A COMPUTER APPROACH TO ORDINARY LIFE ASSURANCE POLICY VALUATION A NEW LOOK AT CONCEPTS by GRAHAM WHITTAKER 1. Introduction. This paper has been written as an 'Expository Article' and, as such, is not meant to be complete. Its main object is to describe useful concepts and methods that are not covered by the standard reading for the Institute's examinations. As the title of the paper implies not all of these concepts are new. Some, in fact, are first principles which are particularly useful when performing an actuarial valuation using a computer. The title also suggests that the paper is confined to Ordinary Life business. However, much of what is said could also be applied to Industrial or even Non-life business. 2. What are the requirements of a system for valuing Ordinary Life insurance policies? 2.1. An economic system. 2.2. A fast system. 2.3. The results should be produced in reports which are clear and useful to management and also in a form suitable for statutory returns. 2.4. A simple system, as far as possible. 2.5. The system should be capable of providing results by different methods of valuation such as the net premium method, the gross premium bonus reserve method, or by methods where the net premium is adjusted using Zillmer-type techniques. 2.6. The system should allow revaluation on different bases. 2.7. The results should be as accurate as the economic constraints will allow. 2.8. A desirable feature would be that the valuation could be made as at any day of the year. 2.9. The system should be capable of producing additional items necessary for an analysis of surplus. 169
170 GRAHAM WHITTAKER 3. What are the features of traditional valuation systems in the U.K.? 3.1. Policy valuation data are grouped by (for example) currency, policy class, bonus group, an integral age, an integral duration, etc. 'In the early days, contracts were valued individually, reference being made to the original papers for the necessary information. Soon, however, the large number of contracts and the need to reduce work led to the practice of valuing in groups and the use of a separate file of valuation data.' 13.2. Actuarial Practice of Life Assurance by H. F. Fisher and J. Young. The use of exact ages and durations would involve prohibitive work, and various assumptions are made.' 13.4. ibid. 3.2. Each group is valued. 3.3. The results are combined for management purposes and statutory returns. 3.4. The process is comparatively slow. 3.5. The policy valuation data are continuously updated, in their valuation groups using 'the continuous method', or the valuation file is updated, and sorted into valuation groups for the valuation. 'In a small office using punched-card equipment the cards can be sorted into valuation groups and tabulated at each valuation but in a small office using hand-written cards or in a large office this is not practicable. Most offices therefore obtain their data by a continuous method ' 16.10. Life and Other Contingencies by P. F. Hooker and L. H. Longley-Cook. 4. What are the alternative possible features of a valuation system using a computer? 4.1. A traditional grouping method could be used or the valuation could be made policy by policy the 'individual valuation method' is feasible once again. In the case of the individual method it is necessary to accumulate valuation results into management and statutory groupings. Figure 1 illustrates this concept. The individual method of valuation is a practical method for a large file of policies for the following reasons: 4.1.1. Using processors which are extremely fast a valuation can be performed in a time comparable with the time it takes to read the valuation file into the computer's main storage. A technique termed 'overlap' may be used to process a policy, read the next policy and
ORDINARY LIFE ASSURANCE POLICY VALUATION 171 write the valuation results for the previous policy, all at the same time; this is possible as each operation is independent. Whether the ultimate is achievable depends upon whether the processing time is less than the 'read time'. Valuation Result for a single policy ] Sterling, E/A, NP 1 120 103 1400 395 1295 900 Array of accumulators Valuation item Valuation Grouping. Number of Policies Annual gross Prems Annual Net Prems Sums Assured P.V.of Future Net Prems P.V. of Sums Assured Reserve Sterling, E/A, WP ' Sterling, E/A, NP Sterling, W/L, WP Sterling, W/L, NP Dollars, E/A, WP Dollars, E/A, NP Dollars, W/L, WP Dollars, W/L, NP Marks, E/A, WP Marks, E/A, NP Marks, W/L, WP Marks, W/L, NP E/A = Endowment Assurance W/L = Whole of Life WP = With profits NP = Without profits FIG. 1. Simplified example of individual policy valuation results accumulation 4.1.2. Because of new technologies the cost of computer equipment is reducing at a very fast rate; labour costs continue to rise. Economies of scale can be gained also, by using larger capacity hardware. It should be noted, however, that systems should be flexible and easily modifiable because of increasing reprogramming costs. As the concepts of the traditional grouping method are well known, and can be applied also to a computer system, the remainder of this paper is confined to the individual method. 4.2. The accumulated results are printed. Further combinations of results could be made during this process. 4.3. It may be necessary to produce the valuation data file from a larger file.
