Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol69.Issue1.2001:

Int. J. Production Economics 69 (2001) 65}83

Cycles optimization: The equivalent annuity and the NPV
approaches
Elisa Luciano!,*, Lorenzo Peccati"
!Dipartimento di Statistica e Matematica Applicata, University of Turin, Piazza Arbarello 8, I-10121 Turin, Italy
"Istituto di Metodi Quantitativi, Universita` **Luigi Bocconi++, via Sarfatti, 25, I-20136 Milan, Italy
Received 20 April 1998; accepted 23 December 1999

Abstract
The paper discusses the use of a generalized version of the equivalent annuity principle, which takes into account
interest rate #uctuations over time. When dealing with applications over a sequence of cycles, e.g. plant replacement ones,
the equivalent annuity is usually de"ned with reference to each single cycle or to the whole sequence. First, we study the
correspondence between net present values and equivalent annuities as de"ned above in optimization problems. We
show that only the second de"nition is appropriate for optimization, whenever the length of the cycle is a choice variable.
However, also the second is not necessarily correct, when the horizon is "nite. Then, we discuss the corresponding
problems of optimality, both with an in"nite and a "nite sequence of cycles. Applications to a simple plant replacement
problem are illustrated: they show how di!erent the optimal decisions can be from the equivalent annuity ones. ( 2001
Elsevier Science B.V. All rights reserved.
Keywords: Net present value; Equivalent annuity; Plant renewal

1. Introduction
Many problems in Production Economics turn out to consist of the "nancial evaluation of a "nite or
in"nite chain of cash #ow cycles. The "nancial evaluation generally consists in computing the net present
value of the cash #ows of the whole chain. However, another approach}the equivalent annuity one}is
sometimes introduced. The equivalent annuity is the constant cash #ow intensity which has the same present
value of the original in#ow and out#ow stream. Due to the character of the de"nition itself, it is generally
argued that using the net present value or its equivalent annuity is the same.
This paper aims to show the possible error in this statement, for optimization purposes. In detail, we are
reminded that the equivalent annuity can be de"ned either cycle by cycle or for the whole chain. We show the
lack of correspondence between net present value and equivalent annuity when the length of the cycles is

* Corresponding author.
E-mail address: luciano@econ.unito.it (E. Luciano).
0925-5273/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 1 5 - 3

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E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

a choice variable and either the horizon (i.e., the number of cycles) is "nite, or the cycle by cycle annuity is
considered.
The argument is relevant for practical purposes, since decisions taken with the two approaches can di!er
signi"cantly. We show this with reference to a plant replacement problem: via equivalent annuity we are led
to strategies which are not the same as the net present value ones and have a signi"cantly smaller value today.
In the numerical examples, the di!erences between the values produced by the two policies range from 10%
or 14% to more than 100%: in one of our illustrative cases, the equivalent annuity leads to a negative value
for the replacement plan, while via net present value we have positive worth.
The consequence of our argument is that the equivalent annuity is useful in order to represent a given cash
#ow stream; it is not for optimization purposes in general. In the latter case, on the contrary, it can produce
suboptimal decisions.
In the previous literature, as far as we know, the limits of application of the equivalent annuity principle
were not known, as the corresponding cases had not been studied.
The objectives of the paper are as follows. In Section 2 we describe the usual equivalent annuity setup, with
constant interest rates, and put into evidence on the one side the correspondence between maximizing (or
minimizing) net present values and annuities over the whole chain, on the other the lack of correspondence
between maximizing present values and equivalent annuities over a single cycle.
In Section 3 we analyze the problem in a general framework, corresponding to variable interest rates. As
for the in"nite horizon case, the results obtained in the constant rate case still hold. As for the "nite horizon
one, the correspondence over the whole chain breaks down. In Section 4 we illustrate the previous

correspondence arguments with reference to a simple plant replacement problem. In Section 5 we study the
optimization technique for the in"nite horizon case. In Section 6 we do the same for the "nite horizon one.
Section 7 applies the techniques of Section 6 to the plant replacement example. Section 8 concludes and
outlines further applied research.

2. Standard setup
Let us consider the case of plant replacements or of inventory management, as presented in Thorstenson
[1] or Babusiaux [2], under the usual assumption of constant interest rates. We are given:
f a sequence of increasing maturities t , t , t ,2,
0 1 2
n
t "0; t "z ; t "z #z ;2; t " + z ;2,
0
1
1 2
1
2
n
s
s/1

which de"nes a chain of consecutive cycles with respective lengths z '0, s"1, 2,2;
s
f the sequence of the corresponding discounted or net present values (from now on NPVs) G , computed
s
at the beginning of the cycle (t ). To be more precise, let us assume that the sth cycle, which starts at
s~1
time t , requires a discrete cash out#ow of amount I and provides continuous cash in#ows
s~1
s
with intensity o (t) at t3(t , t ) and a "nal payo! J . In this case the NPV for the sth cycle, computed
s
s~1 s
s
at t , is
s~1

P

G "!I #
s

s

ts

ts~1

o (t)e~d(t~ts~1 ) dt#J e~d(ts ~ts~1 ), s"1, 2,2,
s
s

where d is the force of interest, which has both the meaning of opportunity cost of equity, cyclically reinvested
in some production processes and that of rate of return on re-investments.

