Hi guys, I seem to understand what the rate of payment is but when i look at the example in the notes then i kind of get lost. Chapter 5, Section 1.2 example. The rate of payment at any time is the derivative of the total payment up to that time.So, in that example, the total amount after t years is 5,218,000t and in my understanding, the rate of payment should be the derivative of that which should give 5,218,000. Kindly correct me if am wrong.
I'm still confused Hi John, I'm sorry but I still don't understand. If jm_kinuthia is correct then is the solution to Question 5.3 wrong? If the rate of payment is the derivative of 5,218,000t which is 5,218,000 then the answer to Question 5.3 should be 5,218,000 x 3 = 15,654,000, which is what I initially thought. However, the solution at the end of the chapter integrates 5,218,000t first, and then subs in 3. Why do you have to integrate?
Apologies Greg. I actually made a mistake when responding (that'll teach me for reading things too quickly). Whilst the rate is correct - the total income received so far is not correct. The 10,000 policies sold are sold evenly over the year. This means that the rate of receipt of premium income is increasing over time (with t), as the rate increases as the number of policies sold increases. A very simple example to illustrate this would be to imagine that 1 new policy is sold each year at the beginning of each year and the annual premium is £10. So in the 1st year 1 policy sold = £10 premium, in the 2nd year there are now 2 policies sold so £20 in that year - so total premium = £30 and so on. However, the policies are sold continuously over the year and so the equivalent of summing is integrating. Hence we integrate to get the total summation. So
Thanks John, After reading the question again a few times and thinking about it more I finally understand exactly what's happening. Technically the office only receives income from all 10,000 policies in the last week of the first year and so it couldn't have received £100,000 x 52.18 in the first year as this would mean it had income from 10,000 policies every week. The integration makes perfect sense now. Thanks
stuck Sir The use of integration process is justifiable but still i am stuck for the other way of thinking. The life office get £5218000 in the first year(evenly over the year) and then £5218000 x 2 in the second year and £5218000 x 3 in the third year. So the total amount should be the addition of all the three i e £31308000.
Life office doesn't gets 5218000 in the first year. It would have been the case if all the 10000 policies are sold at the start of the year. But policies are sold are evenly over the year so we've to integrate 5218000t from 0 to 1 to get the premium income for 1st year. 5218000t is the rate of premium income, not premium received until time t. Let's work in discrete time to understand it better. 10000 policies are sold evenly over the year. So approx. 10000/52 = 192 policies are sold every week. 192 people, who were sold policies in 1st week will pay full 52 premiums of £10 192 people, who were sold policies in 2nd week will pay 51 premiums of £10 and so on So, Total premium collected in the 1st year would be 192 * 52 * 10 + 192 * 51 * 10 + ................. + 192 * 1 * 10 = 2645760 This is not very far off from 2609000 (which is what you'll get after integrating " 5218000t " from 0 to 1), when considering the fact that we've done so many rounding offs and worked in discrete time instead of continuous Similarly you can do this for other years as well.
