INTERNATIONAL ACTUARIAL NOTATION 123
INTERNATIONAL ACTUARIAL NOTATION
ВУ
F. S. PERRYMAN
The Council of the Society announces that the revised International Actuarial N otation, particulars of which are given below, will Ье adopted in the Proceedings of the Society as from the present Number. The re vised Notation will also Ье used Ьу the Examination Committee for all examinations subsequent to those in Мау, i950. The revised Notation has been adopted Ьу the Society of Actuaries and also Ьу the Institute of Actuaries in England, the Faculty of Actuaries in Scotland and Ьу actuarial societies in various other foreign countries. Before giving an account of the circumstances leading up to the present revision and а complete account of the Notation it is desiraЫe to state here the principal changes which are now being introducd:
" ( 1) Interest. In place of the symbol j (m) =the nominal rate of interest convertiЫe m times in а year when the effective rate is i, the symbol j(m) has been substituted; the former defini- tions of i (m) and [ as the effective rate of interest when interest is convertiЫe m times а year and momently respec tively have been abolished.
"(2) Annuities-due. In place of the symbol а for an annuity under which the first payment is to Ье made at once the symbol а has been adopted. This adoption of the trema " as а symbol of acceleration has permitted the introduction of s for the ac cumulated amount of an annuity-due at the end of the term, for which no symbol previously existed.
"(3) Mortality. The use of Q for the probaЬility of dying within а longer term than 1 year has been omitted.
" (4) Compound symbols. The use of D in а compound symbol to indicate decreasing benefits has been introduced.
"(5) Commutation columns. The old definition of Nx=Dx+i +
Dx+ 2 + Dх+з etc. has been abolished in favour of Nx = Dx+ Dx+1 +Dx+2 etc. This renders the use of Nx and Nx for the latter series un necessary." The complete statement of the revised Notation and its history follows: 124 INTERNATIONAL ACTUARIAL NOTATION
The existing international actuarial notation was founded on the ' Кеу to the Notatioп' given in the ]nstitute of Actuaries Text-Book, Part 11, Ltfe Con tingencies, Ьу George Кing, and is embodied in an exp\anatory statement adopted unaпimously Ьу the Second International Actuarial Congress held in London in Мау 1898 and printed оп рр. 618-40 of the Transactions of that Coпgress. At the Third lnterпatioпal Congress held iп Paris iri Juпe 1900 а further statemeпt Ьу Dr Sprague was submitted which rearranged the symbols iп different orders and grouped them оп differeпt principles but did поt iпtro duce апу chaпges iп the symbols themselves Ьеуопd two slight additioпs. This statemeпt is printed оп рр. 622-51 of the Transactions of that Congress. At the Eleventh Iпternational Congress held in Paris in Juпe 1937 an Iпternatioпal Committee was appoiпted to coпsider the question of пotation апd giveп the duty of submitting defiпite proposals to the пехt Congress. This Committee met iп Brussels iп July 1938 апd again in July 1939, апd certaiп chaп.,ges would have been propbsed for adoption at Lucerne iп l 940. 0·.ving to the war this could nQt Ье dопе, but in or.der not to lose the benefit of these changes for an indefinite · period the Iпstitute of Actuaries апd the Faculty of Actuaries have decided to adopt at а date to Ье mutually arranged those recommeпdatioпs which had received uпaпimous or substaпtial support at the meetiпgs of the Committee. As тапу studeпts апd members of the professioп have not ready access to the above-mentioned Transactions it has Ьееп thought desiraЬle to prepare апd issue the followiпg statemeпt of the Iпterпational Actuarial Notatioп after embodying iп it the chaпges arisiпg from the recommeпdations mentioned above.
The geпeral principles оп which the system is based are as follows: То each fuпdamental symbolic letter are attached signs апd letters each haviпg its оwп sigпification. The lower space to the left is reserved for signs indicatiпg the coпditions relative to the duratioп of the operations and to their position with regard to time. The lower space to the right is r•:served for sigпs iпdicatiпg the conditioпs relative to ages and the order of succession of the everits. The upper space to the right is reserved for sigпs indicatiпg the periodicity of the events. The upper space to the left is free, and in it сап Ье placed signs corresponding to other notions.