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

67

The NPV of the whole chain is then
`=
(2.1)

G" + e~dts~1 G
s
s/1
provided that the series converges, as usually occurs in the models having practical interest.1
In order to obtain a quantity with a more natural interpretation than G or G, it is sometimes suggested
s
(see, for instance, the works of GrubbstroK m and Thorstenson [3] or GrubbstroK m and Thorstenson [4]) to
compute the equivalent annuity of a given cash #ow. The equivalent annuity is the continuous and constant
cash #ow intensity which is "nancially equivalent to the NPV.
In principle there are two possibilities:
1. to determine the equivalent annuity C for the sth cycle, with NPV G , through the equation
s
s
zs
C e~dt dt"G NC "G a(z , d), s"1, 2,2,
s
s
s
s s
0

P

(2.2)

$ d/(1!e~dz) is the continuous repayment instalment due in order to extinguish a unitary
where a(z, d)"
debt in z years with force of interest d, usually denoted with the actuarial notation a6 6 ;
z@(d)
2. to determine the equivalent annuity C for the whole sequence of cycles with NPV G through the equation

P

`=
Ce~dt dt"GNC"Gd.
(2.3)
0
The two possibilities have di!erent consequences when dealing with optimization problems, for the following
reason. The NPVs G and G are generally a function of several variables, within which there may be some
s

elements of the maturity sequence Mt N: in most applications then we are asked to maximize or minimize G or
s
G through a correct choice of the underlying variables, in particular of (some or all) the cycle lengths Mz N.
s
s
As concerns C, Eqs. (2.2) and (2.3) show that C is proportional to the NPVs, so that maximizing (or
minimizing) G and maximizing (or minimizing) C is the same, as already noted by GrubbstroK m and Thorstenson [3] and Thorstenson [1].
f As concerns the C , on the opposite side, the two strategies: maximizing (or minimizing) G and
s
s
f maximizing (or minimizing) C
s
are not the same, since2 C "G a and a is a function of z , a"a(z , d). In other words, the correspondence
s
s
s
s
between NPV and equivalent annuity holds for the in"nite chain but not for the single addenda: in the former
case switching from one to the other is appropriate, in the second is not.

1 A special case which is frequently encountered in the applications is that of lattice maturities t "sq, with q'0, and identical G ,g.
s
s
In such a situation the NPV of the whole chain, G, is given by G"g/(1!e~Lq)
2 The key argument for the former result is easily focused. Reconsider Eq. (2.2) and suppose that you are maximizing G through the
s
choice of some variables, one of which is z itself. Suppose that G is di!erentiable. The "rst order necessary condition requires that at
s
s
interior (positive) maxima
LG /Lz "0
s s
holds, while the corresponding condition on C "G a(z , d) requires
s
s s
La(z , d)
LG
s
s a(z , d)#G
"0.

s
s Lz
Lz
s
s
The two conditions coincide i! G "0. A simple consequence is that maximizing (or minimizing) G , in general, is not the same as
s
s
maximizing (or minimizing) C .
s

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E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

This paper is di!erent from previously published works in that it considers a more general context, of
variable interest rates. It studies the equivalent annuity and the corresponding optimization problems both
for the "nite and the in"nite horizon (or number of cycles) case. It analyzes the optimization problems both in
general and by using as an example a plant replacement case.

3. A general framework
Let us set up a general framework, within which we shall study the problem.
First of all, let us suppose that interest rates are variable over time: formally, that the discounting rate d is
a function of time, d"d(t). The discount factor e~dt will be replaced with exp(!:t d(u) du). Throughout this
0
paper we shall always assume that d(t) is continuous and positive: the "rst assumption guarantees that
discount factors are well de"ned for "nite t, the second}which is natural in nominal terms}makes them
decreasing over time. Also, we will assume that lim
exp(!:t d(u) du)"0: the discount factor is not only
0
t?`=
decreasing, but also in"nitesimal as time increases.
In general, the present value at x of 1$ available at y*x will be

A P

B

y
d(u) du .
x
For the sake of simplicity, we shall write /(y) for U(y, 0):
U(y, x)"exp !

A P

$ U(y, 0)"exp !
/(y)"

y

B

d(u) du .

(3.1)

0
We call Generalized Net Present Value (GNPV) a NPV computed with these discount factors.
Secondly, we should keep in mind that the (G)NPV implicitly assumes that investments are "nanced with
equity only. In general a mixed capital structure can be expected: therefore we need an extension of the NPV,
in order to cope with these problems. We shall use the framework introduced in the works of Peccati [5],
GrubbstroK m and Ashcroft [6], Gallo and Peccati [7] and, recently, with reference to inventory problems, in
Luciano and Peccati [8,9]. From a practical viewpoint the extension consists in replacing the standard
(G)NPV with the algebraic sum of the (G)NPVs of investments and "nancing operations. As a consequence,
in the sequel the term (G)NPV refers to this algebraic sum.
Since d now is a function of time, the (G)NPV for the sth cycle, computed at t , is
s~1
ts
o (t)U(t, t ) dt#J U(t , t ), s"1, 2,2.
(3.2)
G "!I #
s
s~1
s s s~1
s
s
ts~1
With an in"nite number of cycles, Eq. (2.1) provides us with the global result of the whole chain and becomes

P

`=
G" + G /(t )
(3.3)
s s~1
s/1
where G is given by Eq. (3.2). We shall assume in what follows that the above mentioned series converges.
s
Also, in order to de"ne equivalent annuities, we assume that :`=/(t) dt(R.
0
With reference to the new meaning of G and to the equivalent annuity, the presence of a time-varying
s
interest intensity implies that
1. the equivalent annuity for one cycle must satisfy the equation

P

ts

ts~1

C U(u, t ) du"G
s
s~1
s

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

69

whence
G
s
;
C"
s :ts U(u, t ) du
s~1
t
s~1

(3.4)

