Sistemul Actuarial International de Notatii

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INTE RNATIONAL ACTUARIAL NOTATIONTHE existing int ernat iona l actua rial notat ion was foun ded on th e Key to theNotat ion given in th e Inst itu te of Actu ar ies Text-Book, Pa rt II, Life Con-tingencies, by George King, an d is embodied in an explan at ory stat emen tadopted un an imously by th e Second Int ern at iona l Actu ar ial Congress held inLondon in May 1898 an d print ed on pp. 618-40 of the Tra nsa ctions of th atCongress. At th e Third Int ern at iona l Congress held in Par is in J un e 1900 afur th er stat ement by Dr Sprague was submitted which rear ra nged th e symbolsin differen t order s a nd grouped th em on different principles but did not int ro-duce an y chan ges in th e symbols t hem selves beyond two slight additions. Thissta tem ent is print ed on pp. 622-51 of th e Tra nsa ctionsof t ha t Congress. At th eEleventh Intern at iona l Congress held in Pa ris in Ju ne 1937 an In tern at iona lCommittee was appointed to consider the question of notation and given theduty of submitting definite proposals to the next Congress. This Committeemet in Bru ssels in J uly 1938 an d again in J uly 1939, an d certa in cha nges wouldha ve been proposed for adoption at Lucerne in 1940. Owing to th e war thiscould not be done, but in order n ot t o lose the benefit of these cha nges for anindefinite period th e Inst itut e of Actu ar ies and th e Fa culty of Actu ar ies ha vedecided to adopt those recommendations which had received unanimous orsubstan tial support at the meetings of th e Comm ittee.

The change is to take effect with Vol. LXXVIof th e J our na l an d with th eMay 1950 examina tions of th e Inst itut e of Actu ar ies.The genera l pr inciples on which th e system is based ar e as follows:To each fun dam ent al symbolic lett er ar e at ta ched signs an d lett ers each

having its own signification.The lower spa ce to th e left is reser ved for signs indicat ing th e cond itions

relative to the duration of the operations and to their position with regard totime.

The lower space to the right is reserved for signs indicating the conditionsrela tive to a ges and t he order of succession of the event s.

The upper space to th e r ight is reserved for signs indicat ing th e periodicityof the events.

The upper spa ce to the left is free, a nd in it can be pla ced signs corr espondingto other notions.

In wha t follows th ese two conven tions ar e used:A letter enclosed in bra cket s, th us (x), denotes a per son a ged x. A letter or

nu mber enclosed in a right an gle, th us n or 15 , denotes a ter m-cert ain of years.

FUNDAMENTAL SYMBOLIC LETTERSInterest

i= th e effective ra te of int erest, na mely, th e total inter est ear ned on 1 in a yearon th e assu mpt ion th at th e a ctu al inter est (if receivable other wise tha nyear ly) is invested fort hwith as it becomes due on th e same term s as th eoriginal p rincipal.

JIA 75 (1949) 0121-0129


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122 International Actuarial Notationth e present value of 1 due a year hence.

th e discoun t on 1 due a year hen ce.th e force of int erest or th e force of d iscoun t.

Mortality Tablesl = nu mber living.d = nu mber dying.p = pr obability of living.q = pr obability of dying. = force of mor tality.m = central death rat e.a = present value of an a nn uity.s = amoun t of an a nn uity.e = expecta t ion of life.

A = presen t value of an a ssur an ce.E = presen t value of an en dowment .

= premium per an nu m. P genera lly refers to net prem iums, to specialpremiums.

V = policy value.W = paid-up policy.

The met hods of u sing the foregoing principal letter s and their precisemeanin g when added t o by suffixes, etc., follow.

Interestth e nominal ra te of inter est, convert ible m times a year.

th e value of an an nu ity-certa in of 1 per an nu m forn year s, th e payment s being ma de at the end of each year.

th e value of a similar a nn uity, th e pa yment s beingma de at th e beginn ing of each year.

th e am oun t of an an nu ity-cert ain of1 per an nu m for n years, th e payments being m ade at t he end of each year.

th e am oun t of a similar an nu ity, th epaymen ts being m ade a t th e beginn ing of each year .

The diaeresis or tr ema above t he lett ers a an d s is used as a symbol ofaccelera tion of paym ent s.

