Fundamentals of Actuarial Mathematics, 3rd Edition

8 Continuous payments

8.1 Introduction to continuous annuities

In this section we consider annuities where payments are made continuously. Naturally, this is not physically possible, but we can picture these as a limiting case of mthly annuities as m goes to infinity. Suppose, for example, that you are to receive a total of 36 500 units each year. This could be done by paying 100 units per day, or 4 1/6 units per hour, or 0.001157 units per second and so on. If you can imagine payments coming in every nanosecond, you may get some feeling for what a continuous annuity would be like.

This may seem as a somewhat artificial concept, but there are many uses for continuous annuities. They can be used to approximate mthly annuity values for large values of m. Moreover, we will show that insurance contracts with the realistic provision of benefit payments at the moment of death, can be viewed as continuous annuities. For insurance contracts purchased by continuous premium payments we can derive some interesting mathematical relationships that are analogues of those that appeared in Chapters 5 and 6.

In the continuous case, we cannot specify an actual payment at any point of time. As shown by the figures above, this approaches zero as the frequency of payment increases. Instead we must speak of the periodic rate of payment. Consider a monthly annuity. If the payment in 1 month is 100, we could describe this by saying that the annual rate of payment for that month is 1200. This would ...