2. the equivalent annuity for the whole chain must satisfy the equation

P

`=
C/(t) dt"G
0
so that it is
G
C"
.
(3.5)
:`=/(t) dt
0
The non-correspondence between maximizing (or minimizing) C and maximizing (or minimizing)
s
G through an appropriate choice of the length z , which we illustrated in the introduction for d constant,
s
s
a fortiori holds true for d non-constant.3
On the other hand, it is easy to show that maximizing (or minimizing) G, the in"nite sequence NPV,
described in Eq. (3.3), is the same as maximizing (or minimizing) the equivalent annuity for the whole
sequence, C. The non-correspondence between optimizing C and G on the one side and the correspondence
s
s
between optimizing C con"rms the correspondence/disagreement results obtained under constant d.
However, with non-constant d we also want to analyze what happens with a "nite number of cycles, i.e.
with a "nite horizon: G
"G
"G
"2"0 in Eq. (3.3). The global result of the "nite chain of
n`1
n`2
n`3
cycles, which we denote with GM , is
n
n
GM " + G /(t ).
(3.6)
n
s s~1
s/1
Having excluded the usefullness, for optimization purposes, of using the equivalent annuity over single cycles,
we de"ne for this "nite horizon case only the equivalent annuity for the whole chain. We denote it with CM .
n
Since CM must be such that
n
tn
CM /(t) dt"GM
n
n
0
we have

P

CM "GM /
n
n

P

tn

/(t) dt.

(3.7)

0
It follows from the de"nition of CM itself and it is demonstrated in Appendix A that maximizing (or minimizing)
n
GM through an appropriate choice of the length z is not the same as maximizing (or minimizing) the present
n
s
value of CM : in the "nite horizon case then optimizing with respect to the NPV or with respect to the
n
equivalent annuity gives di!erent results, even if we take into consideration the whole sequence of cycles.
We sum up our results in Table 1, before switching to the plant replacement application.

3 We thank an anonymous referee for having pointed out the a fortiori argument. However, a formal proof can be found in
Appendix A.

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E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

Table 1
Summary of correspondence results
Horizon

Interest rate

Expression for annuity

Correspondence

In"nite
In"nite
In"nite
In"nite
Finite

Constant
Constant
Variable
Variable
Variable

(2.2)
(2.3)
(3.4)
(3.5)
(3.7)

No
Yes
No
Yes
No

4. A plant replacement application
Let us now turn to an example. Consider a simple plant replacement problem under variable interest
intensity d(t).
We assume that each cycle corresponds to the construction of a new plant. The plant installed from t
to
s~1
t , which characterizes the sth cycle, has an initial cost A(t ), paid at t , which depends on the actual
s
s~1
s~1
payment epoch (costs are supposed to vary over time). It has also a "nal or residual value, which depends on
its working life, R(t !t ). In addition production over the sth life cycle of the investment gives rise to
s
s~1
(continuous) gross margins with intensity P(t!t ) at time t. These margins are gross of maintenance costs,
s~1
which in turn are continuous with intensity m(t!t ). As a result, the present value at t
of the cash-#ows
s~1
s~1
due to the sth plant, corresponding to Eq. (3.2) above, are

P

ts~1 `zs

(P(t!t )!m(t!t ))U(t, t ) dt# R(z )U(t #z , t ).
s~1
s~1
s~1
s
s~1
s s~1
ts~1
The result of the whole sequence of chain, under the in"nite horizon, G, is
G "!A(t )#
s
s~1

C

(4.1)

D

P

ts~1 `zs
`=
[P(t)!m(t)]/(t) dt#R(z )/(t #z ) ,
(4.2)
+ !A(t )/(t )#
s
s~1
s
s~1
s~1
s~1
t
s/1
which corresponds to Eq. (3.3) above. The optimal replacement policy, if it exists, is the one which maximizes
the NPV G in the in"nite horizon case, and its counterpart GM in the "nite horizon case, by a proper choice of
n
the sequence Mz N.
s
Whenever the gross margins are not a choice variable, the corresponding problem consists of minimizing
the total costs due to plant replacement:

C

P

D

ts~1 `zs
`=
+ A(t )/(t )#
m(t!t )/(t) dt!R(z )/(t #z ) .
s~1
s
s~1
s
s~1
s~1
ts~1
s/1
We will now analyze the former case, both under "nite and in"nite horizons. Because of this let
$ P(t)!m(t).
r(t)"
For the time being, for these two horizons we consider only the correspondence between NPV and equivalent
annuities for optimization purposes. In Section 7 below we will analyze their solutions too.
In the in"nite horizon case we give evidence both of the non-correspondence between maximizing a single
cycle NPV (G ) and its equivalent annuity (C ) and of the correspondence between maximizing the whole
s
s
chain NPV (G) and its equivalent annuity (C). In the "nite horizon case we give evidence of the noncorrespondence between maximizing the whole chain NPV (GM ) and its equivalent annuity (CM ).
n
n

