Fundamentals of Actuarial Mathematics, 3rd Edition

7 Fractional durations

7.1 Introduction

Up to now, we have considered cash flows where the payments were at integer times. We now deal with the case where cash flows can occur at fractional durations. For example, if the basic time unit was a year, and payments were made monthly, these would be at times that are multiples of 1/12. In practice this is a common occurrence. Purchasers of life annuities often want the income to be paid monthly. Many purchasers of insurance policies wish to pay premiums monthly, or possibly quarterly or semiannually. One obvious method of handling this feature is to change the time unit. If we are dealing with annuities with monthly payments, we could just take our unit of time as 1 month, and payments would be at integer times. This option was not always feasible in pre-computer days. For cash flows discounted at interest only, calculations were done from tables showing yearly rates of interest, and elaborate formulas were developed to handle the fractional durations. This is no longer necessary, and the change-of-period approach is the modern way to handle the valuation of cash flows at compound interest.

For life annuities however it is still common to keep the time unit as a year. There are several reasons for this. In the first place, the year has been so ingrained as a measure of age that it seems hard to break away from this tradition. Another more important reason is that insurers often do not calculate premiums exactly for the different ...