172 GRAHAM WHITTAKER 'Current developments in the use of electronic equipment might make it possible once again to assemble the data direct from the primary file, even eliminating the intermediate use of cards....' 13.3. Fisher & Young. These developments, of what is termed a 'consolidated file', have been or are being made in some companies at the time of writing. Each policy record of the consolidated file holds such information as basic policy information, money owed to or by the policy holder, loan information, accrued bonuses, valuation data, etc. Where such a file is used it is usual to produce a smaller valuation file by means of a program which assesses sequentially each record of the consolidated file, picks off the relevant information, formats this information into a valuation record, and produces a file of these records on a magnetic tape. This program would be run whenever a valuation was required. For policies with multiple benefits this program could produce one or more valuation records. 4.4. It may or may not be necessary to sort the valuation file before the valuation is performed. It depends on the valuation system itself. In some cases the sorting overhead will not be justified by the reduction in run time of the valuation program; in other cases sorting is a prerequisite to the valuation. 4.5. The following are alternative methods of producing actuarial valuation factors: (i) The computer program calculates the necessary actuarial factors for each policy individually. This method is flexible, but can be slow, as the processing time for each policy can be excessive. (ii) The computer program calculates a set of actuarial factors necessary to value a continuous stream of policies each time there is a 'control break' (for example, a change in policy class). The actuarial factors would be held in a table in the computer's main storage. It is evident that the policies would require sorting. This method saves having to recalculate factors for every policy, and is also flexible. (iii) The computer program reads into main storage a set of actuarial factors necessary to value a continuous stream of policies each time there is a control break. The factors would have been calculated previously and stored on an external storage medium such as a magnetic tape, disk or drum.
ORDINARY LIFE ASSURANCE POLICY VALUATION 173 This method is flexible and is very similar in concept to that described in (ii) above. It is used extensively in the U.S.A. where statutory valuation bases are fixed. However, it is also highly suitable for countries where flexibility of basis is required. The policies normally require sorting. (iv) The actuarial factors are carried on the policy record, having been 'updated' previously. The updating would probably take place once a year for each policy. This method is inflexible as it is equivalent to valuing on a fixed basis. No sort is required. (v) There are many possible variations of the above methods. For example, it is theoretically possible to hold all the valuation factors in main storage. In practice main storage space is limited and so it might be decided to hold 'core resident' the factors for the largest valuation classes only. The advantage of this type of approach is that it is not necessary to sort a large file of valuation records. 4.6. Alternative methods of accumulating the individual results are as follows: (i) The results are accumulated into groups in the computer's main storage as each result is produced. An array of accumulators similar to that shown in Figure 1 would hold the results. The program would decide for a particular policy which 'line' of the array to add into. The program would print the accumulators after processing the file of policy records. (ii) The results are accumulated into groups on an external 'direct access' device, such as a magnetic disk or drum, as each result is produced. The results are printed from the storage device by a subsequent program. This method is slow. (iii) The results are written individually onto an external device in the same sequence as the valued policies, in preparation for a second 'accumulation and print' program. The device would probably be a magnetic tape. 4.7. (i) The actuarial factors could be produced initially as at the valuation date. If the factors are produced at integral durations, this method is imprecise and requires adjustment. If the factors are produced at non-integral durations the technique is similar to that described in (ii) below.
174 GRAHAM WHITTAKER (ii) The actuarial factors could be produced initially as at the policy anniversaries straddling the valuation date and the factors as at the valuation date found by interpolation. This method takes into account the exact duration of the policy, and allows the valuation to be made on any day of the year. 5. The following section assumes that the less well-known method described in section 4.7(ii) is used. It elaborates on section 4.5 by describing the central valuation program. 5.1. Factors calculated on a control change (see 4.5(ii)) Read Policy Record The policies would have previously been sorted. Control Change No Reserve Routine Yes Calculate Factors Routine This routine would also calculate net premiums. It could be class-dependent, or mainly independent of class if it uses a table of benefits and premiums. It could use iterative methods. This routine is probably independent of policy class as its main function is to interpolate and multiply. What this routine does depends on the system required. If the first system is used, the program will finally print the accumulators after valuing all the policies. If the second system is used, a subsequent program would read, accumulate and print the results. Accumulate Results OR Write Results onto Sequential Device 5.2. Sets of factors held on an external storage medium (see 4.5(iii)) The policies would normally be sorted before the valuation programs. The 'calculate factors' routine would be replaced by a routine which reads in a set of factors from the external storage into main storage. A preceding program would produce the factors on the external storage, possibly by an iterative process.
