,

618

THE UNIVERSAL NOTATION.
Explanatory Statement of the Principles underlying the 8yt>tem of
Universal Notation for Life Contingencies, adopted unanimouslyby the Second Inter')iational Actuarial Congress on May 19, 1898.
Prepared by the Committee, (consisting of M. Am. Begault,Brussels; Dii. J. Karup, Qotha; Mr. George King, London;
M. Leon Marie, Paris; and Dr. T. B. Sprague, Edinburgh)appointed for that purpose.

The Universal Notation, as embodied in the following Statement, hasbeen taken, almost as it stands, from the "Key to the Notation" givenin the Listitute of Actuaries Text-Book, Part II, Life Contingencies.
The simple principles on which the system is based are explained,examples are given to make the application of theseThose portions of the Notation of the Text-Book havebeen omitted which are not of an International character, and, in
accordance with a wish expressed by several at the Congress, emphasis
has been laid on the advantages, in many cases, of using the symbols
P, V, and W, in conjunction with other symbols, instead of alone;although, to meet the views strongly held by others, the option has
been retained of employing these letters by themselves in simple cases,Similarly, attention
where confusion could not thereby be caused.
has been specially called to the use of accents aft'ecting such symbols
as P, V, &c., to denote office premiums, valuation premiums, &c., andandsufficient

principles clear.

the corresponding values of other functions.
The only changes that have been introduced into the Notation offor
the Text-Book are, to substitute the hitherto unappropriated letter(FP), to represent the Paid-up Policy, such a change having been
urged by more than one speaker at the Congress and to limit q, theprobability of death, to a term of one year, Q being employed forlonger terms.
The Notation for Selection is that which was officially adopted
by the Institute of Actuaries of London for its two publications,
Select Life Tables and Joint Life Annuity Tables.

W;

:

Of('.

TH£

UNIVERSITYOFThe Universal Notation.
Interest :619

—i= theeffective rate of interest, namely,
actually realised on 1 in one year.

theinterest\^/(,„>,= m\ (1 +*)'"— 1 =the nominal rate of interest, convertibleiin times in a year, when the effective rate is i.

^(»i)

t_ ^i^g effective rate of interest, Avhen interest is convertible
w times in a year.
= *'*^ = the effective rate of interest when interest isconvertible momently.

= (1 +«*)"' = the present value of 1 due a year hence.
d = 1— = the discount on 1 due a year hence.
S == loge(l + = — logt(l — rf) = the force of interest orV^"

i)

j/*^,

the force of discount.
+ f^ + i'^ + &c. + 6'" = the value of an annuity for n years

tt/l!

=s^.

= l4-(l-f^)+(l-fi'certain.
/)2 + &c. + (l + i)'*-' = the amount of anannuity for « years certain.

Mortality Tables—A letter enclosed inbrackets,thus{x),denotes"apersonaged X."For each class of functions a principal letter is used, and its
meaning in any particular case is defined by suffixes, &c., as
circumstances may require.

We have= Number living-.
d = Number dying.
L = Population.
p = Probability of living.
9} — Probability of dying, q being used for a term limited to one
year, and Q for a longer term.
Qi ~= Force of mortality.
VI = Central death-rate.
e = Expectation of life.
E = Endowment.
a = Annuity, first payment to be made at the end of one period ofI\(/L6payment.

a = Annuity, first payment to be made at once.

A = Assurance, or Single Premium.
V = Policy Value.
W = Paid-up Policy.*,=Premium per annum, where P
premiums, andj'refers

generally tonet

premiums, for instance,premiums for assurances with return of premiums.
irtospecialThe ages of the lives involved are denoted by letters placed assymbol; and if twosuffixes at the lower right corner of the principal* In the Text-Book of the Institute of Actuaries this is denoted by (FP)

.

620

TTie

Universal Notation.

or more letters in a suffix are not distinguished by any special mark,joint lives are intended.

Thus,= Number living at age x according to the Mortality Table= Number dying between the ages x and x+l.
Px = Probability that {x) will live one year.
qx = Probability that {x) will die within the year.
mx = "Central death rate'^ for the year x to x-\-\, =iJbx+\
Ix

do!

approximately.
ax= Annuity, first payment at the end of a year, to continue

aj,= A similar annuity, first payment, however, to be made at

axyz= Annuity, first payment at the end of a year, to continue

during the life of(<»).

once.

during the joint lives of(<r),(t/),

and {z).

= Assurance payable at the end of the year of the death of [x).
Axyz = Assurance payable at the end of the year of the failure of theAa,

joint lives, {x), {y), and(z).

