INSTITUTE OF ACTUARIES

Transcription

1 INSTITUTE OF ACTUARIES THE VALUATION OF ANNUITY BUSINESS BY J. A. WESTCOTT, F.I.A. Joint Actuary, Sun Life Assurance Society AND E. M. SMITH, F.I.A. of the Sun Life Assurance Society [Submitted to the Institute, 23 March 1959] RECENT years have seen a material growth in the annuity business of British offices and in many instances what was a relatively insignificant part of the offices' activities has now assumed considerable importance. In fact in the case of some offices, the amount of the liability in respect of annuity business exceeds that in respect of assurances. 2. This trend is likely to be accentuated as a consequence of the Finance Act, 1956, since (i) the tax concession to offices in connexion with pension annuity business enables specially favourable rates to be quoted for pension schemes approved under section 379 of the Income Tax Act, 1952, and for retirement annuities for self-employed and certain other classes of persons ; (ii) retirement annuities now attract, within certain limits, full tax relief on the premiums ; and (iii) immediate annuities have been made attractive to the investor by the provision of relief from tax in respect of the 'capital element' of the annuity payments. 3. At the same time, the Finance Act, 1956, brought about for taxation purposes segregation from the main annuity fund of 'pension annuity business', enabling offices to improve benefits or allow corresponding reductions in premiums under existing business of the latter category. 4. These factors, combined with the prospect at some time in the near future of the introduction of electronic processes, have no doubt caused many actuaries to review the valuation bases and principles of the annuity fund. This paper has been written in the light of such a review, setting out some of the problems considered, and in the hope that a discussion of annuity valuations generally will be of interest to members of the Institute. No attempt has been made in this paper to deal with tax considerations, since these were fully dealt with last session in the paper by C. E. Puckridge and the discussion which followed (J.I.A. 84, 166). 5. The amount of time which the Institute has given to the consideration of the problems involved in the valuation of annuities is relatively small compared with the time given to life assurances, but some excellent papers which deal with valuations of group pension business have been written, e.g. A. G. Simons, J.I.A. 71, 375, and Philip and Robson, T.F.A. 21, AJ

2 322 The Valuation of Annuity Business VALUATION PRINCIPLES 6. Although most of the problems arising in connexion with the valuation of annuities relate to deferred annuities effected under staff pension schemes and group pensions, it is convenient to dispose of the other classes first, particularly since the bases for immediate annuities have some bearing on the bases to be used for deferred annuities and group pensions after the expiration of the period of deferment. (a) Immediate single life annuities 7. There is little choice of methods for the valuation of immediate single life annuities. They are ordinarily grouped according to age nearest birthday at the valuation date and valued by irrespective of the frequency of pay- ments. If the annuities are ' with proportion' an additional reserve should be made of one-half of the periodic payment times Āz. 8. For guaranteed annuities, i.e. annuities payable for a fixed period and thereafter throughout life, an additional reserve is required. The method by which this additional reserve is calculated will depend upon circumstances. If there are only a few annuities involved the additional reserve may be calculated by multiplying the annuity per annum by for each policy at each valuation, where z is the valuation age and n the number of years' guaranteed payments remaining. The maximum number of years for which annuities are 'guaranteed' is usually small, and if there are sufficient such annuities it may be desirable to group them according to the number of years' payments remaining, subdivided according to age attained. The total reserve for such annuities would in that case be found by multiplying the annuity per annum by a factor of the form (b) Immediate joint lives and survivor annuities 9. Here again the method to be used will depend upon the number of annuities to be valued. If there are comparatively few, they may be valued individually by similar formulae to single life annuities but using joint lives and survivor functions ; valuation in groups is not usually appropriate unless some approximate method is introduced, owing to the large numbers of combinations of ages involved. Appropriate allowance would be included for with-proportion annuities and those payable for a number of years and thereafter during the joint lifetime and lifetime of the survivor. It is quite common for the annuity to be reduced on either the first death or (if the annuity relates to a male and a female life) on the death of the male life, and these can readily be dealt with by treating them as a combination of a joint lives and survivor annuity and single life annuities. Where a method of grouping is adopted there will usually be some annuities which do not fit into the grouping and which have therefore to be valued individually, including those on three or more lives. 10. Methods of grouping for valuation purposes and approximate methods of valuation have been suggested by Elphinstone and Lindsay in T.F.A. 17, 39, and by G. V. Bayley in J.I.A. 79, 323, and if the number of annuities to be dealt with is large there is material advantage in adopting one of these methods.

3 The Valuation of Annuity Business 323 (c) Reversionary annuities 11. Reversionary annuities may be valued individually by a net premium method; owing to the nature of the risk the theoretical reserve in the early years is frequently small or even negative, and a minimum reserve would usually be held, of say one-half of a year's office premium. Alternatively, if the volume of business is small an arbitrary basis may be adopted based on the yearly office premiums payable plus an appropriate reserve for single premium and all-paid business. 12. There is justification for assuming, when calculating the reserve for a reversionary annuity, that a gross rate of interest will be earned after the death of the life assured but that a lower rate will be earned during the joint existence. An approximate method can be employed to make provision for the two rates of interest but, as generally the reserves are relatively small, for simplicity the lower rate of interest may be used throughout (this of course slightly overstates the liability). The valuation of reversionary annuities issued in combination with retirement annuity policies under the Finance Act, 1956, presents a special problem and is dealt with later in this paper. (d) Deferred annuities and group pensions 13. These contracts comprise mainly: (i) deferred annuities, without return, by single premium, being the deferred annuity portions of 'guaranteed annuities' (effected before the passing of the Finance Act, 1956) which for tax considerations were issued 'split' into annuities-certain and deferred annuities; (ii) deferred annuities by individual policies or by 'master' policies, effected in connexion with staff pension schemes ; (iii) group pensions ; and (iv) retirement annuity policies effected by the self-employed, etc., under the Finance Act, Usually the annuity under a policy of type (i) does not commence until a relatively high age (e.g. where a life aged 65 effected an annuity guaranteed for 15 years of this type, the first payment under the deferred annuity would not be made until age 80) ; it is thought preferable, therefore, to segregate such policies from ordinary deferred annuities, which usually commence between ages 60 and 70, and to value them separately. 15. A suitable plan is to employ functions of the Karup type and to classify according to year of birth. Thus if Y is the annuity per annum, x is the age at entry and n is the number of years in the term of deferment for an annuity payable half-yearly in arrear, the value of the annuity at the end of the term of deferment will be equal to and the Karup constant will be equal to say. If there are no further premiums payable the valuation reserve at age z will be equal to K1/Dz. If there are premiums payable and a net premium valuation is being made the Karup constant for valuing net premiums of Ρ would be equal to 2 say. The value of future premiums at age z would be adjusted for fractional premium corrections. Having regard to the nature of the business, an annuitants' mortality table such as the a (55) should be employed throughout, viz. during the deferred period as well as after the vesting date. 21-2

4 324 The Valuation of Annuity Business 16. As mentioned already, most of the problems in connexion with the valuation of annuities relate to deferred annuities of type (ii) and group pensions and these will be considered separately in the next few paragraphs. Probably few offices have any large number of deferred annuities effected by individual persons (apart from retirement annuity policies effected by the self-employed) and the great bulk of the business therefore relates to deferred annuities and group pensions effected under staff pension schemes. 17. Basically, deferred annuities and group pensions are the same type of contract in that premiums are payable at commencement or over the period of deferment to provide an annuity payable as from a certain date, and sometimes in this paper the term deferred annuities is used to cover both deferred annuities and group pensions. There are, however, distinctive characteristics and these may well lead to the conclusion that it is preferable to classify them separately, but this will be considered later. METHODS OF VALUATION OF DEFERRED ANNUITIES AND GROUP PENSIONS 18. Opinions are likely to differ as much on the methods to be used to value these classes, by far the largest section of the annuity business, as on those for any other classes of life and annuity business. There are a number of methods of valuation which may be employed, for example (i) retrospective valuation by accumulation of the office premiums paid, (ii) prospective valuation using the true net premiums, (iii) prospective valuation using a net premium equal to a proportion of the office premium, and (iv) prospective valuation of the paid-up annuity secured by the premiums already paid. 19. Each of these methods has its merits and demerits, and none is unquestionably the most suitable. Considering them in turn : (i) Retrospective valuation by accumulation of the office premiums paid 20. For with-return contracts, the accumulation method is simple to operate and, provided that the rates of interest employed are at least equal to those involved in any guaranteed return on death or termination, will ensure that the reserve is never less than such return. It also eliminates any question of negative values. 21. This method has the disadvantage, however, that it may produce a liability at the vesting date differing materially from that to be set up on transfer to the immediate annuity class. Even if a relatively high rate of interest is employed this may not be too satisfactory in that it may produce a liability at the vesting date larger than that needed for longer terms of deferment but possibly less than that needed for shorter terms. To overcome this difficulty the expedient may be adopted of taking the liability as the accumulation of (P' + k'), the constant k' being so calculated as to ensure that the full liability is accumulated at the vesting date but negative values of k' being ignored. An obvious disadvantage of this method becomes apparent when it is desired to make a change of valuation basis for immediate annuities and to apply the revised basis in calculating the liability for deferred annuity contracts at the vesting

5 The Valuation of Annuity Business 325 date. This involves a re-calculation of the constant k' and it may be difficult to obtain readily an indication in advance of the effect of the change in basis. 22. In the case of without-return contracts where it is necessary to accumulate the premiums with benefit of survivorship, a retrospective valuation can readily be made by means of functions of the Karup type but usually a prospective basis of valuation would probably be thought more suitable. 23. The purist may consider that a valuation should look to the future and that only a prospective valuation can, strictly speaking, be justified ; however, since the liability will at least reach the 'sum assured' by the vesting date the retrospective method here described has in fact nearly as much to recommend it as the net premium valuation of life assurances, where a change in valuation basis is accompanied by a change in the net premium. (ii) Prospective valuation using the true net premiums 24. In so far as the business comprises group pensions by single premiums, the liabilities brought out by the three prospective bases are necessarily the same, and the valuation itself is relatively simple. 25. The problems arise mainly where periodic premiums are payable. By analogy with life business one would expect that where a net premium valuation basis is thought suitable for the life business, a similar basis should apply to the annuity business, and, but for the fact that periodically it may appear desirable to change the basis, this method would seem to have most to recommend it. 26. A change in the rate of interest involved in the valuation basis is accompanied by a change in the net premium, and at least one office records on the valuation cards for its life business a simple (Pi' Pi") constant to enable a change of the valuation rate of interest to be readily made. A change in the valuation rate of interest in the case of a deferred annuity type of contract may, however, involve the rate of interest during the term of deferment only, the rate of interest after the vesting date only, or both rates, and it is scarcely practicable to record a suitable function. 27. Rates of premium vary materially with variations in the rates of interest or mortality assumed, and in calculating net premiums care is necessary to ensure that they do not exceed the office premium since this would not ordinarily be regarded as satisfactory. 28. A useful formula for valuing annual premium deferred annuities with return without interest is given by Ogborn and Wallas in their paper in J.I.A. 81, Bearing in mind all the many variations possible (viz. in the ways the annuities may be payable, the periods for which the annuities may be guaranteed, the ages at which they may commence), the re-calculation of net premiums is a major operation (unless a computer is available) and accordingly a net premium equal to a proportion of the office premium is frequently employed as described in the next sub-section. (iii) Prospective valuation using a net premium equal to a proportion of the office premium 30. A common expedient to enable a prospective valuation to be made without calculating pure net premiums is to take the net premium as k times

