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March 1976 – Volume 10, No. 3 PaEe Four THE ACTUARY March, 1976

Unruh THE INTERNATIONALACTUARIAL NOTATION (Continued from page 1) by Frank P. Di Pa010 is on a particular track, its future values should be based on that track. When it In the December 1972 issue of The Actuary, John Boermeester reviewed succinctly changes tracks, future values should the origin and history of the International Actuarial Notation, also known as the :!range with the new track. The expense Halo Notation. According to John Boermeester, the Halo Notation was invented by allowance calculated at issue should be David Jones who first used it in The YaZue of Annuities and Reversionary Payments based on the automatic track. Future published in Great Britain in 1844. With minor modifications, this notation was changes producin, u an increase in net officially adopted by the Institute of in 1872 when it appeared in an Ap- premium should allow a related increase pendix to the Institute of Actuaries Life Tables. In 1898 the Halo Notation was in expense allowance. However, no de- adopted by the International Congress of Actuaries and, except for a number of crease in expense allowance should be minor changes which were made in 1954 during the XIV International Congress required if a change results in a de- of Madrid, it is still in use today. crease in net premium. The Halo Notation is basically an ideograph. The symbol representing the (2) For open policies that provide function is in the centre of the ideograph, while the symbols representing the para- contractually for the possibility of meters are shown, in accordance with well defined rules, as suffixed or prefixed sub- changes in benefits, premiums, or pre- (m) mium-paying period, but do not specify scripts and superscripts. For example, in the case of nla,, a is the function symbol when the change is to occur or what it and x, m and n are the parameters. is going to be, a traditional prospective The Halo Notation is very concise and unambiguous and therefore it can be adjusted premium approach is suggested. easily read and quickly understood. A more diffused notation would be somewhat This approach should assume changes time-consuming to read and may not be as quickly understood. Indeed, this is not stated numerically in the policy perhaps the main reason why the Halo Notation has gallantly survived over 130 would not occur. When a change in pre- years. However, the Halo Notation cannot be easily typed with common typing miums or benefits does occur, new ad- devices nor can it be easily incorporated into computer programmes. justed premiums should be calculated In an attempt to remedy the shortcomings of ‘the Halo Notation with respect such that their equals the to programmability, a number of actuaries have suggested linearizing the actuarial present value of future benefits less the notation. In fact, since 1968, at least nine papers have been published ,on the subject. current minimum cash value plus any These papers are: new expense allowance resulting from 1. Proposals for an International Actuarial Publication Language and its the change, again assuming no further Representation in Computer Compatible Form by Carl Boehm (Germany), Georg changes beyond the point of recalcula- Reichel (Germany) et al. published in the Transactions of the XVIIZ International tion. This procedure would be used for Congress of Actuaries, Munich 1968. minimum values for most life-cycle and 2. A New Actuarial Notation by the General Functions Sub-Committee of the cost-of-living policies. Institute of Actuaries of Australia and New Zealand, published in October 1971. ‘I’he Committee looked beyond the st.~ccial plans that have been discussed 3. ACT, An Actuarial Programming Language by David R W. Jamieson (Can- and recommended that for unique poli- ada), published in the January 1972 issue of ‘The Actuary. It must be noted that cies not yet developed, a central techni- ACT is not meant to be a publication language but only a programming language. cal entity (such as the NAIC Central 8. Some Proposals for a Revision of the Internationul Actuarial Notation by Office) could be empowered to review P. J. Turvey (Great Britain) published in the Journal of the Znstitute of Actuaries policies to determine if they are actu- Students’ Society March 1972. arially srund and meet certain standards 5. Proposals for an International Actuarial Publication Langwge by Carl Boehm of equity and to approve other technical (Germany), J. Adams (Belgium) et al. published in the Transactions of the XIX aspects. This approach would facilitate International Congress of Actuaries, Oslo 1972. approval of complex policies and would 6. A Contribution to the Discussion on a New International Actuurid Notation promote feasibility of product design by J. Engelfriet (Netherlands) and A.I.M. Kool (Netherlands) published in the while decreasing risk of abuse. October 1973 issue of the BLATTER. This reviewer is somewhat disappoint- 7. An Actuarial Notation Based on Symbolic Logic by G. C. Taylor (Australia) ed that the Committee did not recom- Research Paper No. 50 published by the Macquarie University in April 1974. mend also allowing the use of an assump- 8. Thoughts on the Harmonization of Some Proposals for a New International tion of a percentage increase in the in- Actuarial Nolation by C. Boehm (Germany), J. Engelfriet (Netherlands) et al. pub- dex in the case of cost-of-living increases lished by the authors in 1975. so that minimum values could be deter- mined at issue. The proposed minimums 9. A Revised Actuurid Notation. A suggestion by the Notation Sub-Committee