Transcription
FORMULAE AND TABLESFORA C T U A R I A L .EXAM I N A T I O N S(INSTITUTE OF ACTUARIES)T H E INSTITUTE OF ACTUARIESANDT H E FACULTY OF ACTUARIES1980
,'*IFirst published 19x0Reprinted 1983Reprinted 19XXRcpriiited 1990Reprinted 199 IPi-inled in G r a r Drit,iinthe 4lden Press. Oul'ord:it
FORMULAE A N D TABLESFORACTUARIAL EXAMINATIONS( I N S T I T U T E OF. A C T U A R I E S )
INSTITUTE OF ACTUARIESFormulae for the use of Candidates at the ExaminationsSEE ALSO HEADINGSTO 1NDIVIDUAL rABLESThe list given below is intended to help candidates with formulae whichmay be found hard to memorize. Inclusion in the list does not mean thata proof may not be required.11. F I N I T E D I F F E R E N C E SNewton’s formula: ux ,,,, u ri l)Au, n(z)A2u, . . .Newton’s divided-difference formula :U, u, (x-cI) Au. (X-(I)(X-b)A%, . . .bbrLagrange’s interpolation formula :U,(x-a)(x-b). . .(x-k)-1U,(a-b)(a-c). . .(a-k)’x-a1.(b-k)’x-b ( 6 - a ) ( b - c ) .ub .Gauss’s forward formula:U, U O . . x )Au x( ,A1 (x u - 1)(3)A3u- 1 (x l),,)Gauss’s backward formula :U, UO x(l)AU- 1 (x I)(,) AZu-1. ( X ) ( ) A U ( X ) ( . , A - .aISummation by parts:1(U, Au,)-E [ux 0,]1’1(l A)’E& A%,du, - AuX-- A%, -dx(U, 123.*Au,)
NegativeBinomialThere are two formulations, one typically for Subject Iand one typically for Subject 5.For subject 1 :Parameters: k, a positive integer; 0 p 1 with q 1- pPF. P ( X X) (iI:)‘,pkqx- x k, k 1, k 2 , . . .LkE ( X ) -,Var(X)kqP2 -PFor subject 5:Parameters: k 0; 0 p 1 with qPF. P(X x) (“ ;-l)Pv. 1-px 0, I , 2,”,,)”PGF. G(s) (Ik9kqE ( X ) -, Var(X) PP2The two formulations are of course connected, eachdiffering from the other by a shift in location. Inparticular, if X , is as in the subject 1 formulation and X ,as in the subject 5 formulation with an integer value fork,then X , X , k .GeometricNegative Binomial with k 1
2.4.CONTINUOUS DISTRIBUTIONSParameters: - cc p Normal00, 0 0(aMGF. M ( t ) exp pf - a2t2E ( X ) p, V a r w ) n21Gamma.Parameters: c! 0 , A 0.AaXaPDF.f(x)*-I -e p A x ,xr(a)MGF.M(t) ( ;)-’I-- O,t E.Exponential Parameter: 1 0PDF. f(x)E(X)Chi-square Ae-A“, x 01/i.Var(X), I/;?12 is Gamma with a n2 and I -.21-where n is a positive integerBetaParameters: a 0, j 0MGF.aE ( X ) -Var(X) c! p’(a B ’(a P 1)
LognormalParameters: - CO p exp{PDF.f(x)C O , CJ 0-i(%)2},x o XCJ&MGF. -. { 3E ( X ) exp p-0' a'}., Var(X) exp(2p[exp{a2)- I]LParetoParameters: 2 0, ,I 0PDF. f@) A a i " ( l x ) - - ' ,x 0MGF. -E ( X ) L/(a- I), Var(X) a,12/{(a-Generalised Parameters: tl 0, E. 0, k 0ParetoT(cr k)l"xk IPDF.f(x) -I)'(a-2)}x or(a)r(k)(i x ) 'MGF E [ X ] i k / ( a - I), Var[X] i 2 k ( k a- I ) / { ( a - I)'(cr-2)}WeibullParameters: c 0, y 0PDF.f(x) cyx'-lexp{ -cxy}, x 0MGF. -BurrParameters: a 0, ,I 0, y 0 PDF. f ( ) ayi"xS-'(EL xY)-.MGF. -I,x 0
2.5.COMPOUND D I S T R I B U T I O N SCompoundPoissonN-Poisson (A), { X , } zI i i d.NX,( OifN O)Y I 1MGF. of Y. M,,(t) exp{l(M,(r)-1))where M,(t) is the MGF of X,L2.6.PARAMETRIC INFERENCE, NORMAL MODELThe random sample (x,, x2, . ?2 ,k,) has mean R and variance32 cx2- (Cx)2/nn-1(a) For a single sample of size n under the normal modelX-N(P, a*)(ii)- x,-(n- 1)S2U21(b) For two independent samples of sizes rn and n under the normalmodels X N(px,,):aY N(p ,,,a’, respectively-(ii)-under the additional assumption :a a’,
2.7.N O R M A L L I N E A R R E G R E S S I O N MODELY-N(a Sx, a’)The usual estimates, based on bivariate data ( y , x ) of size n, area y-/jxB x,lsxxli-lr-s.e. ( j )2.8.t a p , where s.e.A N A L Y S I S OF V A R I A N C E(a) Single factor normal model Y,,i 1, 2 ,- N(p (6) [C? /S J’T,, a’). . . , k ; j I , 2, . . . ,n,;n Zn,;&,T,with 0Under the appropriate null hypothesis:ss,--where1ss, ZZ(y,-j.)2ss, Zn;(jj;.- y ,11.)’ C-y’.--y?ninSSR ss,-SS, cZy:--y?n
-(b) Two factor no interaction normal model Y ,withi 1,2,N ( p 7 i b,, a2). . . , k;j I, 2 , . . . , b; Z 7 i Xj?, 0Under the appropriate null hypotheses:*whereFb- l,(b- I)(k- I).SST CC(y,; -j . . )' Ixzy; -yf .bk1ss, b C ( j i .- j . .)* -cy:.b1--1SS, k C ( j . , - J . . ) Z -xyt,-bky'1k2.9.NONPARAMETRIC INFERENCE(a) Single sample, size nWilcoxon signed rank testUnder the appropriate null hypothesis:E(W) 0Var ( W ) , n(n 1) (2n 1 )6(b) Two samples, sizes n and m(i) The Run TestUnder the appropriate null hypothesis:2mnE ( R ) - Im nVar(R) 2mn (2mn -rn -n)( m n)*(rn n- 1)