If one of the letters in the suffix is enclosed in a right angle,
the symbol denotes a term-certain, and not the age of a life. Thus,= Annuity to continue during the joint duration of the life ofax^=A^,-;-{x) and a term of n years certain; that is, a temporary
annuity for n years on the life of {x).
Assurance payable at the end of the year of the death of (x)if he die within n years, or at the end of n years if {x) bethen alive; that is, an endowment assurance for n years.

The suffix may consist only of a letter enclosed in a right angle,
in which case a term-certain only is indicated.

Thus,= Annuity for n years certain.
= Assurance payable at the end of n years certain.
A^ =a^v'"'

If a perpendicular bar separates the letters in the suffix, then thestatus after the bar is to follow the status before the bar.

ayx
Ae\xy

Thus,

= Annuity on the life of (x) after the death of {y).
= Assurance payable on the failure of the joint lives (x) and{y)provided these lives both survive (?).

a horizontal bar appears above the suffix, then survivors of thenot joint lives, are intended. The number of survivors isIfdenoted by a letter or number over the right end of the bar.
that letter, say r, is not distinguished by any mark, then the meaning
but if it is enclosed in square brackets, [r],is, at leaiit r survivors;then the meaning is exactly r survivors. If no letter appears over thebar, then unity is supposed, and the meaning is at least one survivor.
Thus,If

lives, ander*""*^'"^p^'"^""[r]= Expectationof life ofr survivors of them.

themlives

and thelast= Probability that exactly r lives out of m lives willsurvive a year.

a^ = Annuity on the last survivor of (x), (y), and (z).

The Universal Notation.

621

When numerals are placed above or below the letters of thedesignate the order in which the lives 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 lives involved are to fail.
Thus,suffix, they%']__ Total probability that /""''*'

w-ii-JId"= Assurance
payableiZd.

"*"'<' *'''•''« ''™'*-;;;;

at the end of theyear of the deathof (w) if he die last of the four lives, the other lives
having failed in the order (z) first, (y) second, and^u-j-ffz'-'(x) third.

_ Annuity to [x) after the failure of the survivor of {y)Jorand {z), provided (z) fail before (y).

\"!/Z,.T^Fy.z — Assurance payable at the end of the year of the death"of the survivor of{x)and (y) if he die before (z).

When, for the sake of distinctness in the symbol, it is desired toseparate the letters in the suffix, a colon is placed between them.
colon is used instead of a point or comma, so as not to confuse withdecimals when numbers take the place of letters. Thus, we write

A<^x+n:y + n) and A35;4o.

A letter at the lower left corner of the principal symbol denotes
the number of years involved in the probability or benefit in question.
Thus,
jil^x

„Ej;

= Probability that {x) will live n years.
= Value of endowment on {x), payable if he survive n years.

If the letter comes before a perpendicular bar, it shows that adeferred period is meant ; while if the letter comes after a perpendicular bar, it shows that a temporary period is meant.
Thus,n\qx= Probability that (x) will die in a year, deferred n years; thatis,that he will die in the {n + \)th. year.

= Probability that [x) will die within n years.
„\ax = cin\\x = Annuity on (x) deferred n years; that is, the first„Q^payment to be made at the end of [n+l] years.

—= Temporary annuity on {x) for n years.
f^a^ = Intercepted annuity, that is, an annuity on {x), deferred
Infixo,xii't

years, and after that to run for n years.

A letter in brackets at the upper right corner of the principal
symbol shows the number of intervals into which the year is to bedivided.
Jm)

Thus,_ Annuity payable by m instalments of — each throughout
the year.

AJ"*'= Assurance payable at the end of that fraction — of a year
in which {x) dies.

.

622

The Universal Notation.
If

the year be divided into an infinite

number ofa bar is placed over the principal symbol.

(oo ),

infinitesimalm become indefinitely great, then, instead of writing

parts, that is, if

Thus,

a = Continuous or momently annuity.
A = Assurance payable at the moment of death.
A small circle placed over the principal symbol shows that thebenefit is to be complete.

Thus,= Complete expectation of life.
= Complete annuity.

e«

In the case of Reversionary Annuities, distinction has sometimes
to be made between those where the times of year at which payments

are to take place, are determined at the outset ; and those where thetimes depend on the moment of failure of the preceding status. Thus,= Annuity to [x], first payment at the end of the year of theayj.

death of(7/),

or,on the average,sixmonthsafter hisdeath.
dy\^%y\^= Annuity to
first payment one year after the death of [y).
= Complete annuity to
first payment one year after the(<r),(.t),

death of (j/).

