When we calculating the EPV of benefits, the solution given is : 24500A[35]:30+5000(IA)[35]:30+5000(D65/D[35]) I am quite confused of the last part:5000(D65/D[35]), why we have this? Thanks!
That final term appears because the bonuses vest at the end of the policy year (see part (i) of the question). So if the policyholder dies in the first year, they will not have been alive at a bonus declaration point, so no bonus will have been added to the policy and the SA will be £250,000. If they die in the second year, they will have survived through one bonus declaration point, so the sum assured will be increased by £5,000. And so on for each future year of death. If the policholder dies in the final year, they will have passed 29 bonus declaration points, so the sum assured will have been increased by 29*5000=£145,000. So the total sum assured will be £395,000. This is covered by the first two terms of your expression below (245,000 from the first term and 30*5000 from the second). The final term is there to ensure the survival benefit is valued correctly. The survival benefit will be paid immediately following the bonus declaration at the end of the final year. So on survival, 30 lots of £5,000 will have been added to the sum assured. (Since the policy is a 30 year policy it's appropriate that on survival for the full 30 years, you'd have 30 bonuses awarded.) Hence the survival payment is 250,000 + 30*5000 = 400,000 (note this is different to the payment if the policyholder dies within the final year). We need the final term in the expression to value the "extra" £5,000 on survival (the first term gives 245,000, the second gives 30*5000, the final term gives the remaining £5,000). The key thing is that because the bonuses vest at the end of the year the payment on death during the final year is different to the payment on survival, which gives the extra term in the expression. Hope this helps.