(ii) Wilcoxon-Mann-Whitney TestUnder the appropriate null hypothesis:Var ( W,) 2.10.mn(m n 1)12B A Y E S I A N METHODS.f(0lx) ccf(xlf))f(@)rFor x a random sample of size n from N ( p , 0 2 ) ,a2known, and a N(po,ai)prior for p, thenPI&2.1 1.-N(P*P,3EMPIRICAL BAYES CREDIBILITYModel 1 . Data { { X , , } , f , } , lX,J represents the aggregate claims in thej-th year fromthe i-th risk.n'Ft XJnJ IN8 f,/NI 1Parameter estimation:QuantityEstimatorNnNV"6)l1 (X,-X)'-( N - 1)-1i IN
Yi, represents the aggregate claims in thej-th year fromthe i-th risk; P, is the corresponding risk volume.NncP i , P,1 PiP 1I 1.NP* ( N n - l ) - '1P,(I-P;/P)I- IParameter estimation:QuantityEstimatorx)IE"0nh'E[s2(@13 . COMPOUND INTEREST1-U"urn -sm (l i)"-1iam - nu"(Ia), j--1ai,nndial1 -1-ism(Du), n-a,aiam1- ( i ' - i )%Iwhere akandiis the value of an annuity certain of 1 p.a. payable inarrear for n years to yield remunerative rate 'i after allowing forreplacement of capital by accumulation at reproductive rate i .
Makeham's formula: A K p(I-t)(C-K)gwhere:A is the present value of capital and net interest payments;K is the present value of capital payments;C is the total capital to be repaid (at redemption price);g is the rate of interest expressed per unit of the redemption price;t is the rate of tax on interest.Value of annuity .certain net of tax:a;--tg (a&-i -gL03where:g is the original rate of interest;t is the rate of tax on interest.1Capital redemption policies : A, 1 - d aaP ---d"-&4. LIFE A N D O T H E R C O N T I N G E N C I E Sex sz1,Gompertz's law:px BcX; rpx g'Makeham's law:p, A Bc";-ix ex -&pxt - 1 )(rp, s'g' (& - I )where B/log, c -log, g and A -log, s.,v;m;A l-diiA 1-6d1P z-daP ,-sm-1 ,v,:, -P: .2m1ayf:,
PREFACEActuarial tables for the use of students preparing for and sitting examinations were first published by the Institute of Actuaries in 1912 under thetitle A Short Collection of Actuarial Tuhles. In 1952 the Institute ofActuaries a n d the Faculty of Actuaries jointly had published ActuarialTuhles -for E.ramirrution Purpose; (Cambridge University Press), whichcontained certain additional and more up-to-date tables. It is now thoughtdesirable to produce a third set of t'ablks. again using more modern tablesand adding certain others; these generally correspond with tables includedin the textbook Lifb Contingencies by A . Neill (Heinemann, 1977).The main changes from 1952 are: replacement of English Life Tuhk No. 10-Mtrlr,.s bq English Lifk Tuhle N o . I.? Moles; replacement of Hypothetical Select Mortality by A1967 70: replacement of u t ? i Jand u i f ' lannuitants' mortality by ( 5 5I. ,use o f a more modern basis in the PensionF u n d Tables: extension of the range of rates of interest in the CompoundInterest section: omission of Premium Conversion Tables and o f a table ofOfficc Premiums for Contingcnt Assurances: a n d inclusion of additionalStatistical Tables. a statement of the International Actuarial Notation. andTables of Logarithms, Antilogarithms and Reciprocals. The following tables have been printed. by the authority a n d under thesuperintendence of the Institute of Actuaries and the Faculty of Actuaries.in order that candidates presenting themselves for examination may hake ;Icompact means of working out actuarial problems in their studies and inthe examination room. The particular tables which are included have beenselected a s being, o n the &hole. the most suilable for [his special purpose:but the Councils of the Institute and the Faculty desire it to be distinctlyunderstood that they d o not express any opinion whatever a s to thecircumstances in which any of the tables may be suitable for use in practice.The thanks of the Councils are given to those firms of consulting actuarieswho have supplied material for inclusion in these tables. particularly insections V a n d VI.The tables are published simulteneou4y in two versions: Forniirkir r i dTuhlr.s,for.ActuurI 11E.i-urninurion.s for the Institute of Actuaries and Tirhlrs,fOr ,4c turrricilE. riniincrrion.sfor the Faculty of Actuaries. ,\part from a listof formulac the tables are the same.