6 326 The Valuation of Annuity Business the office premium. For without-profit annuities k might well be 95% if premiums are payable yearly, or 92½ % if premiums are payable monthly and a loading of 2½ % is imposed. 31. Such a plan has the merit that it is a simple matter to change the valuation basis, since no calculation of pure net premiums is involved, but care may be necessary to eliminate negative values, which could be substantial, particularly if there have been material changes in the bases of calculating premiums. 32. On the other hand, there may be an initial strain in respect of new business, if the rates of interest and the mortality tables used in the valuation are more stringent than those used in calculating the premiums. Especially may this be so in the case of retirement annuity policies, without profits, for the selfemployed (unless, of course, the valuation basis is adjusted accordingly) since one feels that offices generally went out of their way to quote exceptionally favourable terms for these policies. (iv) Prospective valuation of the paid-up annuity secured by the premiums already paid 33. The amount of the annuity secured by the premiums already paid is ordinarily determined by application of a (1 P'/P') factor, which should strictly be operated by employing office premiums on the basis of the scale of premiums inherent in the contract. The method has the merits that no net premiums require to be calculated and as a change of basis involves merely a change in the valuation factors a change of basis presents no undue difficulty. Negative values, of course, cannot arise. 34. If, however, there have been frequent changes of rates, the calculations in respect of contracts issued under staff pension schemes which involve benefits at various rates of premium may be unduly complicated and it may be thought suitable for the purpose of applying the (1 P'/P') factor to use net premiums on the valuation basis throughout. The effect of using (1 P/P) factors based on net premiums calculated on the same bases as the valuation functions is, of course, to produce a liability the same as the ordinary net premium valuation liability. 35. The method described above suffers, however, from one or possibly two defects. In the first place it does not always operate satisfactorily when interest rates are rising. Consider a without-return annuity of 1 per annum. Let x = age at entry, n = term of deferment, t = duration since entry, a = value of annuity at the end of the term of deferment on the valuation basis. Accented functions refer to the premium basis ; others to the valuation basis. 36. Amount of annuity purchased by premiums already paid (assuming a constant percentage loading for expenses) and liability (1)

7 The Valuation of Annuity Business Liability on a true net premium basis 38. Thus (1) is the same as (2) if the valuation and premium bases during the term of deferment are the same. If they are not the same the liability under (1) will be more or less than under (2) depending upon whether is more or less than Now this function increases with an increase in the rate of interest so that if the rate employed during deferment in the calculation of the premiums is less than in the valuation basis and the mortality basis is the same, formula (1) will produce a smaller liability than formula (2). In the case where the rate of interest employed during deferment in the premium basis is higher than in the valuation basis, formula (1) will produce a higher liability than formula (2) but, of course, in practice the true net premium might have to be reduced to make it not greater than the office premium and a greater liability would result than the true net premium liability. The use of formula (1) based on (1 P'/P') is analogous to valuing recurrent single premium business without considering the effect of rates of premium being guaranteed for future years. 39. Secondly, premiums are not always payable yearly, but are frequently payable monthly, certainly so far as the employees' contributions are concerned, and the (1 P'/P') method is difficult to operate unless the valuation is made as on the policy anniversary. 40. In staff pension schemes, where the employees' contributions are applied on a 'with return without interest' basis, and the employer's contributions applied on a without-return basis, the annual premium payable by the employer may be determined from a 'balance of cost' table and the office records may not show (as it has not been specifically calculated) the actual apportionment of an annuity as between the employee's and the employer's contributions. This does not, however, prevent the application of the (1 P'/P') method of valuation. From the amounts of premium, with and without return, the separate amounts of annuity which will be purchased by future premiums can be calculated and, by subtracting their sum from the total pension, the pension already purchased will be found. Where the employees' premiums are returnable with interest, it is necessary to record separately the annuity secured by the employer's and the employee's premiums since, in the valuation of the employee's annuity, a valuation factor in the form of vt will be employed, incorporating provision for the liability for the death return. (2)

8 328 The Valuation of Annuity Business 41. Whatever the method of valuation, when there is a change in valuation basis there may be a very considerable variation in the total valuation liability if the change is applied to existing business. For this reason possibly and because of the considerable amount of work involved, there is a temptation for an actuary, when introducing new valuation bases, to apply them to new business only and not to disturb the valuation of existing business. It is thought, however, that it would be helpful to have some tables indicating the variation in the valuation liability consequent upon changes in the valuation bases and changes in valuation methods ; this is referred to again later. VALUATION DATE (i) Valuation of individual schemes on policy anniversary 42. Under ordinary life business the principle is that policies are grouped so far as practicable and valued on the valuation date. 43. The overwhelming proportion of deferred annuity business, however, is effected under staff pension schemes and in the early stages of the development of this class of business, only a relatively few schemes were involved and these ordinarily embraced a relatively large number of employees. The practice grew up in some quarters of valuing the schemes individually on the respective policy anniversaries and adjusting for 'ons' and 'Offs' and interest from the policy anniversary up to the valuation date. 44. This plan has the advantage that the durations are ordinarily integral and the same age groupings can be used for costing and valuing, thus assisting in the use of one record for both purposes. Furthermore, the valuation, if made when the costing is being made, can act as a check on the costing ; also the work is spread over the year, helping to reduce the work at the valuation date. The disadvantage of the plan is that the total liability at the scheme anniversary has to be adjusted for movements, including retirements, between the scheme anniversary and the valuation date, and there is in total, one suspects, considerably more work involved in valuing each scheme separately than valuing all the schemes together. 45. The pension age for each sex is of course usually the same for the whole scheme and the benefits similar in nature. Accordingly, only one grouping by age for each sex is necessary. (ii) Valuation of all business on valuation date 46. With the development of staff pension scheme business, there have come into being more and more schemes, until so far as some offices are concerned a very large number of schemes are in force; at the same time, the average number of lives included has tended to decrease, and schemes are in fact now frequently written for relatively few employees. In these circumstances the plan described in the last sub-section loses much of its attraction and it becomes preferable to value the business as a class on the valuation date in groups according to the unexpired term on that date. Whilst under this latter plan the business is dealt with in the same way as all other business, so probably involving less work generally, the valuation does not provide any check on the costing, and separate records are required for the valuation and for costing since, with differing dates, different factors apply.

9 The Valuation of Annuity Business It is fairly common practice under group pension schemes for the cost of the past service benefits to be met over a period of years. Frequently the actual period is not specifically defined and the employer's contributions for this purpose are applied as single premiums, in order of seniority by age, to purchase the pensions outright. No special difficulties of valuation present themselves in this connexion. On the other hand, occasionally the period is fixed and it is specifically provided that the capitalized cost of the past service pensions shall be met by equal instalments of a stated amount payable over a certain number of years. In such cases the past service benefits can be valued with the rest of the business and the value of the employer's future contributions separately valued by an annuity-certain for each scheme. 48. In the case of a valuation on the valuation date there is one adjustment which might be made, viz. in respect of incidence of premium income. The general experience is, we believe, for an overwhelming proportion of schemes to be inaugurated in the early part of the year, and very frequently the cost to the employer is met by premiums payable yearly; the average premium due-date is therefore less than half-way through the year. DEFERRED ANNUITIES AND GROUP PENSIONS CLASSIFICATION 49. At first sight these two types of contract have everything in common, but in practice, at least in the way that the business has developed in some offices, there are considerations which complicate their amalgamation. 50. Group pension business is largely arranged (apart from the past service element) on the basis of the employees becoming entitled to a unit or units of pension in respect of each year of service completed and the employees contributing a fixed amount (e.g. 1s. 0d. or 1s. 3d. weekly) in respect of each such unit of pension. Ordinarily the return on death is limited to the employees' own contributions without interest. Originally, much of the business was arranged on the recurrent single premium plan, although with the steady reduction in the size of new schemes, annual premiums are now becoming more common. The actual proportion of the benefits secured by the employees' and the employer's contributions respectively are not necessarily calculated. 51. Deferred annuity schemes are ordinarily arranged on an annual premium basis (excluding possibly the past service element) and it is not uncommon for the pension to take the form of the product of the length of possible future service and 1/nth of the salary 5 years before the normal retirement date, with the employee contributing a percentage of his salary towards the cost. Further, it is quite common for the employees' contributions to be applied on a 'with return with interest' basis, the employer's contributions being ordinarily applied on a without-return basis. 52. Under group pension business by recurrent single premiums, the straightforward method of valuation is to calculate a liability equal to the sum of (i) the value of the deferred pension, and (ii) the value of the assurance for the return of premiums. 53. In the case of deferred annuity business, the separate portions of annuity purchased respectively by the employees' and the employer's contributions are ordinarily available. As the employees' contributions are frequently returnable with interest a class separate from group pensions seems

10 330 The Valuation of Annuity Business appropriate, and it would probably be suitable to value the portion of the benefits secured by the employees' contributions by a pure interest function during the term of deferment, ignoring mortality. 54. Whilst, therefore, the separate classification of group pensions and deferred annuities under staff pension schemes would be avoided if practicable, it may well be that the particular features of the business may frequently render their separation desirable. Any group pension business which exceptionally is with return with interest could perhaps be valued with the deferred annuities and any deferred annuities which are with return without interest, with the group pensions. PENSION ANNUITY BUSINESS AND GENERAL ANNUITY BUSINESS 55. In accordance with the provisions of the Finance Act, 1956, the interest earned by a life office on the premiums received under (a) individual retirement annuities approved under that Act, and (b) pensions schemes approved under section 379 of the Income Tax Act, 1952, is free from income tax. Such business is known as pension annuity business and the remainder of the annuity business is known as general annuity business. 56. As the taxation position of the two funds was examined fully by C. E. Puckridge, in his recent paper, it is sufficient here to say that a life office can assume in calculating premiums and in valuation that it will receive interest free from income tax, i.e. at a gross rate, in respect of pension annuity business, but that it is appropriate to consider that some tax will normally be payable on the interest received during deferment in respect of the premiums relating to deferred annuities and group pensions included in the general annuity business. 57. For the sake of completeness it might be mentioned here that consequent upon the passing of the 1956 Act, offices generally improved the terms, by increasing benefits or reducing premiums, of most deferred annuities and group pensions in force which had been issued under schemes approved under section 379 and were to form part of the pension annuity business. This improvement in benefits was commonly given effect to by calculating the paid-up annuity secured by the premiums already paid and increasing such paidup annuity by one-half per cent per annum for each remaining year of the term of deferment, any balance of annuity being secured by assuming that it was effected at new premium rates calculated on the assumption that the interest earned would be free from income tax. (The improvement was not allowed in respect of certain earlier contracts based on rates of premium in force many years ago which were calculated on relatively high rates of interest.) This business became part of the pension annuity business section, and the bases on which the amount of the liabilities was calculated largely determined in which of the funds any valuation profit or loss as a result of the transaction appeared. Actual views on this no doubt vary and are affected to some extent by the size of the notional loss which was being carried forward for income tax purposes in the general annuity business section.

11 The Valuation of Annuity Business 331 RETIREMENT ANNUITY POLICIES WITH REVERSIONARY ANNUITY BENEFITS 58. A development of the retirement annuity policy for the self-employed is one providing as an ancillary benefit a reversionary annuity to the annuitant's widow commencing on the death of the husband whenever that may occur, the premium in respect of the reversionary annuity ceasing on the death of the wife during the term of deferment or, if she survives, at the end of the term of deferment. These annuities may of course be valued as a deferred annuity plus a reversionary annuity, but if the annuity is 'guaranteed', complications arise in the valuation of the reversionary annuity as the guarantee under the reversionary annuity is reduced correspondingly by any payments made under the retirement annuity. 59. The benefits may, however, be regarded as (a) a reversionary annuity commencing on the death of the male during the term of deferment leaving his wife surviving (payable for say 5 years and thereafter during the lifetime of the wife) plus (b) a deferred joint lives and survivor annuity guaranteed say 5 years, contingent upon the male life surviving the term of deferment. These benefits are somewhat similar to a special endowment assurance on the life of the male for a sum assured which on death (provided his wife is alive) is sufficient to secure the immediate guaranteed annuity to the wife and on survival of the term of deferment by both lives is sufficient to secure a joint lives and survivor guaranteed annuity. In fact, specimen calculations show that a reasonable approximation to the liability for the combined retirement annuity and reversionary annuity is obtained by calculating the net premium liability as for an ordinary endowment assurance by uniform premiums on the life of the male with a sum assured equal to the value, at the end of the term of deferment, of an immediate joint lives and survivor guaranteed annuity, but using a gross rate of interest instead of the net rate ordinarily employed for endowment assurances. On death of the wife during the term of deferment the annuity would be transferred and valued as an ordinary retirement annuity. RATES OF PREMIUM 60. In order to make a comparison of the liability brought out by the various valuation methods and the effect of changes (as referred to in earlier sections of this paper) it is necessary to have scales of office premiums under businessin-force at the present time. 61. The bases employed for the calculation of office premiums have varied considerably over the last 25 years, i.e. over the period during which the bulk of the business now in force was secured. In order to consider the effect of changing conditions, specimen rates of premium have been calculated on bases which it is thought could be assumed to represent an approximation to those applicable to 5-year periods over the 25 years. The formulae used for male lives are as shown overleaf, the ages being appropriately amended for female lives.

