Appendix A. Commutation Functions
A.I Introduction
In thi s appendix we give an introduction to th e use of commutation functions.
These functions were invent ed in t he 18th cent ury and achieved great popu-
larity, which can be ascribed to two reasons :
Reason 1
Tables of commutation functions simplify th e calculat ion of num erical values
for many actu arial functions.
Reason 2
Exp ected values such as net single premiums may be derived within a det er-
ministic mod el closely relat ed to commutation functions.
Both reasons have lost th eir significance, th e first with t he advent of powerful
compute rs, the second with th e growing acceptance of mod els based on prob-
ability t heory, which allows a more complete und erstanding of th e essent ials
of insurance. It may therefore be t aken for granted t hat th e days of glory for
th e commut ation functions now b elong to th e past.
A.2 The Deterministic Model
Imagine a cohort of lives, all of th e same age, observed over time, and denote
by Lx the number st ill living at age x . Thus dx = Lx - LX +1 is th e number of
deaths between the ages of x and x + 1.
Probabilities and expected values may now be deriv ed from simpl e pro-
portions and averages. So is, for inst ance.
(A.2.1)
,120 Appendix A. Commutation Functions
th e proportion of persons alive at age x + i , relative to t he number of persons
alive at age x, and th e probability th at a life aged x will die within a year is
(A.2.2)
In Chapter 2 we introdu ced t he expected curtate future lifet ime of a life
aged x . Replacing kPx by IX+k/l x in (2.4.3), we obtain
ex
IX+l +----,--
IX+ 2 + .. .
= - - - - (A.2.3)
i.
The num erator in thi s expression is th e total numb er of complete future years
to be "lived" by th e Ix lives (x) , so t hat ex is th e average numb er of complet ed
years left.
A.3 Life Annuities
We first consider a life annuity-due with annual payments of 1 unit , as intro-
du ced in Section 4.2, th e net single premium of which annuity was denot ed
by ax . Replacing kPx in (4.2.5) by IX+ k/l x , we obtain
Ix + Vlx+l + v 21x+2 + ...
ax = (A.3.1)
Ix
or
Ix ax = Ix + Vlx+l + v 21x+2 + ... . (A.3.2)
Thi s result is often referred to as th e equivalence principle, and its interpre-
tation within t he deterministic model is evident: if each of th e Ix persons
living at age x were to buy an annui ty of th e given typ e, t he sum of net single
premiums (the left hand side of (A.3.2)) would equal th e present value of the
benefits (th e right hand side of (A.3.2)).
Multiplying both numerator and denominator in (A.3.1) by v X , we find
(A.3.3)
With t he abbreviations
(A.3.4)
we th en obtain t he simple formula
.. Nx
< : »: x
(A.3.5)
Thus th e manu al calculation of ax is ext remely easy if tables of t he com-
mutation function s D x and N, are available. The function D x is called th e
"discounted number of survivors ".
,A.4. Life Insurance 121
Similarly one may obtain formulas for t he net single premium of a tempo-
rary life annuity,
(A.3.6)
imm ediat e life annuities,
N X +1
ax = ---rJ ' (A.3.7)
x
and general annuities with annual payments: formul a (4.4.2) may naturally
be tr anslated to
E(Y) = roDx + r1Dx+ 1 + T2 D x+2 + ...
(A.3.8)
Dx
For th e special case Tk = k + 1 we obtain t he formul a
(I a)x = ~x ; (A.3.9)
x
here t he commutat ion function S x is defined by
Sx D'; + 2D x+ 1 + :~ Dx+2 + .
N; + N X +1 + N X +2 + . (A.3.1O)
A.4 Life Insurance
In addition to (A.3.4) and (A.3.1O) we now define th e commutation functions
vX+ 1dx,
C x + C x+1 + C X+2 + ,
C x + 2C x+1 + :~ Cx+2 + .
M x + M + 1 + M +2 +
X . X (A.4.1)
Replacing kPxqx+k in equat ion (3.2.3) by dx+k/l x , we obt ain
od; + v 2 d x + 1 + v 3 dx +2 + .. .
Ax =
i,
C x + C x+ 1 + C x+2 + ...
Dx
(A.4.2)
Similarly one obt ains
vd x + 2v 2dx+ 1 + 3v 3 d x+2 + ...
i:
C x + 2C x+ 1 + 3C x+2 + ...
Dx
(A.4.3)
, 122 Appendix A. Commutation Functions
Obviously these formul ae may be derived within the det erministic mod el by
means of the equivalence principle. In ord er to det ermine Ax one would start
with
(A.4.4)
by imagining that Lx persons buy a whole life insurance of 1 unit each, payable
at the end of the year of death, in return for a net single premium.
Corresponding formulae for term and endowment insurances are
Mx - M x +n
Dx
Mx - Mx+n + Dx+n
Dx
C x + 2Cx+ 1 + 3C x+2 + ... + nC x+n - 1
Dx
M x + M X + 1 + M X +2 + ...+ M x+n - 1 - nMx+n
Dx
(A.4 .5)
which speak for themselves.
The commutation functions defined in (A.4 .1) can be expressed in terms
of the commut a tion functions defined in Sect ion 3. From dx = Lx -Lx+! follows
(A.4 .6)
Summation yields th e identities
(A.4 .7)
and
R; = N x - dS x ' (A.4 .8)
Dividing both equa t ions by-D, , we retrieve the identities
1 - d ax '
ax - d(Ia) x , (A.4 .9)
see equat ions (4.2.8) and (4.5.2) .
A.5 Net Annual Premiums and Premium Reserves
Consider a whole life insurance with 1 unit payabl e at t he end of the year of
death, and payable by net annual pr emiums. Using (A.3.5) and (A.4 .2) we
find
(A.5.l)
A.I Introduction
In thi s appendix we give an introduction to th e use of commutation functions.