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

71

4.1. Inxnite horizon plant replacement
In this subsection we start by formalizing the "rst-order necessary conditions of stationarity for the single
cycle NPV (G ) and to the corresponding annuity (C ). We provide numerical examples of the ensuing
s
s
optimal decisions. Hopefully, this will allow the reader to appreciate the di!erence in the two conditions for
the application under examination.
Then, we can study the correspondence between the whole NPV (G) and its equivalent annuity (C). Also in
this case we can provide both general formulas and numerical examples.
4.1.1. Optimization with respect to the single cycle NPV (G ) and to the corresponding annuity (C )
s
s
If we stick to the single addendum and compute the equivalent annuity we have
`zs r(t!t )U(t, t ) dt#R(z )U(t #z , t )
!A(t )#:ts~1
ts~1
s~1
s~1
s
s~1
s s~1 .
s~1
(4.3)
C"
s
`zs U(u, t ) du
:ts~1
ts~1
s~1
With the expressions (4.1) and (4.3) for G and C respectively, the "rst-order necessary condition for
s
s
optimizing G using the zeroes of LG /Lz becomes
s
s s
r(z )U(t #z , t ) dt#[R@(z )!R(z )d(t #z )]U(t #z , t )"0
(4.4)
s
s~1
s s~1
s
s s~1
s
s~1
s s~1
while the "rst-order necessary condition for optimizing C using the zeroes of LC /Lz , turns out to be
s
s s
LG
s !C d(t #z )U(t #z , t )"0.
(4.5)
s s~1
s
s~1
s s~1
Lz
s
In general also second-order conditions are useful because G and C show minimum points too. Let us
consider for instance the following numerical cases, in which the optimizing sequence of replacement dates
Mz N via NPVs G is di!erent from the one obtained via equivalent annuities.
s
s
Example 1. We consider the "rst cycle of a chain and we assume the speci"cations given in Table 2.
The NPV G (z) turns out to be maximized at z K7.142. The equivalent annuity C (z)"G (z)/:z /(s) ds,
0
1
G
1
1
in turn, is maximized at z K4.969. The two optimal durations are remarkably di!erent. In order to evaluate
C
the economic implications of this divergence we can compute the relative loss in the NPV:
G (z )!G (z )
1 G
1 C ]100"10.4.
G (z )
1 G
This one is far from being negligible. Also with the usual assumption of constant d, under which the total
NPV simpli"es to

C

P

D

zs
`=
G(z)" + !A(t ) exp(!dt )# r(u) exp(!d(u#t )) du#R(z ) exp(!d(t #z )) ,
s~1
s
s~1
s
s~1
s~1
0
s/1
(4.6)
the "rst-order conditions on G and C are di!erent. In fact, condition (4.4) on G provides us with signi"cant
s
s
s
results, according to the following numerical example.
Example 2. We consider the "rst cycle of a chain and we assume the speci"cations given as Table 3.
The NPV G (z) turns out to be maximized at z K9.881. The equivalent annuity C (z)"G(z)a(z, 0.15), in
1
G
1
turn is maximized at z K5.977. The two optimal durations are remarkably di!erent. In order to evaluate the
C

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E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

Table 2

Table 3

Variable or function

Speci"cation

Variable or function

Speci"cation

A(0)"A
r(t)
/(t)
R(z)

1000
300!20t
exp[!:t d(s) ds], d(s)"0.15#0.01s
0
0.8A exp(!0.04z)

A(0)"A
r(t)
/(t)
R(z)

1000
300!20t
exp[!0.15t]Nd"0.15
0.8A exp(!0.04z)

economic implications of the divergence we can compute the relative loss in the NPV:
G (z )!G (z )
1 G
1 C ]100"14.435,
G (z )
1 G
far from being negligible.
4.1.2. Optimization with respect to the whole chain NPV (G) and to the equivalent annuity (C)
For this case, we discuss only the coincidence-disagreement of the expressions for G and C, while we
postpone to Section 7 below an examination of the corresponding "rst-order necessary conditions of
optimality.
Given the expressions (3.5) for the equivalent annuities and Eq. (4.2) for the in"nite sequence NPV in the
plant replacement model, one can easily get the expression for C.
`zs r(t!t )/(t) dt#R(z )/(t #z )]
+`= [!A(t )/(t )#:ts~1
ts~1
s~1
s
s~1
s .
s/1
s~1
s~1
:`=/(u) du
0

(4.7)

In the constant d subcase, the perpetual annuity C"Gd is

C

D

P

zs
`=
C"d + !A(t ) exp(!dt )# r(u) exp(!d(u#t )) du# R(z ) exp(!d(t #z )) .
s~1
s
s~1
s
s~1
s~1
0
s/1
4.2. Finite horizon plant replacement problem
As for the "nite horizon case, we can proceed along the lines of the previous subsection, with a restriction,
following our discussion in Section 3: having already excluded the relationship between maximizing the
equivalent annuity for a single cycle (C ) and its NPV (G ), we examine only the correspondence between
s
s
maximizing the discounted annuity for the whole chain and the NPV itself. As in the previous subsection, we
do this "rst in general and then using numerical data. We postpone to Section 7 below an examination of the
"rst-order necessary conditions of optimality.
The NPV of the whole sequence of cycles is

C

P

D

ts~1 `zs
n
r(t!t )/(t) dt#R(z )/(t #z ) .
GM " + !A(t )/(t )#
s~1
s
s~1
s
n
s~1
s~1
s~1
t
s/1
The corresponding annuity is
`zs r(t!t )/(t) dt#R(z )/(t #z )]
+ n [!A(ts~1 )/(ts~1 )#:tts~1
s~1
s~1
s
s~1
s
CM " s/1
+n
n
: s/1 zs /(u) du
0

(4.8)

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

73

so that
LGM
n
LCM
" !CM / + z .
S
Lz
Lz
s
s
s/1
Example 3. With the same data from the numerical examples of the preceding section, we have set n"2,
considering therefore a couple of cycles, and we have performed the two parallel optimization procedures for
both the GNPV and the equivalent annuity.

A

B

The maximization of the GNPV is obtained choosing z"6.116 for the "rst cycle and Z"5.15 for the
second one. The optimized value for the GNPV turns out to be 153.90.
Trying to maximize the equivalent annuity, the maximum is found with a completely di!erent policy (z"0
and Z"4.75). The maximized annuity turns out to be negative! Another argument against the use of the
equivalent annuity principle.