are complex and could be very difficult of the Institute of Actuaries of Australia and New Zealand (Edition 10, October for the consumer to understand. How- 1975). ever, the proposed minimums would The table below shows four of the most commonly used actuarial functions in accommodate a modification of the in- Halo Notation and in linearized notation as suggested in the papers listed above. (Continued OR page 5) (Continued OR page 5) March, I.976 THE ACTUARY Page Five l (ml. Unruh Halo Notation a, AZ ax:4 Ax:;;l (Continued from puge 4) Boehm/Reichel at,.) A fr) a(x;n;m) AE(x;n) dex accumulation method for a cost-of- Australia/New Zealand livine ~ policy where the death benefit (1971) a(,) bfb,n) B(x,n) and cash values escalate with the cost- of-living but premiums remain level. Jamieson ANXNMXNM AXNXN (See Life Bnsed on the Consul Turvey apnm (x,n,m) an(v) mer Price Index, by Bragg and Stone- Boehm/Adams a(v,k) AE(x,d cipher, TSA XXII, p. 351). Presumably Engelfriet/Kool ‘a- (x,n,k) AE (x,4 the minimum cash value defined by Taylor l A(xhLn) (2) above could not be greater than tbe Sinthesis comparable cash value for a level face amount policy at the same duration, and Boehm/Engelfrite (k)a(x:n) A(v) for the attained death benefit at that Australia/New Zealand a(x, #n, f=m) Ab,#n) duration. It would be preferable and (1975) provide a greater flexibility if this type of policy could be allowed to be approv- l Taylor’s paper does not give sufficient information to complete this notation. ed by the suggested central entity set up In 1972, the President of the International Actuarial Association appointed by the NAIC or state regulators. the Sub-Committee on Notation with the responsibility of gathering opinions on the The Committee is aware of the criti- desirability of developing a new actuarial notation. The two basic questions that cism of the current non-forfeiture law, the Sub-Committee must answer ‘are: that it allows certain products in today’s marketplace which meet the letter but (a) Is it desirable to linearize the International Actuarial Notation, and, if so, perhaps not the spirit of the law and the is there sufficient amongst the various national actuarial bodies to Report suggests ways to minimize these replace the Halo Notation with a linearized one? abuses. (b) Whether or not linearization is desirable and/or acceptable, should the The Committee re-examined the ex- scope of the actuarial notation be extended to cover fields such as pensions, pense all,owances in the present law be- l disability, demography, social insurance and non-? cause these were developed more than It would appear to be most desirable to extend the scope of the actuarial thirty years ago. More modern estimates notation to other fields. The linearization issue, however, seems to be quite con- of excess of first year expenses over re- troversial. There are three distinct points of view: newal expenses would likely decrease as 1. The traditionalists feel that the Halo Notation has served the profession well a percentage of face amount and increase since it first was officially adopted by the Institute of Actuaries over 100 years ago. as a percentage of premium. The Com- Furthermore, the notation is concise, it can be easily read and understood and why mittee concludes that appropriate mod- change it simply to accommodate today’s computers? Indeed, computers of five or ern expense factors would be bracketed ten years from now may be able to make use of peripheral equipment which will by the following for a whole life plan: enable them to read and print the Halo Notation easily. Low: $10 per $1,000 plus 90% of 2. The reformists point out that common printing devices cannot cope with 1958 CSO 3%“/0 net premiums the Halo Notation. This shortcoming increases substantially the cost of printing High: $20 per $1,000 plus 150% of actuarial textbooks, papers, etc. Furthermore, today’s computers cannot easily digest 1958 CSO 3%% net premiums the Halo Notation. (Note that the Committee recom- 3. The pragmatists maintain that actuarial writers have faithfully adhered to mends the use of net premiums rather the International Actuarial Notation since 1898 and the large amount of actuarial than adjusted premiums for the per- literature available today is full of halos. Should the actuarial notation be suddenly centage of premium allowance to sim- linearized, it would be somewhat chaotic to cross-reference new actuarial papers plify calculations). and textbooks with old ones. The pragmatists agree that there is a programming The Committee concluded that for problem, but the problem is only of a temporary nature. Then, why not leave the other plans, the percentage of premium Halo Notation as is and perhaps expand its scope and develop a notation vocabu- allowances should equally weight a whole lary where each standard notation would be translated into various programming life premium and the plan premium notations, each compatible with a specified programming language? This last view (subject to a maximum of $55 per is the one favoured bv, most members of the Committee on Standard Notation and $1,000). The Committee tested an ex- Nomenclature. pense allowance of $10 per $1,000 plus The I.A.A. Sub-Committee on Notation has recently circulated a questionnaire 100% of the weighted premium on vari- a to sound out oubhc ooinion on the issues of linearization and exnansion of the 1 ous bases. ‘&hen combined with @/z”/o International Actuarial Notation. The Committee has replied to the questionnaire interest and a more modern mortality saying NO to linearization and YES to expansion. Tbe reply wns made as an ex- table, this relatively low expense allow- pression of opinion of the majority of tbe Committee and was not made on behalf ance provides generally lower minimum (Continued on page 7) (Continued on page 7) March, 1976 THE ACTUARY Page Seven

