Joint Life Annuities—Some Experiments with the A1924-29 Table [with DISCUSSION]

Institute and Faculty of Actuaries Joint Life Annuities—Some Experiments with the A1924-29 Table [with DISCUSSION] Author(s): A. E. King and A. R. Reid Source: Transactions of the Faculty of Actuaries, Vol. 15, No. 137 (1934-1936), pp. 93-140 Published by: Cambridge University Press on behalf of the Institute and Faculty of Actuaries Stable URL: http://www.jstor.org/stable/41218238 . Accessed: 28/06/2014 08:18 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Cambridge University Press and Institute and Faculty of Actuaries are collaborating with JSTOR to digitize, preserve and extend access to Transactions of the Faculty of Actuaries. http://www.jstor.org This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/action/showPublisher?publisherCode=cup http://www.jstor.org/action/showPublisher?publisherCode=actuaries http://www.jstor.org/stable/41218238?origin=JSTOR-pdf http://www.jstor.org/page/info/about/policies/terms.jsp http://www.jstor.org/page/info/about/policies/terms.jsp No. 137] 93 TRANSACTIONS OF THE FACULTY OF ACTUARIES Joint Life Annuities - Some Experiments with the A1924-29 Table. By A. E. King, F.F.A., F.I.A., Secretary, Standard Life Assurance Company, and A. R. Reid, M.A., F.F.A., F.I.A., Investment Secretary, Standard Life Assurance Company. [Read before the Faculty on 10th December 1934. A S3mopsis of the Paper will be found on page 119.] INTRODUCTION. ХНЕ writers of the present paper have recently been engaged on some experimental work connected with the A1924-29 ult. table, the two main objects being : - (1) To ascertain what modifications in Makeham's Law are necessary in order that the principle of uniform seniority may be applied. (2) To ascertain whether some approximate method is readily available for the calculation of joint life annuity values by reference to tables of joint life annuity values in respect of equal ages.* At an early stage it was discovered that the various elements of heterogeneity in the A1924-29 table, whether arising by reason of the admixture of the four classes of assurance or otherwise, Avere of sufficient extent to cause the run of the differences of the official ♦ Although we are concerned in this paper with approximations to the values of joint life annuities with reference to joint life annuities in respect of equal ages, no treatment of the subject of approximating to the value of joint life annuities would be complete without referring to the valuable paper by Mr. Lidstone in J.I. A., xlvi. p. 1. The method described in that paper is of course applicable also to Last Survivor Annuities. VOL. XV. G This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 94 Joint Life Annuities - graduated functions to be very irregular in places. To illustrate these irregularities the fourth differences of log lx and fix, taken at quinquennial points, according to the A1924-29 ult. table are set out below, together with the corresponding figures of the OM(5) table, which table has been chosen because it represents a smooth graduation by a formula and not because its underlying data are free from heterogeneities. -AJÍlogZxxlO5 aJ/íjxIO* Age A1924-29 ult. OM Some Experiments with the Al 924-29 Table 95 . rate to such a table represented in the form of a normal life table - although it has been shown that so far as " generation " mortality is concerned Makeham's Law still holds good.* For very many years past the actuarial profession has been served by tables of assurance mortality which have conformed to Makeham's Law, and it is interesting to investigate the various reasons which account for the fact that modern tables of mortality do not readily submit themselves to Makeham's beneficent rule. This question, however, will not be pursued here. The advantages secured when Makeham's Law holds are so con- siderable that most, if not all, actuaries will agree that it is well worth while making every effort either (1) to force a table into compliance with Makeham's Law if no serious monetary error is thereby introduced, or if this prove impracticable (2) to endeavour to find a modification of the law which will enable the statistics to be fitted and also secure most of the advantages of Makeham's Law. In the case of the A1924-29 table it was unhappily found impossible to apply Makeham's Law in its usual form throughout the length of the table, nor were the results obtained by applying the second modification of Makeham's Law very satisfactory to- wards the end of the table. In view of these facts it seemed worth while to enquire whether Makeham's Law could be applied with modifications without losing all the advantages of uniform seniority, and when Mr. C. L. Stoodley delivered his valuable paper to the Faculty in February 1934 it seemed to us that there were grounds for hoping that some measure of success might be achieved. In the discussion following Mr. Stoodley's paper it was pointed out that if the 5-constant expression for ихЛ viz. A + Bcx - - - SSîHt m+nx could be made to fit a table, the calculation of annuity values on two or more lives could be carried out on similar lines to those adopted in the case of a Makeham table and with but little additional work. * vide (i) T.F.A., xi. pp. 194-206. (ii) " Death Rates in Great Britain and Sweden " ; W. O. Kermack, A. G. McKendrick and P. L. McKinlay, The Lancet, 31st March 1934, p. 698. (iii) " The Development of the Mortality of the Adult Swedish Population since 1800," H. Cramer and H. Wold, Nordic Statistical Journal , vol. 5. t T.F.A., xv. pp. 26-28. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp ф Joint Life Annuities - This is due to the fact that if цх be of the form specified, then 1х=к**д**(т+п*) where k, s, and g are the normal Makeham constants, so that tPx may be expressed in the form ФхгРх + со ФхП^Рх where фх= - - - -, софх=1 - фх = - ¡ - -, m+nx m-'-nx and where the symbol м denotes that the function above it is calculated according to the underlying Makeham table. It immediately follows : - (i) that ax= фхах+ со фха'х M M where a is calculated at rate of interest j =_ t - 1? (ii) that аху=фхфуаху + (фхсо фу+фу со фх)а'ху+со фхсо фуа"х?/ where a has the same meaning as under (i) and where a" is calculated at rate of interest k = -^- 1, and (iii) that axyz and awxyz may similarly be expressed as blends respectively of four and five Makeham joint life annuities at vary- ing rates of interest. The extremely light mortality, however, disclosed at the older ages by the A1924-29 experience made it difficult to fit even a 5-constant function of this form to jjlx over the whole table, and the theory was therefore illustrated in that discussion with reference to a graduation of the data which held up to age 65 only. It was hoped that further trials might yield a more satisfactory fit so that the strikingly successful results obtained in respect of the earlier part of the table might be extended over the whole range, and the following gives a summary of the analysis undertaken. Note. - For the purpose of experiment it was considered sufficient to make comparisons with the official graduated table rather than to refer back to the ungraduated data. It should also be noted that the introduction of the extra deaths which were later intimated to the Mortality Committee by certain offices in connection with the ultimate experience did not sensibly affect the analysis. (1) It has been shown by Mr. Lidstone in connection with his demonstration of the strikingly successful application of the Z method to the new table * that with the values A =-00191 В =-00002078 Iog10e = .0485 * J.I.A., lxiv. p. 497. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table 97 it was possible to fit a table of the A1924-29 ult. force of mortality with the ordinary Makeham function, viz. A+Bc*, up to say age 65, and it was suggested therefore that if these constants were employed in framing the basic Makeham table the problem could be narrowed down to fittine; 0 the deductive term ^- to the 0 m+nx differences between the basic Makeham values of /лх and the official values at the end of the table. After many trials a reasonable fit was secured only to age 80. The values of the constants m and n which give this fit are m =2-õ x IO11 logion= *13 and it may therefore be presumed that by the use of these values, temporary joint life annuity values involving ages up to 80 could be calculated with practically the same degree of accuracy as is shown by the examples given in the discussion to Mr. Stoodley's paper referred to above. The restriction of the range to age 80, however, renders the formula of but little practical use so far as the A1924-29 experience is concerned. It should be noted further that even if the formula did fit the data to the end of the table, the value of the constant n is such that certain large negative rates of interest would be involved in the calculation of annuity values. (2) A natural extension of the method outlined in (1) which suggested itself was to employ a Makeham second development curve to give the basic values and to try to fit the deductive term to the differences between the Makeham second development values and the official values. The following values for the constants in the Makeham second development form of pXi viz. А+Пх+Вс* A = -00233 H =--000023 В = -000036 Iog10c= -045 combined with the values of the constants in the deductive term m =2-55xlOn logl0n= -125 were found to give the best fit, and in this way the graduation could be extended with the same degree of accuracy as in the previous case as far as age 90. The higher values of 'ix yielded after that age by the formula as compared with the official values could not sensibly alter annuity values except where the age at entry was very advanced, but again the practicability of the This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp g8 Joint Life Annuities - method is vitiated, although for a different reason. When the basic table follows Makeham 's second development of Gompertz's Law it may readily be shown that, so far as two lives are concerned, a joint life annuity value may be expressed as a blend of three equal age joint life annuity values at different rates of interest, calculated according to the mortality of the basic table. This statement applies also to the case where the basic table is an ordinary Makeham table but, whereas in the latter case the three rates of interest are related only to the rate of interest at which the annuity value is desired, when we work with a Makeham second development basic table the three rates of interest are related not only to the rate at which the annuity value is desired but also to the differences between the ages. The very large number of rates of interest at which equal age annuities would require to be calculated therefore (the position is further complicated in the case of annuities on three or four lives) renders the extension on the Makeham second development plan of purely theoretical interest.* (3) v A modification of the deductive term from the form °%еП v m+nx to the form r^-^ - °8*n °&c suggested interesting possibilities, m+ncX because if /лх be taken as A+Bc*-nCVlogenloggC m+n* it may readily be shown that lx=ks^(m+ncX) where k, s, and g are the usual Makeham constants, and hence after straightforward analysis that a^-( - ¡~^r*+( ~)a*+h 'т+п Some Experiments with the Al 924-29 Table 99 might be real, the value of n had to be retained within certain limits which rendered it impossible to secure a fit. (4) The points of resemblance between certain of the forms adopted above for /x^ and those employed by Mr. Perks in the paper which he delivered to the Institute of Actuaries a few years ago* suggested another method of attack. In the course of the first discussion on the new mortality experience held in Staple Inn Hall, Mr. Perks put forward a successful graduation of the ultimate data obtained by one of his formulae, and we thought that while it had not been found possible to fit closely a Makeham curve mpdified as above to the official A1924-29 table, better results might be obtained by endeavouring to fit a modified Makeham curve to Mr. Perks's graduated values, since these necessarily followed a smooth curve. Again, however, it was found that the material at the older ages was intractable. Although mainly of theoretical import, it is interesting to note in passing how one particular case of Mr. Perks's general formula could be utilised to produce simple rules for the evaluation of joint life annuities such as we are searching for. If in the general expression for цх which he adopts, viz. (A+Bc^Kc^+l + Dc*), К is put =0, it is shown by Mr. Perks that lx takes the form (A-BVlogec d;/ ¿farA*(l+D¿*)v d;/ so that if in graduating by the modified form for fix, viz. A-f-Bc* Ï+Dc* it were found that the constants were connected by the relationship logec = A-D we should have lx=ke-Ax(l + Dc*) i.e. k=b*(l+D«*) whence tPx^8t____ =Ф* ioo Joint Life Annuities - in which event a life annuity would be a blend of two perpetuities at different rates of interest. Correspondingly simple results would be obtained in the case of annuities on two or more lives. In the case, however, of the graduation of the A1924-29 experience which Mr. Perks carried out the constants were not related even approximately in the manner referred to above. (5) In view of the impracticability of each of the attempts described above, the procedure which next suggested itself was to find what was the simplest extension of the 5-constant expression for fJLx given in (1) which would fit the official data. It seemed essential to adda second geometric term to the denominator of the deductive expression for fix and a fit was accordingly obtained by assuming /% to be of the form X + -Qčx_mX^gem+hnx'ogén f+mx+hnx The corresponding expression for lx is lcsxg^(f+rnx+hnx) so that ax = F(x) ax -f M(x) a!x + N.