CONTEYTSTABULATEDVALUESPageRates of InterestFunction81-87SECTIONv:MANCHESTERUNITYSir KNESS EXPERIENCE1893-97(A,H,J.)COMBINEDTHtWITHMORTALITY RATFS OF THE E N G LISH82-83848586-8789-94909102939495 104L I F ETABLENO. - - M AFS IRat&dI.sickness . . . . .Value of sickness benefits of 1 perweek for the whole of life inperiods . . . , . . .Commutation columns for sicknessbenefit values . . . .Stcrirn V I :TABLESPt\sioxFUNDService table and relative salaryscale . . . . . . . . .Contribution functions . . . .Ill-health retirement functions. .Age retirement functions . . .Functions for payment on deatho r withdrawal. . . . .SECTIOS VII: ISTERYATIOUIACTI.ARIAKOTATIOSL105 1 18S r C T I O U VIII: ST,4TISTl(.hL TABIES106Standard Normal Distribution:values of the densit! functionand of the distribution functionCritical points of the f distribution . . . . . . .F distribution: 5 per cent and 1 percent critical points . . . .Critical pointsof Students' rdistribution . . , . . . .Cumulative Poisson distributionValues of the negative exponentialRandom n u m b e r s . . . . .107,108- 1 I 1I12113-1 161 17118-4)04", 4"04"0
SECTION ICOMPOUND INTEREST TABLESInterestVARIOUS RATES O F INTEREST*.1
C O M P O U N D INTEREST TABLES1 per centnConstants(lfi)"I"'34a4(ad 'n FunctionValue1I . .010000009 975009 963on9 954009 9501.004 9881.002 49 I1.000 830,990 099,995 037.997 516,999 171-009 901,009 926,009 938,009 9461.002 494i.noi 7421-004 5751.004 9921.007 4941.006 2421.005 408,004 321 410111.115 6712I3141.126 831.138 091.149 471160971,172 581,184 30I 196 151.208 I 11.220 19232 39244 12257 16269 73282 43,295 26,308 21,321 29,334 50,347 851.361 331.374 941.388 691.402 581.416 601.430 771.445 081.459 531.474 I 21.488 861.503 751.518 791.533 981.549 321.564 811.580 461.596 261,612 23I628 351.644 631816 70'2006 762 216 122.448 632.704 343536373839404142434445462,010 00,020 10,030 30,040 60.OS1 01,061 52,072 14,082 864 9 3 69104 6234748495060708090100,990 10,980 30.970 59,960 98,951 47.942 05,932 72.923 48,914 34-905 29,896 32,887 45,878 66.86Y 96-861 35,852 82.844 38836 02,827 74,819 54-811 43,803 40.795 44,787 57,779 77,772 05,764 40,756 84,749 34,741 92,734 58,727 30,720 10,712 97,705 91,698 92,692 00,685 15-678 376 7 1 65,665 00,658 42,651 90,645 45,639 05,632 73,626 46,620 26-614 12.608 04,55045-498 31,451 12,40839369 71I ,000 02.010 03.030 14.060 45.101 06.152 07.213 58.285 19.368 510.462 2I 1.566 812.682 513-809 314.947 416-096 917.257 918.430 419-614 720.810 922.019 n23-239 224.471 625.716 326.973 528.243 229.525 630.820 932,129 I33.450 434.784 936,132 737.494 138.869 040.257 741 6 6 0 343.076 944.507 645.952 747 412 34X.886 450.375 25 1.879 053.397 854-931 856.481 I58.045 959.626 361.222 662.834 864463 281669 7100-676 3121.671 5144.X63 3170481 4,990 11.970 42.941 03.902 04-853 45.795 56.728 27.651 78.566 09471 310,367 611.255 I12-133 713.003 713-865 I14.717 915.562 316-398 317.226 0I 8.045 618.857 019.660 420.455 821.243 422.023 222.795 223.559 624.316 425.065 825.807 726.542 327.269 627.989 728.702 71 9 . 4 0 630 107 530-799 531.484 732 163 032.x34 733.499 734.158 I34.810 n35-455 536.0Y4 536.727 237.353 737.9 74 038.588 I39.196 I444.955 050- 168 554.888 259.160 963.028 91.010 0000.507 512,340 022,256 2x1206 040. I72 548,148 628,130 690,116 740-105 582,096 454,088 849-082 415476 901,072 124,067 9454 6 4 258,060 982,058 052.055 415,053 031,050 864,048 886,047 073,045 407,043 869,042 446,041 124,039 895,038 748,037 676,036 671,035 727,034 8404-44 004,033 214,032 468. n i l 761O i l 092,030 456,029 851,029 276.n2x 7'702x 204027 705,027 22K026 771.026 334,025 915,025 5 1 3-022 244,019 933,018 219.0 I6 903,015 090I00