ORDINARY LIFE ASSURANCE POLICY VALUATION 175 5.3. Individual calculation of factors (see 4.5(i)) Read Policy Record It may be desirable to sort the policies prior to the valuation program. Reserve Routine This routine could use sub-routines to calculate net premiums and factors. Accumulate Results OR Write Results onto Sequential Device 5.4. Factors held on policy record (see 4.5(iv)) Read Policy Record No sort wih"be necessary. Reserve Routine This routine is probably independent of class as its main function is to interpolate and multiply. Accumulate Results OR Write Results onto Sequential Device
176 GRAHAM WHITTAKER 6. The ensuing section develops practical valuation formulae suitable for the technique described in section 4.7 (ii), 'the policy anniversary factor method'. 6.1. Definitions: A t = benefits valuation factor at integral duration t years. In general this factor could also value future bonuses. a, = premiums valuation factor at integral duration t years. In general this factor could also allow for future contractual increases or decreases in premium. S = initial sum assured. P = premium valued, per unit sum assured. F = fraction of year from previous policy anniversary to valuation date. k = fraction of year from previous premium-due date to valuation date. k' = fraction of year from valuation date to next premium-due date. F' = fraction of year from next premium-due date to the next policy anniversary. m = number of mode premiums in a policy year. 6.2. Annual policies P.V. of sum assured = S [A,_ t (1- F) + A,F] (1) P.V. of future premiums = SP [(ä t _ 1-1) (1 - F) +ä t,f] (2) Reserve = (1) -(2) i.e.sv,_ 1+F =S[(V,_ 1+ P)(1-F)+V,F] (3) Note that no reserve adjustments are required. Moreover, the formula is insensitive to age approximations and so an approximate integral age could be used. 'The use of the age next birthday approximation is based on the implicit assumption that the birthdays of the lives assured are distributed uniformly over the calendar year. This assumption is a reasonable one and in practice any error which it involves is of no consequence.' 15.2 Hooker & Longley-Cook. 6.3. Non-annual policies The formulae for non-annual policies are not quite so simple, and depend upon whether premiums are true or instalment. Figure 2 shows the growth of policy reserves for an annual policy and also for
ORDINARY LIFE ASSURANCE POLICY VALUATION 177 a non-annual instalment premium policy. The figure is also approximately accurate for non-annual true premiums. If equivalent annual net premiums are valued that is (5) (6) Value sum assured by and value future premiums by The formulae could also be used in practice for true premium policies. 6.4. Combining annual and non-annual formulae. Reserve = and adjustment to Reserve = +SPK' for all policies Alternatively, Reserve = Adjustment to Reserve = SPF' for all policies (including annual) where F' = fraction of year from next premium-due date to next policy anniversary number of modes outstanding number of modes in year 6.5. In passing it is interesting to look at this formula if the traditional 'grouping' assumption is made that on average policies are half-way through the policy year on the valuation date. Substituting F = 1/2 for all policies Reserve = S Adjustment to Reserve = [number of modes in second half of year] number of modes in year (7) (4) (8)
178 GRAHAM WHITTAKER Similarly we could substitute F = F o where policies. for all p V t-1 FIG. 2. Reserve approximation for a quarterly instalment premium policy Note V, is reserve for an annual policy. The premiums shown are (1) The equivalent annual premium; (2) One quarter of the above. For true premiums Whole of Life Endowment It is clear that the approximation,v x (m) = tv x is good for endowment policies, but less accurate for whole of life policies. A suitable further adjustment to the reserve for whole of life policies is 6.6. Iterative formulae for valuation factors The following formulae could be used to calculate sets of factors which are stored on an external storage device (section 4.5. (ill)) or to calculate sets of factors during the main valuation program (section (9)
ORDINARY LIFE ASSURANCE POLICY VALUATION 179 4.5. (ii)). The importance of these formulae is, firstly, that they are iterative and therefore fast, and, secondly, that they are independent of policy class. Definitions: t,v is the reserve value of a policy at the end of the t ib policy year P t, is the premium payable at the beginning of the (t + l)' h policy year S, is the Sum assured payable at the end of the (t + l) th policy year ät, is the annuity factor at the end of the f th policy year A, is the assurance factor at the end of the t th policy year (10) (11) (12) where (13) and (14) The program would either calculate or look up in tables values of P, and S, for the more complicated classes of policy. Similar formulae may be developed if it is assumed death claims are payable part way through the policy year ('immediate payment of claims'). 7. Final thoughts 7.1. It would be useful to compare a traditional valuation not using a computer, a traditional valuation using a computer and a valuation by computer using 'the policy anniversary factor' method, by means of the criteria set out in section 2 of the paper. 7.2. What are the practical advantages of the various methods of producing factors described in section 4.5? 7.3. Is the individual valuation method a practical proposition for a large file of policies if a projection of income and outgo is a further requirement? 7.5. A mortality or exit investigation by the census method could also be carried out by the individual 'valuation' method; the calculations are fewer and simpler, but the number of groupings is probably larger. Censuses could be produced more than once a year if an improvement in accuracy is required. c
180 GRAHAM WHITTAKER 7.6. The valuation of ONs and OFFs for an analysis of surplus can be carried out on similar lines to those described in the paper. 7.7. Year-end Accounting items may be produced using similar systems. BIBLIOGRAPHY FISHER, H. F. and YOUNG, J. Actuarial Practice of Life Assurance, chapters 3, 13, 18. HOOKER, P. F. and LONGLEY-COOK, L. H. Life and Other Contingencies,, chapters 1, 15, 16.