^ a"x M M M where ?(x)=f/(f+mx+ànx) M(x) = m*/(f+ mx+h nx) T$(x)=hnx/(f+mx+hnx) and a and a" are calculated at rates of interest respectively i±i-landl±-¿-l. m n A joint life annuity may similarly be expressed as a blend of Makeham joint life annuities at different rates of interest, but these would be six in number in the case of two lives and ten in the case of three lives. The method, therefore, although the best of the modified Makeham type obtained for fitting the A1924-29 table, only gained its fitting power at the expense of losing practically all the advantages of uniform seniority. As an illustration of the theory, however, a table is appended (Table I.) showing the graduated values of fix obtained by utilising the constants employed by Mr. Lidstone, as described in (1), in forming the basic Makeham table and by utilising the following values for the other constants referred to above: - /= 1-7786X109 Iog10w= -10326 'oglQn= -26385 log10Ä =15.14704. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table ioi A comparison is also given between the actual deaths and the expected deaths according to the table as regraduated (Table IL), and specimen single life annuity values are shown as calculated by the formula given above and compared with the true values (Table III.). The results of the above experiments have been somewhat dis- appointing so far as the A1924-29 table is concerned, but the new method of extending the Makeham formula for цх may yet prove of practical use - perhaps when applied to a mortality table more free from heterogeneous matter than the A1924-29 table. SECTION II. Approximate methods of calculating joint life annuities by reference to tables of joint life annuities in respect of equal ages. Official tables of annuities at various rates of interest in respect of two lives of unequal age based on the A1924-29 table will be published for differences of three years and multiples of three years of age to three places of decimals. Annuity values in respect of two, three and four lives of equal age will also be available at various rates of interest, the three-life and four-life values to two places of decimals. So far as two-life annuity values for unequal ages are concerned, simple interpolation will enable values to be calculated when the ages dealt with do not correspond to the ages for which values are published. In the case of three-life and four-life annuities, however, approximate integration or some other method of approximation will ordinarily have to be employed. In an attempt to find an alternative method to that of approximate integration trials of various kinds were made as described in the following paragraphs. Note. - The examples given in the case of the first three trials relate to two- life annuities y but these are for the purpose of illustration only. The main object of the enquiry was to discover a method applicable to multiple-life annuities, since two-life annuity values will be readily available from the official tables referred to above. (1) As has already been remarked, the A1924-29 table cannot be fitted with an ordinary Makeham formula. In the case of the OM table, however, where Makeham's Law did not hold throughout This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 102 Joint Life Annuities - the whole range of the table, quite fair results were obtainable if an 0M(5) table of uniform seniority were employed in conjunction with joint life annuities of equal age based on the 0M table, and this suggests that a standard table of uniform seniority such as the 0M(5) table might be applied to joint life annuities of equal age based on the new table. The following results were obtained by pursuing this course. A1924-29 ult. 3%. Approximations to Joint Life Annuities using 0M(5) Uniform Seniority Tables. Ages Approximate True №псе 21/87 2445 2-685 -240 39/87 2-423 2-669 | --246 57/87 2-318 2-553 I --235 75/87 1-914 1-991 -077 21/78 4-593 4-621 -028 39/78 4-515 4-575 -060 57/78 4-157 4-259 -102 75/78 2-986 2-989 -003 21/69 7-774 7-497 +-277 39/69 7-546 7-366 +-180 57/69 6-579 6-548 +Ю31 21/60 11-457 11-069 +-388 39/60 10-927 10-725 +-202 57/60 8-917 8-914 +-003 21/51 15-014 14-681 +-333 39/51 14-001 13-901 +-100 21/42 18-017 17-812 +-205 39/42 16-315 16-309 +-006 21/33 20-334 20-233 +-101 30/33 19-417 19-412 +-005 21/24 21-872 21-866 +-006 The above results cannot be said to be consistently good. (2) A consideration of some of the modifications of Makeham's formula led us to test whether better results could be obtained by making use of the relationship Рх + 1*у = %1Чв instead of the OM{5) table of uniform seniority, and the following are the results obtained by finding equal ages in this way. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table 103 A1924-29 ult. 3%. Approximations to Joint Life Annuities using the relationship /za;+/*2/=2/xM,. Ages Aplate Tn. Difference 21/87 2-590 2-685 --095 39/87 2-578 2-669 -091 57/87 2-483 2-553 -070 75/87 1-979 1-991 -012 21/78 4-441 4-621 -180 39/78 4-411 4-575 -164 57/78 4-165 4-259 -094 75/78 2-992 2-989 +-003 21/69 7-306 7-497 -191 39/69 7-209 7-366 -157 57/69 6-474 6-548 -074 21/60 10-760 11-069 -309 39/60 10-512 10-725 -213 57/60 8-912 8-914 -002 21/51 14-562 14-681 - 119 39/51 13-899 13-901 -002 21/42 17-248 17-812 --564 39/42 16-344 16-309 +-035 21/33 19-448 20-233 ! --785 30/33 19-345 19-412 -067 21/24 21-896 21-866 +-030 It will be seen that the results yielded are not substantially better than those obtained by employing the OM(5) uniform seniority tables. (3) Another method which suggested itself was that given in a note appearing in J I.A., xliv. p. 293, under the names of Mr. Elderton and one of the authors of this paper, the principles .underlying which it may be useful to restate here. If Gompertz's Law holds, then v^Pxy^^tPw for all values of t and axy = aw. Even if Gompertz's Law does not hold, there must be some age w for which axy = aw and we can write 42vttpxy=^^vttpw even though values of vttrpXy and vttpw may not be equal for all values of t. Provided that, as is almost sure to be the case, tPxy and tpw progress This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 104 Joint Life Annuities - in a smooth series, we may, for example, have vttpXyvttpw where t is large. It is clear that in such circumstances there must be a crossing point where, for some individual value of /, vttpxy = vttpw and if this value of t could be estimated the single age w to be substituted for the joint status xy may immediately be determined. The method actually adopted, however, in the note referred to above was to deal with values of /x. When Gompertz's Law does not hold (since as before remarked there must be some age w when axy=aw) we may have fJ-x+n+fÁy+n> flw+n where n is small and Some' Experiments with the Al 924-29 Table 105 A1924-29 ult. 3%. Approximations to Joint Life Annuities using the method described in J.I. A., xliv. p. 293. Ages АрРуаЙГ^ True Value Differen i об Joint Life Annuities - the best result in any particular case depended very largely on the older ages in the status. Indeed, after extensive experiments it was found that it would be possible when working with the A1924-29 ult. table to construct an empirical table of double entry by means of which the number of years ahead to be utilised in any particular case could be extracted. The two functions with which the table would be entered would be the oldest age of the status and the second oldest age, the other ages and the rate of interest having a relatively unimportant bearing on the result. Such a table (Table IV.) is given on page 115 and the following example illustrates its use. Example. A1924-29 ult. 3%. Let it be required to find the value of a20 .40 .50 .To. From Table IV. it is seen that the value of n given in the row representing 70 and in the column representing 50 is 6. We then have /'2()+б + /Х40+6+/Х50+б + /А70+б=*11329 one-fourth =-02832 = /z6S.13+6. By interpolation axxxx (where я=58-13) = 6*37, which compares with the value 6 42 yielded by Simpson's rule with ordinates at three-year intervals. The following table gives specimen results in respect of two, three and four-life annuity values obtained by application of the method. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table 107 A1924-29 ult. 3%. Approximations to Joint Life Annuities using Table of Double Entry - fxx+n method. Ages ^у^иГ^ "T™e"Val"e Difference Two lives 21/87 2-696 2-685 + 011 39/87 2-680 2-669 +-011 57/87 2-567 2-553 +-014 75/87 1-992 1-991 +-001 21/78 4-608 4-621 -013 39/78 4-563 4-575 -012 57/78 4-257 4-259 -002 75/78 2-992 2-989 +-003 21/69 7-484 7-497 -013 39/69 7-368 7-366 +-002 57/69 6-572 6-548 +-024 21/60 11-068 11-069 -001 39/60 10-743 10-725 +018 57/60 8-925 8-914 +011 21/51 14-666 14-681 -015 39/51 13-915 13-901 +014 21/42 17-813 17-812 +001 39/42 16-307 16-309 -002 21/33 20-233 20-233 30/33 19-426 19-412 +014 21/24 21-871 21-866 +-005 Three lives 30/30/50 14-17 14-18 -01 30/50/50 1215 1214 +01 20/20/60 10-87 10-84 +-03 20/60/60 8-20 8-21 --01 30/30/70 6-93 6-98 --05 30/70/70 4-64 4-64 30/30/80 4-08 4-06 +-02 30/80/80 2-30 2-30 Four lives 30/30/30/50 13-62 13-60 +02 30/50/50/50 10-51 10-51 20/20/20/60 10-72 10-61 +-11 20/60/60/60 6-65 I 6-66 -01 30/30/30/70 6-83 6-85 --02 30/70/70/70 3-47 3-46 +01 30/30/30/80 4-02 4-01 +-01 30/80/80/80 1-55 1-56 -01 30/30/50/70 6-44 6-49 -05 30/30/75/75 3-28 3-28 40/40/50/70 6-28 6-32 -04 20/40/50/70 6-37 6-42 -05 70/70/70/80 2-10 2-10 60/60/60/70 4-53 4-53 Note. - In the case of three and four-life annuities, what has been tabulated as the "True" Value is the value obtained by Simpson's rule using ordinates at three-year intervals. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp io8 Joint Life Annuities - (5) The tests which have been applied show that the method may be depended upon to give reasonable results, but nevertheless it seemed to us that it would be well to investigate further the alternative method already mentioned in (3) of focussing attention on the value of np rather than of p. The tabulated value of /x involves one year of age only, whereas the function np may be said to take a grip of n years of the table, and where the differences of the graduated values do not run smoothly it would seem preferable to deal with a term of years than with a single year. Looking at the matter in another way, we may represent npx in the form e Jo so that when we are employing npx we are involving the summation of fix over a period of n years. The initial problem is similar to that which we have already encountered, namely to estimate the number of years ahead n where nPxy=nPww or more generally 7iPxy • • • (w) == nPww • • • (my It may here be remarked that in connection with the table of double entry (Table V.) which has been constructed for use in connection with this method* on the basis of the A1924-29 table, we have been helped in its preparation by results of experiments prompted by a priori reasoning (see Appendix III.). As in the case of the рх+п method the prepared table is entered with the oldest and second oldest ages of the status. The number of years ahead n is then extracted and the equal age found by means of relationship nPxy • • • (m) == riPww • • • (m) ОГ log lxy. . .(w)- log nlxy. . .(m) = l0g lWW. - .(w) - log nlww. . .(w). A table of log lx is therefore all that is needed in conjunction with Table V. A specially prepared table (Table VI.), by means of which values of loglw-'oglw+n may be conveniently read off, is appended; the two inner columns, it will be seen, form a duplicate of the outer columns and are intended to be cut out in order to form a movable strip which may be employed in the manner described below. The following examples illustrate the use of Tables V. and VI. and show how readily a reliable approxima- tion to a multiple-life annuity value may be obtained. ♦ The method may be termed the npx method. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table 109 Examples. A1924-29 ult. 3%. 1. Two lives - required to find fisico From Table V. we see that 71= 16. log ¿48= -9516 bg¿48+16= -8533 log ¿во = -8936 bg¿G0+16= -5680 2 log lx= 1 8452 2 log 4+i6= 1-4213 Difference =4239 One half =-2120. By sliding the movable strip of Table VI. so that the value of log lx on the strip appears opposite that of log Ix+ig on the table, it is seen by inspection that the age w such that log/«,- logZw+i6 = *2120 lies between 55 and 56. By interpolation w= 55*73 and from the table of equal age two-life annuity values а„™(м= 55-73) = 10*124. True value of a4s: во =10*135. 