Iп what follows these two conventions are used: А letter enclosed in brackets, thus (х), denotes 'а persoп aged х'. А letter or numberenclosed iп а right angle, thus 7iJ or lsl, dt:пotes а term-certain ofyears.
FUNDAMENTAL SYMBOLIC LETTERS Interest i = the effective rate of interest, пamely, the total interest earned on I in а year on the assumpti<;>n that the actual interest (if receivaЫe otherwise than yearly) is invested forthwith as it becomes due on the same terms as the original principal. INTERNATIONAL ACTUARIAL NOTATION 125 v = (1 +i)-1 = the present value of 1 due а year hence. d = 1 -v = the discount on 1 due а year hence. 3 = lo& ( 1 + i) = - lo& ( 1 -d) = the force of interest or the force of discount.
Mortality ТаЬ!еS' l = number living. d = number dying. р = probaЬility of living. q = probaЬility of dying. µ = force of mortality. т = central death rate. а = present value of an annuity. S' = amount of an annuity. е = expectation of Iife. А = present value of an assurance. Е = present value of an endowment. Р} = prem~um per annum. Р generally refers to net premiums, ~т to special тт premшms. V = policy value. W = paid-up policy. The methods of using tl1e foregoing principal letters and their precise meaning when added to Ьу suffixes, etc., fo\1ow.
Interest i<т> = т{(1+i) 1 fm_1} = the nomiпal 1·ate of interest, convertiЫe т times а year. ani = v + v2 + ... + v" = the valuc of an annuity-certain of 1 per annum for п years, tl1e payments being mad~ at the end of each year. 2 11 1 ап~ = 1 + v + v + ... + v - = the value of а similar annuity, the payments being made at the beginning of each year. 2 11 1 S'nJ = r + ( I + i) + ( r + i) + ... + ( 1 + i) - = the amount of an annuity-certain of 1 pcr annum for п years, the payments being made at the end of each year. sni = ( I + i) + ( I + i)2 + ., . + ( 1 +i)" = the amount of а similar annuity, the payments being made at the beginning of each year. The diaeresis or trema above the lette!"s а and s is used as а symbol of acceleration of payшcnts.
Mortality ТаЫеs The ages of the lives involved are denoted Ьу letters placed as suffixes in the \o\ver space to the right. Thus: lж = the number of persons who attain age х according to the mortality tаЫе. dж = {" -!" ~ 1 = thc number of persons who die between ages х and х + r accordiг.g to the mortality tаЫе. Рж = the pl'ObaЬility that (х) will Iive 1 year. qж = the probaЬility that (х) will die within 1 year. 126 INTERNATIONAL ACTUARIAL NOTATION
Р.х = - ~ ~; = the force of mortality at age х.
тх = the central death-rate fo~ the year of age х to х + 1 = dx / f: lжн dt. ех = the curtate 'expectation of life' (or average after-lifetime) of (х).
ln the following it is always to Ье understood (unless otherwise expressed) that the annual payment of an annuity is 1, that the sum assured in any case is 1, and that the symbols indicate the present values: ах= an annuity, first payment at the end of а year, to continue during the life of (х). iix = 1 +аж = an 'annuity-due' to continue during the life of (х), the first payment to Ье made at once. Ах = an assurance рауаЫе at the end of the year of death of (х). Note. ех =ах at rate of interest i =о.
А letter or number at the lower left corner of the principal symbol denotes the number of years involved in the probability or benefit in question. Thus: пРх = the probaЬility that (х) wШ Iive п years. nqx == the probaЬility that (х) will die within п years. Note. When п = х it is customary to omit it, as shown on page 2, provided no amЬiguity is introduced. nEx == vnnPж = the valu'e of an endowment on (х) рауаЫе at the end of п years if (х) Ье then alive.