These functions were invent ed in t he 18th cent ury and achieved great popu-
larity, which can be ascribed to two reasons :
Reason 1
Tables of commutation functions simplify th e calculat ion of num erical values
for many actu arial functions.
Reason 2
Exp ected values such as net single premiums may be derived within a det er-
ministic mod el closely relat ed to commutation functions.
Both reasons have lost th eir significance, th e first with t he advent of powerful
compute rs, the second with th e growing acceptance of mod els based on prob-
ability t heory, which allows a more complete und erstanding of th e essent ials
of insurance. It may therefore be t aken for granted t hat th e days of glory for
th e commut ation functions now b elong to th e past.
A.2 The Deterministic Model
Imagine a cohort of lives, all of th e same age, observed over time, and denote
by Lx the number st ill living at age x . Thus dx = Lx - LX +1 is th e number of
deaths between the ages of x and x + 1.
Probabilities and expected values may now be deriv ed from simpl e pro-
portions and averages. So is, for inst ance.
(A.2.1)
,120 Appendix A. Commutation Functions
th e proportion of persons alive at age x + i , relative to t he number of persons
alive at age x, and th e probability th at a life aged x will die within a year is
(A.2.2)
In Chapter 2 we introdu ced t he expected curtate future lifet ime of a life
aged x . Replacing kPx by IX+k/l x in (2.4.3), we obtain
ex
IX+l +----,--
IX+ 2 + .. .
= - - - - (A.2.3)
i.
The num erator in thi s expression is th e total numb er of complete future years
to be "lived" by th e Ix lives (x) , so t hat ex is th e average numb er of complet ed
years left.
A.3 Life Annuities
We first consider a life annuity-due with annual payments of 1 unit , as intro-
du ced in Section 4.2, th e net single premium of which annuity was denot ed
by ax . Replacing kPx in (4.2.5) by IX+ k/l x , we obtain
Ix + Vlx+l + v 21x+2 + ...
ax = (A.3.1)
Ix
or
Ix ax = Ix + Vlx+l + v 21x+2 + ... . (A.3.2)
Thi s result is often referred to as th e equivalence principle, and its interpre-
tation within t he deterministic model is evident: if each of th e Ix persons
living at age x were to buy an annui ty of th e given typ e, t he sum of net single
premiums (the left hand side of (A.3.2)) would equal th e present value of the
benefits (th e right hand side of (A.3.2)).
Multiplying both numerator and denominator in (A.3.1) by v X , we find
(A.3.3)
With t he abbreviations
(A.3.4)
we th en obtain t he simple formula
.. Nx
< : »: x
(A.3.5)
Thus th e manu al calculation of ax is ext remely easy if tables of t he com-
mutation function s D x and N, are available. The function D x is called th e
"discounted number of survivors ".
,A.4. Life Insurance 121
Similarly one may obtain formulas for t he net single premium of a tempo-
rary life annuity,
(A.3.6)
imm ediat e life annuities,
N X +1
ax = ---rJ ' (A.3.7)
x
and general annuities with annual payments: formul a (4.4.2) may naturally
be tr anslated to
E(Y) = roDx + r1Dx+ 1 + T2 D x+2 + ...
(A.3.8)
Dx
For th e special case Tk = k + 1 we obtain t he formul a
(I a)x = ~x ; (A.3.9)
x
here t he commutat ion function S x is defined by
Sx D'; + 2D x+ 1 + :~ Dx+2 + .
N; + N X +1 + N X +2 + . (A.3.1O)
A.4 Life Insurance
In addition to (A.3.4) and (A.3.1O) we now define th e commutation functions
vX+ 1dx,
C x + C x+1 + C X+2 + ,
C x + 2C x+1 + :~ Cx+2 + .
M x + M + 1 + M +2 +
X . X (A.4.1)
Replacing kPxqx+k in equat ion (3.2.3) by dx+k/l x , we obt ain
od; + v 2 d x + 1 + v 3 dx +2 + .. .
Ax =
i,
C x + C x+ 1 + C x+2 + ...
Dx
(A.4.2)
Similarly one obt ains
vd x + 2v 2dx+ 1 + 3v 3 d x+2 + ...
i:
C x + 2C x+ 1 + 3C x+2 + ...
Dx
(A.4.3)
, 122 Appendix A. Commutation Functions
Obviously these formul ae may be derived within the det erministic mod el by
means of the equivalence principle. In ord er to det ermine Ax one would start
with
(A.4.4)
by imagining that Lx persons buy a whole life insurance of 1 unit each, payable
at the end of the year of death, in return for a net single premium.
Corresponding formulae for term and endowment insurances are
Mx - M x +n
Dx
Mx - Mx+n + Dx+n
Dx
C x + 2Cx+ 1 + 3C x+2 + ... + nC x+n - 1
Dx
M x + M X + 1 + M X +2 + ...+ M x+n - 1 - nMx+n
Dx
(A.4 .5)
which speak for themselves.
The commutation functions defined in (A.4 .1) can be expressed in terms
of the commut a tion functions defined in Sect ion 3. From dx = Lx -Lx+! follows
(A.4 .6)
Summation yields th e identities
(A.4 .7)
and
R; = N x - dS x ' (A.4 .8)
Dividing both equa t ions by-D, , we retrieve the identities
1 - d ax '
ax - d(Ia) x , (A.4 .9)
see equat ions (4.2.8) and (4.5.2) .
A.5 Net Annual Premiums and Premium Reserves
Consider a whole life insurance with 1 unit payabl e at t he end of the year of
death, and payable by net annual pr emiums. Using (A.3.5) and (A.4 .2) we
find
(A.5.l)
