TRANSACTIONS OF SOCIETY OF ACTUARIES 1954 VOL. 6 NO. 14 MONETARY VALUES FOR ORDINARY DISABILITY BEN- EFITS, BASED ON PERIOD 2 OF THE 1952 INTERCOlVI- PANY STUDY OF THE SOCIETY'S COMMITTEE, WITH 2~ INTEREST MANUEL R. CUETO T E~I Committee on Disability and Double Indemnity presented in 1952 the results of an intercompany study of the experience under certain ordinary disability benefits between the 1930 and 1950 an- niversaries. Monetary values based on the new data have been prepared for the primary purpose of furnishing material which can be used to test reserves for disability benefits (active and disabled lives) in comparison with the reserves based on tables currently in use. The monetary values for each of the Benefits I to 5, inclusive, described in the 1952 Reports are based on the Period 2 graduated rates of disable- ment--first two policy years excluded--and the 1930-1950 graduated termination rates, combined with the Commissioners 1941 Standard Or- dinary (CSO) Mortality Table and %½2 interest. While Period 2 rates of disablement (between 1935 and 1939 anniver- saries) are not exactly midway between those for Period 1 (1930- 1935) which are high, and Period 4 (1946--1950) which are low, they are believed to represent a reasonable approach to the problem of attempting to use what might be considered an average or "long-term" level of dis- ability rates, particularly because rates of disability or disablement tend to reflect directly any changes in the economic cycle especially with re- spect to employment. (In this connection, the period 1946-1950 was one of the most favorable in the nation's economic history and the period 1930-1935, one of the poorest.) Another reason for basing the monetary values on the Period 2 rates of disablement was the statement in the Re- port that this was the only period with sufficient data for every benefit. Finally, it might be mentioned that because of the same considerations the New York Life's recently adopted premium rates for the waiver of pre- mium benefit are based on the rates of disablement of Period 2. There may be various points of view as to what would be suitable rates of disablement and termination, depending upon a company's experience and the individual actuary's iudgment. However, it is felt that Period 2 rates of disablement and 1930-1950 termination rates represent a reason- 801 YRATENOM VALUES FOR ORDINARY DISABILITY BENEFITS 109 able and appropriate starting point for the calculation of monetary values to be used in testing current reserves. BENEFITS As a matter of convenience the benefits studied in the Report are re- stated below. References to the benefits will be made by means of the indicated code numbers. Benefit/.--The "total and permanent" clause, under which the prob- able permanence of the disability had to be established on the merits of each case; the benefit covered waiver of premiums plus a monthly life income during continuance of total permanent disability of $10 per thou- sand of life insurance. Issued chiefly from about 918 to early in 1930. (The data for claim rates on this benefit contained a 3 months waiting period which was a necessary but not a sufficient condition for establishing permanence, with retroactive income benefits and nonretroactive waiver benefits. However, the monetary values herein are based on nonretroactive income and waiver benefits. This matter is referred to in the section on Premium Formulas below.) Benefit 2.--The 90-day "presumptive" clause under which it was a matter of contract that permanence would be presumed where total dis- ability which had continued for at least 90 days still existed; the benefit covered waiver of premiums plus a monthly income of $10 per thousand of life insurance; no monthly income benefits for first three months (except where disability was adjudged permanent without reference to the pre- sumptive clause). Issued chiefly from 1921 to 1930. (The monetary values are based on nonretroactive income and waiver benefits.) Benefit 3.--90-day presumptive clause providing monthly income of $10 per thousand of life insurance and waiver of premiums, as in the case of Benefit 2, but with income benefits retroactive to cover the first three months. Issued chiefly from 1925 to early in 1930. (The monetary values are based on retroactive income and waiver benefits.) Benefit 4.--120-day waiting period clause, under which total and per- manent disability was defined as total disability which had lasted 120 days; providing monthly income of $10 per thousand of life insurance combined with waiver of premiums; no income benefits for the first four months. Issued in 1930, 1931, and a few months in 1932. (The monetary values are based on nonretroactive income benefits but retroactive waiver benefits.) Benefit 5.