Mortality TablesThe ages of t he lives involved a re denoted by lett ers placed a s suffixes in th e

lower space to th e right. Thu s:lx = th e nu mber of persons who at ta in age x according to t he morta lity t able.dx=lx-lx+l = th e nu mber of per sons who die bet ween a ges x and x+1

according to t he morta lity ta ble.px = th e probability th a t (x) will live 1 year .qx = th e probability th at (x) will die within 1 year .

logogooogog


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International Actuarial Notationth e force of mort ality a t a ge x.

the central death-rate for the year of age x tothe cur ta te expecta t ion of life (or a vera ge a fter -lifet ime) of (x).

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12 4 I n t e r n a t ion a l A ct u a r i a l N ot a t i onI f t h e n i n s t e a d of w r i t i n g a b a r is p l a ce d o ve r t h e p r i n c ip a l

s y m b o l . T h u s := a con t i n u ou s o r m om e n t l y a n n u i t y.= a n a s s u r a n ce p a y a b le a t t h e m o m e n t of d e a t h .A s m a l l ci r c le p l a ce d o v e r t h e p r i n ci p a l s y m b o l s h o w s t h a t t h e b e n e fi t i s t o b e

c o m p l e t e . T h u s := a com p l e t e a n n u i t y.= t h e c om p l e t e e x p e c t a t i on o f l ife .N o t e . S o m e con s i d e r t h a t w o u l d b e a s a p p r o p r i a t e a s A s a t r a t e

o f i n t e r e s t i = 0 , s o a l s o t h e c o m p l e t e e x p e c t a t i on o f l ife = a t r a t e o fi n t e r e s t i = 0 .

W h e n m o r e t h a n o n e l i f e i s i n v o l v e d t h e f o l l o w i n g r u l e s a r e o b s e r v e d :I f t h e r e a r e t w o o r m o r e l e t t e r s o r n u m b e r s i n a s u f fix w i t h o u t a n y d is t i n g u i s h -

i n g m a r k , joi n t l iv e s a r e i n t e n d e d . T h u s :

N o t e . W h e n , fo r t h e s a k e o f d i s t in c t n e s s , it is d e s i r e d t o s e p a r a t e t h el e t t e r s or n u m b e r s i n a s u f fi x, a co lo n is p l a c e d b e t w e e n t h e m . A co lo ni s u s e d i n s t e a d o f a p o in t or co m m a t o a v o i d co n fu s i on w i t h d e c im a l sw h e n n u m b e r s a r e in v olv ed .= a n a n n u it y , fir s t p a y m e n t a t t h e e n d of a y e a r , t o con t i n u e d u r i n g t h e

joint l ives of (x), (y) and (z).= a n a s s u r a n c e p a y a b l e a t t h e e n d o f t h e y e a r o f t h e f a i l u r e o f t h e j o i n t

l ives (x), (y) and (z).I n p l a c e o f a l i f e a t e r m - c e r t a i n m a y b e i n v o l v e d . T h u s :

= a n a n n u it y t o con t i n u e d u r i n g t h e joi n t d u r a t i on o f t h e l ife of (x ) a n d at e r m of n y e a r s ce r t a i n ; t h a t i s, a t e m p o r a r y a n n u i t y for n y ea r s o n t h elife of(x).a n a s s u r a n ce p a y a b l e a t t h e e n d o f t h e ye a r o f d e a t h o f (x ) if h e d iew i t h i n n y e a r s , o r a t t h e e n d o f n y e a r s if (x ) b e t h e n a l i ve ; t h a t i s , a ne n d o w m e n t a s s u r a n c e for n y e a r s .

I f a p e r p e n d i cu l a r b a r s e p a r a t e s t h e l et t e r s in t h e s u f fi x, t h e n t h e s t a t u s a ft e rt h e b a r i s t o f o l l o w t h e s t a t u s b e f o r e t h e b a r . T h u s :

a r e v e r s i on a r y a n n u i t y, t h a t i s, a n a n n u i t y on t h e li fe o f (x ) a f t e r t h ed e a t h o f ( y ) .a n a s s u r a n c e p a y a b l e o n t h e fa i l u r e o f t h e jo in t l iv e s (x ) a n d ( y) p r o v id e db o t h t h e s e l iv e s s u r v i v e ( z ).