COMPOUND INTEREST TABLES2 per centConstantsFunctionVdluen(lii)”12020 000019 901019 852019 819019 801I 009 950I 004 963I001 6 5 29x0 792990 148995 062998 1 5 1019 608019 705019 754019 7863456I 014 975I 012 4691 010 801008 600 23441414243444546.472 07.315 58.162 28.982 6,102 178,094 560,088 118,082 602,077 825II1217.293 49.786 810,575 311.348 412 106 212.849 3,485 95-72845,714 I6,700 I6,686 43,672 9718.629 320.012 121.412 322.840 624,297 413.577 714.291 914.992 015.678 516.351 4,073 650,069 970.066 702,063 7824 6 1 1571617181920,515 67,54598,576 90,608 44,640 6 I,659 78,646 84,634 16,621 72609 5325.782 327.299 028.845 017.011 217.658 01 8.292 218.913 919.523 521222324251673421,706 891.741 021.775 841.81 1 361.847591,884541.922 231.960681.999 X Y,59758,585 86,57437,563 I I,552 07541 25-53063-57023-5l003,500 0 3,058 785,056 631,054 668,052 871,051 220,049 699,048 293446 990,045 77822.396 522.937 723 468 323 988 624-498 624.998 64 4 4 6503031327 039 895.080692 I22 302.164 742.208 I14,490 22,480 61,471 10,452 8051.994 454.034 356. I14 958.237 260.402 1)-039 233,038 5070 3 7 X2I,037 1714 3 6 5562.252 21)2 297 242.343 I92,390 0 5,444 01,435 30,426 77,418 40,410 2062-610 064.862 267-I 59 569,502 771 892 725.488 825.969 526.440 626-002 627.355 527.799 5,402 I S,394 27,386 54.37X 96371 5 3,304 78,25003205 I I,168 261,372 79400 24,428 25,456 812.437 X 5607080903.281 0 33.999 564.875445.943 134817.434 38.583 09.154 649SO471-020000n,870 562,486 6 I2-536 342.587072.638 X I2.691 5946’178 526,154 512,136 510,122 515. I 1 1 3271.345873X 5.601 4IS163637(ad5.20401.31948303132333435I .000 02.020 0306044.121 6(1 i G6.308 I1426272829x,887971.10408II12I31 004 975I 007 4691 009 114I 009 967J0.980 41.94 I 62.88393.807 74.713 51 020001.040 401.061 211.082431.126 I61,148691,171 661.195091.218991.243 37I268241.2936178910r”,980 39,961 17,942 32,92385,905 461 9510.949 712,168 713.412 I14.680 315.973 930,421 932.030 333.670935.344 337.051 238.792 241.568 142,370 444.227 046.1 I I h48.033 849.994 520.121 020.706 921.281 .I21.84440.515 050,346 755,262 624,212 158,043 596,042 61 I,041 687,040 819.040 0022x.234 x28.66 I 629.080 020.490 2- 0 3 5 9724 3 5 41 7.I134 890,034 3 X X033 911174.330 67 6 x 17 279 353 581.940 6x4.579 429.X92 330.286 630.673 I31.052 131423 6033 453-033 O I XI132 6024 3 7 204,031 827114.051 5149 977 9193,772 034.760 937.448 639.744 541.586 94 2 8 7hX026 668,025 I61,024 046,023 203247.156 4445‘Kl474X4c,YI61I70xoW1100
COMPOUND INTEREST TABLES2 per centI( 1 i)"III-731.025 001.050 631.076 8941.103 8 11.131 4161 , I59 691.188 69578910II121314IS16171819202117--2324-.' 50607080901001.218 401.248 861.280 081.312 091.344 891 378 511.412 971.448 301.484 511,521 621.559 661.598 651.638 621.679 581.721 571.764611.808 731.853 941.900 291.947 801.996 502.046 412.097 572.150 012.203 762.258 852 315 322.373 211.''3 iilaiil,975 61,951 81,928 60,905 95883 851.000 02.025 03.075 64. I52 55.256 30,975 61.927 42.856 03.762 04.645 8,862 30,841 27,820 75,800 73,781 20,762 14,743 56,725 42,70773690 47,673 62,657 20,641 17,625 53,610 276.387 77.547 48.736 19954511.203 412.483 513.795 65.508 16.349 