12 332 The Valuation of Annuity Business Deferred annuity commencing at age 65 payable monthly in advance, guaranteed 5 years. Let (1 + e) = multiplier to allow for loading required. (a) Single premium, without return (b) Annual premium, without return (c) Single premium, with return without interest (It is appreciated that the loading should strictly be applied to the whole of the premium.) (d) Annual premium, with return without interest 62. The rates have been calculated for deferred annuities commencing at ages 65 (males) and 60 (females) and in order to simplify the calculations it has been assumed that in all cases the term of deferment is an integral number of years. 63. The bases of interest and mortality assumed and the expense loadings included are as shown on the next page. 64. It is probable that for some years the annuity funds of offices actively seeking pension scheme business have been 'C' type for taxation purposes, so that (apart from business since 1956 under section 379 schemes) the rate of interest assumed during the term of deferment should properly have been lower than that assumed to apply thereafter. In practice it is thought that not many offices adopted such a practice, possibly having in mind that a margin in the rate of interest after the vesting date could properly be justified in view of the length of the contract and as an offset against possible adverse trends in mortality and to make provision for the expenses of paying the annuities. 65. The rates of premium for annuities with return without interest and without return are set out in the Appendix and it will be seen that the rates

13 The Valuation of Annuity Business 333 vary considerably according to the basis adopted (far more so in fact than in the case of life assurances). Basis Years to which basis is assumed to apply Interest basis assumed (%) During deferment After deferment Loadings (all terms and ages) (%) During deferment Mortality After deferment A B C D E F /2 3 / / Nil A ult. A ult. A ult. Ages 'rated down' 1 year a (f) and a (m) ult. a (f) and a (m) ult. Ages 'rated down'2 yeares a(5) Note. It is not suggested that any office would have employed precisely these formulae or these bases for calculating rates of premium and, in fact, the A table was not available until It is appreciated, too, that certain offices have employed scales of rates of single premiums with a loading of 5 % scaled down where the term of deferment is less than, say, 5 years, and scales of rates of annual premiums obtained from summing the reciprocals of the single premiums ; for simplicity of calculation of scales of rates for the purpose of this paper, flat loadings of 4% have been employed. In practice the rates were not revised at regular intervals of 5 years. It is thought, however, that the use of the bases shown will not affect the conclusions to be drawn. VALUATION BASES (a) Rates of interest 66. It is not the purpose of this paper to endeavour to establish a basis for the rates of interest to be employed in valuations since these must necessarily depend upon the circumstances of the particular office, always bearing in mind that the future terms of the contracts to be valued may be very long. 67. Certain considerations were discussed in Puckridge's paper and it may be accepted that a gross rate of interest is suitable for immediate annuities. 68. An actuary considering possible valuation bases for deferred annuities would naturally bear in mind that the business ordinarily comprises policies placed on the books over many years. They may have been effected at widely differing rates of premiums, based on, inter alia, varying assumptions as to the taxation position of the office in respect of the policies during their lifetime. The actuary will not normally consider it desirable to deal with each section of the business separately but will consider the office as a whole and fix his bases to take into account the taxation position which he expects to apply. For pension annuity business, a gross rate of interest is suitable throughout for deferred annuities, but for general annuity business during the term of

14 334 The Valuation of Annuity Business deferment a rate somewhere between the net and gross rate should properly be employed. As the process of transferring deferred annuities in the general annuity section to the pension annuity section proceeds, it may well be possible to justify a rate of interest moving nearer towards the gross rate. In deciding upon a rate of interest it is well to bear in mind that part of the excess of the rate of interest earned over that assumed in the valuation is required to cover expenses. Furthermore, it is desirable to make the valuation sufficiently stringent to take care of the cost of options, etc., for which a reserve may not be specifically made. 69. For the purpose of this paper, however, it is proposed to regard reasonable rates for a prospective valuation as being as follows : General annuity Pension annuity business % business % Immediate annuities 4 4 Deferred annuities During deferment 3 4 After deferment Prior to the Finance Act, 1956, a fairly common rate of interest for valuing immediate annuities was 3 % which rate was frequently combined with the employment of a mortality table less severe than the latest tables relating to annuitants. Offices may be reluctant to weaken their valuation by increasing the rate of interest assumed; on the other hand, it is difficult to see why a lower rate of interest should be needed for the valuation of immediate annuities under general annuity business than for pension annuity business. During the last two or three years, however, immediate annuities have been granted at rates of interest materially higher than those previously employed, and the strain which would be incurred by valuing such business at artificially low rates of interest may result in offices deciding to employ more realistic rates of interest in coming valuations. (b) Mortality 71. The mortality assumed for the valuation of annuities should take into account the nature of the business. That applicable to pension schemes after the annuities have become payable might reasonably be assumed to be higher than that applicable to ordinary purchased annuities because the lives are of a different type. They are not people who have purchased annuities because they think that they are good lives and likely to profit by the transactions. (That this is so is borne out by the results of the Continuous Mortality Investigation experience given in J.I.A. 84, 73.) On the other hand, it is important to remember that while the a(55) is a forecast mortality applicable to annuities purchased in the year 1955, in general deferred annuities may not become payable until many years later. Mortality trends seem to be persistently lower and actuaries cannot overlook Gwilt's warning in T.F.A. 25, 92, that improvements in mortality in Great Britain have not kept up with those in Scandinavian countries and that the very great improvement that has taken place there could presumably take place in Great Britain. In these circumstances the a(55) Table does not seem an unreasonable table to employ for the valuation of ordinary purchased annuities and, after vesting, for the valuation of deferred annuities (but see paragraph 114). A somewhat less

15 The Valuation of Annuity Business 335 stringent mortality table, e.g. the a(f) and a(m) Table with ages rated down might be justified for the valuation of vested annuities under pension schemes. 72. During the term of deferment, mortality is usually ignored when premiums are returnable with interest on death. This assumption is tantamount to assuming in the case of a net premium valuation that on death the net premiums will be returned accumulated with interest at the valuation rate, for that is what the reserve will amount to. If therefore a true net premium valuation is to be used consideration should be given to the position which will arise in the event of the death of the annuitant during deferment; whilst an additional reserve could be made, this will seldom be thought necessary. 73. If premiums are not returnable on death it is important to ensure that the mortality is not overstated, since otherwise the liability will refer to too few survivors at the vesting date, i.e. it will be understated. The rates of premium for deferred annuities and group pensions granted in connexion with staff pension schemes have usually been calculated on the basis of assurance mortality during deferment, not distinguishing between sexes. In our view this is scarcely justifiable unless the proportion of females is small in view of the much lighter mortality of females ; a rating of, say, 4 years off the age (during deferment) should properly be made in calculating the the rates of premium for females, and possibly a similar rating should strictly be made for the valuation. 74. In considering the mortality to be assumed during deferment it should not be overlooked that there is a tendency for employees whose health begins to fail to leave the service or retire early and for healthy employees to remain actively employed. The mortality experience during deferment of the lives in a scheme (active service mortality) is thus likely to be lower than that applicable to assured lives generally. Attention was drawn to this feature by A. C. Stalker in a paper in T.F.A. 23, 14, where rates of mortality are given applicable to lives in active service. It is apparent therefore that the mortality assumed during deferment for calculating rates of premium, and calculating the valuation liability, in respect of without-return annuities under pension schemes should be lighter than assured lives mortality unless the surrender value on leaving 'in good health' has a substantial fine. If the return of premiums on leaving service 'in good health' is 90% of the premiums paid accumulated with compound interest and if 'good health' is liberally interpreted, it is suggested that where the A Table is employed for valuation purposes the ages should properly be rated down. In the case of annuities with return without interest the same mortality basis during deferment as for without-return contracts would appear to be appropriate. (The mortality assumed during deferment does not have so much effect upon the liability as in the case of without-return annuities.) (c) Loadings 75. The allowance for expenses (other than commission) in deferred annuity and group pension premiums may take the direct form of a loading in the premiums or the indirect form of the use of a lower rate of interest than could otherwise be justified, or a combination of both. Latterly, the tendency has probably been towards the inclusion of a specific loading in premiums,

16 336 The Valuation of Annuity Business although whether this may be expected to cover all expenses, with the size of schemes growing progressively smaller, is a moot point. 76. In net premium valuations of annual premium contracts the value of the difference between the office premiums and the net premiums is reserved for future expenses and commission ; where the value of only a proportion, say, 95% of the future office premiums is taken into account a reserve for future expenses and commissions is of course automatically made. The practice is not unknown of increasing the value of the annuity at the vesting date by, for example, 1½% or possibly 2% in order to provide a reserve for expenses of payment of the annuity. These expenses may be substantial if offices are unwise enough to allow themselves to be manoeuvred into the position of paying pensions under section 379 funds direct and accounting for tax under P.A.Y.E. This would result in the keeping of records of tax codes for each pensioner, calculating the tax deductible from each payment and accounting for the tax deductible to the Inland Revenue. Even in a fully mechanized system using a large scale computer the additional work involved would be substantial. NET LIABILITIES AND EFFECT OF CHANGE OF VALUATION BASIS USING VARIOUS VALUATION METHODS (a) Single premium business 77. Table 1 shows specimens of the reserve values under single premium deferred annuities, without return. These values have been calculated prospectively by the formula (wherex+(wherex+nis the age at the end of the term of deferment and x+t is the age at the valuation date) on the bases as regards interest and mortality inherent in the various scales of premium (viz.a,b,c,d,e and F)and two further bases (viz.m and N)one being a basis which might be used by offices valuing general annuity business at the present time and the other where a more stringent view is taken of the values of annuities at the vesting date. 78. As these are without-return contracts, the reserve values in columns A,B/C,D and E not only show the reserve values on the various bases but also indicate how those values would be increased if the reserve values for business on other scales of premiums were increased to accord with those inherent in new premium scales as they are introduced. It will be noted that basis M produces values slightly lower than those under scale E, due of course to a lower value for the annuity at the vesting date, resulting from the use of 4% interest instead of 3%. 79. In periods of falling interest rates an office is faced with the problem of increasing reserves for existing business up to those on the bases of valuation adopted for the business on the latest premium scales, unless the actuary considers it justifiable to value the business on various bases according to the respective premium scales. It will be seen that the strain of putting up the increased reserves may be very substantial. 80. It has been fairly general practice for offices when putting into operation group pension schemes by recurrent single premiums to allow some

17 The Valuation of Annuity Business 337 Table 1. Deferred annuities, without return, by single premiums, commencing at exact age 65 (males) or 60 (females) payable monthly in advance and 'guaranteed' 5 years Reserve value per 10 per annum annuity Exact age attained A B, C D Basis E F M N Male lives Female lives l Ο Note. It is appreciated that at some of the younger ages the circumstances in which under group pension schemes the employer may contribute as well as the employee are rare, but the figures for all the ages have been inserted for the sake of completeness. Interest basis (%) Mortality Basis During deferment After deferment During deferment After deferment A-F M Ν See 65 A A a(55) a (55) measure of guarantee as regards the rates which will apply in respect of employees already included in the scheme. Where rates of premium are increasing, therefore, the yearly increases in benefits may require to be calculated on a number of premium bases unless it is possible to reach a compromise with the employer and charge an overall scale of rates representing a rough average of the various scales which would apply. The office may in consequence have a continuous flow of business on the various terms. Where rates of premium are reduced, on the other hand, it is difficult to justify to an employer why single premiums on the old scale should be charged for any of the new business and in practice probably only the current 'low' scale would be used. 81. No attempt has been made to construct a model office with a view to examining the effect of changes in valuation bases, since offices have varied so 22 AJ