5. Optimization over in5nite cycles
Net present value models of investments of the type discussed in the previous section and, more generally,
of the type (3.3), are often studied with optimization aims.
The cycle result G is assumed to be a function of a vector of variables4 hs"[h , h ,2, h ] to be
s
s,1 s,2
s,k
chosen in the set5 Hs-Rk, so that G "G (h ). Such variables can be concretely speci"ed as the duration of
s
s s
each cycle, but also with reference to other choice possibilities, as, for instance, price policies and/or
plant-maintenance policies. The problem consists then in choosing h"MhsN3X`= Hs in order to maximize
s/1
(or to minimize)
`=
G"G(h)" + G (hs)/(t )
s
s~1
s/1
or, since we are in the correspondence case, its corresponding annuity C. We will now examine the
optimization problem directly on NPV, knowing that identical results hold for the discounted annuity C.
First of all, note that h could contain the maturities t , which would mean that the optimizing decision
s
includes the cycle timing. This is the case of the plant replacement problem.
Let us assume "rst that the maturities t }or, equivalently, the durations z }are "xed. It is clear that in this
s
s
case in order to maximize (or minimize) G(h) it is su$cient to maximize (or minimize) each partial result,
G (hs), so that the optimization of the whole chain can be performed trivially. In the maximization case, for
s
instance, we have
max G (hs).
s
hs|Hs
The collapse of the optimal chain problem into a sequence of cycle optimization problems however does
not occur when the cycle timing is also a decision variable.
In order to illustrate this case, let us assume, for the sake of simplicity, that the length z of the sth cycle
s
fully determines its value, G (z ). In this case, h is nothing but the sequence z"Mz N of the cycle durations.
s s
s

4 In the inventory case, for instance, the variables belonging to the vector hs could be the lot size for reorders, the date of
replenishment, the level of inventories under which a new order has to be placed, and so on.
5 We could also consider the case of k depending on s, but the setting with k constant will be su$cient for our objectives.

74

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

"+s~1 z , the value of the whole chain is then
h/1 h
`=
s~1
G(z)" + G (z )/ + z
s s
h
s/1
h/1
or, equivalently,

Since t

s~1

A

B

`=
G(z)" + G (t !t )/(t ).
s s
s~1
s~1
s/1
Let us state the "rst-order necessary conditions for the maximization case. In this paragraph we present the
methodology, while we in Section 7 we apply it to the plant replacement case.
The problem of "nding the best timing of the cycles turns out to be the following discrete optimal control
problem:
`=
sup + G (t !t )/(t )
s s
s~1
s~1
M N
ts s/1
(P) s.t. t *t ,
s
s~1
t "0.
0
The problem P can be studied via functional analysis by introducing the following assumption:
Assumption 1. The functions G ( ) ) depend in the same way on the di!erence t !t :
s
s
s~1
G (t !t )"H(t !t ).
s s
s~1
s
s~1
Under this assumption, it can be proved a sequence Mt N attains the supremum in P if the corresponding
s
NPV is the solution v : RPR of the functional equation
v(x)"sup [H(y!x)/(x)#v(y)]
(5.1)
ywx
at x"0. The converse also holds, provided that the solution to (5.1) converges to 0 as time tends to in"nity.
In this case we can identify the solution to our problem and the one of Eq. (5.1). We present an outline of the
results in Appendix B, which draws upon Stokey and Lucas [10].
Also, we can study the properties of (P) by resorting to the so-called Euler equation, in the following way.
First of all, we observe that if a sequence MtHN solves problem (P), then, since every tH appears in two addenda
s
s
of G only, tH solves
s
max H(x!tH )/(tH )#H(tH !x)/(x)
s`1
s~1
s~1
x
(Q) s.t. tH )x)tH o
s`1
s~1
If we assume that the function H is di!erentiable and the constraint in (Q) is not binding, the "rst order
necessary condition for the above problem, in correspondence to the sequence MtHN, is the Euler equation
s
H@(tH!tH )/(tH )!H@(tH !tH)/(tH)#H(tH !tH)/@(tH)"0.
s
s
s`1
s
s
s`1
s~1
s~1
s
Since de"nition (3.1) of / implies /@(tH)"!/(tH)d(tH), this condition can be re-written as
s
s
s
H@(tH!tH )/(tH )!/(tH)[H@(tH !tH)#H(tH !tH)d(tH)]"0.
s
s
s`1
s
s`1
s
s~1
s~1
s

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

75

It is a second-order di!erence equation, for which the boundary condition is the transversality one:
lim tH/(tH)[H@(tH !tH)#H(tH !tH)d(tH)]"0.
s
s
s`1
s
s`1
s
s
s?`=
If the functions H, H@ and d are "nite and the sequence MtHN diverges, the transversality condition is satis"ed,
s
due to the fact that
lim tH/(tH)"0.
s
s
s?`=
In our context the Euler equations are not su$cient for "nding a solution. They are only necessary, since they
assume that an optimal sequence exists. They will be applied to the plant replacement model in Section 7.
6. Optimization over a 5nite number of cycles
Consider now the case of n cycles. As above, we present "rst the methodology and postpone to Section
7 the applications. Since with n cycles, i.e. a "nite horizon, optimization of the NPV GM has been argued above
to be di!erent from optimization of the equivalent annuity CM , we study the former problem only. Also, we
distinguish the general, variable interest intensity d case from the constant d one.
6.1. Variable interest intensity
The value to be optimized, GM , is a function of the n variables z , z ,2, z , which we collect in the vector z:
1 2
n
n
s~1
GM (z)" + G (z )/ + z .
s s
h
s/1
h/1
Let us assume that the G are di!erentiable.
s
As usual in "nite horizon dynamic models,6 the set of the corresponding "rst order necessary condition
conditions can be treated recursively, if t is given.
n
Since

A

A

B A

B

B

u~1
s~1
+ z "/ + z U(t
,t )
h
h
u~1 s~1
h/1
h/1
u~1
and t
"t # + z , the partial derivatives of GM are:
u~1
s~1
h
h/s
/

s~1
n
u~1
s~1
LGM (z) dG (z )
" s s / + z # + G (z )/ + z U@ t # + z , t
,
h
u u
h s~1
h 1 s~1
dz
Lz
s
s
h/1
u/s`1
h/s
h/1
s"1, 2,2, n!1,

A

B

A

B A

B

dG (z ) n~1
LGM
" n n / + z ,
h
dz
Lz
n
n
h/1

A

B

where U@ is the partial derivative of U with respect to its "rst argument.
1
6 With reference to net present value problems, this has been pointed out in a more restrictive framework by Schneider [11] and
Trovato [12,13].