2. Four lives - required to find ̂20:40:00:70 From Table V. we see that n= 1 1. logZ20= -9913 bgZ20+11= -9801 logZ40= -9685 bg ¿4o+ii= "9424 log /50= -9457 log ¿5o+ii= '8849 log Z70 = -7529 log /70+11 = -3140 2 log /* = 3-6584 2 log k+ii = 3-1214 Difference =-5370 One-fourth = '1342. By sliding the movable strip of Table VI. so that the value of log lx on the strip appears opposite that of log lx+n on the table, it is seen that the age w such that log lw - log lw+'' = '1342 lies between 58 and 59. By interpolation w=. 58*01 and from the table of equal age four-life annuity values аг™™™(и>=58-01) = 6-42 which is also the value of a2o:4O:5O:7o yielded by Simpson's rule with ordinates at three-year intervals. The following table gives specimen results obtained by this method. VOL. XV. H This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp no Joint Life Annuities - A1924-29 ult. 3%. Approximations to Joint Life Annuities using Table of Double Entry - npx method. AQES |¿PP™te| "True- | Difference Two lives 21/87 2-682 2-685 -003 39/87 2-667 2-669 -002 57/87 2-542 2-553 -Oil 75/87 1-992 1-991 +-001 21/78 4-623 4-621 +-002 39/78 4-581 4-575 +-006 57/78 4-256 4-259 -003 75/78 2-992 2-989 +-003 21/69 7-476 7-497 -021 39/69 7-356 7-366 -010 57/69 6-564 6-548 +-016 21/60 11-068 11-069 -001 39/60 ! 10-739 10-725 4-4)14 57/60 8-921 8-914 +-007 21/51 14-685 14-681 +-004 39/51 13-928 13-901 +-027 21/42 17-788 17-812 -024 39/42 16-314 16-309 +-005 21/33 20-219 20-233 -014 30/33 19-414 19-412 +-002 21/24 ; 21-863 21-866 -003 Three lives 30/30/50 I 14-18 14-18 30/50/50 1214 1214 20/20/60 10-84 10-84 20/60/60 8-20 8-21 -01 30/30/70 6-96 6-98 -02 30/70/70 4-64 4-64 30/30/80 4-06 4-06 30/80/80 | 2-31 2-30 +-01 Four lives ' 30/30/30/50 13-59 13-60 -01 30/50/50/50 10-51 10-51 20/20/20/60 ! 10-63 10-61 +-02 20/60/60/60 | 6-67 6-66 +-01 30/30/30/70 | 6-83 6-85 -02 30/70/70/70 | 3-47 3-46 +01 30/30/30/80 I 4-01 401 I 30/80/80/80 ¡ 1-56 1-56 .. ! 30/30/50/70 I 6-48 6-49 | -01 30/30/75/75 3-29 3-28 +-01 40/40/50/70 6-32 6-32 20/40/50/70 6-42 6-42 .. 70/70/70/80 2-10 2-10 60/60/60/70 4-57 4-53 +-04 Note. - In the case of three and four-life annuities, what has been tabulated as the "True" Value is the value obtained by Simpson's rule using ordinates at three-year intervals. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table ш It may be remarked that further investigation may show that the nPx method, with certain modifications, may be applied satisfactorily to the calculation of values of Last Survivor annuities. CONCLUSION. In addition to the various experiments described in Sections I. and II. of this paper, some trials were made to see whether other types of formulae would fit the curve of ̂ x in respect of the A1924-29 ult. table without requiring of them that they should furnish some special aid in the calculation of joint life annuities. One expression which was considered was /ха.=А + Вса5+Мт1а;. This formula is perhaps worthy of some attention, for not only is a modified law of uniform seniority preserved - although, it is true, a form which is of little practical use,* - but an interesting relation- ship exists between the constants с and n. This relationship is easily determined by simple analysis : thus, if the basic values of the force of mortality are summed for groups of 15 ages, say, 35-49, 50-64, 65-79 and 80-94 and the respective differences between the group totals represented by a, ß, and 7, then it is readily seen that we must have /tf - etc10' If we write с for n this expression becomes a quadratic in cVo ; in other words, for two values of с the formula fix=A + 'BdB+Mnx resolves itself into the ordinary Makeham form /ха.=А+Всж. Moreover, it will be seen that if n = 0 or n=l two further values may be obtained for с such that the formula assumes the ordinary Makeham form. For four values of c, therefore, we can assume that if the ordinary Makeham formula gives an indifferent fit so also will the formula fix= A + Bc^+Mn*. It has been found by experiment, however, that the best results are obtained when the values of с and n are very nearly equal - in other words, when they lie close to the roots of the quadratic above mentioned - and very reasonable fittings may be obtained over the greater portion of the A1924-29 table. When low values are given to с and n the least satisfactory portion of the curve is in respect of the early ages, whilst, on the other hand, if high values be given to с and n the later part of the table leaves most to be desired. * vide The Theory of the Construction of Tables of Mortality, G. F. Hardy, p. 68. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 112 joint Life Annuities - Other types of formulae were dealt with, but none of these involving five constants or less gave as satisfactory results as those yielded by the valuable formula of Mr. Perks already referred to in Section I., and no good purpose would therefore be served in dwelling on these alternative types. In conclusion a practical point should perhaps be mentioned. Past investigations have shown that on account of some species of adverse selection (particularly where husband and wife are concerned) joint life assurance mortality has proved heavier than comparable single life mortality. Those who propose to employ the A1924-29 table for the calculation of joint life assurance benefits will no doubt keep this point in mind. We wish to take this opportunity of recording our indebtedness to Mr. D. W. A. Donald, who has given much valuable help in connection with the numerical work involved in the experiments undertaken. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-2 9 Table 113 APPENDIX I. Table I. Comparison between /% as obtained by the formula of p. 100 and the official A1924-29 ult. values. AGE A of *** * 1()5 Ia* X 105 Vm' Biff AGE A aw ^x X 1()5 ** X 10* Diff im' AGE A of formula A1924-29ult. Vm' Biff AGE A aw formula A 1924-29 ult. Diff im' 25 225 235 -10 65 3074 3084 -10 26 229 235 - 6 66 3405 3405 27 233 235 - 2 67 3772 3779 - 7 28 238 235 3 68 4178 4202 -24 29 244 237 7 69 4629 4676 -47 30 250 239 11 70 5127 5195 -68 31 257 243 14 71 5677 5760 -83 32 265 249 16 72 6283 6369 -86 33 274 258 16 73 6951 7023 -72 34 284 268 16 74 7684 7724 -40 35 295 279 16 75 8487 8478 . 9 36 307 294 13 76 9363 9292 71 37 320 311 9 77 10320 10171 149 38 336 330 6 78 11357 11120 237 39 353 353 .. 79 12481 12141 340 40 372 377 - 5 80 13690 13240 450 41 393 - 402 - 9 81 14986 14418 568 42 417 427 -10 82 16366 15671 695 43 444 453 - 9 83 17823 17000 823 44 474 481 - 7 84 19344 18409 935 45 506 512 - 6 85 20905 19896 1009 46 544 546 - 2 86 22468 21463 1005 47 586 584 2 87 23976 23112 864 48 632 629 3 88 25359 24846 513 49 684 678 6 89 26548 26670 - 122 50 742 736 6 90 27520 28588 -1068 51 807 80Ó 7 91 28362 30604 -2242 52 879 871 8 92 29332 32725 -3393 53 960 951 9 93 30872 34955 -4083 54 1050 1040 10 94 33511 37302 -3791 55 1151 1141 10 95 37724 39774 -2050 56 1263 1256 7 96 43836 42380 1456 57 1389 1387 2 97 51989 45132 6857 58 1529 1538 - 9 98 62238 48042 14196 59 1685 1707 -22 99 74563 51124 23439 60 1859 1894 -35 61 2053 2094 -41 62 2269 2308 -39 63 2509 2541 -32 64 2777 2795 -18 This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp г 14 Joint Life Annuities - Table II. Comparison of actual deaths with expected deaths by the graduation described on p. 100. A1924-29 ult. Age Actual Expected Actual-Expected Group Deaths Deaths Deaths + 2Ц-2Ц 1505 1556 51 ЪЦ-ЪЦ 2325 2454 129 Щ-Щ 3725 3811 86 Щ-ЬЦ 5733 5500 233 Щ-Щ 7618 7774 156 ЪЦ-ЪЦ 9953 10030 77 ЪЦ-ЪЦ 11801 11733 68 Щ-Щ 12491 12257 234 Щ-Щ 13526 13659 133 70^-741 15607 15447 160 ЧЪ'-ЧЦ 14159 14435 276 80|-84¿ 9183 9535 352 8б|-89| 3919 4006 87 90|-94| 999 931 68 95|-99| 122 165 43 Total . 112,666 113,293 763 1390 Table III. Comparison of annuity values calculated according to the formula on p. 100 with the official A1924-29 ult. 3% values.* Age . Value by Official lva. Age . formula - Value lva. difference 25 23-602 23-616 -014 45 17-521 17-520 -001 65 9-201 9-214 -013 85 3-037 3-063 -026 * In the case of single life annuities the value by means of the formula ax = F(x)Cbx + M(x)tt'a? + N(x)tt"x M M M is most readily obtained by calculating ax at the rate of interest required direct from the modified Makeham table, which was the method adopted in finding the above values. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table 115 APPENDIX II. Table IV. fix+n method. A1924-29 ult. Table showing values of n (see p. 106) to be used in conjunction with ¡ix+n method. Second Oldest Age. ; OldestAge* 40 or under 45 50 55 60 65 70 75 80 85 90 ; 40 or under 29 ! 41 28 42 27 43 25 44 24 45 23 23 46 22 22 47 21 21 48 21 20 i 49 21 20 50 22 20 18 51 22 20 17 52 21 20 16 53 20 19 16 54 19 19 15 55 17 17 15 14 56 16 16 14 13 57 15 15 14 12 58 14 14 14 12 1 59 14 14 13 12 | 60 14 14 13 12 8 61 13 13 12 11 7 62 13 12 11 11 7 63 12 11 11 10 6 64 11 10 10 9 6 65 10 9 9 9 5 4 66 9 9 9 8 5 4 67 8 8 8 7 5 4 68 7 7 7 7 4 4 69 7 7 6 6 4 4 70 7 765444 71 6 655444 72 6 555444 73 б 555443 74 б 555443 75 5 5554433 76 4 4444433 77 4 444443 3 78 4 4444433 79 4 4444432 80 4 44333322 81 4 44333322 82 3 33333322 83 3 33333222 84 3 33333222 85 3 333332222 86 3 33333222 2 87 3 333332222 88 3 333332222 89 3 322222222 90 3 3222222222 This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp nó Joint Life Annuities - Table V. nl>x method. A1924-29 ult. Table showing values of n (see p. 108) for use in connection with npx method. Second Oldest Age. Olde&tAge. 40orunder 45 50 55 60 65 70 75 80 85 90 45 or under 31 30 46 30 29 47 30 28 48 29 27 49 28 25 50 27 24 22 51 26 23 21 52 25 22 20 53 24 22 20 54 23 21 19 55 22 20 19 18 56 22 20 18 17 57 21 19 18 17 58 20 18 17 16 59 19 18 17 16 60 18 17 16 15 14 61 18 17 16 15 14 62 17 16 15 14 13 63 16 15 15 14 13 64 15 15 14 13 12 65 14 14 14 13 12 12 66 14 14 13 12 11 11 67 13 13 13 12 11 11 68 12 12 12 11 11 10 69 11 11 11 10 10 10 70 11 И И 10 10 9 7 71 11 11 11 10 10 9 7 72 10 10 10 9 9 8 7 73 10 10 10 9 9 8 7 74 9 998887 75 9 9988776 76 9 8877766 77 8 8877766 78 8 8777666 79 8 7766665 80 7 77666555 81 7 76666555 82 7 66655555 83 6 66555555 84 6 65555555 85 6 555555444 86 5 555444444 87 5 544444444 88 5 444444444 89 4 444444444 90 4 444444444 4 This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp ii6a Table VI. npx method. A1924-29 ult. Table of log lw to 4 decimal places* AOE АОБ it 17 w w it ^glw log^ 17 20 20 -9913 -9913 21 21 -9903 -9903 22 22 -9893 -9893 23 23 -9883 -9883 24 24 -9873 -9873 25 25 -9862 -9862 26 26 -9852 -9852 27 27 -9842 -9842 28 28 -9832 -9832 29 29 -9821 -9821 30 30 -9811 -9811 31 31 -9801 -9801 32 32 -9790 -9790 33 33 -9779 -9779 34 34 -9767 -9767 35 35 9756 9756 36 86 -9743 -9743 37 37 -9730 -9730 38 38 9716 -9716 39 39 -9701 -9701 40 40 -9685 -9685 41 41 -9669 -9669 42 42 9651 -9651 43 43 -9631 -9631 44 44 -9611 -9611 45 45 -9590 -9590 46 46 -9567 -9567 47 47 -9542 -9542 48 48 -9516 -9516 49 49 -9487 -9487 50 50 -9457 -9457 51 51 9424 9424 52 52 -9387 -9387 53 53 -9348 -9348 54 54 -9305 -9305 55 55 -9257 -9257 56 56 9205 -9205 57 57 -9148 -9148 58 58 -9084 -9084 59 59 -9014 -9014 60 60 -8936 -8936 61 61 -8849 -8849 62 62 8754 8754 63 63 -8649 -8649 64 64 -8533 -8533 65 65 -8405 -8405 66 66 -8265 -8265 67 67 8109 -8109 68 68 -7936 -7936 69 69 -7743 -7743 70 70 -7529 -7529 71 71 -7291 -7291 72 72 -7028 -7028 73 73 6737 -6737 74 74 -6417 -6417 75 75 -6066 -6066 76 76 -5680 -5680 77 77 5258 -5258 78 78 -4796 -4796 79 79 -4291 -4291 80 80 -3740 -3740 81 81 -3140 -3140 82 82 2486 -2486 83 83 1777 1777 84 84 1009 1009 85 85 0177 0177 86 86 - 0721 -0721 87 87 - 1688 -1688 88 88 -2729 -2729 89 89 -3848 -3848 90 90 -5047 -5047 91 91 -6332 -6332 92 92 -7707 -7707 93 93 -9176 -9176 * For convenience the published values of log lx have been replaced by log 1XX 10 ~6. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table 117 APPENDIX III. Those familiar with the late George King's Text Book Part IL will remember that the vie probable in respect of any age x represented the period of years which would elapse before a body of lives, all aged ж, were reduced to one half the number ; in other words, represent- ing the vie probable by (vp)x one may write (vp)x=n where npx=h The vie probable often closely corresponds in value to the expectation of life, although in most tables (vp)x is greater than ex in early life and less than ex in later life. The difference in value between the two functions in any well-graduated table progresses in a smooth manner. The problem considered was to find an age w such that exy.,.(m)=eww...(my It seemed a reasonable assumption that a close approximation might be obtained by employing age w obtained from the relationship {vp)xy . . .(m)=(vp)ivw -(my The value of w is readily obtained by inspection from a table of log lw bearing in mind that loglw-loglt0+n=-'og 2 Ttl the value of n having been obtained by means of the relationship log lxy... (m) - log 2 = log Hxy . . . (m). Example. - For what age w does (vp)54.66 = (vp)U4Vrl log¿ñ4 = 6-9305 logZw= 6-8265 13-7570 log 2 -3010 13-4560 = log ̂ 64.66 whence, by inspection of a table of log lx, n=9'3l logZ61 = 6-8849 logZ62=6-8754 ¿(log 2)= -1505 ¿(bg 2)= -1505 6-7344 = log >ЧСЛ 6-7249 = log HG2 whence w = 9-78 whence n = 9*16. .