47,170 17.97098.752 I9.514 210.257 810-983211.690 912.381 413.055 013.712 214.353 414.978 915.589 2,595 39,580 86,566 70,552 88,539 39,516 23513 40500 88-488 66,476 74,465 1 12.432 542.493 352.555 682-619 572.685 062.752 192 821 002.891 522.963 813037903 113853 191 70j.271 493.353 283.437 I 1,453 77,442 70,431 91,421 37.41 I 09,401 07,391 28,381 74,372 43,363 35,354 48,345 84,337 4032917321 1531331305 67,298 22,290 944.399 795.632 107.209 579.228 8611.813 72,227 28,177 55,138 70,108 36,0846515.140416.519 017.931 919-380220.864 722.386 323.946 025.544 727.183 328.862 930.584432 349 034 157 836.011 737,912 039.859 X41.856 343.902 746.000348.150 350.354 052.612 954.928 216,184 516.765 417.332 I17.885 018.474 418-950619.464057.301 459 733 962.227 364.783 067.402 670.087 672.839 875.660 878.552 381,516 I84.554087.667 990 859 694,131 197.484 319.964 920.453 520.930 321 3 9 5 421 849 272.291 922.723 823 I45 223.556 323.957 324.348 624.730 325.102 825.466 125.820 626.166 426-503 826.833 027,154 227.467 527.713 228.071 428.362 3135.991 6185.284 1248.382 7329.154 3432.548 730.908 732.897 934.451 835.665 836.614 1(UiilV'1.025 0000.518 827,350 137,265 8 18.2 15 247- 1 8 1 550. I57 495139 467- 1 2 5 457I I4 259,105 106097 487.09 I 048,085 537,080766-076 599,072 928-069 670,066 76 I,064 147061 787,059 647,057 696,055 9 I3,054 276,052 769,051 377,050 088,048 891,047 778,046 739045 768044 U59044 007043 206,042 452,041 741,041 070040 436039 836-039 268-038 729,038 217,037 730,037 268036 827,036 407,036 006,035 623, 0 3 5 258,032 353,030 397,029 026,028 038,027312nConstantsI2345678910II12I31415Value,025 000,024 846,074 7694 2 4 7 18,024 6931.012 4231.006 1921.002 060,975 610987 730993 846.99 7 9441617181920,024 390024 541,024617,024 667211.006 2 1 I1.009 3271.01 1 40777--2324252627282930313233341.0124491.018 7 1 11.015 5771.013 491,010 723 975363738394041424344454647484950607080901005
COMPOUND INTEREST TABLES3 per centnConstants,030 000,029 778,029 668,029 595,029 5591.014 8891.0074171.002 466.970 874.9X5 329,992 63X,997 540(l i)"I23Value4.-)56789.,10.II121314151617m y 116,029 341-029 450,029 522201 007 4451.011 1 8 111221 4 1 3 6771.014 9262324252627281.022 44451.018 681I016 17718I9I.Yl6 10,521 89506 69491 93,477 61, 4 6 3 69450 19,437 08,424 3541 1 99,399 99,388 34,377 03,366 04,355 3834503334983252331575306 56,297 6328896280 54,272 3726444256 74,249 26,242 fl0234 95228 11,169 73,126 30093 98,069 93,052 0340413.359 YU4243444546413.460 703-564 523.671 453.781 603-895 044.01 1 904. I32 254.256 224.383 914x4950605.891 6070807.917 8210.640 8914.300 4719.218 63901006,970 87942 60915 14,88849,86261,837 48,813 09,789 41,766 42,7440972242701 3868095661 12641 86,623 17,605 0258739,570 29553 68,537 551.973 591-03? 792.093 782 . I56 592-221 292.287 932,356 572.427 262.500 OX2.575 ox2.652 342.731 912.813 862 898 282 985 233 074783 167033 262043132333435363738391.''1.030 001-06090L.092 731.125 511.159 271.194 051.229 871-266 771.304 771.343 9238423,425 76.46853512 59,55797,604 71,652 85,702431.753 511.806 1 I1.860 29sifI ,000 02-030 03.090 94.183 65.309 16.468 47.662 58-892 310.159 I11.43912-807814.192015617 817.086 318.598 920. IS6 921.761 623.414 425.116926.870 428676 530.536 832.452 934-426 536459 338.553 040 709 642.930 945.218 947.575 450.002 752.502 855.077 857.730 260.462 163.275 966. I74 26Y 159 471.134 275 401 378.663 382.023 285 483 