18 338 The Valuation of Annuity Business considerably as regards the development and conduct of the business that it is thought that this would be of little benefit, but the tables given above and later may enable a rough but reasonable indication to be obtained of such changes. 82. Table 2 shows specimens of the reserve values under single premium deferred annuities, with return without interest. These values have been calculated prospectively by the formula (60 being substituted for 65 in the case of females) on the same bases as regards interest and mortality as those inherent in the premiums. The table also shows for the respective premium scales the corresponding reserves if calculated on bases M and N. 83. In this instance, of course, the reserve values on any particular basis show only the values for annuities actually effected on the particular premium scale corresponding to that basis ; the values for other annuities would differ very slightly on account of the variation in the premiums returnable on death. It will be seen from the values under bases M and N, however, that values do not vary greatly on this account. 84. These tables do raise one problem, viz. to what extent an office committed to granting single premium business on very favourable terms (such as scale A rates of premium) should set up a reserve, when premium rates increase, in respect of future business which will accrue on those terms. If the rates are such as could be justified by ignoring all margins, possibly such a liability could be ignored ; otherwise an extra reserve might properly be set up, approximating to the difference between the value of future pensions and the value of future premiums. (b) Annual premium business 85. Apart mainly from the problem mentioned in the previous paragraph, the valuation of single premium business is relatively simple as compared with annual premium business and the effect on the liability of changes in valuation bases more readily assessable. On the other hand, the matter is simplified in one respect as regards annual premium business in that it is believed that the rate guarantees ordinarily given under annual premium schemes were for only short periods and consequently the business at the various scales of rates will have been written in certain limited periods of years, and have limited spans of durations and ages at entry, somewhat on the lines indicated below : Basis of rates Possible calendar years of issue Possible number of years duration as at 31 December 1958,say Possible range of ages at entry, if retirement ages 65 (males) and 60 (females) Males Females A ΒC D Ε F I onwards years years years 9 13 years 4 8 years Up to 3 years Up to 41 Up to 46 Up to 51 Up to 56 Up to 61 All Up to 36 Up to 41 Up to 46 Up to 51 Up to 56 All

19 The Valuation of Annuity Business Table 3 shows specimens of the reserve values under annual premium deferred annuities, without return, calculated on valuation methods (ii), (iii) and (iv). 87. Where the rates of interest and mortality inherent in the premiums are employed, the liability by valuation method (iv) is identical with the net premium liability, [method (ii)], and the values in column (11) are therefore identical with those in column (4). 88. It is interesting to note that the liabilities by method (iv) employing the bases M and Ν exceed the corresponding net premium liabilities except where the net premium valued is reduced in order that it may not exceed the office premium. 89. Table 4 shows the corresponding specimen reserve values for annual premium deferred annuities, with return without interest. In this case the reserve values on method (ii) have been calculated by the modified formula where x = age at entry, n = term of deferment, t = duration since entry, Ρ' = the office premium payable. 90. In the case of female lives 60 and 65 are substituted for 65 and 70 respectively. 91. Similar features to those commented upon in connexion with deferred annuities without return appear. 92. In certain circumstances it may be thought that method (iv) represents a simple and attractive method of valuation, and it is instructive to consider what variations appear in the proportions (1 P'/P') when calculated by (i) with return without interest premiums, and (ii) without return premiums on the basis of the last scale of rates (basis F) for male lives : Age at entry x Duration in years t Value of (1 -P'/P') If calculated on the basis of with-return premiums If calculated on the basis of without-return premiums

20 340 The Valuation of Annuity Business

21 The Valuation of Annuity Business 341

22 342 The Valuation of Annuity Business

23 The Valuation of Annuity Business 343

24 344 The Valuation of Annuity Business

25 The Valuation of Annuity Business 345

26 346 The Valuation of Annuity Business 93. Bearing in mind the very wide differences in values which may be obtained by variations in the rates of interest or mortality, it would seem reasonable to employ the proportions shown in the last column in the table in 92 for the whole of the business. EFFECT ON LIABILITIES OF IMPROVEMENT IN BENEFITS AND CHANGES IN VALUATION BASES CONSEQUENT UPON THE FINANCE ACT, As mentioned in 57, following the Finance Act, 1956, offices generally improved the benefits or reduced the premiums under certain business effected in connexion with section 379 schemes. A similar improvement was allowed in connexion with the section 388 schemes modified to be eligible as section 379 schemes. In general, the improvement (which applied only to deferred annuities and group pensions which had not vested) took the form of increasing the benefits purchased by the premiums already paid, at the rate of ½ % for each year of future service, subsequent to the date of the change, and increasing the pension corresponding to future premiums to that which would be secured by such premiums on the basis of the new rates of premium. Alternatively, future premiums were reduced to those necessary in future on the new scale to secure the balance of the pension required. 95. Where liabilities are calculated by a (1-P'/P') type formula [method (iv), future premiums and the improved benefits secured thereby do not enter into the calculations. Similarly, in the case of a net premium valuation where, after improvement of the benefits, the liabilities are calculated on a basis as regards mortality and interest identical with those inherent in the revised premium scale, the liability in respect of future benefits would be exactly zero at the date of transfer. 96. Table 5 shows specimens of the benefits and reserve values prior to the changes and specimens of the revised benefits and reserve values after the changes consequent upon the Finance Act, No calculations have been made in respect of basis A rates of premium, because these are already on 4% interest and no improvement in benefits is applicable. 97. It will be seen that in all cases the reserve values in column (9) are less than the reserve values in column (7) despite the improvement in the benefits. A similar feature occurs in the case of annuities with return without interest. WITH-PROFIT ANNUITIES 98. Bonus distributions in the annuity fund present features and problems different from those in the life fund. Whether or not there will be a mortality profit is conjectural and the valuation 'profits' from discontinuance are less than in the life fund. In effect the bonus is very largely provided from the excess of the rate of interest earned over that inherent in the premiums together with the normal addition for the right to participate in profits. So far, apart from retirement annuity policies, the issue of annuities with participation in profits has been largely restricted to policies effected in conjunction with staff pension schemes.

27 The Valuation of Annuity Business 347 Table 5. Deferred annuities of 10 per annum, without return, by annual premiums, commencing at exact age 65 (males) or 60 (females) payable monthly in advance, guaranteed 5 years Basis of Premiums Age at entry x Age attained x+t Unexpired term n-t Reserve Annuity value corre- on basis sponding inherent to in premium premiums already (true net paid premium reserve) Reserve value on valuation basis (see below) Revised benefits consequent upon the Finance Act, 1956 Reserve value on revised valuation basis (see below) (1) (2) (3) (5) (6) Male lives (4) (7) (8) (9) B C D I E O Text Female lives B C I D E I The reserve values in column (6) have been calculated by valuing the annuity shown in column (5) on the bases as regards interest and mortality corresponding to those inherent in the respective premium scales. The values in column (7) have been similarly calculated by valuing the annuity shown in columns (5) on basis M (see Table I). The reserve values in columns (6) and (7) are thus the same as those shown in columns (11) and (12) of Table 3. The values in column (9) have been calculated by valuing the annuity shown in column (8) on the same basis as the values in column (7) excepthat 4% interest was employed throughout.

28 348 The Valuation of Annuity Business (a) Retirement annuity policies 99. Bonuses granted in connexion with retirement annuity policies ordinarily take the form of an addition to the basic yearly annuity and are payable from the normal vesting date, such addition being either a simple addition calculated by reference to the basic annuity or a compound addition calculated by reference to the basic annuity and any existing bonuses The cost of providing a uniform annuity bonus increases from year to year, as does the cost of providing a uniform reversionary bonus under an endowment assurance. In the case of endowment assurances by periodic premiums provision for future bonuses may conveniently be made by valuing the basic contract on a net premium basis at a relatively low rate of interest, which has the effect of controlling the rate of release of surplus. Similarly, provision may be made during the term of deferment for future annuity bonuses by making a net premium valuation at a low rate of interest, and there appears to be no need to demonstrate this. (b) Group pension and deferred annuity policies 101. There is considerable scope for the design of methods of declaring bonuses. These may take the form of simple or compound additions to the basic annuity or (at any rate in so far as the employer's policy is concerned) the form of periodic cash refunds. Bonuses may accrue only during the period of deferment or there may also be a final bonus (additional annuity or cash refund) at the vesting date. (There are also plans under which rebates of premium are allowed calculated by reference to the rate of interest announced by the office as earned on its funds.) 102. Here again where the bonuses are declared as additions to the basic annuity, the net premium valuation at a low rate of interest during the term of deferment may commend itself as a suitable basis, with the modification that if the policy includes provision for a final bonus, provision for this may also conveniently be made by employing a relatively low rate of interest to value the annuity at the vesting date Where bonuses take the form of periodic cash refunds, it is customary for the annuities to be secured by single premiums and for bonuses during deferment to be based upon the amounts of such single premiums and the period for which they participate. If the rate of bonus is to be constant, the surplus which should be released each year during the period of deferment should strictly also be constant. The practical expedient of valuing at a low rate of interest has the effect of releasing surpluses increasing from year to year. Provided the rate of interest is chosen so that the initial reserves and the reserves at the vesting date are adequate, therefore, the surplus released in the early years of the deferred period will tend to be on the low side and in the later years too high. Hitherto, however, business has normally been effected on the basis that the pensions are purchased in order of seniority by age so that in practice most of the business is deferred for a relatively short period. The effect of the increasing rate of release of surplus is therefore not very important and is on the safe side, and a suitable rate of interest can be fixed by ensuring that the reserves at entry are adequate to provide the cost of the benefits and all bonuses during deferment.

29 The Valuation of Annuity Business Where a final cash bonus is to be paid at the vesting date this may conveniently be provided for by assuming, when valuing annuities during deferment, that the interest rate after deferment will be lower than would be appropriate for without-profit business. The lower rate would be fixed so that the difference between the liabilities at the vesting date on the two rates of interest would be equal to the cash sum which it is desired to provide as bonus It may be helpful to illustrate the application of these principles relating to cash bonuses by some examples. Ignoring loadings for simplicity, let it be assumed that the single premium to provide an annuity of 1 per annum is determined from a formula derived from the equation : where the A table with 4% interest is used during deferment and the a (55) (males) table with 2½% interest thereafter The value of the annuity at the vesting date has been taken at 2½ %, the difference between the values at 2½% and 4% being a provision for the final cash bonus The following are specimens of the single premiums per 10 per annum annuity, payable monthly in advance, from age 65 : Age Male lives Single premium The liabilities could be calculated at rate i (say 4%) by the following approximate formula which makes provision for cash bonuses during deferment of say 1.5% per annum of the single premium: where interest at rate i is used during deferment and at rate 2½% thereafter (the same mortality tables being used as stated in 105) In practice it may be adequate to calculate liabilities by a simpler formula of the form (2) below employing a lower rate of interest, i', such that i i' is equal to the rate of bonus it is hoped to provide. Thus if a bonus of 1.5% per annum during deferment is to be provided, the rate of interest of 2½% would be used for all valuation functions The reserve values produced by the two formulae on these bases are given overleaf. (1) (2)

30 350 The Valuation of Annuity Business Age attained Age at entry Reserve value per 10 per annum Formula (1) Formula (2) In each case provision has been made for a final bonus equal to the difference between the2½%and 4% values at the vesting date Bearing in mind that reserve values of deferred annuities vary so greatly according to the bases adopted for valuing the annuity at the vesting date, it is suggested that the simpler formula (2) produces satisfactory results. MISCELLANEOUS OPTIONS 113. Any option granted under a policy may be expected in certain contingencies, however remote, to cost the office money and strictly speaking some additional reserve should be set up. In some cases the amount involved taking into account all contingencies may be negligible, but in other cases it may be material. (a) Widows' options 114. Possibly the most important option from this point of view is the option commonly available to an employee under pension schemes at or just before the vesting date of forgoing part of his pension in exchange for a pension to his widow commencing after his death, no evidence being called for as to the employee's state of health. It is to be expected that an employee who on approaching the vesting date is in serious ill-health will take full advantage of this widow's option. In practice it is generally recognized that the substitution of a reversionary annuity to a female life in good health for an annuity to an impaired male life (even although such an annuity may be 'guaranteed' for a number of years) may materially affect the ultimate cost to the office ; when determining the terms upon which the option will be permitted some allowance may be made for the possible state of health of the male life in calculating the amount of the widow's pension, as, for example, by equating the value of the reversionary annuity on the female life to 90 % only of the value for an ordinary male life of the annuity forgone, but it is scarcely practicable to quote terms which will fully safeguard the office. It will not be satisfactory therefore when determining the valuation basis to be employed for male deferred annuities after the vesting date to have regard only to the mortality of male pensioners, since it is necessary to take into account the effect of the exercise of widows' options. Although one might after obtaining adequate experience (as to proportion of impaired lives, degree of impairment etc.) possibly obtain some measure of the cost of those options, a practical solution would seem to be to rate down the age of the male life or to use a 'stringent' mortality table at the vesting date; alternatively a lower rate of interest than would otherwise be thought suitable could be employed.