*. by interpolation the value of w corresponding to n=9*31 is 61-76. The method of utilising the vie probable in calculating Joint Life Annuity values based on the A1924-29 table was tested and the results were generally satisfactory. It was found, however, that more satisfactory results could usually be obtained if instead of employing the value n where nPxy-(in)=i the value of n were used where nPxy^imy^h This prompted a further investigation This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp ii8 Joint Life Annuities - into the matter, but the subject is perhaps not of sufficient general interest to warrant any further treatment here. It should be noted, however, that what has been described in the body of the paper as the npx method is in reality a further extension of the vie probable method. That value of n appearing in Table V. in respect of any particular combination of ages was selected to give the best result, and for that value of n, the function nPxy-(m) will have a value, not necessarily J as in the vie probable method nor £ as mentioned above, but a fraction of varying amount depending on the relative ages of tbe lives. It will generally be found, however, that the fraction will not differ greatly from the value J. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al924r-29 Table 119 SYNOPSIS. The two main objects of the paper are : - (1) To ascertain what modifications in Makeham's Law are necessary in order that the principle of Uniform Seniority may be applied to the A1924-29 ult. table. (2) To ascertain whether some approximate method is readily available for the calculation of joint life annuity values according to the A1924-29 ult. table. As regards (1) developments of the suggestions put forward in the discussion following Mr. С L. Stoodley's paper to the Faculty in February 1934 are examined, and the conclusion is reached that, while the results are somewhat disappointing so far as the A 1924-29 table is concerned, some of the formulae brought out may yet prove of practical use when applied to a mortality table more free from heterogeneous matter. In the second part of the paper the results of applying to the Al 924-29 ult. table several standard approximate methods are first examined, and it is shown that such methods are liable to produce systematic errors when applied to this table. Two new methods of approximation are then developed, which are shown to yield much more satisfactory results, and tables are reproduced to facilitate the practical application of these methods. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 120 Joint Life Annuities - DISCUSSION. Mr. R. LI. Gwilt. - It is indeed a pleasure to have the privilege of opening the discussion on a paper of such value and interest as the one which has been submitted to us this evening. The authors are to be congratulated on the skill with which they have achieved the objects they set out to attain, and I do not think I can pay any higher tribute to the delightful lucidity with which the results have been presented than to assure Mr. King and Mr. Reid that they misled me into supposing that I had obtained a clear grasp of the whole subject after a first reading. A second reading, however, has brought home to me more fully how much food for thought they have provided, and I look forward to studying at my leisure the details of the paper and the implications of the various formulae. Before I comment on the experiments the authors describe in their paper, I think I should explain that when I was asked at very short notice to open the discussion this evening I had not yet had an opportunity of reading the paper. That being so, it was with the greatest hesitation and diffidence that I consented to do so, and I hope the authors will forgive me for venturing to contribute to the discussion without having been able to devote to their paper the time and study which it so greatly deserves. In the circumstances, my remarks must of necessity be brief and of a superficial nature. I shall not trespass for long on your time, which will be much better spent in listening to remarks of the other speakers who have perhaps had more opportunity of studying the paper. One has but to turn to page 94 of thè paper and examine the comparison of fourth differences of mortality functions by the new table and the 0M(5) table to realise the difficulties witli which the authors have had to contend. The table on page 110 shows how successfully these difficulties have been overcome. That these results were possible does, I think, suggest that at least some of the advantages claimed for a table based on a modification of Makeham's Law are more illusory than real. Most actuaries will regard the approximate values of joint life annuities obtained by the authors' npx method as sufficiently accurate for practical purposes, and I personally should prefer to calculate the value of a four-life annuity by the authors' method rather than to work out the value by one of the elegant but lengthy blending formulae which would be available under one of the modified Makeham tables. At the same time, a table following some practicable modification of Makeham's Law of the type discussed in the paper possesses advantages other than those connected with the calculation of the values of joint life annuities. Unfortunately - and here we must sympathise with them - the authors have been unable to find a modification which would simplify complicated calculations and would at the same time give a satis- factory fit over the whole range of the A1924-29 table. We shall all, I am sure, share their hope that the new methods will prove of real practical value in the future. The underlying idea is most attractive, and the possibilities are considerable. Briefly, the proposition is that commencing with a basic Makeham table we can represent various different mortality curves by modifying the formula for ¡xx in such a way that single life annuity- values on the modified table can be expressed as a blend of two or more single life annuity values calculated on the basic Makehanl table. By This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp S onte Experiments with the A 1924^29 Table 121 changing the constants in the adjusting term added to or deducted from the basic fix we can produce a large number of different mortality tables for special purposes - perhaps, for example, to express certain types of extra risk - and monetary functions on those modified tables can be readily obtained from the basic Makeham values. May I express the hope that the possibilities of the method in this and other directions will be more fully developed at a later date. An interesting feature of the first modification discussed, namely, the deduction from the formula for ux of the term n °&еП is that however the m + nx basic table may be graduated the blending method will apply to any table which can be represented by modifying the basic fix in this way. The modifications of Makeham's Law leading to the blending method of calculat- ing annuity values are of most practical value when for single life functions the blending involves only two values. The cases of this type dealt with in the paper are of great interest. Among the possibilities we have the particular case of Perks's formula mentioned on page 99 which, in the special T> circumstances when logec= A - -, leads to the result that single life annuity values can be represented as a blend of two perpetuities at different rates of interest. Unfortunately, when the constants satisfy this condition the formula will not satisfactorily represent any normal mortality table. Secondly, we have the modification u - n °%e n, which for single life annuities leads x m + nx to a blend of two annuities on the basic Makeham table for the same age, but at different rates of interest. For this case two, three and four-lite annuities can be expressed respectively as blends of three, four and five Makeham joint life annuities. Then there is the case discussed on page 98, which for single life functions leads to a blend of two annuities on the basic Makeham table at the same rate of interest but for different ages. For joint life annuities, however, the results are more complicated than in the case where for single life annuities we can use the same age and different rates of interest. The expressions for two, three and four-life annuities involve, respectively, four, eight and sixteen Makeham joint life annuities. In reading through the paper it seemed to me that it would be of interest to consider a further case in which the single life annuity value on the modified table might be represented as a blend of two annuities on the basic table at different rates of interest, and also at different ages. An expression for цх which would satisfy this case is A + Be» - r*n *(cX loge n loge c + lo& r' The corresponding expressions for lx and ax are lx = ]cs*gcX (m + rtn**) m rxncX , and ax = - ^ ax -' --- ax+h , m + rxnc m m + rxnc M where the expression for h is the same as that given by the authors on page 98 and a* is at a rate of interest - - 1. I have no doubt this modification has been considered by the authors. The modification they investigated giving a blend of annuity values on the basic table for two different ages was found to be impracticable, as the value of h This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 122 Joint Life Annuities - would have been imaginary for values of the other constants required to fit the experience approximately. Perhaps this would also be the case for the further modification 1 have mentioned, but I have not had time to consider the formula in detail. I shall be most interested to hear the authors' comments on it. The very useful Section II. of the paper will be of great practical value,, and we shall be deeply grateful to the authors for providing, in advance of the publication of the joint life tables, not only a satisfactory approximate method of calculating the values of multiple life annuities, but also the necessary tables for applying the method. Mr. 0. S. Penn. - In what is, I believe, known as the "placer" method of finding tfold, the debris from the bed of a stream is collected together and washed with great perseverance in a manner which disposes of the unessential parts of the debris and leaves the gold behind, generally in minute particles and dust, but sometimes with a small and rarely with a considerable nugget. My technical knowledge is not sufficient to say whether this is an accurate description of the method, but if I may be permitted to use the analogy, Messrs. ' King and Reid seem to have done a very great deal of washing, much of it, perhaps, without results of immediate practical value, but they have persisted until they have found a very real and substantial nugget which is described in Section II. of the paper. I am sorry that the short time available since the paper came into my hands, and the pressure of other circumstances, have prevented me from giving their method more than a very perfunctory trial ; but it seems to me likely that it will fill a most useful function, since the approximate values appear in most cases quite sufficiently accurate to justify its use as against the more laborious method of the summation formula in computing values of three and four-life annuities. If the authors had succeeded in their original aim of fitting a curve of the modified Makeham type to the A1924-29 experience, they would indeed have achieved a miracle in view of the nature of the data, and I venture to suggest that their own feelings would not have been unmixed with surprise. Whether or not they regarded Section II. of the paper as a sideline, it seems to me they have every reason to congratulate themselves on the results they have obtained. There is one point that has occurred to me. I do not know how far it may be possible - there may be practical obstacles in the way - but it seems to me it would be very useful to have Tables V. and VI. at any rate, and possibly also Table IV., printed and added to the official joint life tables which are still in the press. Mr. A. R. Davidson. - I have read this paper with the greatest interest. I know at least one of the authors has been for many years interested in the subject of joint life annuities and how to calculate them, because under his instruction I used to calculate joint life annuities by means of the method which is described in this paper as the ixx+n method. You think of a number and add it to the ages, and so on, and I always felt a certain amount of surprise when my approximations came out so well ! But in any approximation there must, of course, be a certain amount of skill on the part of the approximator. Or, it may be low cunning ! In any case, I have a great affection for that ¡xx+n method, and I was very sorry when I s.iw that the authors thought it necessary ta have something more exact. They have introduced the npx method, and they have supplied the means of applying it to the new mortality table. I should have thought that the approximations on the ¡¿х+п method which they have obtained in the paper were very good approximations considering the circumstances, but possibly something more exact is necessary. I do not This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the A 1924-29 Table 123 myself quite see why ; because, after all, the new mortality table certainly does not represent joint life mortality, whatever mortality it may represent. I think the old fix+n method is probably sufficient for ordinary work, and certainly it seems to me to be just a little more elegant, and certainly a little more generally applicable, than the other method. For accuracy, however, the npx method is a