The staff of a company are subject to two modes of decrement; death and withdrawal from employment. Decrement due to death take place uniformly over the age in the associated single decrement table; 50% of the decrements due to withdrawal occur uniformly over the year of age and the balance occurs at the end of the year of age, in the associated single-decrement table. You are given the independent rate of mortality as 0.001 per year of age and independent rate of withdrawal as 0.1 per year of age. Calculate the probability that a new employee aged exactly 20 will die as an employee at age 21 last birthday.
Hi Joe, Thank you so much replying. I don't understand how the withdrawal decrement is being applied, and how its transition rate is being calculated. My calculations: Required: (aq) D = (mu/sigma+mu) * (1 - (ap)) mu = force of mortality sigma = withdrawal transition rate (ap)20 = (1-0.1/2) * (1-0.1/2) * (1-0.001) = 0.9015975 mu = - ln(1-0.001) = 0.001 sigma = - ln( (1-0.1/2) * (1-0.1/2)) = 0.1025866 (aq)D = 0.00095 The (aq)D mentioned in the solution is 0.000879. Once again thanks!
Hi, I think the main issue here is that we're looking for the probability of death at age 21 last birthday rather than 20. In your approach above you have also used a combination of a uniform distribution approach and constant force approach. We can use the rates given in the question unadjusted. So if we project the multiple decrement table: aq(d)_20 = 100,000 * 0.001 * (1 - 0.5*0.05) = 97.5 withdrawals in year age 20: 100,000 * 0.05 * (1 - (0.5*0.001)) = 4997.5 withdrawals end year: 100,000 * 0.95 * 0.999 * 0.05 = 4745.25 Hence lives at start of age 21 = 100,000 - 97.5 - 4997.5 - 4745.25 = 90,159.75 Hence deaths in year age 21: 90,159.75 * 0.001 * (1 - (0.5*0.05)) = 87.91 So required probability is 87.91 / 100,000 = 0.000879 Joe
Hi! Thank you so much for explaining. I still have some doubts. - I don't know what is meant by ; the probability that a new employee aged exactly 20 will die as an employee at age 21 last birthday. - According to uniform distribution wouldn't the following give the probability in the middle of a year; (1 - 0.5*0.05) and (1 - (0.5*0.001)) I really appreciate you giving me your precious time. Thanks.
It's the probability that the life survives to age 21 exact but then dies before age 22 exact so: (ap)20 * aq(d)21 The probabilities presented there are indeed probabilities of surviving those particular decrements to the middle of the year. So we can multiply those probabilities by the death rate or withdrawal rate respectively to give the probabilities of dying or withdrawing mid-year.
Also, please note that multiple decrement tables and probabilities with decrements occurring uniformly across a year do not appear to be on the syllabus any more. Questions are now framed in terms of forces of decrements (mortality, sickness, withdrawal etc). I note that there are a couple of older questions in IAI exams on this but I can't see that this type of question is examinable anymore. At ActEd we have adapted older questions written in this way to be in terms of forces of decrements. Joe