I f a h o r i z on t a l b a r a p p e a r s a b o v e t h e s u f fi x t h e n s u r v i v or s o f t h e l iv e s , a n dn o t joi n t li ve s , a r e i n t e n d e d . T h e n u m b e r o f s u r v iv or s ca n b e d e n o t e d b y a l e t t e ro r n u m b e r o v e r t h e r i g h t e n d o f t h e b a r . I f t h a t l e t t e r , s a y r , i s n o t d i s t i n g u i s h e db y a n y m a r k , t h e n t h e m e a n in g is a t least r su rv ivors ; bu t i f i t i s e n c lose d ins q u a r e b r a c k e t s , [r ] , t h e n t h e m e a n i n g is exactly r s u r v i v or s . I f n o l e t t e r or


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International Actuarial Notation 125number appears over the bar, then unity is supposed and the meaning isat least one sur vivor. Thu s:

an an nu ity payable so long as at leas t one of the thr ee lives (x), (y) and(z) is a live.an a nn uity payable so long as at least two of th e th ree lives (x), (y) and(z) a re a live.probability that exactly two of the three lives (x), (y) and (z) will sur vivea year.probability th at th e sur vivor of the t wo lives (x) and (y) will die withinn yearan a ssur an ce payable at th e end of t he year of deat h of t he sur vivor ofth e lives (x) an d (y) pr ovided th e deat h occur s with in n year s.

When num era ls ar e placed a bove or below th e lett ers of th e suffix, th eydesigna te th e order in which th e lives a re to fail. The nu mer al placed over th esuffix point s out th e life whose failure will fina lly determine th e event ; an d th enu mer als placed u nder th e suffix indicat e th e order in which th e oth er livesinvolved a re to fail, Thu s:

an a ssur an ce payable at th e end of th e year of deat h of (x) if he diesfirst of the two lives (x) and (y).an a ssur an ce payable a t the end of t he year of dea th of (x) if he diessecond of the three lives (x), (y) and (z).an assu ra nce payable a t th e end of t he year of deat h of (x) i f he diessecond of the three lives, (y) having died first.an a ssur an ce payable a t the end of t he year of dea th of th e survivor of3 (x)and (y) if he dies before (z).an assu ra nce payable at the end of the year of deat h of (x) if he dieswith in a ter m of n years.

an annuity to (x) after the failure of the survivor of (y) and (z),pr ovided (z) fails before (y).

Note. Sometimes to mak e quite clear th a t a joint -life st at us is involved asymbol is placed above th e lives included. Thu s joint -life t empora ry assu ra nce on (x) and (y).

In th e case of reversiona ry an nu ities, distinction ha s somet imes to be madebetween t hose where th e times of year at which paymen ts ar e to ta ke place ar edetermined at the outset a nd those where the t imes depend on th e failur e ofth e preceding sta tu s. Thus:

an nu ity to (x), first paymen t a t th e end of th e year of th e dea th of(y) or,on th e average, about 6 mont hs after his death .an nu ity to (x), first paymen t 1 year after the deat h of (y).

complete an nu ity to (x), firs t paymen t 1 year a fter th e deat h of(y).


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International Actuarial NotationANNUAL PREMIUMS

The symbol P with the appropriate suffix or suffixes used in simple cases,where no misunderstanding can occur, to denote the annual premium for abenefit. Thus:= the annual premium for an assurance payable at the end of the year ofdeath of (x).

= the annual premium for an endowment assurance on (x) payable aftern years or at the end of the year of death of (x) if he die within n years.= the annual premium for a contingent assurance payable at the end ofthe year of death of (x) if he die before (y).

In all these cases it is optional to use the symbol P in conjunction with theprincipal symbol denoting the benefit. Thus instead of we may writeIn the more complicated cases it is necessary to use the two symbolsin this way. Suffixes, etc., showing the conditions of the benefit are to beattached to the principal letter, and those showing the conditions of paymentof the premium are to be attached to the subsidiary symbol P. Thus:= the annual premium payable for n years only for an assurancepayable at the moment of the death of (x).= the annual premium payable during the joint lives of (x) and (y)for an assurance payable at the end of the year of death of (x).

= the annual premium payable for n years only for an annuity on(x) deferred n years.= the annual premium payable for t years only, by m instalmentsthroughout the year, for an endowment assurance for n yearson (x) (see below as to P[m]).

Notes. (1) As a general rule the symbol P could be used without the principalsymbol in the case of assurance where the sum assured is payable atthe end of the year of death, but if it is payable at other if thebenefit is an annuity, then the principal symbol should be used.(2) A point which was not brought out when the internationalsystem was adopted is that there are two kinds of premiums payable mtimes a year, viz. those which cease on payment of the instalment im-mediately preceding death and those which continue to be payable tothe end of the year of death. To distinguish the latter the m is some-times enclosed in square brackets, thus P[m].