wonderful one as applied to this table. I gather that before you apply it to any other table you will require to calculate some- thing corresponding to Table V. again, so that that makes it a method which will require the attention of the authors or some other energetic and ingenious persons whenever a new mortality table is published. One cannot but sympathise with the authors, because when this new mortality table came out one felt sure that they could hardly refuse to accept the challenge in the matter of joint life annuities, and really it must have been a very difficult matter for them. If I had been given the Life Office data and allowed to leave out what I wished and put in what I wished, and to move the deaths from one age to another, I could have prepared something that would have puzzled the authors even more than this table ! The authors have, however, in spite of the difficulties with which they have been faced, brought forth a method which produces most wonderful approximations, and which can be simply enough applied to any new table that may be published by calculating Table V. Towards the beginning of their paper the authors refer to the question of forecasting mortality, and therefore, perhaps, I shall not be blamed, and perhaps there may be some excuse if I say that I have not yet got it quite clear in my mind why anyone should wish to know the value of a joint life annuity - or any annuity for that matter- on the basis of the present table. The table seems to me to represent no consecutive life mortality, which is the only thing of any importance, either past, present or to come, and if the authors had addressed themselves to the problem of finding the present value of an annuity which was to be calculated according to their best estimate of the mortality which is going to be experienced in the future, they would not, I think, have been faced with anything like the difficulties which they have encountered in connection with the new table. You will excuse me interpolating that remark, but I have not made a speech at the Faculty for years without saying that ! Before I sit down I should like to say what great pleasure it has given me to read the paper and to see the determination and the skill with which the authors have gone through with so many experiments to bring out something so very successful and really so beautifully effective in the end. I think it is a master stroke to have been able to find such a very exact approximation of joint life annuities on this table, a table which is really a very difficult table upon which to base approximations, for the reason mentioned by Mr. Gwilt and by the authors. I think the new method pla-es a great many of the difficulties of the calculation of joint life annuities finally behind us, and I trust that the Institute and the Faculty when they publish their next table, which will no doubt be of a similar nature to the one which has just been completed, will not be able, in spite of their best efforts, to provide a problem in the calculation of joint life annuities which it will be beyond the ingenuity of the authors to solve. Mr. C. L. Stoodley. - I am glad of the opportunity to make a few remarks on the interesting paper before us to-night. The first section of the paper has a particular appeal to me not only by reason of the fact that, in a sense, I can claim a parental interest in some of the formulae evolved, but also, and principally, because I am a firm believer in the doctrine that the force of mortality is, amongst other natural forces, capable of expression in a simple mathematical form. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 124 Joint Life Annuities - The authors' attempts to fit a curve to the A1924-29 official values were made in the full knowledge of the inherent difficulties of the task, and it is somewhat of a problem in the circumstances to decide whether they merit more commendation on the tenacity they have displayed in their search for a formula which, above all other things, should preserve the principle of uniform seniority, or on the ingenuity they have shown in devising formulae to this end. During the comparatively short time at my disposal I have been able to do little more than consider in a general way the various formulae and suggestions put forward by the authors, and although I shall have certain concrete suggestions to make and results to submit in the course of my remarks, much that I am about to say must, of necessity, be in the nature of conjecture and is meant only to indicate possible lines of approach to the general question of fitting curves of the type described by the authors. The various formulae for 'ix consist of one or other of the Makeham expressions and deductive terms whose indefinite integrals are the logarithms of their respective denominators. In the case of both the root 5 constant formula and its extension to a 7 constant formula, the Lidstone graduation by Makeham over the range up to 65 has been adopted and the authors have confined their attention to the fitting of the deductive term to the differences between the basic Makeham values and the official values beyond that age, and indeed suggest on page 97 that the problem could be narrowed down to the mere fitting of this deductive term. While it is true that the fitting of this deductive term is part of the problem it is by no means the whole, and I must confess that I am at a loss to understand why the Lidstone constants giving a fit up to age 65 have been so rigidly retained in the fitting of the two formulae under discussion. Each of these formulae should, I suggest, be considered as a whole, since it appears to me that the crucial problem is to ascertain whether a series, to which the deductive term can be fitted, can be formed of the differences between the official values and any Makeham series. It is not necessary that the basic curve should coincide with the "official" curve over any section or make contact on any point, and, in fact, the Makeham constants may, within certain limits, be arbitrary. Further, it is not imperative that the Makeham curve and the "official" curve should converge at the early ages. The two curves may converge at the extreme ages, and for an example of this type of fitting I would refer to my reply to the discussion on my last paper in which I demonstrated that the Ow and OM(5> /a's which coincide from age 85 could be related by a deductive term of the form °£e n • ™ . I do not suggest that by working on the lines I have indicated it is ipso facto possible to secure a better fit than that obtained by the authors. Had time permitted I would have pursued the question beyond the theoretical stage even in the face of the knowledge of the vast amount of experimental work necessary in the method of " trial and error " by which alone these modified Makeham curves can be fitted. I did, however, investigate the particular case of the difference series formed by assuming the following Makeham constants : - A ='00185 В = -1)0002018 log10 с = '05 and the results may possibly be of interest to the meeting. Efforts to fit the authors' deductive term 0&еП'п proved abortive and, m + nx although the analogous form of - ̂ - '- offered greater potentialities of This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the A1924-29 Table 125 üt, it was not pursued to any extent as it led to the unsuitable form of -^Lfork m-nx Further investigation disclosed that the difference series was approxi- mately in geometrical progression and indicated that a curve of the form A + Bcx-Mnx would provide a reasonably good fit, and with values of M = -00000023388 and Iog10w= Ю68484 this was found to be so. To give the meeting some idea of the degree of fit in comparison with that secured by the authors' 7 constant formula I would state that the net accumulated deviations at the quinquennial ages 25 to 95 inclusive are -01123 compared with - Ю1741 by the authors' formula, whilst the accumu- lated square deviations are in the ratio of approximately 100 to 178. This particular form of A + Be* + Mnx for 'ix is mentioned by the authors in the conclusion to their paper, but they do not give any indication that in their experiments they considered the term Mnx in a deductive sense. While it is true that the law of uniform seniority does not hold, except in a modified way, in the case of a table graduated by this formula, the formula does, nevertheless, appear to offer in conjunction with the methods given in Section II. of the paper, facilities for the calculation of contingent functions. Turning now to Section II. of the paper. In a sense it is a boon that the •authors were unable to fit their modified Makeham curves to the A1924-29 table, since had they been able to do so, the profession might not have seen the interesting new npx method of evaluating multiple joint life functions. From the examples given by the authors it cannot be disputed that their method gives remarkably good approximations to the values obtained by approximate integration methods, and the only possible criticism is that, following the sustained elegance of the first section of the paper, the method might possiby be considered a little cumbrous in operation. Elegance of expression is, however, not necessarily synonymous with •accuracy of result, and while the results obtained by the method I am about to outline may be considered sufficiently accurate for all practical purposes, they are not so good as those by the method submitted by the authors. My •only justifications for bringing the method to the notice of the meeting are, firstly, that the method involves the blending principle pursued by the authors in the first section of the paper, and, secondly, that I am of the opinion that when the method is applied toa table which does not follow Makeham's Law but which has a greater claim to a smooth graduation than the A1924-29 table, very satisfactory results will be obtained. Now in the case of a table graduated by Makeham's Law, the value of /uw where w is the equivalent equal age is the arithmetical mean of the con- stituent ft's. The authors have established that the application of this relationship to the A1924-29 table does not produce satisfactory results, but from an inspection of the deviations, it occurred to me that it might be possible to apply a series of weights to the constituent p's and by dividing by the sum of the weights obtain the p corresponding to the equivalent equal эде. Attempts to express these weights in a simple mathematical form proved fruitless in the case of the A1924-29 table, and the only possible method of obtaining them appeared to be by extensive solution of equations based on the official joint life annuity values. In the case of the status 20 : 60 the /z corresponding to the equivalent equal ages is -00998, whence, assuming a unit weight of 100 in respect of the older age, we have 100/x60 + Кд20 = (100 + К) -00998 whence К the weight in respect of fi2o= 117, VOL. XV. I This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp I2Ó Joint Life Annuities - Applying these weights in the case of the three-age status 20 : 20 : 60 we obtain '00732, which corresponds exactly with the "true" value. Other results are as follows : - Status Method "True" 20:60:60 -01282 -01291 20 : 20 : 20 : 60 -00603 -00604 20:60:60:60 -01428 -01443 This brief outline will, I hope, convey some indication of the principle of the proposed method. In concluding my remarks may I say how much I have enjoyed the paper. Not only is it of inestimable practical value, but should prove a source of inspiration and example to the profession. Mr. K. K. Weatherhead.- I did not expect to be called on to speak, and I am afraid I have not had time to look into the paper very thoroughly. It occurred to me, however, on reading it over, that the authors may have limited themselves unnecessarily by making comparisons with the official graduated table instead of the rough data and by adopting Mr. Lidstone's constants for their basic Makeham table. With regard to the problem of finding values of ахуг, from the purely practical aspect the values are required either for the purpose of calculating premiums or for valuation, and it may be that the old-fashioned Simpson's rule, with the modification suggested by Milne, gives sufficiently accurate values. As the necessary two-life values were not available I could not make any tests, but on many of the existing tables quite reasonable approximations are obtained. I was very interested to learn from Mr. Penn that the joint life annuity values were " in the press." I have been wondering for some time where they had got to. Mr. A. M. M 'Allist er. - The authors have confined their investigation to joint life annuities payable throughout life, and it may therefore be of interest if I give a few figures relating to temporary annuities.