0 Unruh values allow an interest rate higher than decreased depending on the extra mor- (Conhued from pnge 5) annual premium policies. tality but where minimum cash values would be identical to a standard policy, cash values than those provided by the The Committee analyzed numerous and (8) removal of anomalies that occur current law at 31/2% interest. It would other aspects. Various recommendations due to the uniform percentage of gross hardly seem appropriate to make a are made which are designed to simplify premium requirement. change in the expense allowance with- the law, to make the law more flexible out at least increasing the interest rate and consistent, and to eliminate certain We are indebted to Mr. Unruh and so as to more closely approximate the technical flaws. These include: (1) ex- the members of the Committee for this rate used in company asset shares. The panding the law to include deferred thorough, comprehensive, and practical Committee makes no specific recommcn- annuities during the buildup period on report and we hope that its final recom- dation for a more modern expense al- an accumulation of percentage of pre- mendations will induce appropriate re- lowance, mortality table, or interest rate; mium basis, (2) exemption of accident visions in the Standard Non-forfeiture however, these were studied and dis- and health insurance for non-forfeiture Law. Especially needed is additional cussed in detail. It is recommended that purposes (except possibly for plans with flexibility in developing and marketing a single set of minimum values be de- return of premium benefits), (3) treat- new products which provide protection fined based on the highest permissible ment of term riders as separate policies against erosion of insurance coverage interest rate and that the present link- under a “severability” principle, (4) due to inflation. age between the policy valuation and treatment of renewable and convertible Meantime this reviewer would again non-forfeiture interest rates be elimi- term policies as a series of short term urge all members of the Society to read nated. It is recommended that guaranteed policies, (5) exemption of policies that the report and to send their comments paid-up insurance non-forfeiture options never give rise to significant values by and suggestions to : should be those purchased by the cash a direct test for triviality, (6) expansion value at a rate as high as that used for of esemptions for term plans, (7) an CFIA’RLES GREELEY cash values. It is also recommended that alternate treatment for substandard poli- Metropolitan Life single premium policy minimum cash cies so that the death benefit could be 1 Madison Avenue come Tax and Reserve Valuation New York, N. Y. 10010 q (Continued from page 6) Even though the deduction for the “tabular interest” declines as the earnings rate rises above Fraser’s “optimal” point of 61/z%, the after-tax yield continues to I COMPETITION No. 6 I rise until the earnings rate reaches 12%. The marginal tax rate exceeds 50% after To our dismay we realized that actuaries the deduction reaches its maximum, but does not exceed 100% until the after-tax as a group have no plans to honor the yield passes its “optimal” level of 12%. With the “10 for 1” approximation rule, U.S. BicentenniuZ. Our Canadian mem- the optimal level for the after-tax yield occurs when the current earnings rate equals bers have an excuse, of course, but we lOl/,% plus one-half of the original assumed reserve interest rate: are shamefaced about the lack of patri- Let CER be the current earnings rate otic spirit among our U.S. members. I, be the original assumed reserve interest rate We invite suggestions for something the Society or its members might do to After-tax gain = 1.1 CER - [l.l CER - CER (1 + 10 I, - 10 CER)] ‘/ make an actuarial contribution to the = (1.05 + 5 I,) CER - 5 (CER) a Bicentennial celebration. Taking the partial derivative of the after-tax gain with respect to CER and set to zero. Examples : CER = lOl/,$TJ + 1/L A mortality table ending at age 200. It would seem that the peaking ‘deduction’ phenomenon described by Mr. Full retirement benefits after 13 years Fraser is only incidental because the marginal tax rate increases at a constant rate. of service and attainment of age 50. A declining afte.r-tax income occurs for a company only at a much higher earnings Declaration of Actuarial Indepen- rate (above 12% in this illustration). cl dence.

Actuarial Notation (Continued jron page 5) Perhaps the Society’s motto or emblem of the Society. Furthermore, the Committee has made it clear that if during the next could be given a Bicentennial theme. few months it will become evident that the prevailing opinion amongst the Society’s For each of the three submissions we embership is contrary to that already expressed by the Committee, then the I.A.A. like best we will award a delightful ub-Committee will be so notified. book, “American Journeys”, an anthol- @ Thus, the Committee on Standard Notation and Nomenclature would welcome 0~7 of travel in the United States (from the opinion of any member of the Society on the question of linearization and ex- John Alden to Neil Armstrong). pansion of the International Actuarial Notation. If you wish to express your opinion, The usual Rules will obtain and the you can do so by writing to me at Confederation Life, 321 Bloor Street East, Tor- final mailing date is May 17. onto, Canada, M4W IHI. 0 C.E.