989.048 492.719 996-501 5100.396 5104.408 4108.540 611 2.796 9I63453 4230.594 I321.363 0443.348 9607.287 7aii(urn)-'0.970 9 1.030 0001 9 1 3 5 0 522 61 I2.828 6,353 530,269 0273.717 I4.579 7,218 3555.417 2,184 5986,230 3,160 5067.019 7,142 4567.786 1,128 434X.510 2. I 17 231-108 0779 252 69.954 0.I0046210.6350494 03011,296 I,088 52611,9379,083 767,079 61 I12.561 I13,166 1.075 953,072 70913.753 514 323 8,069 81414.877 5,067 21615.4150,064 8721 5-936 9,062 74716.4436.060 X 14-059 04716 935 5,057 42817.413 I17.876 8,055 938-054 56418 327 018 764 I,053 29319,188 5,052 I 1 519.600 4.051 01920.000 4,049 999449 04720.388 820.765 8,048 1564 4 7 32221,131 821,4872,046 539,045 80421,832 3,045 I 1222 167 222.492 5,044 45922.808 2,043 X44-043 26223 1 1 4 823.412 4,042 71223 701 4,042 19223 981 9-041 698,041 23024 254 324,518 7.040 7x5-040 36324 775 425,024 7,039 9614 3 9 57825.266 7,039 2 1325 501 725.729 8-038 8654 % 1.3427.675 h29.123 4034 337-033 I I '30 200 83 1.002 4,032 256.03 I 64731 08090100
COMPOUND INTEREST TABLES3-j per 95060708090I001.035 001.071 231,108 721,147 521.187691.229 261.272 281.316 811 362 901.410 601.459 97I 511 071.563 961.618 691.675 351.733 991 794 681.857 491,922 501.989 792,059 432,131 512.206 112.283 332.363 242.445 962.531 572.620 172,711 882.806 792.905 033.006 713.1 11 943.220 863.333 593.450 273.571 033.696 013.825 373 - 9 9 264.097 834.241 264-389 704.543 344,702 364.866 945.037 285.213 595.396 065.584 937-878 0911.1128315.675 74Z2.112 I S31.191 41L'"s7iiaid,966 18,933 51,901 94,871 44.X41 97,813 50,785 99,759 41-733 73.708 92,684 95,661 78,639 40,617 78,596 89.576 71,557 20,538 36,520 16,502 57,485 57,469 15,453 29,437 96,423 15,408 84,39501,381 65,368 75,356 28,344 23,332 59.32 1 34,310 48,299 98,289 83,280 03-270 56,261 41,252 57,244 03,235 78-227 81,220 10,21266-205 47,198 52-191 81,185 32,179 05-126 93,08999,063 79,045 22,032061.000 02.035 03,106 24.214 95.362 56.550 27-179 4Y.051 710.368 511.731 413.142 014.602 016.1 I3 017.677 019.295 720.971 022.705 024.499 726,357 228,279 730.269 532.328 934.460 436666 538.949 941.313 143.759 146.290 648,910 X51 622 754.429 557.334 560.341 263.453 266 674 070.007 673.457 977 028 980.724 984,550 388.509 592.607 496.848 6101.238 3105.781 7110-4840115.351 0120.388 3125.601 8130.997 9196-5169288.937 9419.306 8603.205 0862.61 I 70.966 21,899 72.801 63.673 I4.515 I5.328 66.114 56.874 07.607 I8.316 69.00 I 69.663 310.302 710.920 511.517412.094 112.651 313.189 713,709 814.212 414.698 015,167 115.620 416.058 416-481 516.890 417.285 417.667 018.035 818 392 018.736 319.068 919-390219,700 720 000 720.290 520.570 520-841 I21.102 521.355 I21 599 I21.834 922.062 722,282 822.495 522 700 922,899 423.091 223.276 623,455 624.944 726400 426.748 827,279 327.655 4(ad '1.035 0000.526 400,356 934-272 251.22 I 48 1. I R7 668,163 544,145 417,131 446,120 241, 1 1 1 092103 484,097 062,091 571,086 825,082 685,079 043,075 8 I7,072 940070 361,068 037,065 932,064 019,062 273,060 674,059 205,057 8 5 2,056 603,055 445,054 37 1,053 372,052 442,051 572,050 760,049 998,049 284,048 6 13.047 982,047 388,046 827-046 298,045 798,045 325,044 878,044 453.044 05 1.043 669,043 306,042 962,042 634,040 089,038 461.037 385.036 658,036 j('2)n(I ;)!( I i): (I 3343536373839I.d(2'(/141( 121,',(21i.i(41,/i(lZ)in1,d2ji44)IloglO(l i)Value.035 000,034 699,034 550,034 451,034 40 I1.0173491.008 6371-002871,966 184,982 946,991 437,997 137,033 8164 3 4 101034 254,034 3521.008 6751.013 03 I1 0159421,017 4001426 1751.021 7811,018 859,0149403404142434445464748495060708090I007