* Temporary annuities will not appear in the published volume, and if in addition we are not given commutation functions for unequal ages it may be thought desirable to- adopt some approximate method of arriving at these annuities for valuation and other purposes. The number of cases investigated was, of course, too- small to allow of any definite conclusions being reached as to the limitations of the methods, but the results may indicate to some extent the suitability of the methods for practical purposes. The figures have been based on the new mortality ultimate experience 3%. In the third column of the table are given the "true" values. In the fourth column are given the approximate values arrived at by entering conversion tables inversely with the endowment assurance net premium obtained by means of the equation * xyn' = * xn' + ±yn' ~~ -t nl- The results of the whole may be regarded as satisfactory. The differences between the true and approximate values range from "002 to *012 in the * See table on page 127. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the A1924-29 Table 127 15-year group, from *003 to 4)14 in the 19-year group, from 4)02 to 4)24 in the 24-year group. In the 36-year group the differences become rather large, and it would seem advisable to restrict the use of the method to a shorter term depending on the maturity age. Approximate values were also calculated by reference to temporary annuities for equal ages. In the first case the equal ages were derived from the average value of the /¿'s half- way through the annuity term. In the second case the npx method was used as described on page 108 of the authors' paper, the value of n being either the annuity term or the value given in Table V., whichever were the smaller. The differences between the true and approximate values were irregular in the case of the ц method, but the differences in the npx method compare favourably with the corresponding figures under the Vxyñ' method and are superior to the latter in the case of the very long term annuities. Both the equal age methods depend on the use of temporary annuities and may not therefore be regarded at present of much practical use. The final method tried was Gompertz as described on page 272 of the Text Book. The results, however, do not compare well with those obtained by other methods. ах„п' : A1024-29 ultimate 3% Term of д " True " Рж„п| м- nPx Gompertz years A«es Value Method Method Method Method 15 21/30 11-478 11-480 .. 11-484 11-476 21/42 11-209 11-211 11-233 11-215 11-197 21/51 10-642 10-649 10-639 10-649 10-616 21/60 9-332 9-344 9-385 9-359 9-316 30/42 11166 11169 11173 11-158 30/51 10-603 10-610 10-590 10-602 30/60 9-301 9-313 9-345 9-317 42/51 10-366 10-377 10-359 10-375 19 27/33 13-505 13-502 13-515 13-508 13-490 30/33 13-460 13-464 13-464 13-459 13-460 33/39 13177 13-184 13174 13172 13165 33/42 13-016 13002 12-997 13003 12-971 42/48 12-164 12174 12157 12170 45/48 11-970 11-982 11-967 11-971 24 21/30 15-841 15-845 15-847 15-839 21/42 15059 15065 15085 15064 21/51 13-553 13-557 13-549 13-584 „ 21/60 10-834 10-845 10-805 10-878 30/42 14-911 14-915 14-883 14-905 14-799 30/51 13-437 13-433 13-407 13-457 13-324 30/60 10-766 10-764 10-705 10-798 10-707 42/51 12-881 12-857 12-917 12-899 36 21/30 19-372 19-370 19-402 19-381 21/51 14-624 14-570 14-627 14-645 30/42 17-060 16-963 16-977 17-048 30/51 14-425 14-270 14-410 14-440 42/51 13-579 13128 13-542 13-598 12-933 This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 128 Joint Life Annuities - The Secretary then read the following communication from Mr. G. J. Lidstone, LL.D. - It is always a pleasure to listen to Mr. King and Mr. Reid, and when they put their heads together we may expect something specially good. We have certainly got it to-night, for the paper is exceedingly interesting and instructive, and written with the most admir- able clearness. Looking a little below the surface we can see that it must have involved a prodigious amount of work, and though they make little of that we must be grateful to them for it. We must also be grateful that they have given us their comparative failures as well as their successes. Most research- workers have their failures, " And if they should be written every one, I suppose that the world itself could not contain the books that ¿should be written " : but in this case the authors' less successful experiments are full of interest and instruction, and as they point out the methods suggested may well be more successful as applied to some more tractable experience. Actuaries of my generation have perhaps been a little spoiled by working for many years with the Нм (Text Book graduation), (W5) and CM select tables which followed Makeham's Law and gave us all the advantages of a perfectly smooth mathematical graduation and all the practical conveniences that follow from that particular law. It is perfectly clear that the A1924-29 ultimate table does not follow that law through the whole of its course, and so we cannot at present enjoy the whole of those advantages. But in this connection I may perhaps be allowed to repeat some remarks which I made (T.F. 4., x. 339-41) in discussing Dr. Buchanan's valuable "Notes on Graduation." I said : The modern trend of actuarial work is markedly in the direction of the introduction of mathematical approximations - interpolations, ¡ approximate integrations, and group methods of valuation, etc. These depend largely for the goodness of their results on the mathematical smoothness of the Life Table. ... In such matters there are great advantages if the tables can be adequately repre- ! sented by a continuous mathematical formula, even apart from such ' advantages as the method of uniform seniority which is available ' when Makeham's Law can be used. ; I am not for a moment suggesting that the official table is ill-graduated, and I think the authors' test by fourth differences, on page 94 of the paper, is perhaps a little severe. But when I consider (I) the way in which the data were collected, and the known careless- ; ness of some of the offices in making their returns ! (2) the possible disturbances introduced by the Census method, : especially in association with the forcible truncation of the period i of selection after three years ! (3) the increased roughness due to the inclusion of duplicates, and the I possibility that the very low mortality at the highest ages may be partly due to the inclusion of unclaimed paid-up policies in the : existing, and their exclusion from the deaths (4) the heterogeneous character of the data and the mixture of different classes of assurance (5) the fact that few, if any, of the offices will find that the table exactly respresents their own experience - I when I consider these facts, while I welcome the new table as an up-to-date working tool, I cannot think that its shape is" so exactly defined that anything would have been lost by a little grinding-down in order to produce a sharper mathematical edge. I should have used with equal confidence either the authors' graduation given in Tables I., IL, and III., or Mr. Perks's admirable graduation by a simple formula of the Perks family. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the A1924-29 Table 129 I think perhaps that in their experiment (1), pages 96 and 97, the authors may have handicapped themselves by following too closely the values of the Makeham constants which I used for illustrative purposes in my recent paper on the application of the Z method to the new table. For that purpose I had only to consider the tract of ages 20 to 65, and even that I fitted by rather rough methods, adequate for that particular purpose. I think the constants I used, though satisfactory over the tract of ages with which I was concerned, might probably produce too high values at higher ages, and that may have thrown too great a strain on the authors' supple- mentary curve. Generally I think that where we have to force a fit on rather unwilling material it is important to use rather elaborate methods such as moments or least squares. And since an error of given amount is more important at the younger ages than at the highest, I am inclined to wonder whether it might be better, in such cases, to apply the method of least squares to the logarithm of /x or qx. I think the effect of this would be to make a minimum of the squares of the ratios of the deviations to the true values. I will try to make some remarks on the second part of the paper in the printed discussion. The Chairman (Mr. Hugh W. Brown).- Before I ask the authors to reply to the discussion, I am sure you will wish to thank them very cordially for the admirable paper they have submitted to us to-night. They have dealt with their subject very thoroughly. They have made a co-ordinated survey of the possibilities of the new table in regard to the calculations of joint life annuities. As a result of persevering and ingenious experiments they have discovered that approximate methods of calculation are available. As has been said before, the modern trend in practical actuarial work is towards short and therefore approximate methods. While it may be regretted that the new table has proved to be of such a nature that it is not practicable to apply the principles of Makeham's Law of uniform seniority in this connection, the authors have succeeded in providing us with a valuable instrument to arrive at reasonable actuarial values of joint life annuities in that table. I ask you to join with me in offering our very grateful thanks to Mr. King and Mr. Reid for their paper. Mr. A. R. Reid, in replying to the discussion, said : The discussion has been a most interesting one, and I should like to deal very briefly with one or two points which have arisen. Mr. King also has some remarks to make, and it may be that on fuller consideration we shall wish, with your permission, to make a written reply to some of the points which have been raised, but in any event a few remarks here may not be out of place. We were very much indebted to Mr. Gwilt for his contribution to the discussion. He made a suggestion of a further experiment based on the expression for lx given in Section I. (3) of the paper, indicating that another factor might be added, giving as a result lx = ksxgcX(m + rxncX), which ex- pression, being a blend of those given in Sections I. (2) and I. (3) respectively, might serve the purpose. I may say, however, that we did use that formula and had great hopes of it, but unfortunately it does not seem to work. It is difficult to say shortly what happens, but the function rx seems to try to- shoulder the whole burden itself and gives an absolutely impossible rôle to the function ncX. Mr. Stoodley, in his important contribution to the discussion, raised a point (also mentioned by Mr. Lidstone in his written remarks) with refer- ence to the adoption of the Lidstone graduation for the earlier part of the table and the fitting of the deductive term in the form of 'ix to the residue. I admit that the paper does rather suggest that we tried that and nothing This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 130 Joint Life Annuities - else, but actually, although only five results are shown there, I should think that we had at least twenty-five different attempts, and it would have been out of the question to put them all into the printed paper. The ideal would be to try to fit the expression for /аж or for lx as a whole to the data, but in view of the rather complicated nature of the expressions we adopted the principle of fitting in two stages. There are two different ways of doing this. You may tit the Makeham expression for ¡ix to the »later part of the table, and subsequently try to fit the supplementary term to the resulting residues thrown off at the earlier part, or conversely fit the Makeham expression to the earlier part and deal separately with the residues thrown off at the older ages. We found that by far the most successful results were yielded by the second process. Further, we tried several different values for the А, В and c, of the Makeham form, but after extensive experi- ment it was found that the constants adopted by Mr. Lidstone were the best. Mr. Lidstone states that they were calculated rather roughly for a particular purpose, but as one might expect coming from such a source they do seem to be the best. With regard to Mr. Stoodley's remarks on the double geometric formula mentioned on p. Ill of the paper I may say that we carried out about a dozen graduations based on that formula for different values of с and n, one of these involving very nearly the same constants as mentioned by Mr. Stoodley. It might perhaps be of interest to show graphically the general relationship between с and n involved in the equation given in the middle of p. 111. This may be done most readily by plotting the values of log10n against those of log10c thus : - LoglOn j jl •05 • ' ^* This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp S onte Experiments with the A 1924-29 Table 131 It will be seen that the pair of values adopted by Mr. Stoodley, viz. Iog10c = *05 Iog10n = *068 approximately satisfy the relationship represented by the graph. Other points of interest to note in connection with the diagram are ( 1 ) that there are certain real values for log10c for which there are no corresponding real values for log10n and vice versa, and (2) that there are two points where log10c and log10u are equal. Mr. Stoodley has compared the graduation by his double geometric formula with the graduation given in Tables L, II. and III. of the paper, but it should be remembered that we did not put forward the graduation referred to as a specially good graduation of the data, but merely u as an illustration of the theory." (See p. 100.) With Mr. Davidson's remarks on the form which should be taken by an ideal mortality table, of