The Effect of a Falling Rate of Interest on the Values of Certain Actuarial Functions

Synopsis

The paper first discusses in a general way the methods hitherto devised for giving effect in actuarial calculations to a falling rate of interest, which methods it is suggested are cumbersome and inelastic and so of little practical value. A method is put forward which it is claimed facilitates the study of the many important and interesting problems involved in the assumption of a declining rate of interest.

Where i_t is the rate of interest in the tth year, the assumption is made that

where a, c and k are constants to be determined.

By making certain substitutions the following equation for the value of it is derived :–

It is demonstrated that under certain conditions this equation produces a decreasing series, and that by giving the constants suitable values a wide range of interest curves of the same general character may be obtained.

The formula is developed in combination with actuarial functions, and it is shown that where f′(x) represents the value of a function at a decreasing rate of interest determined by the above formula

where is calculated at rate k and at rate l.

The method is applied to the evaluation of annuities, reversions and other actuarial functions under four sets of interest conditions and the results are analysed and compared with values on a 4 per cent. basis.

The cognate question of an increasing rate is also touched upon.