COMPOUND INTEREST TABLES4 per centConstantsValue,040000,039 608,039 414,039 285.039 22 I1.019 8041.009 8531.003 274(I ;)"nI1.040 002345678910-081 60,124 86,169 86,216 65265 32315 93368 57423 31480 24,539 45,601 03,665 07,731 6X800 94,872 98,947 90,025 82,106 8 52.191 122.278 772.369 922.464 722.563 302.665 842-772 472.883 372.998 703.118 653.243 403.373 133.508 063.648 383.794 323.946 094.103 934.268 094.438 814.616 374.801 024.993 065.192 785.400 505.616 525.841 186.074 826.317 826.570 536.833 357,106 6810.519 6315.571 6223.049 8034.1 19 3350.504 95II.961 538,980 581,990 243,996 737.038 4 6 2,038 839-039 029,039 157I .009 9021,014 8771,018 2041.019 8691.029 9021.024 8771.021 537lOgj,(l i),017 033 1 54,924 56,889 00854 80,821 93790 31759 92730 69702 59675 56,649 58,624 60,600 57.,577 48-555 26,533 91,513 37.493 63,474 64-456 39,438 83,421 96,405 73,390 12,375 12,360 69,346 82,333 48,320 65,308 32.296 46,285 06,274 09,263 55.253 42,243 67,234 30.225 29,216 62,208 29,200 28,192 57,185 17,178 05,171 20,164 61-158 28,152 19.I46 34,140 71.OY5 06.064 22,043 38,029 31,019 801.000 02.04003.121 64.246 55.416 36,633 07.898 39.214 210.582 812.006 113.486 415.025 816-626 818.291 920.023621.824 523.697 525645 427.671 229.778 13 1.969 234.248 036.617 939.082 641.645 944.31 1 747.084 249.967 652.966 356.084 959.328 362.701 566.209 569.857 973-652 277,598 381.702 285.970 390.409 I95.025 599.826 5104.819 6110.0 I2 4115.412 9121.029 4126.870 6132.945 4139.263 2145.833 7152.667 I237,990 7364.290 5551.245 0827.983 3I 237623 7aa(a&'0.961 5 1.040 0001.886 1 0.530 1962.775 I360 3493,629 9275 4904.451 8,224 627190 7625.242 16.002 I,166 6106.732 7,148 5287.435 3,134 4938-I10 9,123 291. I I4 1498.760 59.385 I,106 5529-9856-100 144,094 66910,563 111.1184 ,089 94111.652 3085 82012.165 7,082 19912.659 3,078 99313.133 9,076 13913.590 3,073 58214.029 2,071 28014.451 1,069 19914.856 8 ,067 30915.247 0,065 58715.622 I,064 01215.982 8,062 56716.329 6,061 23916.663 1,060 0 I316.983 7,058 88017.292 04 5 7 83017.588 5,056 85517.873 6,055 94918.147 6.OS5 10418.411 2-054 31518.664 6,053 57718,908 3,052 887,052 24019.142 619.367 9,051 63219.584 5,051 061,050 52319,792 819,993 I,050 017-049 54020.185 69 4 9 09020.370 820,548 8,048 665.048 26220.720 020,884 7,047 88221.042 9,047 52221.195 1,047 18121.341 5,046 85721.482 2.046 55022.623 5,044 202,042 74523.394 523.915 4.041 81424.267 3 .04I 20824.505 0-040 8090100
COMPOUND INTEREST TABLES4 per 50607080901001.045001.092 031.141 71.622851.695881.772201.851 941.935 282,022372.113 382,208482.307862.411 712.520242.633652.752172.876013.005433.140 683-282013.429703,584043,745323.913 864-089984.274034.466 364,667354.877385.096 865.326225.565 905.816 78.271468.643679.0326414.0274121.784 1433.8301052,5371 181.588 52c".956 94,91573,87630,838 56-80245,76790,73483,70319,672 90,64393,61620,58966,56427.53997,516 72,49447,47318,45280,43330,41464,39679,37970,363 35,34770,332 73,318 40-30469.29151,27902,26700,255 50.24450,23397,22390.2I425,205 03,196 20,18775.I7967,17193,16453,15744.I5066.I44 17-13796,13202,12634,12090. I 15 69-110 71,071 29,04590-02956,01903,012 26 3uii(4' nConstants1.00002.04503 137 1215.464017.159918.932 120.784 38.937041.68920,9569 1.04500011,8727 0.533 