course I am in entire agreement, but at the same time the ideas put forward in the paper, I think he will admit, are an attempt to ascertain the results which can be secured by mortality tables in their present form. We were glad to have the results of the temporary annuity values obtained by Mr. M'Allister. Close approximations, 1 think, may be expected if a method which yields good results for annuities throughout the whole of life be applied to temporary annuity values. We will examine Mx. M'Allister's results with much interest. Mr. A. E. King, in replying to the discussion, said : I have been very interested in listening to Mr. Lidstone's remarks, and I look forward to seeing them in print, together with the further material he proposes to put in. Mr. Davidson asks: "Is it necessary to have both the fj.x+n and npx methods ? " Well, one gets a certain amount of comfort from a cross check, and if I, in practice, wanted a four-life joint annuity I should ask one assistant to use the /xx+n method and another the np~ method. Both methods, I think, properly applied, will give fairly good results, and one can, as I have said, gather comfort if practically the same answer is arrived at from two different sources by two different methods. That, I think, is the justification for giving two methods. Mr. Reid has dealt with Mr. Stoodley's important contribution to the discussion. I would like to say a few words about the method he has suggested for finding joint annuity values by weighting the values of px. This is a most ingenious method. It, of course, presupposes that two-life annuity values are available. As I understand the method, one has first to take an annuity value in respect of the two oldest ages in the status and to find the equivalent equal ages from the annuity table ; secondly, by simple operation on values of /xx one calculates the "weight" to be attached to the younger of the two ages, reckoning the "weight" of the oldest age as 100. The " weight " ascertained for the younger of the two ages is then, I take it, applied to all the ages in the status saving the oldest in order to derive the equal ages for the whole status, but as I am not quite sure of this I will defer con- sideration of the method until Mr. Stoodley's remarks appear in print. Mr. Weatherhead has raised two points : one with regard to the fact that we had not gone back to the original data, and the other expressing his opinion that some form of Simpson's rule might give sufficiently good results for practical purposes. With regard to the second point, I think it will be found that Simpson's rule is not very dependable when used with modern tables, and with regard to the first, I am afraid that it was sheer This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 132 Joint Life Annuities - laziness which drove us to operate on the official values. There seemed to be sufficient work ahead even when dealing with the actual official values to- keep us busy without our going back to the rough data. Further, it must be remembered that the rough data were smoothed by a summation formula,, and for this reason there was perhaps less risk of distortion being imported into our results when working on the official table than had the rough data been graduated in a different way. Mr. President, ladies and gentlemen, thank you very much indeed for your kind reception of this paper and for the attention you have given us. [Mr. Lidstone has since contributed the following additional remarks. - Ed. T.F.A.] The interesting results given at the top of p. 98 are said to be "readily shown," but they are far from obvious and it would be useful if the authors would place the proofs on record. As regards the form of uniform seniority discussed on p. Ill, I flo not agree that it "is of little practical use." It involves (1) a substituted age v' which öfters no difficulties and (2) tabulation of annuity-values, for equal ages, based on A + k(p-A) instead of /x, where к depends on the ages involved. Here all that would be necessary would be to tabulate for interval-values of к so arranged that first-difference interpolation would be adequate. There would of course be a double interpolation, for w and /c, but this would offer no practical difficulty. The constants w and к may be found as follows (cf. Hardy's Lectures, p. 68), the procedure being the eame for any number of lives, say m. After t years the décrémentai force will be : - In respect of the status xyz (m) 8 + raA + B(cx + t + ď+t + сг+ г + ...) + M(nx + * + n"+t + nz+t...) and in respect of the status www . . . (m) with A 4- 1c(fi - A) for /x 8 + mA + mkBcw+t + mkMnw+t. These, and therefore the annuity-values, will be equal if rc* + i + cy+t+ cz+t+ _ =mJccw + t for all values of t. This will be true if cx + t + cy + t + cz+t+ _ / c yH-t nx+t + nv+t + nz + t+ . .. '"гГ/ or сх + су + сг+ ... =(- Y" (independent of О or w = (log 2cx - log 2nz)/(log с - log n) and h- 2c* = ~~ ̂ "X ~ - Bs^ + Ms^g _ ~ S^-mA mcw ~~ mnw ~ m( Hcw + Mnw) ~ m(¡iw - A) these values being independent of £, B, and M. The values of к and w for practical values of the disparities between the ages would be tabulated, and it would probably be best to treat the disparities as deductions from the oldest age, not additions to the youngest. (Cf. Mr. D. C. Fraser's remarks on tables of uniform seniority, J.I. A., xlvi. 55.) As an example take a: = 30, ?/ = 45, z = 60 ; c = l'10, n=l#15. Then we shall find w = 60 -2*94 = 30 + 27 *06 = 57*06; and /c=*572. Thus in this case the common age is very near the oldest, and the exaggeration of 'lxvz о which this produces is corrected by к being less than - . It would be interesting to have the results of similar trials based on the values of This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 924-29 Table 133 с and n which the authors find to give the best approach to the A1924-21> ult. table. In determining the constants for the form px = A + Bcx + Mnx the authors have proceeded experimentally, assuming trial values for с and then determining the remaining four constants from four relations derived from the official table. They may have been led to this course by consider- ing the analogous method commonly followed in fitting a Makeham curve^ In that case the reasons for adopting this plan are twofold, viz. : - (1) From previous experience we have a very good idea of the range within which с is likely to fall. (2) Having fixed a trial value of c, the remaining constants A and В enter linearly, and we can readily find them from equations- involving the actual exposed-to-risk and deaths. Neither of these reasons (the second of which is specially important) exists, in the case of the above formula, and I think that it is both simpler and more satisfactory to find definite values for each of the five constants by making them satisfy five suitable relations - not necessarily of the form used by the authors. The authors' reasoning on p. Ill requires very careful consideration if we are to avoid serious misapprehension. It is, I think, not true in general that "an interesting relationship exists between the constants, ç and ?i," nor that "for four values of c, therefore, we can assume that if the ordinary Makeham formula gives an indifferent fit so will the formula цх = А + Всх + Мпх." These statements apply at most only to the authors7 experimental method of fitting, above referred to, and I am not at all sure that they apply even in this case. There are five constants, and only four equations which are therefore indeterminate. They can be made determinate by introducing an additional condition, and if this is that с has an assumed value the equation in the middle of p. Ill gives a single determinate value of n. The authors seem to suggest that an alternative additional condition is that c - n' but in general this is impossible because the formula then resolves itself into the ordinary Makeham form A + B'cx involving only three constants which cannot in general satisfy the original four equations.. For the same reason it does not seem that we can make corn equal to 0 or 1 . It wouîd seem to be a course of doubtful validity to use the original four equations to find a single relation between с and n and then proceed as though that relation stood alone, independently of satisfying those equations. The authors find that the best results are obtained when с and n are very nearly equal. In that case let n = c(l + k) where к is small. Then nx = cx(l + k)x Ц c*(l + xk) leading to px = A + Vcx + Nsc* N = Ш M a form which might perhaps be considered, since assuming c>l this form would put a slight drag on the geometrical increase at the highest ages- without greatly affecting the beginning and middle of the table. Passing to Section II. of the paper I may perhaps not improperly suggest that the case of the A1924-29 table is one in which consideration might be given to the suitability of the method mentioned in the authors' footnote on p. 93. The method has the great advantage that it applies the same systematic method to last survivor as well as joint life annuities. The method as originally, put forward applies to a wide range of disparities of age ; the range can be extended in the case of two lives in the way suggested by Mr. Todhunter and illustrated by Mr. A. E. King, and similar extensions. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 134 Joint Life Annuities - may be possible in the case of three, and maybe four, lives. Perhaps some of the younger members of the profession may be moved to make the necessary tests, which will be much helped by the knowledge of the "true" annuity-values as given by the authors. As regards more ad hoc methods, reference should also be made to those suggested by G. F. Hardy, J.I.A., xxiii. 279-83 (1882), (giving in this part of the paper two methods not assuming Makeham's Law). A. W. Evans, J.I.A., lxii. 126 (1931). I hardly agree that the results given in the table on p. 105 (based on a previously known method) "can be claimed as fairly satisfactory approxi- mations," for they show the important drawback of unreliability, some of the errors being quite inadmissibly large in rather normal and unexpected cases. On the other hand, the authors' new methods as exhibited in the tables on p. 107 and p. 110 are quite remarkably and consistently successful, and the authors are to be warmly congratulated upon them. The methods are partly -empirical, but the authors have been guided by theoretical considerations of great interest and they are certainly too modest in saying, on p. 118, that their further investigation "is perhaps not of sufficient general interest to warrant any further treatment here." I hope they will be induced to repent of this statement and give us the investigation. ¡So far as they have founded on the analogy between the expectation of life and the vie probable, the case of the annuity -value would seem to be similarly associated with the point n at which i = vHnp9 not np, and this would mean np>i, whereas the authors find np = J nearer the mark. It would be interesting to have some explana- tion of this seeming anomaly. It is helpful to consider the curve of vnnpx (where x is a single or joint life .status), in the way suggested by Mr. A. E. King himself (T. F.A., ix. (1922), p. 225 and diagram). For any status X the curve has the following properties : - (1) It starts at 1, continuously falls and eventually reaches 0, the extreme tail contributing little to the area. (2) For ordinary rates of interest the curve is convex to the £-axis during the whole of its course for a single life, and nearly, if not quite, the whole for joint lives. (Let unaccented letters refer to the original status and accented letters to a substituted status.) Then, if we knew nothing as to the law of mortality and merely make ¡i = /x' at the point 0 where in any case the curve is falling least quickly, we have evidently done little to ensure that the curves shall run close together and have nearly the same areas (annuity- values). And if we merely make /x = /x' at a later point n, this does not secure that the two curves of vnnp have either the same height or the same slope at щ and again we cannot be certain that we have done much to ensure the corre- spondence of the curves and their areas. But if, as the authors happily say and suggest, we " take a grip of the curve" by making the ordinates equal at a suitable point n in the body of the curve (as well as at the point 0 and at the end of the curve), we are reminded of Mr. King's happy illustration of ■"tying the parcel" {T. F.A., ix. 227 f.n.) : and we have evidently done much to ensure the general correspondence of the curves and their areas. Looking at the matter jrraphieally, it is hard to draw two curves having the attributes (1) and (2) and having another common point at n in the rump of the curve, without making them correspond pretty closely in their general course and This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the A 1924-29 Table 135 their area. What is the most "suitable point" the authors determine practically and most successfully by a quasi-empirical process. It may be pointed out that - by the common device of changing the rate of interest - we can, if desired, "tie the parcel" at two places m and n, i.e. make v'nnp' = vnnp and v'mmp' = vmmp. Let ò' = ò f С and M be the modulus "43429 ... ; then we shall have to satisfy the equations (taking common logs) {logT- lognr + nMC= log°Z- log'ftZ = A say {logT- log T - log тГ + mMC = log °l - log ml = В say whence, eliminating C, (n - m) log °V + m log H' - n log mV = nB - mA where м, m, and nB - mA are known. Hence by trial we may easily find X', the status corresponding to ¡x and I', whether a single life or joint lives ; then from either of the original equations we find С and we have The numbers m and n are in any case somewhat arbitrary, and the authors suggest to me it is convenient to make n = 2m. I find on investigation that this avoids the necessity of a method of " trial and error." For the above equation for finding the substituted status X' then becomes m log °Г + m log 2ml' - 2m log mV = 2mB - mA or A2mlogZ' = 2B-A where A2m means that the interval of differencing is m. Hence if A2m log I is tabulated for convenient values of m the equation can be easily solved by ordinary inverse interpolation. By analogy this might " be called " the " mp and np method." It seems possible that when thus " tying " in two places the actual positions of those places may be less important, and the results therefore less sensitive, than when " tying " in one place only. But it is unsafe to prognosticate in such matters, and some actual experiments would be very interesting. The following written communication has been received from Messrs. King and Reid :- We are greatly indebted to the various contributors to the discussion for the interest they have shown in the paper. We first propose to comment further on two points raised in the course of the oral discussion. (1) Mr. Stoodley's suggestion of a weighted /x method contains interesting possibilities. It presupposes that one has available complete tables of two-life annuity values, so that for any status xy those relative weights, say hx and hy, may be found which would determine the equal age to give an exact result. When the status is increased by another life, y, the method presupposes that the use of the weights lcx, hy and hz would give a good approximation to the trne result. Mr. Stoodley does not define in his remarks what exactly the procedure would be in the case of, say, a three-life status where the ages were all different, say ас, у, and z. If, however, we have understood correctly what he has since informed us, it would appear that (considering x as the oldest age) he would find two series of weights (1) kx and hy and (2) h'x and h'3i the use of each pair of which would reproduce exactly the appropri- ate two-life annuity and then use for the three-life status 1, -77, and ̂A From one or two trials which we made it did not seem to us that the method as propounded could be trusted to give dependable results This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 136 Joint Life Annuities - so far as the A1924-29 table is concerned, but if after more extensive trials a reliable scheme is forthcoming, no doubt the profession will be given the benefit of it. (2) Mr. McAllister's experiments with joint life temporary annuities are worthy of careful attention. A life annuity payable for a term of, say, m years can of course always be expressed as the difference between two annuities for the whole term of life, one being multiplied by a vmmp factor. As temporary joint life annuity values for equal ages will apparently not be included in the official joint life annuity volume, this method will require to be resorted to in practice ; should, however, temporary joint life annuity values for equal ages be published, one would turn to a method such as that described by Mr. McAllister. Mr. Minister's modification of the npx method, viz. to give to n the value appearing in Table V. or the term of the temporary annuity m, whichever is the less, certainly seems to give satisfactory results. It must be borne in mind, however, that reasoning in a manner analogous to that described in Section II. (3) of the paper, if tPxytPww when "¿" is large andifS^^S™^, then there must be some value of t within the term of m years where tPxy = tPww It therefore follows that the best results are to be ex- pected if the value of n be taken within the term of m years. It has not been possible for us in the short time available to go into the matter thoroughly, but it would appear that very good results may be expected if n be given the value of the nearest integer to two-thirds of the annuity term m, or the value given in Table V., whichever is the less, as the following results have been obtained on this basis : - ^ Te™ SU T- Value 21/30 15 11-476 11-478 30/42 11165 11166 30/51 10-599 10-603 27/33 19 13-511 13 505 33/42 13012 13016 21/60 24 10-832 10-834 42/51 36 13-589 13-579 This problem merits further investigation. We will now pass to points raised by Mr. Lidstone in his very interesting further communication to the Transactions : - (i) He has suggested that it would be of use if we gave a demon- stration that (see top of p. 98) " when the basic table follows Makeham's second development of Gompertz's Law it may readily be shown that, so far as two lives are concerned, a This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the A 1924-29 Table 137 joint life annuity value may be expressed as a blend of three equal age joint life annuity values at different rates of interest, calculated according to the mortality of the basic table." The following demonstration will perhaps suffice : - If fc-A + tte + Be--?*1«*-" 771 + IIх tPx=stw(2tx+tW-l)m4-nX r * + t r * m + nx where s, w and g have the usual significations = фх*и№*+ «»у (e* - 1) + со фЖ^Ч*** + ř VX(C' " 1] where фх = - - - ïix and соф = т + пх = фхгРх + ™фхП1{рх 2М 2М where 2м denotes that the function appearing above it is calculated according to the basic second development Makeham table, • *• tVxy = ФхФугРху + (ФжСО фу + CO фхфу)пгфху + софхсофуП2г1рху. 2М Now under a second development Makeham table tVxy = ™(d-2mtVuu 2M 2M where d = y -x 1c = u-x and cu=l(cx + cv' .'. under the modified plan which we are considering ^ tPxy = фхфу&мЧ'1 - 2*)}ř tPuu + (фхсо фу + со фхфу){м)2(а~ 2khiytpuu + со фхсо ф^^2^" 2kWytpuu 2М = фхфуЪиРии + (Ф*СО фу + СО фа;фу)^2«|>«п + СО фжСО фуЪиРии 2 M where Vj is at rate jx = ^.^ - 1 . 1 + г _., V2 " " h~nwW-2k) V3 J> >» IS" n2w2(d-2k) L' .-. ажу = а blend of three equal age second development Makeham annuities at difieren t rates of interest. (ii) It is useful to have for reference a clear demonstration and full details of the modified uniform seniority plan which would be applicable when цх takes the double geometric form This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 138 Joint Life Annuities - A + Bcx + МиЛ On the point as to whether the form of uniform seniority yielded by this formula is of practical use or otherwise, however, we see no reason to modify the comment made in the paper ; it may also be noted that Mr. Gr. F. Hardy when discussing this matter made a comment in somewhat similar terms, although he was dealing with an even simpler expression - one without a constant term. (in) We must confess that it was an omission on our part not to have stated on p. Ill of the paper that we had adopted a trial value of с ; we must also confess, however, that the adoption of any other course did not occur to us in view of the manifest advantages which are secured by experimenting with trial values of c. The principal advantage of their employment is that the operator is able to see clearly the effects produced by variations in the constants, and particular attention can then be given to the important lower portion of the mortality curve. We further venture to suggest that few actuaries would find the five constants from five equations and adopt them rigorously without modification, and it will be appreci- ated that immediately a modification is made the process is rendered analogous to that of adopting trial values. As regards the opening sentence of the final paragraph on p. 11 1 of the paper, we are of opinion that the statement made is not inaccurate. (iv) It is certainly to be hoped that for the benefit of the pro- fession some actuary, who may be looking for scope to employ his talents to a useful purpose, will re-study the paper referred to in the footnote on p. 93 of the paper and test the method described with the material now available. (v) In regard to the results on p. 105, although the average error is about one half of the average error brought out by the two previous tests, we agree that "fairly satisfactory" was too favourable a verdict. Our main purpose in inserting those results was to provide a stepping-stone to the last two methods described. (vi) There is not in our view an inherent "anomaly" in the fact that the crossing point of two curves of the form v'p will generally be found somewhere in the region where np = $. On setting out the curves of npxy for various values of x and у in separate graphs and superimposing the curve of nPww (where exy = eww' it was observed that, while at the region where np equalled ¿ the values of npxy and „^^ were rapidly approaching each other, the curves (in respect of most entry ages) actually crossed in the region where np changed direction - and np in general changed direction in that portion of the curve where np equalled approximately 1/3. Further, it was found that the point where the curve of v'pxy crossed the curve of vnnp^w (where axv = aww) rarely differed very greatly from the point where the curve of npxy crossed the curve of ̂„^ mentioned in the preceding para- graph ; this is perhaps not surprising, for if one considers a series consisting of the various values of vn and then, produced from this, two new series, one obtained by weighting the This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Some Experiments with the Al 92^-29 Table 139 successive values of vn with npxy and the other by weighting with npw (at the same time securing that 2vnnpxy=2vnnpww) ; it is reasonable to assume that, if one were considering where the curves of v'pxy and vnnpww would, if superimposed, cross, one another, one would have particular regard to the form taken by the weights. For example, if the weights themselves were regular in progression but the curve of weights had a point of inflection, this latter would be an important point to« bear in mind seeing that the curve of vn has no point of inflection. When therefore considering v'p it would seem that the vital function for the purposes of the investigation before us is np. (vii) The ingenious mp and np method given by Mr. Lidstone contains within itself the power to yield very accurate approximations, as the fact of the curves being made to cross twice should automatically ensure that the areas contained between them and the horizontal axis (i.e. the annuity values) will not be widely divergent. Unless, however, it is applied in the shape of the particular form where n is taken as 2m, the method would appear to be rather cumbrous for practical use. In the special case where n = 2m we should still be faced with the problem of the best value to select for m for each example, and the values given^in Table V. would probably be of service ; if, for example, the value selected for m were two-thirds (to the nearest integer) of the appropriate value of n given in Table V., this would ensure that the "parcel," instead of being tied in one place, would be tied in two different places equidistant from the single place which yielded a satisfactory result. The method (in common with the npx method itself) would appear to possess the great advantage that it is not dependent for its success on the selection of one particular value of the period m. The following example of the method will probably be of interest. Example. A1924-1929 ult. 3%. Let it be required to find the value of а39:оо- From Table V. it is seen that the value of "n" which would be employed in connection with the 1грх method is 18, and values of "m" and "n" which, therefore, suggest them- selves for use are : - m=12. n=*24. We then have, following the notation used by Mr. Lidstone, A = log ¿39:60 "log ¿63:84 = '8979 B = log ¿39:60 -ЬД ¿51 = 72= -2185, .*. 2B-A=-4609 one-half = --2304. Now Ai22logř53= --2204 and л 122l°g?54= --2429, by interpolation w = 53-44. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp 140 Joint Life Annuities Further С = в - (lQg Iww - log lw+v.w+12) 12 x M = •00409, .'. ô' = -02956 + -00409 = •03365. Now aww at 3% (ô = -02956) = 11-102 and aww at Z'% (ô = -03440) = 10-651 where w = 53-44, . * . by interpolation ато(Ь = -03365) = 10-723, while, as seen from the table on p. 110 of the paper, the true value is 10-725, and the value yielded by the npx method is 10-739. This content downloaded from 91.213.220.135 on Sat, 28 Jun 2014 08:18:32 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Article Contents p. 93 p. 94 p. 95 p. 96 p. 97 p. 98 p. 99 p. 100 p. 101 p. 102 p. 103 p. 104 p. 105 p. 106 p. 107 p. 108 p. 109 p. 110 p. 111 p. 112 p. 113 p. 114 p. 115 p. 116 p. 116A p. 117 p. 118 p. 119 p. 120 p. 121 p. 122 p. 123 p. 124 p. 125 p. 126 p. 127 p. 128 p. 129 p. 130 p. 131 p. 132 p. 133 p. 134 p. 135 p. 136 p. 137 p. 138 p. 139 p. 140 Issue Table of Contents Transactions of the Faculty of Actuaries, Vol. 15, No. 137 (1934-1936), pp. 93-140 Joint Life Annuities—Some Experiments with the A1924-29 Table [with DISCUSSION] [pp. 93-140]