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International Actuarial Notation 127Note. As a genera l ru le V or W can be used a s th e ma in symbol if th e sum

assu red is payable at the end of th e year of death an d th e premium ispayable periodically th roughout th e dur at ion of the assu ra nce. If thepremium is payable for a limited num ber of year s, say n, th e policy valueafter t year s could be writt en , or, if desired

In investigations where modified prem iums an d policy values ar e in questionsuch modifica tion ma y be den oted by adding accent s to the symbols. Thu s,when a premium oth er th an th e net premium (a valuat ion premium) is used ina valua tion it ma y be denoted by P and the corr esponding policy value by V.Similar ly, the office (or comm ercial) pr emium ma y be denoted by P and thecorresponding paid-up policy by W.

COMPOUND SYMBOLS(Ia) = an ann uity(IA) =an assurance comm encing at 1 a nd increasing 1per annum.

If the whole benefit is to be temporary the symbol of limitation is placedoutside the brackets. Thus:

= a tempora ry increasing ann uity.= a tempora ry increasing assura nce.

If only t he increa se is to be t emporar y but th e benefit is to cont inue t her eafter,th en th e symbol of limita tion is placed imm ediat ely after th e symbol 1. Thu s:

= a whole-life an nu ity= a whole-life assu ra nce increasing for n years and thereafter stationary.

If the benefit is a decreasin g one, th e corr esponding symbol is D. Fr om t hena tu re of th e case this decrease mu st ha ve a limit, as oth erwise negat ive valuesmight be implied. Thus:

= a tem pora ry a ssur an ce comm encing at n an d decreasing by 1 ineach successive year.

If the benefit is a varying one, th e corr esponding symbol is v. Thu s:(va) = a varying a nn uity.

COMMUTATION COLUMNSSingle lives


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128 International Actuarial NotationWhen it is desired to const ru ct th e assu ra nce column s so a s to give directly

assu ra nces payable at th e moment of deat h the symbols ar e distinguished by abar placed over them. Thus:

which is an approximation to

Joint lives

SELECTIONIf t he su ffix to a symbol which denotes th e a ge is enclosed in squa re bra cket s,

it indicat es the a ge at which th e life was selected. To th is ma y be a dded, out sideth e bra ckets, th e nu mber of years which ha ve elapsed since selection, s o tha t thetotal suffix denotes t he presen t age. Thu s:

= t he n um ber in th e select life ta ble who were selected at age x and ha veat ta ined age x + t .

a[x]= value of an annuity on a life now aged x and now select.a[x-n]+n = valueof an an nu ity on a life now aged x an d select n years ago at

age x - n .

and similarly for other functions.When Dr Sprague presented his stat ement, he ment ioned th at an objection

ha d been ra ised tha t th e nota tion in some cases offers th e choice of two symbolsfor t he sam e benefit. For insta nce, a temporar y an nu ity ma y be denoted eith erby or by This is, he says, a necessar y consequen ce of th e pr inciplesun derlying th e system, a nd n eith er of th e altern at ive form s could ha ve beensuppressed with out injur y to t he symm etry of th e system.

The following ar e th e chan ges which ha ve now been int roduced:(1) In ter est . In place of th e symbol j(m) = he nomina l ra te of int erest con-

vertible m t imes in a year wh en th e effective ra te is i, th e symbol i(m)ha s beensubstituted; the former definitions of i(m)an d as th e effective ra te of inter estwhen inter est is convertible m times a year a nd m omen tly respectively ha vebeen abolished.


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International Actuarial Notation 129(2) Ann uities-due. In place of th e symbol a for an annu ity un der which th e

first paymen t is to be m ade at once t he symbol has been a dopted. This adop-tion of the tr ema . as a symbol of accelera tion has permitt ed th e int roduction of

for th e accum ulat ed a mount of an ann uity-due at th e end of the ter m, forwhich no symbol previously existed.(3) Mort ality. The use of Q for t he probability of dying with in a longer ter m

th an I year has been omitted.(4) Compound symbols. The use of D in a compoun d symbol to indicat e

decrea sing benefits has been intr oduced.(5) Comm ut at ion colum ns . The old definition of

etc. ha s been abolished in favour of etc. This ren dersth e use of an d for th e latt er series un necessar y.

Though th e use of j(m) o repr esent th e nominal ra te of inter est is supersededby i(m) s an intern at iona l symbol, it was recognized by the Comm ittee th at its u selocally, where it ma y be a ma tt er of convenience a nd cann ot be misun derstood,is not pr event ed. A suggestion th a t U should be int roduced in to the officialintern at iona l notat ion as a symbol for th e sur ren der value of a policy was notadopted, though it was agreed that, if a symbol were considered necessary,U should be chosen.

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