Hello, For this question, this is my initial attempt: The actual answer is 11.011 which I think is reasonably close to my answer as well (to 2dp). However, I am not sure if this is a fluke or if my attempt is actually legit. I think any inaccuracy would arrive in my third line where the value of adue:73.25 is estimated using the weighted average of adue:73 and adue:74. If this happens to be a fluke, may I know which step of my attempt is invalid? Thanks!
Hi, Annuity factors are not linear (with the application of interest and survival probabilities) and so linear interpolation would not be the way to approach this unless we needed something very approximate. The clue to answering this is the line in the question: assume that the force of mortality is constant between ages 73 and 74 only With this line we can calculate this fully accurately: 0.25 * v^0.25 * (p73)^0.25 + 0.25 * v^0.5 * (p73)^0.5 + 0.25 * v^0.75 * (p73)^0.75 + (p73)^0.75 * a_74(4) Or we can combine the final two terms together (rather than 0.25 paid in 0.75 years plus an annuity in arrears from age 74 we can just treat this as an annuity in advance starting from age 74): 0.25 * v^0.25 * (p73)^0.25 + 0.25 * v^0.5 * (p73)^0.5 + (p73)^0.75 * adue_74(4) Hope this helps Joe
Thanks Joe, that clear things up! May I also ask when we can use the weighted average calculation (linear interpolation) like I did in my attempt? I am sure I have seen it used before somewhere in the online classroom when we had a non-integer in the subscript. Thanks!
Hi, We can use linear interpolation to calculate probabilities with non-integer ages but only if we have been told to assume uniform distribution of deaths. The uniformity allows us to linearly interpolate. Had we been asked to assume that here we could have calculated e.g. 0.25p73.25 = l73.5/l73.25 = (0.5 * l73 + 0.5 * l74) / (0.25 * l74 + 0.75 * l73). Joe