---6 months waiting period clause, providing waiver of pre- miums only; issued only when applied for, subject to a specific extra pre- 110 MONETARY VALUES FOR ORDINARY DISABILITY BENEFITS mium. (The similar benefit which is automatically included by some com- panies in all policies issued was not studied in the Report.) Issued after 1931. (The monetary values are based on retroactive waiver benefits.) RATES OF DISABLEMENT The rates of disablement used were for males and females combined. The Report stated that distinction by sex was not practical because in most cases the exposure to disability was obtained from records main- tained for valuation purposes which made no separation by sex. The Period 2 graduated rates of disablement for each of the Benefits 1 to 5, inclusive, were shown in the Report for attained ages 18 to 59 inclu- sive. Because of the sparseness of the data at the youngest ages covered by the experience, it was evident that no rates could be developed therefrom for the purpose of extension to the younger ages. However, in the interest of making the monetary values as complete as possible without serious inaccuracy, it was decided to extend the rates of disablement for Benefits 1 to 4 back to age 51 (the earliest age at which these benefits were gen- erally issued) by arbitrarily using the rates at age 18 for ages below 18. Since Benefit 5 is currently being issued by various companies below age ,51 it was thought advisable to adopt rates of disablement at least down to age .01 The rates of disablement shown in the Report decreased as the age decreased. At attained age 18, the youngest age for which a rate was given, the rate of disablement under Benefit 5 was 16. per thousand. For attained age 71 the Committee actually obtained a graduated rate of .52 per thousand but this rate was not published. Below this age the data available in the study were not sufficient to produce reliable results, and there were no other statistics at young ages which were deemed acceptable for this purpose. Because of these facts, and also because the rates of dis- ablement at the young ages are relatively small so that any changes in these rates would have only a minor effect on the net premiums, it was considered reasonable to use the rate of disablement of .52 per thousand for attained ages 10 to 71 inclusive under Benefit 5. Rates of disablement are based on the date of disablement, ~.e., the date upon which the life actually became totally disabled provided the life con- tinued disabled during the waiting period. The use of a date of disable- ment with its distinct advantages is discussed on page 74 of the Report. Applying the continuous function D~ to the rates of disablement made it unnecessary to first calculate probabilities of disablement r, from the absolute rates of disablement r-' given in the Report, before proceeding to the computation of commutation columns. To illustrate, making the as- YRATENOM VALUES FOR ORDINARY DISABILITY BENEFITS 111 sumption heretofore followed in the preparation of commutation columns for disability benefits that ~l values can be used, instead of l~, to obtain the exposure to disability, and applying the continuous function b~ to the rates of disablement r', values of ~C emerged directly as may be seen below. ~D "rf---~ v~/" (1- ½qx) .Dx. 1 ~r~q½- DISABLED LIFE SEITlVNNA Tables of disabled life annuity values were prepared for each of the five benefits based on the 1930-1950 graduated termination rates and 2½% interest. The termination rates were for males and females combined. For convenience, the continuous form of disability annuity was calculated for both income and waiver benefits although it was recognized that tradi- tionally this form is used only for the waiver benefit, a monthly disability annuity being used for the income benefit or the income and waiver benefit combined. However, net premiums and reserves may be obtained for the income benefit from the commutation columns given herein by appropri- ate adjustment in the applicable formula therefor. The termination rates used were those published in the Report, which had a select period of 51 years. The rates, monthly for the first and second years and annually thereafter during the select periods for each benefit, were available only in quinary age groups according to the insurance age at the policy anniversary preceding the date of disablement. Since the age at disablement was on the average ½ year higher than the age at the pre- ceding policy anniversary, these rates were assumed to apply to an age year older than the central age of each quinary age group. Hence there were in effect available select graduated termination rates for quinquen- nial ages at disablement, 17½, 22½, etc. Because it was thought desirable to obtain tables of disabled life annuities for all ages an interpolation was made. It was found easier and more expedient to base the interpolation on the termination rates rather than first compute disability annuities at quinquennial ages and then interpolate for the values at other ages and periods. The interpolation was made by means of the Karup-King third difference interpolation formula employing the electronic calculator (I.B.M. 604). Under the central age assumption mentioned above, the earliest age at 112 MONETARY VALUES FOR ORDINARY DISABILITY BENEFITS disablement for which termination rates were available for each benefit was .