31 The Valuation of Annuity Business 351 (b) Optional retirement dates under retirement annuity policies 115. It sometimes happens, especially in the case of retirement annuity policies for the self-employed, that the policy contains a schedule quoting amounts of annuity according to the age at which the annuitant decides that the annuity should commence. With changing interest rates, it may well be that the valuation liability will vary according to the actual age selected. If the policy is valued on the basis of retirement at the latest age in the range (which gives the highest amount of annuity) some additional reserve should strictly be set up to provide for the contingency of an earlier retirement age with a higher liability being selected. (c) Surrender and paid-up annuity options 116. In making his valuation the actuary will naturally consider the position which arises on the surrender of annuities or (if subject to premiums) their conversion to paid-up annuities. Except possibly during the early years of a deferred annuity, surrender values are unlikely to exceed the valuation liability and may well be much smaller. Paid-up annuities are sometimes granted on very favourable bases, and may even be equal to the full value by a (1 P'/P') formula. From Tables 3 and 4 an indication can be obtained of the differences between the liability under the original policy on a pure net premium or modified net premium basis and the liability under the policy if calculated by reference to the benefits secured by the premiums paid to date Ordinarily the effect of the above options on the liability will be unimportant but the position might be different if, for example, as a result of the introduction of a comprehensive national pension scheme a large proportion of existing schemes were to be discontinued. In such an event it might even be necessary for offices to reduce staffs and change their organizations in the light of the new circumstances. The cost to the office of the possible exercise of the options in present conditions is not in general specifically provided for in the valuation but this is one of the factors normally taken care of by ensuring that the valuation basis is inherently strong, e.g. that there is a satisfactory margin between the rate of interest which it is expected will be earned in future and that assumed in the valuation. AUTOMATION 118. At the time of writing this paper it is believed that no life office in this country has had practical experience of the use of a computer for work in connexion with the valuation of annuities, etc., although some offices have had the experience of employing computers for carrying out specific jobs, for example the calculation of reversionary bonuses, and probably most actuaries are by now aware of the possibilities, and some of the limitations, of the machines From the information available at the present time it would seem that most offices who have placed an order for a computer have ordered one of the small machines designed to operate with punched cards, and that only a few offices have decided to purchase a large-scale computer using magnetic tape. Probably when experience has been gained on the small machines more offices will feel justified in purchasing a large one ; such a machine will,

32 352 TheValuation of Annuity Business however, involv e conception s not covere d in thi s paper, includin g a greate r emphasis on individual valuations. Therefore for the present purpose the smal l machin e is the one considered Most small computers have very restricted storage capacity and can only include a relatively small number of instructions in the 'programme'. They can, however, very satisfactorily carry out a large repetitive job which is not unduly complicated. It is not economic to use a computer for the processing of tiny batches of cards since the time taken to set up the machine, feed in the storage information, pass the cards through the machine and tabulate them is greater than would be taken to make the calculations manually. It would be possible to use a computer for the costing of large schemes but it may not be worth while doing so for small schemes, say under lives, unless several schemes with identical, or nearly identical, features could be dealt with together It might appear that the use of a computer for valuation of group pensions would increase the advantages of making the valuation at the renewal date. This may well be so in the case of a large machine where the storage and programme capacity are sufficient to enable all the necessary work to be done at one time. In the case of a small machine, however, it would seem unlikely, especially if the mortality and interest bases used for valuation differ from thos e used in calculatin g the curren t premiums, that it woul d be possibl e both to cost the scheme and to value it on one run of the cards. Furthermore, schemes with special features, e.g. unusual pension ages, would require special functions for the costing and would thus generally best be done manually; if however the suggestion made in 129 is adopted, these schemes would be valued on the basis of a fixed pension age and could therefore be valued together with the rest of the business If a true net premium valuation is to be employed the punched cards prepared for valuation purposes could be passed through the computer and the net premiums calculated and punched on to the cards. If a 'fixed maturity age ' is to be employed the net premium will depend upon the nature of the returns, if any, on death, the term and the sex, and the necessary tables of net premiums could probably be stored. The machine would pick the appropriate net premium per unit from the store, multiply by the annuity per annum, compare the result with the office premium to ensure that it does not excee d it, and punc h the net premiu m into the card If a(i -P'/P') method dis to be employed ed where e ethe(i -P'/P')is based on the equivalent annual premium at current rates on a without-return basis, it would be necessary to ohave as a constant on the card the equivalent without-return annual premium and this could be calculated in a similar way to the net premium Other constants required, for example the Karup constants for the valuation of without-return deferred annuities effected in connexion with guaranteed annuities, could also be calculated by the computer. These would involve the storage of a table Nx so that constants of the form ofp x N x+n and annuityx N x+n could d be calculated The amount of work involved as at the end of the year in making a valuation manually is not very large and it may be thought that the use of the computer is not justified. It is, however, quite feasible to make the valuation

33 The Valuation of Annuity Business 353 on a computer. The punched cards containing the business-in-force for each year of a class would be passed through the machine, which would have the valuation functions stored. The appropriate functions would be picked out from the store and punched into the cards. The various multiplications would be made by the computer and the products also punched into the cards. The cards would be tabulated to form a valuation showing the business-inforce, the valuation functions and the products, with appropriate totals. Thus the actuary would have the same facility for making a scrutiny of the work as under the normal manual method One great advantage of having a computer available will it is hoped be that, if a change of valuation basis is desired, this can be achieved with comparatively little trouble and delay It is conceivable that with the assistance of a large-scale computer an office would be able to calculate the valuation liability individually for each policy, store the totals of business-in-force and liability in valuation groups as required and print the valuation results. Any actuary responsible for a valuation would, however, naturally require the results to be in such a form that he could make a proper check by a scrutiny of various stages of the calculation. CONCLUSION 128. Whilst for completeness in this paper we have made reference to adjustments and additional reserves which should properly be made in calculating an office's liability, a study of the tables demonstrates very clearly that the reserve held in respect of a deferred annuity contract varies so greatly according to the bases employed that any great refinements in methods are scarcely called for For simplicity and to enable a change of valuation basis to be accomplished with relatively little work a prospective valuation using for nonprofit deferred annuities a net premium of a proportion of the office premium may be satisfactory in most circumstances. The same value of annuity at the vesting date could be employed whatever the frequency of payment of the annuity (the actual value chosen depending upon the spread of the business between the frequencies). Equally it might well be satisfactory to value all deferred annuities as if guaranteed five years, if this will be correct for the great preponderance of the business. In the case of scheme deferred annuities and group pensions, if all but a few of the annuities commence at ages 65 (males) and 60 (females), then all might be valued as if they commence at these ages, as was suggested by Simons in his paper There only remains for us to express our appreciation of the help received from those colleagues who made valuable suggestions in connexion with this paper, and assisted with the calculations. 23 AJ

34 354 The Valuation of Annuity Business APPENDIX Rates of deferred annuity office premium Deferred annuity of 10 per annum commencing at exact age 65 (males) or 60 (females) payable monthly in advance, guaranteed 5 years. For bases of interest mortality and loadings-see $65. For formulae-see $61. Age at entry Term Basis x t A B C D E F Male lives Without return Single premiums With return, without interest Without return Annual premiums With return, without interest I Female lives Without return Single premiums

35 The Valuation of Annuity Business 355 Age at Term APPENDIX (cont.) Basis entry x t A B C D E F With return, without interest Without return Annual premiums With return. without interest

36 356 The Valuation of Annuity Business35 ABSTRACT OF THE DISCUSSION Mr J. A. Westcott, in introducing the paper, said that it contained little which was new, being largely a restatement of principles and practice in such a way as to form the basis of general discussion. The authors recognized that in many instances practice might vary from that stated. He referred to the scales of premiums used in the paper. Largely for simplicity, they had decided to employ rates determined from the bases set out in 65. They realized that the bases were not wholly representative. For example, in many offices business effected over a short period about 1947 had been on rates of premium based on 2½ % interest or even less. The amount of business on such a basis was, however, probably small in view of the effect of rate guarantees for existing schemes and arrangements made with employers to use 'composite' rates of premium. Similarly, the rate of 3½% assumed as being representative of the period was possibly on the high side and a rate of 3 % might well be more appropriate. It was believed that following the passing of the Finance Act, 1956, some offices, but not all, had made reductions in premiums charged under deferred annuities effected in connexion with existing section 388 schemes as well as reductions in connexion with section 379 schemes. Such reductions might be difficult to justify when the general annuity fund remained ' type C ', except perhaps as a practical measure. Possibly they might be described as ' for the time being' and could be revoked at any time in the event of a general fall in interest rates. The reductions had not been referred to in the paper since the way the office would deal with them would depend entirely on how the concession was made. A criticism might be made that it would have been preferable in bases M andt?mploy N to e during the period of deferment the A table with arating-dow n of the ages. Whilst a few offices publishing valuation returnsduring the past two years had followed that course, more had employed thetabl e as it stood and the authors had therefore considered it more helpful to useth e table without adjustment.h e was sorry to find that in one respect the paper was out-of-date even beforei t had been submitted. From announcements in the insurance press, it seemedtha t one office at least had had practical experience, and with considerablesuccess, of the application of a large-scale computer to the recosting and valua-tio n of group pension schemes.m r W. F. J. Rowlandson, in opening the discussion, proposed to confine hisremark familia s to group pensio n scheme s on a non-profi t basis. To those who were not r with group schemes the paper gave some indication that the businesswa s no longer run on the comparatively simple lines described in earlier papers,eve n when the basis was non-participating, and the addition of profit-sharingproduce d further difficulties. Premium rates alone were quite a complicatedsubject. Against six scales in the paper, the pioneers probably had at least tenduring the period reviewed. They were no doubt still using many of them inthei r daily work, and having to employ such a variety of rates for current re-costing s was quite an administrative problem. The early scales had laid a heavyburde n on the offices, especially since the normal guarantees then given providedtha t the rates to be used for all benefits for any employee were fixed when hejoine d the scheme, and the initial rates for a scheme applied to all entrants in itsthe

37 The Valuation of Annuity Business 357 first five years. That type of guarantee was often given to annual premium schemes, and 85 should therefore be subject to reservations. From the administrator's point of view the early type of guarantee was very convenient. It meant that he had only to divide the members of a scheme into groups according to date of entry and all benefits of each group were costed on the same scale. Valuation calculations would be correspondingly simple, even if the valuation basis varied with the rates. Complications crept in when it became possible for a member to be subject to two or more scales of rates, as occurred when schemes were extended, and particularly when the scope of the guarantee of rates for new schemes was restricted soon after the end of the war. That did not cause much trouble for level annual premium costings but it was easy to appreciate the complexities under the single premium plan where every member's benefits were recosted each year and records were based on members, not the separate items of benefits. Valuation was liable to be as complicated as the costing, often more so. When a member had benefits costed on more than one scale of rates every one of the four methods mentioned in 18 was liable to produce awkward problems, even if a common valuation basis for all premium scales was applied, and that might not be possible. It might also happen, as it certainly had in the preceding ten years, that the actuary considered it desirable to maintain two or more valuation bases in order to keep a control over the emergence of strains and releases, which would not be possible with a common valuation basis. Tables 2, 3 and 4 made it plain that for most of the period reviewed the offices which were early in the field had had to cope with a continuous succession of strains. Judging from the pre-war Board of Trade returns, the general approach seemed to have been to value the accrued pensions on a basis at least as stringent as the current premium basis, and sometimes to take the full strain of future deficiencies under annual premium schemes. No office, however, appeared to have provided for the strain of future deficiencies on the single premium basis. Perhaps the suggestion in 84 that the older rates could be justified by ignoring all margins was felt to be applicable, but that was rather unlikely in view of the considerable differences between scales A and B, which understated the real differences, and it seemed more likely that the offices had had to let those deficiencies emerge gradually in view of the large amounts involved. He was surprised that most of them still valued, under single premium schemes, only the accrued pensions, particularly where the full expected pensions under annual premium schemes were prospectively valued on a basis that was more stringent than the premium basis. In earlier days the additional liabilities were, no doubt, difficult to finance immediately, but at the time of speaking the amounts required should be comparatively modest because most of the business would be on recent scales, which were comparable with a valuation basis suited to current conditions. Although he had suggested that the amount involved was currently small and, indeed, might be ignored on a short-term view, some actuaries clearly considered that something a little more stringent than the premium basis should be held, otherwise they would not use the basis he had mentioned for their annual premium schemes. If the rates were guaranteed to apply up to pension age, a single premium scheme was only a series of individual contracts subject to increasing annual premiums on a scale determined at inception, and the same principles should be applied as for level annual premiums. After re-reading Puckridge's paper, it

