then die in the 25th.

    (IA):[40](1):<25> = v*q[40] + 2 * v^2 * p[40] * q[40]+1 + ... + 25 * v^25 * 24p[40] * q64

    Your expression above does not quite represent this final term as you have added a year to the ages.

    Hope this helps.
    Joe
     
    Han likes this.

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