½71 These rates were extended back by extrapolation to age ½51 for Benefits 1 to 4, inclusive, and to age ½01 for Benefit 5 (in a few cases there would have been anomalies in results if some slight adjustments had not been made). The disability annuities were then computed from these rates. It will be noted that the above method of extending the termination rates to the younger ages differed from the arbitrary method used with respect to rates of disablement. It was felt that the more precise treatment accorded termination rates was warranted because of the fact that the dis- ability annuities produced therefrom were the direct basis for disabled life reserves. For Benefit 1 the disability annuities were based on the Benefit 1 ter- mination rates given in the Report, i.e., monthly rates for the first two years of disability and annual rates thereafter. For each of Benefits 2 and 3 the disability annuities were derived from the monthly termination rates applicable to the respective benefit for dis- ability year ,1 the combined monthly rates for disability year 2, and the combined annual rates thereafter. In this connection the Report stated that the termination data for Benefits 2 and 3 were merged after the first disability year because preliminary graduations indicated that the ter- mination rates for these benefits were very similar thereafter. For Benefit 4 the disability annuities were derived from the Benefit 4 monthly termination rates for disability year I and the termination rates (monthly and annual) for Benefits 2 and 3 combined for all later disa- bility years. Here again the Report indicated that the termination rates for this benefit after the first year were not greatly different from those for Benefits 2 and 3 combined and hence were not published. For Benefit 5 the disability annuities were based on the Benefit 5 monthly termination rates for the first two years and the annual ter- mination rates for the balance of the select period of 51 years; thereafter the termination rates for Benefits 2 and 3 combined were used because no tables of ultimate termination rates were prepared for this benefit on ac- count of lack of data. It should be pointed out that, as emphasized in the Report, the ter- mination rates for Benefit 5 are very largely based on the experience of the 1940's which included perhaps abnormally high rates. As a matter of fact this word of caution also applies to the ultimate experience for ter- mination rates of all the Benefits. Because of this it may be that in the calculation of gross premiums under Benefit 5, any net premiums based on these new data may require an additional margin. MONETARY VALUES FOR ORDINARY DISABILITY BENEFITS 113 Disabled life annuities were computed on the continuous basis since the extensive tables of termination rates available for the first time facilitated the calculation of this more practical form. Furthermore, the employment of electronic equipment to obtain both the disabled life commutation col- umns and the annuities made it just as convenient to compute continuous annuities as monthly annuities. It may be well to mention that disabled life annuities calculated from termination rates based on the date of disablement instead of the date of disability took on a different form from that heretofore used. The disabled life annuity formula for an n-payment policy for, say, a waiver benefit with a six months waiting period and no provision for retroactive benefits was as follows: 1 dlx÷ll2+ll~:U-x-1 D* +r:l ,/2)+,/5 i 1 i -~- . . . + D~+ t/~l+23/1~-~-~Dx+l121+2) D ~ . a~ } "~ x{ + l/...+_ ° x+ l/2l+2:u-x--2 77-1 ~{2/,+I.,/,+,ID 1 'D'C -- -- x~ z+,/.~+6/,2 +' Dx+42+7/,~ + -tq- Di J-iDi • . . x+i/2+23/12~v2 x+,/2+2 / + (~Dtx+,/-%+~+ i i D 3+l~/~+~ ~t + . . " + D ~ l~+~/q+--~-~ ) } where x = the age at policy anniversary preceding disablement, u = issue age plus premium-paying period, and u < y, the limiting age of the coverage. In the foregoing formula the value of di.+~/2~+2:._._2 tY/-~ has been taken as equal to D i. + 1/2 +u-x- 1 alz+ ,/S +u--z--, :l~-t i D +=i }~-__/I +,-~-?~ 1 D Ix+l/21+--*-i ", 1(1 = a ix+ ,/21 + ~-:~--~'q + DI~+,/~I+ ~ / 4 2 Dix+,m+.o = +i~..~_~:_.+j~/,+~i~ ½ on the assumption that az +~-l/2+ .... Sl/'~-1:1 equal to .½ This represents only a slight overstatement. 114 MONETARY VALUES FOR ORDINARY DISABILITY BENEFITS Similar formulas were used for determining the disability annuities for other waiting periods, thus producing disabled life annuities which in the case of the above example, but for a 3 months waiting period, were ex- pressed as 2/~+,~a .~W:Y:---x-u:zx/a+l In preparing commutation columns involv- ing these annuities it was necessary to discount the annuity value back to the beginning of the waiting period. This will be seen from the notation shown below. It should be noted that in discounting the annuity values interest only was involved, since those disabled have survived the waiting period to qualify for the benefit. NOTATION In the preparation of commutation columns for each benefit the nota- tion used herein is similar to that employed by Mr. Phillips in TASA XXXIII, 8, except that the continuous form of disabled life annuity is used instead of the monthly form and the disabled life annuities in the functions involving these values are discounted back to the beginning of the waiting period. ~C = ~)-i • ~r t (see section on Rates of Disablement) x 1/-1 ~M~ = ~ ~C (coverage to age y) x ~C" - "~"( v °-~/" ,,,~-l.