38 358 The Valuation of Annuity Business seemed clear to him that it was undesirable that by omitting a reserve for future deficiencies, the actuary should produce a surplus that would suffer tax on being transferred out of the annuity fund, when the recoupment of that tax might be long-delayed or never achieved at all. For a single premium scheme it was troublesome to produce a reserve for future deficiencies, since there were complications where the employee's contributions secured more than the scale pension. A reasonable approximation should not be unduly difficult to devise. One method could be to value future deficiencies for a unit pension for each year of future service, irrespective of any immediate excesses over scale pension, costing the whole unit on, say, the non-return scale. The result should be quite close to the true figure since all pensions were eventually determined by the unit of accrual. Offices which had only recently entered the pensions market would have been spared the large-scale strains to which he had referred. They were still likely, however, to find quite a number of strains or releases in the ordinary processes of valuation discussed in the paper even though they valued on the premium basis. An obvious case occurred where the valuation premium was, say, 95 % of the office annual premium, as suggested in 30. A figure of 5 % had become almost the traditional loading for group rates, but it had usually been applied to the net premium, and the authors had followed a long line of precedents in their loadings for office premiums. The valuation premium would be ¼ % less than the true net premium : enough to produce a strain of up to 5 % of the new annual premiums. In addition, the custom of scaling down the last few loadings on single premiums, from which annual premiums were then derived, meant that the loadings on annual premiums never reached the nominal 5 %, causing a further strain. Adding 2 % for monthly premiums and taking 92½ % as the net premium carried that process a stage further. He estimated that those strains could amount to 10 % of the new annual premiums, a significant proportion of the surplus in a young fund. Even an office using a strict net premium basis was liable to produce initial strains in respect of annual premiums returnable without interest. He drew attention to a difference between the office premium quoted in 61 and the net premium in 89, namely that the latter allowed for returning the loadings on death. Omission of that comparatively minor factor could using basis F result in an initial strain of between 5 % and 10 % of the relevant new premium income. Failure to make the further adjustment required when adding 2½ % for monthly premiums increased the strain by about 60 % of itself. An office making a gross premium valuation could suffer that strain in addition to those already mentioned. Similar strains occurred with single premium schemes but in that case they arose from all the business on the books, since the valuation would treat each year's accrual as if it were new business. He expected the total strain eventually to be of the same order as the initial strain on annual premium business. With the loadings customarily used, valuing the paid-up pension produced a reserve rather less than the prospective net premium reserve for annual premium contracts. The loading included in P' gradually decreased with increasing age so that (1 P'/P') was always slightly less than would have been derived from the corresponding net premiums. The resulting paid-up pension was what would have been secured by applying each year's premium at the single premium rates, so that the single premium loadings were charged. The valuation, therefore, released the single premium loading percentages for each age, with the result that slightly more was available for expenses in the early years, offset later

39 The Valuation of Annuity Business 359 by the scaling-down in the last five years, in contrast to the level loading throughout which was released on a net premium basis. Too much was made of the adjustments to 31 December where valuation was on the policy anniversary. Was there the need for all the minute detail required to produce complete accuracy in the final reserve? He doubted whether they achieved the accuracy aimed at, and many before him had questioned whether it was wanted. An office which followed the practice of making debits and credits at the next policy anniversary for all movements could well ignore everything except interest to the end of the year, premiums outstanding at 31 December and pensioners transferred to the immediate annuity schedule. Offices which paid out surrenders and death returns as they occurred could also deduct the actual payments. That suggestion simply meant deferring into the next year surrender profits and death releases on approximately half a year's ' offs '. There would be some offset from late notification of ' ons ', but ' offs ' were invariably larger than late ' ons ', so the approximation produced would be on the stringent side. One advantage of valuing schemes individually was that it provided a more detailed check on the valuation. Quite small errors could be detected by building up an approximate reserve, and it was naturally easier to track down errors within the confines of one scheme. At least one office went further by making an analysis of surplus for each scheme, and found the extra work worth while. It would readily be appreciated that the high proportion of movements and the recosting every year made group business liable to more errors than ordinary business and any procedures which provided additional checks should be welcomed. Although he had many schemes renewable on 1 January, he had found that the average interest adjustment applied to policy anniversary valuations had, over the last few years, produced an accumulation period varying between 192 and 197 days, making the average renewal in the third week in June. Although not exactly comparable with distribution of premium income, it suggested that the adjustment proposed in 48 would not make a great deal of difference. In studying the current valuation bases F, M and N, and, in effect, the nonprofit premium bases of many offices, he was struck by the absence of any attempt to make specific allowance for improving mortality. Pensioners' mortality made an allowance by the rough-and-ready method of using an average basis which was too stringent for immediate pensions and was probably inadequate for the distant future. The mortality before pension age seemed to do no more than reflect current experience. Even basis F with a year's ratingdown on A might already be inadequate. The combined result was a premium scale that overcharged at the older ages and undercharged at the younger ages. For any one scheme the net result depended on the age distribution, and at the quotation stage competition with offices using a more sophisticated approach to rates could have unfortunate results. Since reserves were normally concentrated into the last few years before pension age, it was almost certain that a valuation on one of those bases would be more stringent than was necessary. To avoid the necessity of trying to estimate averages, he would prefer to make a realistic assessment of the mortality appropriate to an immediate pension and to incorporate an allowance for improving mortality. They could make a reasonable forecast for the few early years where the big liabilities arose and there was no need to be unduly concerned if the long-term forecast was

40 360 The Valuation of Annuity Business wide of the mark. It should give a better estimate than current practice did, and in any case on past experience most of the premiums paid at young ages would be refunded. The few youngsters who stayed until retirement would only be entitled to comparatively small benefits secured on current rates calculated on such a basis. A new attitude to guarantees and valuation would be required. He did not think that the alternatives of a guarantee to apply a certain scale to premiums received over a moderate term of years or a guarantee to maintain the initial scale for the whole term up to pension age, even if restricted to benefits entered upon immediately, could properly be applied to the same scale of single premium rates. In the second case the rates at the younger ages would have to be specially loaded to make up for the deficiencies in later years, when the rates would only be sufficient to provide for mortality appropriate to an earlier period. The valuation basis would have to be chosen so as to avoid releasing that loading. The problem had some similarity to a net premium valuation for withprofit policies, and a specially reduced rate of interest could be employed. The object, however, was to ensure that the loading was not released at all and it would be necessary to increase the value of the pension year by year to absorb what would otherwise emerge as an apparent interest surplus. To the charge that those ideas were too complicated for immediate practical application, the answer was the use of computers. Even a small machine should have the capacity to handle the calculations required, and there was nothing he had suggested which could not easily be dealt with by existing methods. He congratulated the authors on bringing together such a mass of information, on all aspects of annuity valuation, which would be of permanent value to students of all ages. Mr P. R. Smith referred to 33, on the prospective valuation of the paid-up annuity secured by the premiums already paid. His view was that it was undesirable to value only the paid-up annuity on the premium basis in the case of annual-premium schemes, and the annuity already purchased in the case of single-premium schemes, where there were rate guarantees. He preferred to value the whole of the prospective pension, and if the result was to throw up a new business strain which was not manageable, what needed to be looked at was the valuation basis and not the method. He appreciated that if the basis was weakened too much there might be trouble with guaranteed returns on withdrawal, but if necessary an extra reserve could be held against that which might be determined either arbitrarily or, if the experience justified it, by assuming some withdrawal rates in the early years. In 32 the authors suggested, particularly in the case of retirement annuity policies for the self-employed, that the new business strain involved in adopting a net premium valuation basis might be uncomfortably large. The right answer, again, was to revise the valuation basis. It was inequitable to penalize the existing with-profit policyholders by throwing onto the current surplus a heavy strain in respect of new business simply because it was desired to adopt a stringent valuation basis. A similar point was referred to in 70 in relation to the effect of Finance Act, Although in the past it might have been justifiable to adopt very stringent valuation bases which held back surplus, it was questionable whether it was still proper to do so. Certainly if offices were to continue to sell with-profit contracts of the normal type as a hedge against inflation and in competition with contracts of the ' variable ' type variable annuities and variable assurances they would have to liberalize their views on the distribution of

41 The Valuation of Annuity Business 361 surplus so as not to penalize the existing generation of policyholders unduly in order to set up large reserves on new business. The authors had also referred to with-profit annuities, but he had been disappointed that they had made no reference to what seemed to him a crucial problem. If an office issued with-profit assurances and with-profit annuities how did it decide the correct relative rates of bonus? His own office issued assurances, group pension policies and retirement annuities, all on a participating basis, and consequently the problem was in the forefront of his mind. Some offices had thought it right to link the rate of bonus on with-profit retirement annuities to that on assurances, but he did not think that correct, for at least two reasons. In the first place, the two types of business were subject to widely differing taxation bases, and there was no reason for supposing that those bases would remain unaltered. Although it might conceivably be right for the time being to declare the same rates of bonus on both types of business, if there was a change in the taxation basis of one or both of those types the position would have to be reviewed. The second reason for his doubt lay in the question of mortality. Actuaries had grown so accustomed to regarding interest as the main item of surplus and mortality as quite a minor item that they might not have adjusted their minds to the idea that with a large and increasing volume of annuity business the future might be very different. The authors mentioned annuitants' mortality and sounded a warning note, but the last sentence of 71 was surprising, seeming to suggest that it was necessary to defend the use of such a stringent basis as the a(55) table for valuation. His experience suggested that that table was by no means too stringent for immediate annuities, and the only question was whether it would prove stringent enough for deferred annuities, particularly those of long term. The problem of holding the balance evenly between different classes of withprofit policyholders would be increasingly difficult. He could not see that the traditional net premium valuation method helped very much in finding the right answer. Something on the lines of a bonus reserve valuation was necessary, although he did not suggest that that would provide a complete answer. Mr R. J. W. Crabbe was prompted to speak by a sentence in 23 'The purist may consider that a valuation should look to the future and that only a prospective valuation can, strictly speaking, be justified '. He was a purist, because he held that a valuation was concerned solely with the extent to which a fund was able (or more than able) to meet its future commitments, and that was dependent solely on the future. He equally held that an actuary was justified in using retrospective methods, or others which might be described as hybrids, but justified only on certain conditions, which he would discuss. To him, a prospective valuation basis meant one decided by reference to the future course of all the factors which might affect the fund : mortality, interest, expenses, and profits distribution in so far as there was any legal or moral obligation to maintain a minimum level of profits. The obvious case of such an obligation was where the policyholders were paying for profit-sharing rights by means of a bonus loading. Other factors might be involved, such as rate guarantees. He suggested that the report on the valuation should make it clear that that was the way in which the basis had been decided and should show how effect had been given to it in the calculations. Those principles had been obscured by the fact that for many years the prospective method of valuation

42 362 The Valuation of Annuity Business epitomized by the form A - na had been almost universal in both calculation and presentation. It had led to a stereotyped method (enshrined in the fourth schedule to the Board of Trade returns) of expressing the reserves for expenses and profits. That method did not comply automatically with his definition of a prospective valuation. The apparent provision for future expenses and profits might be profoundly modified in either direction by the mortality and interest assumptions, and, moreover, adjustments for such things as the cost of guarantees were not automatically provided for. As the paper had shown, other methods of valuation in current use ranged from different mathematical representations of the A-na formula to methods which were frankly retrospective. The first group included the ' policy value ' method commonly used in North America, as well as the authors' formula (2) in 37. A half-way house was formula (1) in 36. Both formula (1) and the purely retrospective methods were acceptable provided that before a method was adopted it was tested against a basis which gave full play to the future factors affecting liability, and provided that the results of the test were indicated in the valuation report. Where there were compelling practical reasons for adopting such a course, a valuation method which did not automatically demonstrate the provision made for expenses, etc., should be admissible, provided it was regarded merely as a practical means of giving effect to the underlying valuation assumptions, which should be purely prospective. In his report the actuary should justify his methods, whatever they were, by statements of the underlying prospective assumptions sufficient to indicate how his methods gave effect to them. Mr J. Edey said that the authors had referred in their first paragraph to the growth in annuity liability, and it was salutary as Mr Smith had pointed out to think of the future growth in the cost of the bonus on annuities by comparison with the cost of the bonus on assurances. It might give a different picture from that with which they were familiar. He wished to underline some points with which the authors had dealt. They had referred to expenses. It was important in any annuity valuation to remember the cost of paying annuities. In his view it was not sufficient to leave an interest margin and to consider that that would meet the cost. It was only necessary to consider the position which would arise in a closed fund to see that an interest margin was not adequate. In 9 and 10 the authors had referred to joint life and survivor annuities, which in the past had usually been regarded as a relatively small class. In group pension schemes a large number of such annuities were emerging. The authors had suggested a convenient way of dealing with them; alternatively it was possible that when the computers were working the annuities would be valued individually and summed, instead of being sorted and valued in groups. Mr Rowlandson had drawn attention to the fact that annual premiums could be regarded as a series of level single premiums, and recurrent single premiums as variable annual premiums. It was convenient from some points of view to think of both types as recurrent single premiums, and to treat the annual premiums as such even in the office records. If the premium basis was, say, 2½% and the valuation was at 3 %, there was no value in the guarantee of future premium rates. The surpluses could be released as the premiums were paid, but in fact the premiums probably could not be collected in those circumstances. If, on the other hand, the premium basis was 3½ % and the valuation was at 3 %,

