Present Values and Accumulations

ANGUS S. MACDONALD
Volume 3, pp. 1331–1336

In

Encyclopedia Of Actuarial Science

(ISBN 0-470-84676-3)

Edited by

Jozef L. Teugels and Bjørn Sundt

 John Wiley & Sons, Ltd, Chichester, 2004

 Since (1 + i)n ≥ (1 + ni) (i > 0), an astute investor
Present Values and will turn simple interest into compound interest just
Accumulations by withdrawing his money each year and investing
it afresh, if he is able to do so; therefore the use
of simple interest is unusual, and unless otherwise
Effective Interest stated, interest is always compound.
Given effective interest of i per year, it is easily
Money has a time value; if we invest $1 today, we seen that $1 invested for any length of time T ≥ 0
expect to get back more than $1 at some future time will accumulate to $(1 + i)T . This gives us the rule
as a reward for lending our money to someone else for changing the time unit; for example, if it was more
who will use it productively. Suppose that we invest convenient to use the month as time unit, interest of
$1, and a year later we get back $(1 + i). The amount i per year effective would be equivalent to interest
invested is called the principal, and we say that i is of j = (1 + i)1/12 − 1 per month effective, because
the effective rate of interest per year. Evidently, this (1 + j )12 = 1 + i.
definition depends on the time unit we choose to use.
In a riskless world, which may be well approximated
by the market for good quality government bonds, Changing Interest Rates and the Force of
i will be certain, but if the investment is risky, i is Interest
uncertain, and our expectation at the outset to receive
$(1 + i) can only be in the probabilistic sense. The rate of interest need not be constant. To deal
We can regard the accumulation of invested money with variable interest rates in the greatest gener-
in either a retrospective or prospective way. We may ality, we define the accumulation factor A(t, s) to
take a given amount, $X say, to be invested now be the amount to which $1 invested at time t will
and ask, as above, to what amount will it accumulate accumulate by time s > t. The corresponding dis-
after T years? Or, we may take a given amount, $Y count factor is V (t, s), the amount that must be
say, required in T years’ time (to meet some liability invested at time t to produce $1 at time s, and clearly
perhaps) and ask, how much we should invest now, V (t, s) = 1/A(t, s). The fact that interest is com-
so that the accumulation in T years’ time will equal pound is expressed by the relation
$Y ? The latter quantity is called the present value of
$Y in T years’ time. For example, if the effective A(t, s) = A(t, r)A(r, s) for t < r < s. (1)
annual rate of interest is i per year, then we need
The force of interest at time t, denoted δ(t), is defined
to invest $1/(1 + i) now, in order to receive $1 at
as
the end of one year. In standard actuarial notation,
1/(1 + i) is denoted v, and is called the discount 1 dA(0, t) d
factor. It is immediately clear that in a deterministic δ(t) = = log A(0, t). (2)
A(0, t) dt dt
setting, accumulating and taking present values are
inverse operations. The first equality gives an ordinary differential equa-
Although a time unit must be introduced in the tion for A(0, t), which with boundary condition
definition of i and v, money may be invested over A(0, 0) = 1 has the following solution:
longer or shorter periods. First, consider an amount
 t 
of $1 to be invested for n complete years, at a rate i A(0, t) = exp δ(s) ds
per year effective. 0


 t 
• Under simple interest, only the amount originally so V (0, t) = exp − δ(s) ds . (3)
0
invested attracts interest payments each year, and
after n years the accumulation is $(1 + ni). The special case of constant interest rates is now
• Under compound interest, interest is earned each given by setting δ(t) = δ, a constant, from which we
year on the amount originally invested and inter- obtain the following basic relationships:
est already earned, and after n years the accumu-
lation is $(1 + i)n . (1 + i) = eδ and δ = log(1 + i). (4)
2 Present Values and Accumulations

The theory of cash flows and their accumulations recipient survives – but if they are guaranteed regard-
and present values has been put in a very general less of events, the annuity is called an annuity certain.
framework by Norberg [10]. Actuarial notation extends to annuities certain as
follows:

Nominal Interest • A temporary annuity certain is one payable for

a limited term. The simplest example is a level
In some cases, interest may be expressed as an annual annuity of $1 per year, payable at the end of each
amount payable in equal instalments during the year; of the next n years. Its accumulation at the end
then the annual rate of interest is called nominal. of n years is denoted sn , and its present value at
For example, under a nominal rate of interest of the outset is denoted an . We have
8% per year, payable quarterly, interest payments
of 2% of the principal would be made at the end 
n−1
(1 + i)n − 1
of each quarter-year. A nominal rate of i per year sn = (1 + i)r = , (5)
i
payable m times during the year is denoted i (m) . r=0
This is equivalent to an effective rate of interest of 
n
1 − vn
i (m) /m per 1/m year, and by the rule for changing an = vr = . (6)
time unit, this is equivalent to effective interest of r=1
i
(1 + i (m) /m)m − 1 per year.
There are simple recursive relationships between
accumulations and present values of annuities
Rates of Discount certain of successive terms, such as sn+1 = 1 +
(1 + i)sn and an+1 = v + van , which have very
Instead of supposing that interest is always paid at
intuitive interpretations and can easily be verified
the end of the year (or other time unit), we can sup-
directly.
pose that it is paid in advance, at the start of the
• A perpetuity is an annuity without a limited term.
year. Although this is rarely encountered in prac-
The present value of a perpetuity of $1 per year,
tice, for obvious reasons, it is important in actuarial
payable in arrear, is denoted a∞ , and by taking
mathematics. The effective rate of discount per year,
the limit in equation (5) we have a∞ = 1/i. The
denoted d, is defined by d = i/(1 + i), and receiv-
accumulation of a perpetuity is undefined.
ing this in advance is clearly equivalent to receiving
• An annuity may be payable in advance instead of
i in arrears. We have the simple relation d = 1 − v.
in arrears, in which case it is called an annuity-
Nominal rates of discount d (m) may also be defined,
due. The actuarial symbols for accumulations and
exactly as for interest.
present values are modified by placing a pair of
dots over the s or a. For example, a temporary
Annuities Certain annuity-due of $1 per year, payable yearly for n
years would have accumulation s̈n after n years or
We often have to deal with more than one payment, present value än at outset; a perpetuity of $1 per
for example, we may be interested in the accumula- year payable in advance would have present value
tion of regular payments made into a bank account. ä∞ ; and so on.
This is simply done; both present values and accu- We have
mulations of multiple payments can be found by
summing the present values or accumulations of each n
(1 + i)n − 1
individual payment. s̈n = (1 + i)r = , (7)
r=1
d
An annuity is a series of payments to be made
at defined times in the future. The simplest are level 
n−1
1 − vn
annuities, for example, of amount $1 per annum. The än = vr = , (8)
d
payments may be contingent on the occurrence or r=0
nonoccurrence of a future event – for example, a 1
pension is an annuity that is paid as long as the ä∞ = . (9)
d
 Present Values and Accumulations 3

• Annuities are commonly payable more frequently s̈n − n

(I s̈)n = , (17)
than annually, say m times per year. A level annu- d
ity of $1 per year, payable in arrears m times a än − nv n
year for n years has accumulation denoted sn(m) (I ä)n = . (18)
d
after n years and present value denoted an(m) at
outset; the symbols for annuities-due, perpetu- (I s)(m)
n (and so on) is a valid notation for increas-
ities, and so on are modified similarly. We have ing annuities payable m times a year, but note that
(1 + i)n − 1 the payments are of amount $1/m during the first
sn(m) = , (10) year, $2/m during the second year and so on,
i (m)
not the arithmetically increasing sequence $1/m,
1 − vn
an(m) = (m) , (11) $2/m, $3/m, . . . at intervals of 1/m year. The
i
notation for the latter is (I (m) s)(m)
n (and so on).
(1 + i)n − 1 • In theory, annuities or other cash flows may be
s̈n(m) = , (12)
d (m) payable continuously rather than discretely. In
1 − vn practice, this is rarely encountered but it may
än(m) = (m) . (13) be an adequate approximation to payments made
d
daily or weekly. In the international actuarial
Comparing, for example, equations (5) and (10),
notation, continuous payment is indicated by a bar
we find convenient relationships such as
over the annuity symbol. For example, an annuity
i of $1 per year payable continuously for n years
sn(m) = sn . (14) has accumulation s n and present value a n . We
i (m)
have
In precomputer days, when all calculations
involving accumulations and present values of  n  n
annuities had to be performed using tables and sn = (1 + i)n−t dt = eδ(n−t) dt
0 0
logarithms, these relationships were useful. It was
only necessary to tabulate sn or an , and the ratios (1 + i) − 1 n
= , (19)
i/i (m) and i/d (m) , at each annual rate of interest δ
 n 
needed, and all values of sn(m) and an(m) could be n