If the letter or number comes before а perpendicular bar it shows that а period of deferment is meant. Thus: nlqx == the probaЬility that (х) will die in а year, deferred п years; that is, that he will die in the (п + 1 )th year. ~Jax == an annuity on (х) deferred п years; that is, that the first payment is to Ье made at the end of (п + х) years. nltax = an intercepted or deferred temporary annuity on (х) deferred п years and, after that, to run for t years.
А letter or number in brackets at the upper right comer of the principal symbol shows the number of intervals into which the year is to Ье divided. Thus: а 1 ~ 1 = an annuity on (х) рауаЫе Ьу т instalments of i/m each throughout the year, the first payment being one of 1/т at the end of the first r/mth of а year. а 1 ~> =а similar annuity but the first payment of х/т is to Ье made at once, so that
1 1 1 i'i '; = х/т +а ~>. N';1 == an assurance рауаЫе at the end of that fraction х/т of а year in which (х) dies. INTERNATIONAL ACTUARIAL NOTATION 127
If т~оо then instead of writing (оо) а bar is placed over the principal symbol. Thus: ii = а continuous or momently annuity. А = an assurance рауаЫе at the moment of death.
А small circle placed over the principal symbol shows that the benefit is to Ье complete. Thus: 8 = а complete annuity. ~ = the complete expectation of life. Note. Some consider that ё would Ье as appropriate as ~. As ех = ах at rate of interest i = о, so also the complete expectation of life == lix at rate of interest i = о. When more than one life is involved the following rules are observed: If there are two or more letters or numbers in а suffix without any distinguish ing mark, joint lives are intended. Thus:
[Х/1 = fx Х / 11 , dZll = lx 11 -lx+I :ll+l' Note. When, for the sake of distinctness, it is desired to separate the letters or numbers in а suffix, а colon is placed between them. А colon is used instead of а point or comma to avoid confusion with decimals when numbers are involved. aXJ1z = an annuity, first payment at the end of а year, to continue during the joint lives of (х), (у) and (z). А3'11• = an assurance рауаЫе at the end of the year of the failure of the joint lives (х), (у) and (z).
In place of а life а term-certain тау Ье involved. Thus: Dz:nJ = an annuity to continue during the joint duration of the Iife of (х) and а term of п years certain; that is, а temporary annuity for п years on the Ые of (х). Ax:1i1 = an assuraпce рауаЫе at the end of the year of death of (х) if he die within п years, or at the end of п years if (х) Ье then alive; that is, an endo\vment assurance for п years.
If а perpendicular bar separates the letters in the suffix, then the status after the bar is to follow the status beforc the bar. Thus: avix =а reversionary annuity, that is, an annuity on the life of (х) after the death of (у). А, 1 ху = an assurance рауаЫе on the failure of the joint lives (х) and (у) provided both these lives survive (z).
If а horizontal bar appears above tl1e suffix then survivors of the Iives, and not joint lives, are intended. Tl1e number of survivors can Ье dcnotcd Ьу а Ietter or number over the right end of the bar. If that letter, say r, is not distinguished Ьу any mark, then the meaning is at ,least r survivors; but if it is enclosed in square brackets, [r], then the meaning is exactly r survivors. If no Ietter or 128 INTERNATIONAL ACTUARIAL NOTATION number appears over the bar, then unity is supposed and the meaning is at least опе survivor. Thus: ахи•= an ~nn~ity рауаЫе so long as at least one of the three lives (х). (у) and (z) 1s al1ve. a"'ll; = an annui!J рауаЬ!е so long as at least two of the three lives (х), (у) and (z) are al1ve. Рх~: = probabllity that exactly two of the three lives (х). (у) and (z) will survive а year. nЧЖУ = probabllity that the survivor of the two lives (х) and (у) \Vill die within п years = nЧх х nЧи· nAжv = an assurance рауаЫе at the end of the year of death of the survivor of the lives (х) and (у) provided the death occu.rs within п years. When numerals are placed above or below the letters of the suffix, they designate the order in \vhich the Iives are to fail. The numeral placed over the suffix points out the life whose failure will finally determine the event; and the numerals placed under the suffix indicate the order in which the other Iives involved are to fail. Thus: А1 11 = an assurance рауаЬ!е at the end of the year of death Of (х) if he dies first of the t\vo lives (х) and (у). A~yz = an assurance рауаЫе at the end of the year of death of (х) if he dies sccond of the three lives (х), (у) and (z). А~уо = an assurance рауаЫе at the end of the year of death of (х) if he dies 1 second of tl1e thгee lives, (у) having died first. Ажv:о = an assшance р:~уаЬ!е at the end of the year of death of the survivor of " 3 (х) and (у) if he dies before (z). A1:ni = an assurance рауаЫе at the end of the year of death of (х) if he dies within а term of п years. a;µlx} d'r = an annuity to (х) after the failure of the survivor of (у) and (z), afu- x provided (;r) fails before (у). 1 Note. Sometimcs to make quite clear that а joint-life status is involved а symbol ,...... , is placed above tl1e livcs included. Thus A;1v:ii1 =а joint life temporary assurance on (х) and (у).