43 The Valuation of Annuity Business 363 there was a valuation strain and something had to be reserved for the guarantee of rates. Another point arose on 116. In the case of an increase in the valuation rate of interest a test should always be made to see that the reserves on the new basis were not less than the guaranteed surrender values. Mr M. I. Douse wished to refer to some of the mechanical aspects of valuation which other speakers had indicated were important, starting with the question of valuing individual schemes at annual renewal dates. That was a valuable method which he would not yet discard, although the position might be different if and when a computer was available. The same tabulation of information could be used both for the annual renewal estimate of the premiums and for the valuation. Each scheme was being valued on its own, and it was possible to do a remarkably detailed analysis of the surplus or strain. That formed a valuable audit of the working of the scheme as a whole. If on the other hand all the data were aggregated at the valuation date, the analysis of surplus and the tracing of errors would be much more difficult, and, he was inclined to think, would become almost impossible. Referring to the prospective net premium method of valuation, especially in connexion with single premium business, he said that while it was perfectly satisfactory in theory, the business was often large in volume; records were expensive to establish and even more so to alter. He would not like to have to use a strict net premium method. It would be better to value a proportion, say 95 %, of the office premium. He agreed that because of the way the business had developed in the past there were obsolete rates of premium in force and guarantees for the future on terms unfavourable to the offices, and in those cases there was certainly a strain on new business. Apart from the premiums expected in future years on the existing benefits, there might be incremental benefits on the same terms. There was a strain, and it would be easy to make a valuation which would completely ignore or hide that strain. But by valuing on annual renewal dates, as he had indicated, it was possible to estimate the strains incurred on the various bases of rates and to make a satisfactory approximation to the strains arising as a result of the guarantee. It should be added that most of the offices in the field had only limited amounts of benefit on those obsolete rates still on their books and, as a result of Finance Act, 1956, much of it was being eliminated as section 388 schemes were converted to section 379 form. The extent of the problem was narrowing. He agreed with Mr Crabbe that while the actuary had to say how he was providing for liabilities of that nature, he should be left free to decide how he was to make the provision, and to furnish the results of his valuation, in particular in the Board of Trade returns, accordingly. Mr M. E. Ogborn held the view that in certain circumstances the only way of making a prospective valuation of deferred annuity business was to make it retrospectively an unorthodox but a ' purist's ' approach. In a situation, which some of them knew, where the future of pension business was largely unknown, where the liability depended on rates of withdrawal as well as of mortality, where the age at which a man would retire was unknown and where business had been effected on different tax assumptions and it was uncertain which of the assumptions to allow for in the valuation rate of interest, where also there was the overriding consideration that guaranteed surrender values might be

44 364 The Valuation of Annuity Business exercised against the office at any time, there would be a set of widely different prospective valuations according to the assumptions made. It seemed to him that in that situation the suitable course was to go back to the original terms of the contracts, and to make a valuation (which could conveniently be done retrospectively) on the basis of those terms ; if the liability was calculated both at the valuation date and at the normal age of retirement it would be possible to make an estimate of the prospective valuation on any assumptions that might be desired for the period of deferment. In 12 the authors referred to a reversionary annuity where one rate of interest was assumed during the deferred period and another while the annuity was in possession. It could be shown that that was conveniently dealt with at one rate of interest about half-way between the two. In the authors dealt with valuation by means of the paid-up annuity, represented by the formula (1 P'/P') ; it might be easier for those calculations in some respects if it was observed that that formula was also the ratio of the annuities due or where there was no return at death. In 53 the authors said that where there was a return of premiums at death mortality could be ignored. That was so if the return was with interest at the same rate as used in the valuation, but not otherwise. It was interesting to compare col. 6 with col. 12 in Table 3 ; they showed that the valuation of the paid-up policies was very similar to a net premium valuation, as would be expected on general grounds. There was a similar correspondence between cols. 8 and 13. With regard to 98 on with-profit business, Mr Smith had asked how they decided the relative rates of bonus. The problem was made easier if the premiums included about the same relative bonus loading. Another awkward problem was that, before the issue of with-profit annuities had commenced, the with-profit assurances had shared in the whole surplus, including profits on non-participating business and miscellaneous profits. If the with-profit annuities were given a similar share there would be a watering of those profits, which could become serious if there was a large amount of with-profit annuity business. It was aggravated because in pension schemes large units might be coming in, and it would only be justice if at first a greater share of miscellaneous profits went to the classes which originally had them and less to the new annuity class, although that situation might change with the years. He had a suggestion to make with regard to automation. It was natural to think in terms of doing a full valuation by automatic methods ; but he wondered whether it would not be better to use the system to produce characteristics similar for each class of business valued so as to permit, for example, a simple n-point valuation at the end. It was a curious idea to use an instrument as powerful as automation to produce an approximate valuation, but such a valuation reduced the problem to calculations of average ages and durations and standard deviations which in principle were the same whatever data went in, whether for pension or assurance business. A few characteristics were produced, and at the end all that was necessary was to multiply by the appropriate factors, whatever the basis chosen. It would give great power over the data. Mr A. K. Tudor said that before Finance Act, 1956, reversionary annuities were usually sufficiently uncommon and the reserves sufficiently small for arbitrary methods of valuation to be adequate in most offices, but the terms of the Act had encouraged the issue of reversionary annuities for widows in con-

45 The Valuation of Annuity Business 365 nexion with ' self-employed ' pensions by enabling them to be written on more favourable terms. Moreover, it had become the rule for premiums to be limited to the working life of the male, so increasing the individual reserves. It had therefore become necessary to evolve methods of valuing those limited premium reversionary annuities on a fairly large scale. In 58-9 of the paper the authors had discussed the question briefly on the assumption that the retirement annuities and reversionary annuities could be combined for valuation, but it was not always convenient to do so for example when the associated retirement annuity was with profits. He did not think that a guarantee period was often required if there was a widow's pension, and the complications referred to in that connexion might be comparatively rare. One method was to group the annuities according to the age of the male, taking a weighted mean age difference between husband and wife for each such group, and to use a PxN type of constant for the limitation of premiums. That method had limitations, but he believed that the range of age-differences was sufficiently narrow for it to be reasonably accurate. In the section on the valuation of with-profit deferred annuities, the authors were clearly predisposed towards net premium valuations. He wished to put on record that the bonus reserve method as used for assurances was equally suitable for deferred annuities. A constant rate of future annuity bonus could be valued at a rate of interest approaching the expected rate. The surplus would emerge in a form reasonably closely matched to the uniform annuity bonus and the level bonus loading in the premiums which paid for a decreasing bonus. If surplus arose predominantly from interest, it would emerge, under a net premium valuation at a low rate of interest, in a steeply increasing form. In his paper on participating deferred annuities in 1956 (J.I.A. 82, 179) Mr Edey had referred to the increasing bonuses which ordinary net premium methods produced and had suggested that because of that the method of participation by the addition of simple bonus was not suitable for an annual premium contract. Conversely, if a simple bonus system was adopted, the net premium method of valuation was not suitable. With regard to widow's options, which were dealt with in 114, presumably the terms of conversion from single life into joint life and last survivor annuities would be fixed in order to impose little or no valuation strain on the office at retirement, and the cost to the office of the option would therefore arise from the low mortality of the male single life annuities left to come into possession. The authors' suggestion that a light mortality basis for the male lives after deferment should be used therefore seemed particularly appropriate. In fact, a suitable mortality basis might be that appropriate to immediate annuities purchased at that date, which of course was many years ahead. The authors suggested a stringent table, but was there a table stringent enough? If a (55) was used, then a rating down in age or a percentage loading was certainly called for where such widow's options existed on a large scale. Of course, particularly in the new class of pension annuities, they were influenced by the premium basis, but a gradual strengthening of the valuation basis until the required stringency was reached should be possible. Mr N. Williams commented on the reference in 127 to the checks at various stages on the results obtained from a large-scale computer. He would expect that the process of scrutiny would be fed in the same way as the process of valuation so that the machine itself would be programmed to scrutinize the

46 366 The Valuation of Annuity Business results, having first worked them out. It depended on what was involved in scrutinizing the results, but in so far as there was a test of whether the answer was likely to be right, it seemed to him that that, at any rate, could be put in with the rest of the system. The machine could carry out almost the whole process, printing only a fairly small amount of data. Mr E. J. W. Dyson congratulated the authors on the immense amount of work they had put into the paper. He felt strongly that a paper with arithmetical results was much easier to follow and he commended the authors for the tables which they had presented. Referring to Mr Tudor's remarks on valuing reversionary annuities, he said that, even if the contract was with profits, a joint life and survivor annuity could be treated as the sum of two single-life annuities minus a joint life annuity. The with-profit deferred annuity and the non-profit reversionary annuity could then be readily valued, using, for example, a Karup constant. Similarly, by the Karup method it was possible to allow for return of premiums with or without interest and whether or not the whole premium was returned. Another general point on valuation technique arose if a benefit could be split into a principal part and a subsidiary part for example, a joint life and survivor annuity guaranteed for a period of years, taking the joint life and survivor annuity as the principal benefit and the guarantee as the subsidiary benefit. That led to a very accurate valuation simply by valuing the principal benefit, finding the reserve for the subsidiary benefit at the inception of the contract and holding a proportion of that at the valuation date. It was similar to the method commonly used for valuing single premium term assurances. Mr A. S. Clarke commented on two points which had already been referred to in the discussion, the first being with regard to the valuation of retirement annuities with profits. It had been suggested in the discussion that the death knell of the net premium method had been sounded by the gross premium method, but that overlooked the fact that the method which was advocated in the paper, and which was generally used, was to value on a net premium basis with a lower rate of interest than that involved in the premium quoted, which levelled out the emergence of surplus. He had been disappointed to note that in discussing retirement annuities the authors had not referred to the fact that final bonuses were also being employed. They had mentioned such bonuses in 102 in connexion with the valuation of with-profit group pension business, but not in relation to individual business. The method described in the group section could equally be used for individual contracts. It had been a surprise to him that more offices had not adopted a final bonus in connexion with retirement annuity with-profit contracts. He agreed that they were comparable with endowment assurances during the period of deferment, but at maturity an entirely different situation arose and the final bonus provided some hedge against the possible improvement of annuitant mortality, which they all knew might be a very serious problem. Mr Douse had drawn attention to the question of conducting a valuation on the revision dates rather than on one particular valuation date. The speaker agreed that it was most important that an analysis of surplus should be attempted and if possible completed. His office had adopted the fixed valuation date method of valuation, not the revision dates, and, in spite of Mr Douse's suggestion that it would be impossible, they had carried out surplus investigations,