76

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

Using the previous result and keeping in mind that the discount factors are positive, the "rst order
necessary condition +G"0, where 0 is the null vector, provides us with the system:
n
dG (z )
s s # + G (z )U@ (t
, t )"0 for s"1, 2,2, n!1,
u u 1 u~1 s~1
dz
s
u/s`1
dG (z )
n n "0.
dz
n
Under suitable conditions the last equation will give us the optimal duration of the last cycle zH. If we insert
n
this value in the equation with s"n!1, the latter becomes
dG
(z
)
n~1 n~1 #G (zH)U@ (t
#z , t
)"0,
n~1 n~2
n n 1 n~2
dz
n~1
or
dG
(z
)
n~1 n~1 !G (zH)U@ (t !zH, t !zH!z
)"0
(6.1)
n
n~1
n n
n n 2 n
dz
n~1
,t
"t
#z
"t !zH.
since t
"t !zH!z
n
n
n~1 n~1
n~2
n~1
n
n~2
n
Again, under suitable conditions and for given t Eq. (6.1) provides us with the optimal value zH . By
n~1
n
going backwards in this way all the optimal durations can, at least in principle, be computed.
We note also that the "rst order necessary condition for zH can be re-written as
n~1
) dz
.
(6.2)
)d(t !zH!z
dG
(z
)"G (zH)U(t !zH, t !zH!z
n
n~1 n~1
n
n~1 n
n n
n
n~1 n~1
n n
This formulation has an important "nancial interpretation: the optimal duration zH must be chosen so as
n~1
to equate the variation of the value of the (n!1)th cycle (value which is computed at t
) to the marginal
n~2
variation * also evaluated at time t
"t !zH!z
* of the nth cycle. According to the properties of
n~2
n
n
n~1
the interest intensity, namely the fact that it is the simple interest rate maturing instantaneously on any
operation, the marginal variation of the nth cycle is computed by applying to its value at
) * the interest intensity which holds at time t
, namely
t
* G (zH)U(t !zH, t !zH!z
n
n~1
n~2
n n
n
n~2
n n
), and multiplying times the di!erential of the n!1st duration, dz
.
d(t !zH!z
n
n~1
n~1
n
Similarly, for s"1, 2,2, n!2, the backward approach to solving the "rst-order conditions yields

B

A

u~1
n
dG (z )
s s ! + G (zH)U@ t
,t
!z ! + zH "0.
i
2
u~1
u~1
s
u
u
dz
s
i/s`1
u/s`1
Since t
"t !+n z , this condition can be re-written as
i/u i
u~1
n
dG (z )
n
n
n
s s ! + G (zH)U@ t ! + zH, t ! + zH "0.
(6.3)
i
i n
u u 2 n
dz
s
i/s`1
u/s`1
i/u
Let us denote with tH the optimal maturity, as given by the backward procedure:
s
n
$ ! + zH.
tH"t
i
s
n
i/s`1
Exploiting the di!erentiability of U and using the de"nition above, condition (6.3) can in turn be transformed into

A

n
dG (z )" + G (zH)U(tH , tH!z )d(tH!z ) dz
s
s
s s
u~1 s
s s
u u
u/s`1
which can be interpreted analogously to (6.2).

B

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

77

6.2. Constant interest intensity
In the particular case when the interest intensity d is constant we do not need to know the "nal date t in
n
order to solve the optimization problem, since, if we have a solution zH to dG (z )/dz "0, we can substitute
n
n n
n
it into the equation with s"n!1, as before. However, since the discount factors depend on the length of
time over which they are applied only, U(t
,t
)"exp[!d(t
!t
)]"exp(!dz
), and the
n~1 n~2
n~2
n~1
n~1
necessary condition for s"n!1, (6.1) above, becomes
dG
(z
)
n~1 n~1 !dG (zH) exp(!dz
)"0
n~1
n n
dz
n~1
in which t does not appear. Analogously, for s"1, 2,2, n!1, the "rst-order condition (6.3) becomes
n

C A

dG (z )
u~1
n
s s !d + G (zH) exp !d + zH#z
i
s
u u
dz
s
i/s`1
u/s`1

BD

"0

in which t does not appear.
n

7. Optimization in the plant replacement model
7.1. Inxnite number of cycles
Let us consider the in"nite horizon maximization problem studied in Section 5 for the plant replacement
case:

C

P

ts
`=
sup + !A(t )/(t )#
r(t!t )/(t) dt#R(t !t )/(t )
s~1
s
s~1
s
s~1
s~1
M N
ts s/1
ts~1
(P) s.t. t *t ,
s
s~1

D

t "0.
0
Let us suppose that the initial costs of the plants (before discounting) and the interest intensity are constant,
so that A(t )"A for every s, and U(t, t )"U(t!t )"exp(!d(t!t )). Under these hypotheses the
s
s~1
s~1
s~1
optimal control problem (P) can be rewritten in terms of durations z :
s
zs
`=
s~1
sup + !A# r(u) exp(!du) du#R(z ) exp(!dz ) exp !d + z
s
s
h
M N
0
h/1
zs s/1
(P) s.t. z *0.
s

C

P

D A

This problem satis"es Assumption 1, since

P

G (z )"H(z )"!A#
s s
s

zs
0

r(u)U(u) du#R(z )U(z )
s
s

B

78

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

so that it can be re-written, according to Section 5, in the Q-form:

C
C

P
P

D

x~t*s~1
max !A#
r(u)U(u) du#R(x!tH )U(x!tH ) /(tH )
s~1
s~1
s~1
0
x
t*s`1
r(u)U(u) du#R(tH !x)U(tH !x) /(x)
# !A#
s`1
s`1
0

D

(Q) s.t. tH )x)tH .
s`1
s~1
The H function in Q is di!erentiable, with derivative
H@(x!tH )" [r(x!tH )#R@(x!tH )!R(x!tH )d] exp (!d(x!tH )).
s~1
s~1
s~1
s~1
s~1
It follows that the Euler condition for Q is
[r(x!tH )#R@(x!tH )!R(x!tH )d]
s~1
s~1
s~1
! [r(tH !x)#R@(tH !x)!R(tH !x)d] exp(!d(tH !x))
s`1
s`1
s`1
s`1
tHs`1 ~x
! d !A#
r(u) exp(!du) du#R(tH !x) exp(!d(tH !x)) "0.
s`1
s`1
0
Let us suppose that both the di!erence between gross margins and maintenance costs r(z ) and the residual
s
value R(z ) are linear functions, i.e. r(z )"r !r z and R(z )"R !R z , with r r , R and R positive
s
s
0
1 s
s
0
1 s
0, 1 0
1
constants. In this case the Euler equation admits a stationary solution zH i!:

C

D

P

[r !dR !R !(r !dR )zH][1!exp(!dzH)]
0
0
1
1
1
zH
!d !A# (r !r u) exp(!du) du#(R !R zH) exp(!dzH) "0.
0
1
0
1
0
With some algebra, this equation becomes

C

D

P

d2(R !A)
0
exp(!dzH)"!dzH#1!
r !dR
1
1
which admits a unique positive solution provided that
d2(R !A)
0
(0.
r !dR
1
1
In turn, as the costs of the plant (A) should be greater than the "nal value, even if the plant is not used
(R(0)"R ), the condition for the existence of a stationary duration zH'0 becomes r !dR '0.
0
1
1
The transversality condition is satis"ed by the stationary solution, since the sequence MtHN diverges and
s
lim tH exp(!dtH)"0.
s
s
s?`=
7.2. Finite number of cycles
In the "nite chain case, the NPV GM (z) can be maximized by a proper choice of the sequence Mz N. If A and
n
s
d are constant, it is

C

P

D A

B

zs
s~1
n
+ !A# r(u) exp(!du) du#R(z ) exp(!dz ) exp !d + z .
s
s
h
0
h/1
s/1

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

79

Since the interest intensity is constant, we do not need to know the "nal date in order to solve this
maximization problem. We treat it recursively, as suggested in Section 6.2 above, starting from zH, which
n
must satisfy the condition
dG (z )
n n "[r(z )#R@(z )!dR(z )] exp(!dz )"0.
n
n
n
n
dz
n
This condition can be simpli"ed into
r(z )#R@(z )!dR(z )"0
n
n
n
which, under linearity of r and R, becomes
R !r #dR
0
0.
(7.1)
zH"! 1
n
r !dR
1
1
With r !dR '0, the "nite horizon problem has then a solution provided that !R #r !dR '0. In
1
1
1
0
0
this case in fact we obtain a positive "nite length zH, which can be inserted into the necessary condition for zH :
n~1
n
dG
(z
)
)
0" n~1 n~1 !dG (zH) exp(!dz
n~1
n n
dz
n~1

C

"exp(!dz
) r(z
)#R@(z )!dR(z
)
n~1
n~1
n~1
n~1

P

z

H
n

D

r(u) exp(!du) du#dR(zH) exp(!dzH) .
n
n
0
Using again the linearity of r and R the latter condition can be re-written as
#dA!d

zHn
(r !r u) exp(!du) du!(R !R zH) exp(!dzH) "0
n
0
1
0
1 n
0

C P

r !r zH !R !d(R !R zH )# d A!
1
0
1 n~1
0
1 n~1

D

and easily solved for zH as a function of zH:
n
n~1
1
r !dR !r /d
R !A
0
1 !zH .
zH " !d 0
#exp(!dzH) 0
(7.2)
n~1 d
n
n
r !dR
r !dR
1
1
Substituting for zH, we get zH and can proceed backward to zH for s)n!2, according to the last formula
s
n~1
n
of Section 6.

A

B

Example 4. We consider n"3 cycles of a chain and assume the conditions given in Table 4.
Using (7.1) and (7.2), the NPV G(z) turns out to be maximized at
z"[0, 4.926, 4.914, 5.667].
If, by maintaining the same data, we increase the number of cycles up to n"20, all the "rst "fteen optimal
lengths are equal * and amount to 4.925 * while the length itself increases steadily only during the last "ve
Table 4

Table 5

Variable or function

Speci"cation

Variable or function

Speci"cation

A
r(t)
/(t)
R(z)

1000
700!100t
exp[!0.1t]Nd"0.1
900!100z

A
r(t)
/(t)
R(z)

1000
700 exp(!0.2t)
exp[!0.1t]Nd"0.1
900 exp(!0.1z)