~/• l~xli~ ,,,+61-~-,:_~1, i~7~ (where m represents the number of months of waiting period) u-- 1 ~t--r Mz ---~ x (if u is greater than y, the limiting age Mz --~ x of coverage) r. ~C~ = Cr .,.nil2 -i (where m represents the number of X'~ "a X.fl/O~m/12 months of waiting period) cohr Mz x y--1 to--r ~C~ (coverage to age y) .r MONETARY VALUES FOR ORDINARY DISABILITY BENEFITS 511 NET PREMIUMS Tables of net premiums were prepared for each of the Benefits 1 to 5, inclusive. It was found more convenient by means of the electronic equip- ment to calculate net premiums for all ages rather than for quinquennial ages only and interpolate. To provide the exact disability benefit of the various types, the formu- las for disability net premiums required only minor adjustments in view of the fact that the rates of disablement and the termination rates were based on the type of disability provision contained in each benefit. For example, where the waiver benefit was taken as retroactive in the calculation of monetary values, as in the case of Benefits 3, 4 and 5, the formula for the net premium for such benefit under an Ordinary Life policy for waiver of $100 of annual premium was l '~-. mv,~/~2 ~M I N~- N~ where m is the number of months of waiting period, and y the limiting age of the coverage, the values of N~ being based on the CSO 2½% Table. ~m 21/~'v The factor 21 • makes provision for the retroactive payment and is taken as v~ ,4/~ ~v a/~ or ~ lvl/~ , depending upon whether a 3 months, 4 months or 6 months waiting period is involved. The formulas for the income benefit also included an adjustment. Since disabled life annuities were prepared on the continuous basis, an adjust- ment was made to change these values when used for an income benefit from the continuous to the monthly form. Also, where the income benefit was retroactive--for example, in the case of Benefit 3--an adjustment was made similar to that in the formulas for the waiver benefit. To illustrate, the premiums under Benefit 3 for a retroactive income benefit providing for the payment of 830 per thousand of life insurance at the end of the waiting period of m months, and thereafter a payment of 810 at the end of each month for life during disablement, were computed by the following formula: • t~ .1Vi x N,- N~ where the disability commutation functions were based on those com- puted for Benefit 3 and m was, of course, equal to 3. The term v"h2.~M~ 120 (2-~) N,- N~ 116 MONETARY VALUES FOR ORDINARY DISABILITY BENEFITS makes the necessary adjustment to convert from the continuous to the monthly form of disabled life annuity and the term N,- ~N makes provision for retroactive income benefits on the basis of completed months of disability. PREMIU~ FORMULAS For Waiver Benefits The net premium formulas for waiver of $100 of annual premium were as follows: Benefits 1 and 2 (nonretroactive waiver benefits) Ordinary Life 100 ~M~ n-year period 100 ~M x+n= u <y N~- ~N uhr 1 O0 ~,I~ x+n=u~y N.- N~ Benefits ,3 4 and 5 (retroactive waiver benefits) where m si the number of months of waiting period. Ordinary Life 0 o~--r m m/12 ~rr N~-- ~N n-year period l~-r m ,,/~.o. M, f x+n=u<y 100 M~+-2v ~ m x+n=u~y N~- Nv MONETARY VALUES FOR ORDINARY DISABILITY BENEFITS 711 For Income Benefits The net premium formulas for income benefits of $10 per month were as follows: Benefils 1, 2 and 4 (nonretroactive income benefits) where m is the number of months of waiting period. For $10 monthly income for life: Ordinary Life and n-payment life, 120t~,~'~_ 1 r -~1/" ~M'. 1 t ylVIx ~ ~ " x+n=uNy N~- Ny n-payment life x+n=u<y N~- N, For $10 monthly income for the balance of n years n-payment n-year endowment 1 2 0 { *z--r ~x + 1 m/12 .ulYlz . .r } ~1 x+n=u<y N,- N, ~)f~{u~r~ 1 T m/t2 M r x+n=u~.y NX- N~/ For Benefit 3, which provided for retroactive income benefits, the above formulas for income benefits were modified to include in the numerator the term 120(m -- 1)/12v 21',*" • ~Mv as has already been explained in the section on Net Premiums. In the case of an n-year endowment where u < y, the ,M~ term would be replaced by ~M~. With regard to Benefit ,1 it should be noted that some of the companies issued this benefit with the provision for retroactive income payments. As was mentioned in the section on Benefits, the data for claim rates on this benefit were based on such experience. However, no adjustment was made in the above formula for this fact--that is, the income benefit was treated as nonretroactive. In applying the foregoing formulas y, the limiting age of the coverage, was taken as 60. Electronic equipment was used in the calculation of net premiums as in the other calculations.