47 The Valuation of Annuity Business 367 although they had found them very difficult. They were, however, most important in view of the numerous changes which took place from day to day in group practice and which were necessarily reflected in the valuation results. Mr W. Perks objected to the word 'purist' in 23, since that paragraph was concerned with basic principles of valuation. He was surprised that the paper and the discussion so far had failed to bring out a point which had been made several times in Institute discussions in recent years that a valuation served two purposes : to test solvency and to provide a basis for distribution of surplus. For solvency, a retrospective valuation might be perfectly sound but it needed justification from a prospective point of view. Similarly, for distribution a prospective valuation might be perfectly sound but in its turn it needed justification from a retrospective point of view. The great strength of the net premium method was that it looked both ways and served both purposes at the same time. A prospective gross premium valuation for distribution, such as a bonus reserve valuation, might in fact be thoroughly unreliable and it needed to be judged from the point of view of whether it produced negative values or strains at the early durations. To his mind that was an important weakness of the method. In the United Kingdom, life office actuaries lived under the influence of the 1909 Act, which was based upon the principle of freedom with publicity. In his view the major purpose of the valuation returns under the Act was to provide a clear demonstration of solvency, and if the actuary was to retain his freedom in those matters he must provide adequate publicity in the Board of Trade returns. A valuation for the purposes of the Board of Trade returns not only had to be sound but had to be seen to be sound. Mr F. H. Wales, in closing the discussion, said the authors had produced a paper which would set many thinking and which would be a useful work of reference for students. They had put on record the results of a large number of calculations in such a form that those who wished could fairly easily proceed to approximate estimates of the cost of changes in the basis of valuation in their own business. He had been interested in the discussion on the merits of prospective and retrospective valuation, and he had found Mr Crabbe's exposition most interesting. It had been clear and helpful. In 7 the authors referred to the valuation of annuities 'with proportion', which had reminded him how few of those contracts they saw. He had always felt that there was bound to be ill-feeling among not-very-well-educated people who, as next-of-kin of deceased annuitants, found that there was no further payment, possibly for the burial expenses which had been incurred. That had arisen from the keenness to quote rates as low as possible, and it would have been much better if the standard quotation had been for 'with proportion' annuities. The references in to retrospective valuation led him to remark that the difficulty about such a valuation was that it could be justified only if the rate of interest employed to accumulate the premiums and reserves met two conditions first, that it was enough to build up the annuity reserves required at vesting and to cover any guaranteed surrender values in the meantime ; and secondly, that it was a rate which the office could reasonably expect to earn during the rest of the period of deferment. It was the second consideration

48 368 The Valuation of Annuity Business which was not always easy to fulfil, and taxation questions bedevilled the problem. It might be of interest if he quoted the case of an office in which a part of the deferred annuity business (with return with interest) had formerly been valued by the accumulation method, i.e. retrospectively. The business had been divided into two parts that issued before 1936, which was accumulated at 4¼% (and had been valued at as high a rate as 5 %), and post-1935 business, at 3½ %. At the end of 1956 a change had been made to a prospective valuation at 3 % during deferment and 3½ % thereafter, using a(55) mortality, the annual loading being 1½% of the annuity. The retrospective liability for the pre-1936 business had been 1.9 m. and for the post-1935 business 2.5 m. Over all there had been a strain of 108,000 in the change to the new basis of valuation by the prospective method. It would have been interesting to analyse the strain into its component parts more stringent mortality, lower interest (particularly for pre-1936 business), and loading but unfortunately that had not been done, and he mentioned the figures only to show what could be hidden in a retrospective reserve which looked reasonably strong. He was left with an uneasy feeling with regard to 57. As he had understood the position, the valuation liability in respect of schemes approved under section 379 was separated from the rest of the valuation liability in the annuity fund, and transferred to the new pension annuity fund, with a share of any notional loss being brought forward. When the reserve was in the new fund a calculation was made, on whatever basis was deemed equitable, to readjust the benefits or premiums, and the business was revalued on a basis appropriate to a tax-free fund. That would normally produce a profit against which would be set the new fund's share of the loss brought forward. From what the authors had written he felt that he had perhaps been mistaken; it would be helpful if they would amplify the last two sentences of their paragraph. For the benefit of students, he would have preferred the authors to set out their formulae properly in 61 and to use, for example, as the denominator in formula (c). Mr Rowlandson had referred to the same point in another connexion and had referred to the strain that arose in the valuation through not using the theoretical formula. The premiums which the authors had produced in their Appendix were frightening : a horrid warning of what could happen when mortality and interest combined against them. In life assurance business they had had mortality moving in their favour, and their interest earnings had not been affected by anything comparable to the changeover to the Type C position. For comparison he had looked at what had happened to non-profit premiums for assurances during the period referred to in 65. The current annual premium for a wholelife policy at age 28 was about 5 % higher than would have been quoted in 1931; that for an endowment assurance, effected at the same age to mature at 65, was up by about 12 %. In the authors' table, the premium for an annuity at 65, with return, for a man aged 28 had risen by 62 %. As would be expected, the increase was relatively less for older entry ages. In 69 he was not sure that the authors were correct in using the same rate of interest for their prospective valuation of deferred annuities after vesting as for immediate annuities. A feature of the valuation of immediate annuities was that all the investing had been done, provided the investments had been suitably chosen as regards dates of redemption. They knew what rate of interest they had got on their money. Recently it had been a fairly high rate! With deferred annuities they did not know the rate of interest at which they would be able to

49 The Valuation of Annuity Business 369 invest future premiums, if any, and future accumulations up to the vesting date of the annuity. In theory, at least, therefore, a rather lower rate should be assumed for the deferred annuity after vesting. Another argument for doing so was that it was necessary to forecast rates of mortality over a very much longer period in the future. He felt that the 1956 Act had completely changed the picture with respect to mortality under purchased annuities. Before the Act the effect of self-selection had been great. The prospective gain in spendable income by sinking capital in an annuity was so small compared with the loss to the estate on early death that it had been only after very careful consideration of the state of health that the plunge had been taken. Since the Act became law a large part of the annuity was tax-free and anyone with a large unearned income had to think very carefully whether it would not be wise to buy an annuity regardless of whether he or she was one of the ' super ' lives who would live for a very long time. In 84 the authors raised an important question. He knew of one scheme probably there were many like it under which until about 1942 the premium scale then current was guaranteed for the rest of the working lives of all members once they had been admitted. That guarantee had then been withdrawn from new entrants, but despite the normal losses of staff due to withdrawal and retirement which had reduced by 55 % the number of employees to whom the guarantee applied, the income and reserves for the survivors were still increasing. A similar problem arose in connexion with retirement annuities under Finance Act, 1956; some keen rates were guaranteed, in some cases for long periods. They ought to have in mind the liability which might be incurred under those policies. Mr Tudor had mentioned with-profit annuities, as had the authors. The latter had pointed out, quite rightly, in 98 that the bonus was largely provided from the interest surplus. Few offices seemed to add further bonuses after the annuity had vested. It was possible to estimate fairly accurately at the vesting date what the future interest profit was likely to be. By then the reserves were already invested, but he imagined that no actuary would wish to distribute more than a limited proportion of that estimated profit. Although it went against the grain to go on declaring bonuses to people after they had outlived their expectation of life, so much of the profit arose from interest surplus that he preferred to go on adding bonuses to the annuities despite the practical difficulties involved. In conclusion, he underlined the remarks in 127. It would be unfortunate if the adoption of computers led to inadequate information being publicly available, or indeed available to the actuary responsible for valuation. As Mr Perks had said, insurance in Britain had developed in a healthy atmosphere of freedom with publicity. He hoped it would continue to do so. The President (Mr F. M. Redington), in proposing a vote of thanks to the authors, said that they had rendered a service in collecting together and surveying enough problems to fill several papers. Basically, the more important of those problems were old ones which had been puzzling the profession since its foundation, but they had been accentuated by the exceptional length of some deferred annuity contracts, by the break in conditions at retirement and by the sheer size of annuity business, much of it non-profit. In recent years events had been in their favour and the more difficult problems had not been weighing heavily on their shoulders. They had had high rates of interest both on their funds and on their new money and the problem of asset valuation, which would 24 AJ

50 370 The Valuation of Annuity Business usually arise in those circumstances, had been considerably moderated by the rise in equity values. If rates of interest dropped sharply, those valuation problems might become acute. The use of a basic valuation rate of interest of 4 %, from which he was in no position to dissent, could in the course of time become inappropriate. The dangers arose when they had a low rate of interest, and then the profession came under test. Although the practical problems might look severe, he believed that in principle they were simple. Moreover he did not believe that even in practice they were very difficult, given two things : first, that the actuary saw the situation as a whole, in other words that his view embraced both assets and liabilities ; and secondly, that he was candid in his official report. In the President's view the actuary's duty was clear and simple. He had, at least to his own satisfaction, to value all his commitments with his face turned to the future ; that was what the word ' liability ' meant. In practice the actuary should not have great difficulty in demonstrating that he had done that. To illustrate the point he considered what an actuary might do if the rate of interest dropped from its current high rate to 2½ % and stayed there for a long time. There were several ways in which the actuary could solve his problems. He could write up the assets, by a sufficient amount to change his valuation basis to 2½ %. Or he could gradually change his rate of interest from 4 % to 2½ %. Or he could value his old business at 4 % and his new business at 2½ %, thereby avoiding all strains. In the last two cases, however, the actuary should satisfy himself that the use of a higher rate than the 2½ % he was earning on new money was justified, for example by the possession of sufficient assets at higher interest rates ; and he should, if the point was important, make reference to it in his official report. He added a word on recurrent single premium business under group pensions. The actuary could and should make tests of his guarantees and if necessary make a statement in his report. If the actuary had a large volume of future premiums guaranteed on a 5 % basis but was earning only 2½ % on new money, it was his duty to make some comment, and he would not be true to his profession if he did not. They were grateful to the authors for the enormous amount of work they had put into the paper, the extensive analyses they had made of the problems, and their contributions towards a solution. Mr J. A. Westcott, in acknowledging the vote of thanks, said that he had expected that he might have to defend the method of valuing with-profit business set out in the paper, as opposed to a bonus reserve valuation, and he had, therefore, been greatly relieved to have the authority of Mr Perks on the weaknesses of bonus reserve valuations. He agreed that the paper would have been better had it set out the dual purposes of valuation namely, the test of the financial position of the office and, secondly, the distribution of surplus. There were two aspects of the subject which he felt merited more discussion, although the President had enlarged on the question of rate guarantees. It seemed to the authors that if an office had incurred, under rate guarantees, a liability to accept future premiums on terms more favourable than could be justified on the valuation date if quoting for new business, an appropriate reserve should be made. He thought that that was just as necessary as it was under annual premium contracts to take into account future premiums and the benefits which they would secure. The same thing applied, to a more limited

51 The Valuation of Annuity Business 371 extent, to retirement annuity policies by single premium, where an office had committed itself to accept future business on specified terms. He apologized for the fact that the paper had not mentioned retirement annuity policies with a final bonus, but he confessed that he had not realized that such policies were being issued. He understood Mr Ogborn to suggest that with-profit annuity rates should be loaded for the same proportion of the bonus as assurance rates. That was a matter, presumably, to which actuaries would have to give consideration, but, bearing in mind that assurance business had miscellaneous sources of profit which were probably not available to annuities, a somewhat higher proportion seemed to him to be suitable for the latter. Mr G. C. Crook sent the following written contribution to the discussion : The liability in respect of guarantees of premium rates for group deferred annuity business might well have been considerable a few years ago. The rapid increase in business at higher premiums after the last war eased matters but final rescue came with the Finance Act, While not wishing to dwell on shortcomings in the valuation of this business in the past I think it is of interest to consider the possible effect in the future of premium rate guarantees in respect of schemes included in the pension annuity business. The additional liability in respect of a pension of 1 per annum for each year of future service of an employee aged χ could be found as follows : where SP(x + t) is the office single premium payable at age (x+t) to purchase an annuity of 1 per annum at retirement age (x+n), not returnable on death before age (x + n). Accented symbols are on the valuation basis, others on the basis assumed in the premium rates ; e is the proportion of the office premium reserved for future expenses and profits. If the valuation mortality basis during the deferred period is the same as that for premium rates, the expression reduces to a simple form to enable calculations to be made more readily: where is calculated at a rate of interest such that It can be shown that a reasonable approximation for the additional liability, where premiums are returnable on death before normal retirement date, can be found from the same formula, SP(x) having the meaning defined therein. The following table shows the net single premium and the additional liability when rates are guaranteed for the whole of the future service of employees in respect of the level of benefits existing at the valuation date. The premiums are based on the A Table during deferment, the a(55) Table thereafter, and a rate of interest of 4 % throughout. The valuation is based on the same mortality tables, but the rate of interest is 3 % throughout. The table also shows the additional liability likely to arise where the guarantee of premium rates includes the right to vary rates up to a certain maximum after say 7 or 10 years. A 10 % increase has been assumed. The deferred annuities, without return, commence at exact age 65 (males) or 60 (females) payable monthly in advance and ' guaranteed ' 5 years. 24-2