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

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 t th 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.