In the case of reversionary annuities, distinctil)n has sometimes to Ье made between those where the times of year at which payments are to take place are determined at the outset and those where the times depend on the failure of the prcceding status. Thus: а 111 х = annuity to (х), first payment at the end of the year of the death of (у) or, 011 the average, about 6 months after his death. Ь111 х = annuity to (х), first payment 1 year after the death of (у).
8111" = complete annuity to (х), first payment 1 year after the death of (у). INTERNATIONAL ACTUARIAL NOTATION 129
ANNUAL PREMIUMS
The symbol Р with the appropriate suffix or suffixes is used in simple cases, where no misuпderstaпdiпg сап occur, to denote the annual premium for а beпefit. Thus: Р ж = the аппuаl premium for an assuraпce рауаЫе at the end of the year of death of (х). Р:i::7il = the annual premium for an endowmeпt assuraпce ·оп (х) рауаЫе after п years or at the епd of the year of death of (х) if he die withiп п years. Р~и = the anпual premium for а coпtingeпt assurance рауаЫе at the end of the year of death of (х) if he die before (у).
Iп all these cases it is optional to use the symbol Р in conjunction with the principal symbol deпoting the benefit. Thus instead of Рж:nт we may \Vrite P(Ax:ni)· In the more complicated cases it is necessary to use the t\ю symbols in this v.·ay. Suffixes, etc., showing the conditions of the benefit are to Ье attached to the priпcipal letter, and those showing the coпditions of payment of the premium z.re to Ье attached to the subsidiary symbol Р. Thus: nP(Ax) = the annual premium рауаЬ!е for п years only for an assuraпce рауаЬ!е at the moment of the death of (х). Рх 11 (Ах) = the aпnual premium рауаЫе during the joiпt lives of (."() and (у) for an assurance рауаЫе at the end of the year of death of (х). nP(nJa") = the annual premium рауаЫе for п years only for ап annuity оп (х) deferred п years.