found. In modern times this trick is superfluous, an = (1 + i)−t dt = e−δt dt

0 0
since, for example, sn(m) can be found from first
1 − vn
principles as the accumulation of an annuity of = , (20)
$1/m, payable in arrears for nm time units at δ
 ∞  ∞
an effective rate of interest of (1 + i)1/m − 1 per a∞ = (1 + i)−t dt = e−δt dt =
1
. (21)
time unit. Accordingly, the i (m) and sn(m) notation 0 0 δ
is increasingly of historical interest only.
• A few special cases of nonlevel annuities arise Increasing continuous annuities may have a rate
often enough so that their accumulations and of payment that increases continuously, so that at
present values are included in the international time t the rate of payment is $t per year, or that
actuarial notation, namely, arithmetically inc- increases at discrete time points, for example, a
reasing annuities. An annuity payable annually rate of payment that is level at $t per year during
for n years, of amount $t in the tth year, has the tth year. The former is indicated by a bar that
accumulation denoted (I s)n and present value extends over the I, the latter by a bar that does
denoted (I a)n if payable in arrears, or (I s̈)n and not. We have
(I ä)n if payable in advance.

n−1  r+1
s̈n − n (I s)n = (r + 1) (1 + i)n−t dt
(I s)n = , (15) r
i r=0

än − nv n s̈n − n
(I a)n = , (16) = , (22)
i δ
4 Present Values and Accumulations


n−1  r+1 or more individuals. The simplest insurance con-
(I a)n = (r + 1) (1 + i)−t dt tracts such as whole life insurance guarantee to
r=0 r pay a fixed amount on death, while the sim-
än − nv n plest annuities guarantee a level amount through-
= , (23) out life. For simplicity, we will suppose that
δ
 n cash flows are continuous, and death benefits are
sn − n payable at the moment of death. We can (a) rep-
(I s)n = t (1 + i)n−t dt = , (24)
0 δ resent the future lifetime of a person now age
 n
a n − nv n x by the random variable Tx ; and (b) assume a
(I a)n = t (1 + i)−t dt = . (25) fixed rate of interest of i per year effective; and
0 δ
then the present value of $1 paid upon death is
Much of the above actuarial notation served to the random variable v Tx , and the present value
simplify calculations before widespread computing of an annuity of $1 per annum, payable continu-
power became available, and it is clear that it is now ously while they live, is the random variable a Tx .
a trivial task to calculate any of these present values The principle of equivalence states that two series
and accumulations (except possibly continuous cash of contingent payments that have equal expected
flows) with a simple spreadsheet; indeed restrictions present values can be equated in value; this is
such as constant interest rates and regular payments just the law of large numbers (see Probability
are no longer important. Only under very particular Theory) applied to random present values. For
assumptions can any of the above actuarial formulae example, in order to find the rate of premium P x
be adapted to nonconstant interest rates [16]. that should be paid throughout life by the person
For full treatments of the mathematics of interest now age x, we should solve
rates, see [8, 9].
E[v Tx ] = P x E[a Tx ]. (26)
In fact, these expected values are identical to the
Accumulations and Present Values Under
present values of contingent payments obtained
Uncertainty by regarding the life table as a deterministic
There may be uncertainty about the timing and model of mortality, and many of them are repre-
amount of future cash flows, and/or the rate of sented in the international actuarial notation. For
interest at which they may be accumulated or dis- example, E[v Tx ] = Ax and E[a Tx ] = a x . Calcu-
counted. Probabilistic models have been developed lation of these expected present values requires a
that attempt to model each of these separately or in suitable life table (see Life Table; Life Insurance
combination. Many of these models are described in Mathematics). In this model, expected present
detail in other articles; here we just indicate some of values may be the basis of pricing and reserv-
the major lines of development. ing in life insurance and pensions, but the higher
Note that when we admit uncertainty, present val- moments and distributions of the present values
ues and accumulations are no longer equivalent, as are of interest for risk management (see [15] for
they were in the deterministic model. For example, an early example, which is an interesting reminder
if a payment of $1 now will accumulate to a ran- of just how radically the scope of actuarial science
dom amount $X in a year, Jensen’s inequality (see has expanded since the advent of computers).
Convexity) shows that E[1/X]  = 1/E[X]. In fact, For more on this approach to life insurance math-
the only way to restore equality is to condition on ematics, see [1, 2].
knowing X, in other words, to remove all the uncer- • For more complicated contracts than life
tainty. Financial institutions are usually concerned insurance, such as disability insurance or income
with managing future uncertainty, so both actuarial protection insurance, multiple state models were
and financial mathematics tend to stress present val- developed and expected present values of extre-
ues much more than accumulations. mely general contingent payments were obtained
as solutions of Thiele’s differential equations
• Life insurance contracts define payments that (see Life Insurance Mathematics) [4, 5]. This
are contingent upon the death or survival of one development reached its logical conclusion when
 Present Values and Accumulations 5

