Commutation Functions

ANGUS S. MACDONALD
Volume 1, pp. 300–302

In

Encyclopedia Of Actuarial Science

(ISBN 0-470-84676-3)

Edited by

Jozef L. Teugels and Bjørn Sundt

 John Wiley & Sons, Ltd, Chichester, 2004

Commutation Functions show how they may be used:

Dx = v x lx (3)
The basis of most calculations in life insurance is ∞
!
the expected present value (EPV) of some payments Nx = Dy (4)
made either on the death of the insured person, or y=x
periodically, as long as the insured person survives. ∞
!
The primary computational tool of the actuary was Sx = Ny (5)
(and often still is) the life table that tabulates lx at y=x
integer ages x, representing the expected number (in
a probabilistic sense) of survivors at age x out of a Cx = v x+1 (lx+t − lx+t+1 ) (6)
large cohort of lives alive at some starting age (often, ∞
!
but not necessarily at birth). From this starting point, Mx = Cy (7)
it is simple to develop mathematical expressions for y=x
various EPVs, assuming a deterministic and constant ∞
!
rate of interest i per annum effective. For example, Rx = My . (8)
defining v = 1/(1 + i) for convenience: y=x
∞
! lx+t − lx+t+1 The most elementary calculation using commuta-
Ax = v t+1 (1)
lx tion functions is to note that
t=0
Dx+t v x+t lx+t
is the EPV of a sum assured of $1 payable at the end = x = v t t px , (9)
of the year in which a person now aged x dies, and Dx v lx
∞
! lx+t+1
∞
! lx+t which is the EPV of $1 payable in t years’ time to
ax = v t+1 = vt (2) someone who is now aged x, provided they are then
lx lx
t=0 t=1 alive. It is clear that an annuity payable yearly is a
sum of such contingent payments, so by the linearity
is the EPV of an annuity of $1 per annum, payable
of expected values we can write equation (2) as
at the end of each future year provided someone now
aged x is then alive. These are the simplest examples ∞
! ∞
! Dx+t+1
lx+t+1 Nx+1
of the International Actuarial Notation for EPVs. ax = v t+1 = = . (10)
We make two remarks: t=0
lx t=0
D x Dx

• Although the summations are taken to ∞, the Moreover, annuities with limited terms are easily
sums of course terminate at the highest age tabu- dealt with by simple differences of the function Nx ,
lated in the life table. for example;
• In the probabilistic setting, the life table lx is just a
convenient way to compute survival probabilities, n−1
! lx+t+1
and we would most naturally express EPVs in ax:n = v t+1
lx
terms of these probabilities. It will become clear, t=0

however, why we have chosen to express EPVs n−1

! Dx+t+1 Nx+1 − Nx+t+1
in terms of lx . = = (11)
t=0
Dx Dx
Although it is simple to write down such mathe-
matical expressions, it is only since modern comput- is the EPV of an annuity of $1 per annum, payable
ers became available that it has been equally simple in arrear for at most n years to someone who is now
to compute them numerically. Commutation functions aged x.
are an ingenious and effective system of tabulated Assurances and annuities whose amounts increase
functions that allow most of the EPVs in everyday at an arithmetic rate present an even greater compu-
use to be calculated with a minimal number of arith- tational challenge, but one that is easily dealt with
metical operations. We list their definitions and then by commutation functions. For example, consider an
2 Commutation Functions

annuity payable annually in arrear, for life, to a per- Therefore, the arithmetic was reduced to a few oper-
son now aged x, of amount $1 in year 1, $2 in year ations with commutation functions and then some
2, $3 in year 3, and so on. The EPV of this annu- simple adjustments. These approximate methods are
ity (giving it its symbol in the International Actuarial described in detail in textbooks on life insurance
Notation) is simply mathematics; see [1–3].
Sx+1 We may mention two specialized variants of
(I a)x =. (12) the classical commutation functions described above
Dx
(see [3] for details).
The commutation functions Cx , Mx , and Rx do for
assurances payable at the end of the year of death, • They may be adapted to a continuous-time model
exactly what the functions Dx , Nx , and Sx do for instead of the discrete-time life table, allowing
annuities payable in arrear. For example, it is easily annuity payments to be made continuously and
seen that equation (1) can be computed as death benefits to be paid immediately on death.
∞
! ! Cx+t ∞ • They may be extended to the valuation of pension
lx+t − lx+t+1 Mx
Ax = v t+1 = = . funds, which requires a multiple-decrement table
t=0
lx t=0
D x Dx (e.g. including death, withdrawal from employ-
(13) ment, age retirement, ill-health retirement, and so
on) and payments that may be a multiple of cur-
Assurances with a limited term are also easily accom- rent or averaged salaries.
modated, for example,
n−1
! Modern treatments have downplayed the use of
lx+t − lx+t+1
A1x:n = v t+1 tables of commutation functions, since clearly the
t=0
lx same EPVs can be calculated in a few columns of
n−1 a spreadsheet. As Gerber points out in [2], their use
! Cx+t Mx − Mx+n
= = (14) was also closely associated with the obsolete inter-
t=0
Dx Dx pretation of the life table in which lx was regarded as
a deterministic model of the survival of a cohort of
is the EPV of a temporary (or term) assurance payable
lives. We have presented them above as a means of
at the end of the year of death, for a person now
calculating certain EPVs in a probabilistic model, in
aged x, if death occurs within n years. EPVs of
which setting it is clear that they yield numerically
increasing assurances can be simply computed using
correct results, but have no theoretical content. Ger-
the function Rx .
ber said in [2]: “It may therefore be taken for granted
Combinations of level and arithmetically increas-
that the days of glory for the commutation functions
ing assurances and annuities cover most benefits
now belong in the past.”
found in practice, so these six commutation func-
tions were the basis of most numerical work in life
insurance until the advent of computers. However, References
the assumptions underlying them, that annuities are
paid annually and that insurance benefits are paid at [1] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A.
the end of the year of death, are not always realistic, & Nesbitt, C.J. (1986). Actuarial Mathematics, The Soci-
rather they reflect the simplicity of the underlying ety of Actuaries, Itasca, IL.
life table lx , tabulated at integer ages. There is no [2] Gerber, H.U. (1990). Life Insurance Mathematics,
Springer-Verlag, Berlin.
theoretical objection to setting up a life table based
[3] Neill, A. (1977). Life Contingencies, Heinemann, London.
on a smaller time unit, in order to handle annuities
payable more frequently than annually, or assurances
payable soon after death; but in practice, the EPVs (See also International Actuarial Notation)
of such payments were usually found approximately
starting with the EPVs based on the usual life table. ANGUS S. MACDONALD