At the beginning of the year an insurer issued 100000 polici

At the beginning of the year, an insurer issued 100,000 policies expected to experience first year mortality of q_[40]. At the end of the year, 80% of the survivors are expected to experience mortality of q_[41] in the second year.

You are given:

(i) q_[40] = 0.006                   (ii) q_[41] = 0.007                       (iii) q_[40]+1 = 0.008

Calculate the expected mortality rate for the remaining 20% of the survivors.

Solution

At the end of the first year, there are (100,000)(1-q_[40]) = 99,400 lives remaining. 80% of these, or 79,520, experience q_[41], which means at the end of the second year, there are (79,520)(1-.007) = 78,963.36 of these lives remaining.

q_[40]+1 tells us that of all 99,400 lives that make it to the end of the first year, (1-.008) of them survive. That means that at the end of the second year, there must be a total of (99,400)(1-.008) = 98,604.8 lives.

If there are a total of 98,604.8 survivors at the end of the second year and 78,963.4 came from the \"80%\" group, then 19,641.4 came from the \"20%\" group. The \"20%\" group started the second year with (99,400)(.2) = 19,880 lives and finished with 19,641.4. Therefore, q=1-(19,641.4/19,880) = .012