life histories were formulated as counting [3] Hesselager, O. & Norberg, R. (1996). On probability dis-
processes, in which setting the familiar expected tributions of present values in life insurance, Insurance:
Mathematics & Economics 18, 35–42.
present values could again be derived [6] as
[4] Hoem, J.M. (1969). Markov chain models in life
well as computationally tractable equations for insurance, Blätter der Deutschen Gesellschaft für Ver-
the higher moments [13], and distributions [3] sicherungsmathematik 9, 91–107.
of present values. All of classical life insurance [5] Hoem, J.M. (1988). The versatility of the Markov
mathematics is generalized very elegantly using chain as a tool in the mathematics of life insurance,
counting processes [11, 12], an interesting in Transactions of the 23rd International Congress of
example of Jewell’s advocacy that actuarial Actuaries, Helsinki, S, pp. 171–202.
[6] Hoem, J.M. & Aalen, O.O. (1978). Actuarial values
science would progress when models were
of payment streams, Scandinavian Actuarial Journal
formulated in terms of the basic random events 38–47.
instead of focusing on expected values [7]. [7] Jewell, W.S. (1980). Generalized models of the insur-
• Alternatively, or in addition, we may regard ance business (life and/or non-life insurance), in Trans-
the interest rates as random (see Interest- actions of the 21st International Congress of Actuaries,
rate Modeling), and develop accumulations and Zurich and Lausanne, S, pp. 87–141.
[8] Kellison, S.G. (1991). The Theory of Interest, 2nd
present values from that point of view. Under
Edition, Irwin, Burr Ridge, IL.
suitable distributional assumptions, it may be [9] McCutcheon, J.J. & Scott, W.F. (1986). An Introduction
possible to calculate or approximate moments to the Mathematics of Finance, Heinemann, London.
and distributions of present values of simple [10] Norberg, R. (1990). Payment measures, interest, and
contingent payments; for example, [14] assumed discounting. An axiomatic approach with applications
that the force of interest followed a second- to insurance, Scandinavian Actuarial Journal 14–33.
order autoregressive process, while [17] assumed [11] Norberg, R. (1991). Reserves in life and pension insur-
ance, Scandinavian Actuarial Journal 3–24.
that the rate of interest was log-normal. The
[12] Norberg, R. (1992). Hattendorff’s theorem and Thiele’s
application of such stochastic asset models differential equation generalized, Scandinavian Actuar-
(see Asset–Liability Modeling) to actuarial ial Journal 2–14.
problems has since become extremely important, [13] Norberg, R. (1995). Differential equations for moments
but the derivation of explicit expressions for of present values in life insurance, Insurance: Mathe-
moments or distributions of expected values and matics & Economics 17, 171–180.
[14] Pollard, J.H. (1971). On fluctuating interest rates, Bul-
accumulations is not common. Complex asset
letin de L’Association Royale des Actuaires Belges 66,
models may be applied to complex models of 68–97.
the entire insurance company, and it would be [15] Pollard, A.H. & Pollard, J.H. (1969). A stochastic
surprising if analytical results could be found; as approach to actuarial functions, Journal of the Institute
a rule it is hardly worthwhile to look for them, of Actuaries 95, 79–113.
instead, numerical methods such as Monte Carlo [16] Stoodley, C.L. (1934). The effect of a falling interest rate
simulation are used (see Stochastic Simulation). on the values of certain actuarial functions, Transactions
of the Faculty of Actuaries 14, 137–175.
[17] Waters, H.R. (1978). The moments and distributions of
actuarial functions, Journal of the Institute of Actuaries
References 105, 61–75.

[1] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A.

& Nesbitt, C.J. (1986). Actuarial Mathematics, The (See also Annuities; Interest-rate Modeling; Life
Society of Actuaries, Itasca, IL. Insurance Mathematics)
[2] Gerber, H.U. (1990). Life Insurance Mathematics,
Springer-Verlag, Berlin. ANGUS S. MACDONALD