WVfor Policy-Value, andThe symbols P for Premium,for Freeor Paid-up Policy, are usually to be employed in conjunction with theother symbol denoting the benefit. Thus,P(Aj.j,)= The annual premium for a contingent assurance on {x)against ((/).

= Value after years of an endowment assurance on (x).
tW{Aj) = Paid-up Policy after years which is the equivalent oftV{A^:^)tt

an ordinary policy onthat has been(«)

tyears inforce.

showing the conditions of the benefit, are to beletter, andsuffixes, &c., showing theconditions of payment of the premium, are to be attached to theThus,
subsidiary symbol P.
Suffixes, &c.,

attachedtotheprincipal„P(Aa>)= Annual premium, payable for n years only, for an

Vxyi-Ax)— Annual premium, payable during the joint lives ofassurance payable at the moment of the death of (x).
{x) and {y), for an assurance payable at the end of
the year of the death of {x)Premium per annum for t years only, payable by nc
instalments throughout the year, for an endowmentassurance for n years on [x).

t?''^\kxn)=

permissible, however, in simple cases where no confusion
as symbols by
to employ the letters P, V, andthemselves. Thus, we may write Vly for P{AJ.y) ^V^^ji for ^V(Ax»!) ; andIt

canisWarise,;,W, for,W(A,).
In particular investigations, where modified values of functions
are in question, such modification may be denoted by adding accents:::

The Universal Notation.

623

Thus, where a premium otlier than the net premiumis used in a valuation, it may be denoted byP'; and the corresponding policy-value may be denoted by V.
Similarly, the "office" or "commercial'^ premium may be denoted byP", and the surrender- value of a policy by V, and the "office"paid-up policy by
The following compound symbols are used
to the symbols.

premium)(a valuation

W.

= Annuitycommencing at 1, and= Assurance increasing 1 per annum.
(va) = Varying annuity.
(vA) = Varying assurance.

is(la)((I A)(If the whole benefit is to be temporary, the symbol of limitation
placed outside the brackets. Thus.

= Temporary "increasing annuity.
=
assurance.
varyingannuity.
(va)^l =„assurance.
(vA)J,^j =„„{la)xn\{lA)lrti>}

ifIf only the increase or the variation is to be temporary, but thebenefit to be for the whole of life, then the symbol of limitation isplaced immediately after the symbol I or v. Thus,= Whole-life annuity=assurance„"~=(I;na)-r
rr \("^(i^jA)ar/"a \Commutation Columns

D* =^^

assurance„{v;rA)a;({£mcreasmgo tor « jyears.
•.

(1^varyingo for n jyears.
-^—v^'h-^^1= D^ + D^+,-HD^+2+&c.

= M;, + M^+, + Ma:+2+&c.

R^

When it is desired to construct the assurance columns so as
directly assurances payable at the moment of death, thesymbols are distinguished by a bar placed over them. Thus,to giveC^.

= -y^+^d^, approximately.

Rar= Ma; + Ma.+, + M;,+2+&C.

Joint Lives—J)^= V^Cxp=2-ydy = V ^IJy{lxly— lx+\ly+l)

djy + ^

::624

The Universal Notation.

Selection—Square brackets in the suffix to a symbol denote the age at which
was selected. Any additional term in the suffix, not enclosed
in square brackets, denotes the number of years which have elapsed
thelifesinceThe

selection.

total

present age of the life.
ftj-^-ia^_c]+nsuffix,

therefore, denotes, as usual, theThus,= value of an annuity on a life now aged x and now select.
= value of an annuity on a life now aged x + n, and selectn years ago at age x.

ttj;£._„]+„= value of an annuity on a life now aged x, and select nyears ago at age x — n.

Similarly for other functions.

Notes—Dr. Spkaguk, while approving in general of the list of Symbols
and Functions tabulated above, and therefore accepting this Schedule,
thinks that the arrangement and explanations might be improved, andthat some additional Symbols might, with advantage, be included.
He therefore proposes to prepare a Paper on the subject, and submit
itto the Paris Congress of 1900.

Dr. Sprague and Mr. King consider that P, V, and W, should be
used alone, and not in combination with other Symbols (such as Aor a), except in the case of complicated benefits where confusion
might otherwise arise.
Dr. Karup desires to point out that he considers the Symbols i^"^^and i to be superfluous, as ^, j\,„) and 8 sufficed to express all therelations between the effective and the nominal rates of interest.

For from thedefinitionof/,„) itfollows that i=M+'^^^'"M—!_,

and therefore that i'"" and / are in fact identical. And similarly it
follows, from the definition of 8, that i = e* — l = e-'('«)— 1, so that I and iare also identical.