The Paper draws attention to the loss arising on the sickness funds of Friendly Societies as a result of the steady fall in the rate of mortality amongst the general population in recent years. To set against this loss there are possible sources of profit such as a gain from favourable rates of sickness actually experienced or from excess rates of interest earned ; and if a Society grants a funeral benefit in addition to sickness benefit (as is commonly the case) the gain from light mortality under the former will help to offset the loss from light mortality under the latter. The relative periods of full and reduced benefits and the limiting age up to which sickness benefit is payable (whether granted throughout life or restricted to members below a certain age) are also of importance when the question of mortality loss is under consideration.

Notwithstanding the period that has elapsed and the improvement in mortality that has taken place since the date of publication of the tables based upon the experience of the Manchester Unity of Oddfellows during the years 1893-97, these tables are still employed as the standard tables for much Friendly Society work. The object of the Paper is to consider how the loss arising from certain assumed reductions in the rate of mortality may be set against gains from other sources, so that standard reserves may be set up without profit or loss accruing.

When interpolations are made between values of a function by any of the ordinary interpolation formulae, there are generally discontinuities in the slope and curvature of the successive interpolation curves, and when, as often happens in practical work, the given values are themselves derived from rough data, these discontinuities may be considerable. To remove these breaks of continuity T. B. Sprague in 1880 devised his osculatory interpolation formula, and in the present Paper is given a brief review of subsequent contributions to the subject. From the practical point of view, the most important of the earlier contributions were those of Mr. 6. King, who successfully applied the method to the construction and graduation of national and other life tables; but while two sets of national life tables have since been constructed on the lines proposed by him, there has, for the last fifteen years, been no further development in this country either in the theory itself or in its application.

In contrast to this has been the interest in the method shown by American actuaries, particularly in recent years. Their work is briefly reviewed, and special attention is directed to that of Mr. W. A. Jenkins, who, in 1926, put forward a method of greater generality than that underlying the earlier formulae.

The weak point of the osculatory method, regarded as a smoothing agent, rests on the fact that the graduated curve is required to pass through certain predetermined points. The curve will, in fact, be constrained to take a form similar to that assumed by a flexible steel wire which is clamped at fixed points, so that, while the curve is free from discontinuities, any departure of these points from the smooth curve will be reproduced with resulting undulations. To remove this tendency to waviness, Jenkins has devised his modified osculatory method, which, while requiring the successive interpolation curves to have the same slope and curvature at their common points at the ends of each interval, does not require the curves to pass through the points corresponding to the calculated values.

His fifth difference formula has been applied to regraduate the English Life Table No. 9, Males, and in connection therewith it is pointed out that the success of the method must depend to some extent on the provision of a good set of guiding values ; for, while the curve is not required to pass through the points corresponding to these values, the closer they lie to the smooth curve the better is the graduated curve likely to be. The method recommended for adoption is one based directly on summations in fiveyear groups of the unadjusted rates of mortality. By means of a formula derived from the modified osculatory formula quinquennial values are obtained. These are treated as first approximations, and the formula is again applied to them in order to produce rates which are treated as the graduated rates. The results are satisfactory both from the point of view of smoothness and of fidelity to the data, as measured by the agreement between actual and expected deaths.

An interesting point which emerges from the various graduations and from the comparisons of actual with expected deaths is that the crude rates of mortality appear always to exceed the graduated rates for ages centering about 24, 34 . . . and to fall below them for ages centering about 29, 39 . . . ; and it is suggested that this is due to some persistent tendency to misstatement of age, and that it is also largely responsible for the waviness of graduations of population data which is generally characteristic of the application of the ordinary osculatory method.

As a further test of the possibilities of the method, and, in particular, of its smoothing powers, it has been applied to graduate the data of the Government Female Annuitant Experience 1900-1920. These data had proved very intractable to curve-fitting methods, but had been found to show an element of periodicity up to about age 90 with an apparent change in the form of the curve after that age. In connection therewith, formulae are given for applying the method to the data at the ends of the experience where the main method is inapplicable. The result is to produce a curve which is almost perfectly smooth and which closely follows the original data.

The method is essentially a graphic method in which the curves are drawn in accordance with mathematical formulae instead of being drawn by hand, and the work is arranged in such a way that it can be carried out practically from beginning to end by purely mechanical processes.

During the lifetime of most Life Assurance Companies until 1897 the general trend of interest rates was downward. Such conditions proved favourable for the traditional investment policy of these years. From 1897, however, until 1920 the trend of interest rates changed to an upward direction and heavy depreciation in security values was experienced. In recent years much thought has been given to the problems of investment, and modifications in policy have been made with a view to avoiding or at least diminishing future depreciation in capital values.

The principal movements in security prices are the long period trends and the short cycle variations, both of which accompany the general economic movement of all prices and can be studied in their general relationship to the economic cycle. The statistical data needful to such study, especially in this country, is very limited. Although statistics are available in America they are for the most part restricted to the two broad groups of Bond prices and Share prices, while in France the problem most dealt with is the relative movement between fixed interest securities and those having a variable dividend.

A new investigation has been made in respect of the last five years and the results are given in the present Paper. The monthly movement in prices has been taken out for a variety of classes ranging from British Government securities to Industrials, and where such exist a separate index is shown for Debentures, Preference and Ordinary. The prices employed are net prices after allowing for accrued interest.

As the results only show the movements of the last five years, the period covered is not a complete cycle but only the opening phase of the short period cycle. They are in no way intended to afford a solution to the various problems of movement but merely illustrate the position at the present time.

The main conclusion of the Paper is that the prices of securities, like all prices, are related to the general economic cycle. While the evidence cited is in some ways contradictory, it is suggested that the movement of Ordinary securities follows directly that of the economic curve, while the movement of Debentures follows that curve inversely but with a different tempo and interval. At the same time the intermediate fluctuations of the Ordinary are of much greater extent than those of Debentures.

As regards the more fixed type of investment policy in which sales of any magnitude are not contemplated, if the long period trend is downward it would appear inadvisable to enter the Ordinary field at all. On the other hand, the investments in Debentures will give complete security with steady appreciation of capital. When the long period trend is upward, however, Debentures will suffer depreciation, and to balance this a proportion of funds must be held in Ordinary Securities which in such conditions will show considerable appreciation in capital value, quite apart from an increasing yield. With a more mobile investment policy the Ordinary field deserves to receive the same considerations as the Debenture field, the profits of the Ordinary more than balancing the losses of the Debentures in a period of long period upward trend, while in a downward trend the short period cycle gives opportunity for profitable investment in both areas.