99822 749 0 ,363 77333.5875 ,27874444.3900 ,22779255 I57 9 -193878675.892 7 .I69 7016.5959 .I51 61087.2688 ,13757497.9127 ,126379 108.5289 ,117248 119.118 6,109 666129,6829 ,103 275 1310.2228 .097 820 1410,7395 ,093 114 15I 1.2340 .OS9015161 1.7072,085 418 1712.160 0 ,0822371812.5933-079407 1913.0079 ,076876 2013.4047 ,0746n I2113 784 4 ,072546 2214,1478 ,070682 2314.4955 .068 987 2444565 2 14.8282 .067439 2547.5706 15,1466 ,066021 2650-7113 15.4513 ,0647I9 2753.9933 15,7429 ,063521 2857.4230 16.0219 ,0624I 5 2961.0071 16.2889 ,061 392 3064.7524 16.5444 ,060 443 3168.6662 16,7889 ,059563 3272.7562 17.0229 .OS8 745 3377.0303 17.2468 ,057982 3481.4966 17.4610 ,057270 3586.1640 17,666n .OS6606 3691.0413 17.8622 ,055 984 37,055402 3896,1382 18.050n39101.4644 18,2297 ,054856107.0303 18-401 6.OS4343 40112.8467 18.566 1,053 862 41118.9248 18.723 5 ,053 409 42125.2764 18.8742 -052 982 43131.9138 19.0184 ,052 581 44,052202 45138-8500 19.1563146.0982 19.2884 -051 845 46153.6726 19.4147 4 5 1 507 47161.5879 19.5356 -051 189 48169.8594 19.6513 ,050887 49178.5030 19.762n,050 60250289.4980 20.6380 ,048454 60461.8697 21.202 I,047165 70729.5577 21.5653 ,046 371 80I 145.2690 21.7992 ,045873 90I 790,8560 21.9499 ,045558 100Value445 000,044 505,044 260.044 098,044 0171.0222521.011 0651.003 675,956938,978232.989056,996 339.043062,043536,043776,0439361.011 1261-0167201.02046I1.0223351,033 6261.0279701.024211log,,(l i),019116 39
COMPOUND INTEREST TABLES5 per centConstantsFunction1I@)ii4li(l2)6(I I) (l i)i(I i)hI'?.,c.tI.h 2)d(4id(l21I l(2)IIrbId(2'Id'4'I di12'login(l /)Value,050 000.049 390,049 089,048 889,048 7901.024 6951.012 2721.004 074,952 381,975 900-987 877,995 942047 619048 200048 494048 691I 0 1 2 3481018 559I022 7151 024 7971037 148I011 0591 026 881021 189 3n(l 90100101.050 001.102 501.157 631.215 511.276 281.340 101.407 101.477 461-551 331,628 891.710 34I 795 861.885 651.979 932.078 932.182 872.292 022.406 622.526 952,653 302,785 962.925 263,071 523.225 103-386 353.555 673.733 463.920 134.1 16 144.321 944.538 044.764 945-003 195.253 355.516 025-791 826,081 4 16.385 486.704 757.039 997.391 997.761 598.149 678.557 158.985 019.434 269.905 9710.401 2710.921 331 1.467 4018.679 1930,426 4349.561 4480 730 37131.501 26L.".952 38,907 03,863 84.E22 70,783 53,746 2271068.676 84.64461.613 91,58468,556 84.530 32,505 07,481 02,458 11,436 30.415 52.395 73,376 89,35894,341 85.325 57,310 07295 30,281 24,267 85-255 09,242 95,231 38-220 36,209 87-199 87,190 35,181 29172 66,16444,156 61,149 15,142 05,135 28,128 84,122 70,116 86. I I 1 30,106 00,100 95,096 14,091 56,087 20.OS3 54.032 87,020 18-012 39,007 60Siil1.000 02.050 03.152 54-310 15.525 66.801 98.142 09.549 111.026 612,577 914.206 815.917 117.713 019,598 621.578 623.657 525,840 428.132 430539 033.066 035.719 338.505 241.430 544,502 047.727 151.1 13 554.669 158-402 662.322 766.438 870.760 875.298 X80.063 885.067 090.320 395-836 3101.62X 1107.709 5114.095 0120.799 8127.839 8135.231 8142.993 3151.143 0159.700 2168.685 Z178.119 4188.025 4198.426 7209,348 035
Fund Tables: extension of the range of rates of interest in the Compound Interest section: omission of Premium Conversion Tables and ofa table of Officc Premiums for Contingcnt Assurances: and inclusion of additional Statistical Tables. a statement of the International Actuarial Notation. and Tables of Logarithms, Antilogarithms and Reciprocals.