1P Notes. (1) As а general rule the symbol Р could Ье used \Yithout the principal symbol in the case of assuraпces ~1 here the sum assured is рауаЫе at the end of the year of death, but if it is рауаЬ!е at other times, or if the beпefit is ап aпnuity, then the priг.cipal symbol should Ье used. (2) P(;:JJ. А poiпt which was not brought out v.,·heп the interпatioпal system \Vas adopted is that there are two kiпds of premiums рауаЫе т times а year, viz. those \vhich cease оп paymeпt of the iпstalment im mediately precediпg death and those which coпtiпue to Ье рауаЬ!е to the епd of tЬе year of death. То distinguish the latter the т is some times enclosed iп square brackets, thus piml. POI,ICY VALUES AND PAID-UP POLICIES 1V"' = the value of ап ordinary whole-life assurance оп (х) \Vhich has Ьееп t years iп force, the prcmium then just duc being uпpaid. 1W"' = the paid-up policy the present value of which is 1Vx. The symbols V апd W тау, iп simple cascs, Ье used аlопе, but in the more complicated cases it is пecessary to iпsert the full symbol for the benefit thus: 1 1V(mJ(Ax:lii) (corresponding to р<т (Ах:т)), 1V(n1a,J. 130 INTERNATIONAL ACTUARIAL NOTATION Note. As а general rule V or W can Ье used as the main syrnbol if the sum assured is рауаЫе at the end of the year of death and the premium is рауаЫе periodically throughout the duration of the assurance. If the premium is рауаЫе for а limited number of years, say п, the policy value after t years could Ье written 1V[,.P(A)], or, if desired 7V(A). In investigations where modified premiums and policy values are in question such modification may Ье denoted Ьу adding accents to the symbols. Thus, when а premium other than the net premium (а valuation premium) is used in а valuation it may Ье denoted Ьу Р' and the corresponding policy value Ьу V'. Similarly, the office (or commercial) premium may Ье denoted Ьу рн and the corresponding paid-up policy Ьу W". COMPOUND SYMBOLS (Ia) = an annuity } . . . commencшg at 1 and шcreasшg 1 per annum. (IA) = an assurance If the whole benefit is to Ье temporary the symbol of limitation is p\aced outside the brackets. Thus: (Ia)x:1il = а temporary increasing annuity. (IA)~:1il = а temporary increa,sing assurancc .• If only the increase is to Ье temporary but the benefit is to continue thereafter, then the symbol of limitation is placed immediately after the symbol I. Thus: (I-a) = а whole-life annuity } . . . nl "' h l"f шcreasшg forn years and thereafterstatюnary. ( I 711А)" = а w о 1е- 1 е assurance If the bcncfit is а decreasing one, the corresponding symbol is D. From the nature of the case this decrease must have а limit, as othervrise negative values might Ьс implied. Thus: (DniA)~:nJ =а temporary assurance commencing at п and decreasing Ьу 1 in cach successive year. If tl1e bencfit is а varying one the corresponding symbol is v. Thus: (va) = а varying annuity. COMMUTATION COLUMNS Single lives D" = v"'l", N" = D"+Dxн +D"+2+ etc., S" = Nx+N"н +N"-1-2+ etc., с" = vz+l dx, l\i" = с"+ С"н + СХ+2 + etc., R" = Mx+Mx+l +Mz+2+ etc. INTERNATIONAL ACTUARIAL NOTATION 131 When it is desired to construct the assurance columns so as to give directly assurances рауаЬ!е at the moment of death the symbols are distinguished Ьу а bar placed over tliem. Thus: -Сж = vx+1 dx \Vh,ic!J is an approximation to Jl 'lf':+I /Lx+llx+1dt. ---- о Мж = C:i: + С:~:н + Сж+ 2 + etc. Rx = Мж + M:i:+l + M:i:+2 + etc. Joint lives DXJI = vl SELECTION If the suffix to а symbol which denotes the age is enclosed in а square bracket it indicates the age at \\•hich the life was selected. То this may Ье added, outside the bracket, the number of years \Vhich have elapsed since selection, so that the total suffix denotes the present age. Thus: /[ж)+1 = the number in the select life tаЫе who were selected at age х and have attained age х + t. dlж1+1 = l1xJ+1- l1:t1+1+ 1 • aixi = value of an annuity оп а lifc no\V aged х and now select. a1z-nl+n = value of an annuity on а life now aged х and sclcct п years ago at agex-n. N1:i:1 = D1x1 + Drx1н + D1:c1+2 + · ·· · il'rз:J = N1x17 D1:c1 = 1 + й~х1• and similarly for other functions. When Dr Sprague presented his statement, hc mentioned that an objection had been raised that the notation in some cases offers the choicc of two symbols for the same benefit. For instancc, а te·mporary annuity may Ье denoted either Ьу паж or Ьу ax:ni· This is, he says, а necessary consequence of the principlcs underlying the system, and neither of the alternative forms could havc bccn suppressed withoнt injury to the symmetry of the system.