80

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

cycles. In particular, for the last cycle, we have a length of 5.667. The same e!ect can be noticed in the
following example.
Example 5. We consider n"20 cycles and adopt the speci"cations given in Table 5.
Using Eqs. (7.1) and (7.2), the NPV G(z) turns out to be maximized when the "rst "fteen cycles have length
8.124, with optimal length rapidly increasing (up to the last one, 13.581) after.
8. Conclusions and further research
This paper has extended the equivalent annuity de"nition}in the presence of a sequence of cycle
investments}to the case of interest intensity variable over time.
It has examined "rst the correspondence/non correspondence of two procedures: maximizing (or minimizing) the NPV directly or through the equivalent annuity.
Second, it has extended the ensuing optimization problem, with respect to the length of the cycles.
As for the correspondence, we have argued that it holds}in an in"nite horizon framework}provided that
we consider not the annuity over a single cycle and the corresponding NPV, but the equivalent annuity over
the whole chain of cycles and its NPV. In the "nite horizon, the correspondence collapses also for the whole
chain (Section 3).
We have used a plant replacement model as an example (Section 4).
As for the optimal length problem, in the in"nite horizon case we have discussed both functional equation
methods and Euler conditions (Section 5). In the "nite horizon one we have used recursive methods (Section
6). The latter have been applied to the plant replacement model (Section 7).
The plant replacement application is interesting not only per se, but also since it provides evidence of
a substantial stationarity of the cycle lengths in su$ciently long chains. It also gives examples of optimal
nonmonotonic policies.
The aforementioned stationarity of the optimal replacement policy even in "nite chains is currently under
study. It holds under more general assumptions than the ones used in the examples of the present paper. It
seems particularly promising for practical purposes, in that it indicates a simple, general structure for
replacement policies, well grounded in theory. This general structure permits to understand better the
implications of largely used approximations, such as the in"nite horizon one.
The main theoretical conclusion of the paper is that the equivalent annuity principle must be used with
caution, whenever the model has optimization purposes, such as the choice of the optimal length of the NPV
cycles. Our contribution consists exactly in pointing out the limits of its application and recalling methodologies for the corresponding optimality problem.
The main practical contributions of the paper are three. First, lack of care in using equivalent annuities can
be relevant in practice. In the plant replacement model example, for instance, we lose the 10}15% of net
present value already during the "rst cycle, when using the equivalent annuity instead of the net present
value. Furthermore, in one of our examples, the equivalent annuity produces a net loss, while net present
value gives positive earnings. Second, our applications show that, under weak hypotheses, optimal replacement policies are approximately stationary even over a "nite horizon, which is somewhat surprising and
interesting in itself. Last but not least, our setting is able to cope with the time evolution of interest rates,
which these days is becoming more and more important also for non-"nancial users.
Acknowledgements
The Authors wish to thank Robert GrubbstroK m for helpful comments and suggestions. The usual
disclaimers apply.

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

81

Appendix A.
In this appendix, with reference to the variable d case, we prove that:
1. in the in"nite horizon case, maximizing (or minimizing) G is not the same as maximizing (or minimizing) C ;
s
s
2. in the "nite horizon case, maximizing (or minimizing) GM is not the same as maximizing (or minimizing) CM .
Proof. 1. The "rst-order necessary condition for G requires that at interior (positive) stationary points, with
s
G di!erentiable, we have
s
LG /Lz "0.
s s
For t
given, the "rst-order necessary condition on
s~1
ts~1 `zs
ts
U(u, t ) du"G
U(u, t )du
C"
s~1
s
s~1
s s
ts~1
ts~1
is

NP
C

NP

D

LC
G U(t #z , t )
LG /Lz
s"
s s~1 "0.
s s
! s s~1
s~1
s
`z
`zs U(u, t ) du]2
Lz
:ts~1 U(u, t ) du [:ts~1
s
t
ts~1
s~1
s~1
In turn, this condition can be simpli"ed into
LG
U(t #z , t )
s !G
s~1
s s~1 "0.
s :ts~1 `zs U(u, t ) du
Lz
s~1
s
t
s~1
Since U(t , t )O0, the previous condition is the same as LG /Lz i! G "0. This means that maximizing (or
s s~1
s s
s
minimizing) G is not the same as maximizing (or minimizing) C .
s
s
2. The "rst-order necessary condition for a stationary point for GM and CM with respect to z are di!erent. The
s
one for GM is
LGM /Lz "0
s
while, for t
given, the "rst-order necessary condition on
s~1
GM
CM "
:z1 `z2 `2`zn /(t) dt
0
is

(A.1)

(A.2)

LGM
LCM
Lz
GM /(t )
s
n
"
!
"0.
Lz
:z1 `z2 `2`zn /(t) dt [:z1 `z2 `2`zn /(t) dt]2
s
0
0
In turn, this condition can be simpli"ed into
LGM
/(t )
n
!GM
"0.
(A.3)
Lz
:z1 `z2 `2`zn /(t) dt
s
0
Since /(t)O0 for every t, condition (A.3) is the same as (A.1) i! GM "0. It follows that maximizing (or
minimizing) GM is not the same as maximizing (or minimizing) CM .

82

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

Appendix B.
This appendix analyzes problem (P). Following Stokey and Lucas [10], we can de"ne P(0), the set of
admissible sequences of maturities t , which are the nondecreasing ones, starting at 0:
s
$ MMt N`= : t "0, t *t s"0, 1, N
P(0)"
2
s`1
s
s s/0 0
and in general P(t ), the set of admissible (nondecreasing) sequences of maturities t , starting from t *0:
0
s
0
$ MMt N`= : t *t , s"0, 1, N.
P(t )"
2
s
0
s s/0 s`1
$ tI i!
We denote with tI the elements of P, i.e., those nondecreasing sequences which start at t : Mt N`= "
0 s s/0
t *t for every s*0. On the set P(t ) we can de"ne the functions u : P(t )PR, partial sums of the series
s`1
s
0
n
0
G, as follows:
n
$ +
u (tI )"
H(t !t )/(t ).
n
s
s~1
s~1
s/1
We can also de"ne their limit, as n diverges:
$ lim u (tI )
u(tI )"
n
n?`=
$ RXM!R,#RN. Let us denote with vH(t )
which maps P(t ) into the set of extended real numbers RM "
0
0
the sup of u(tI ):
$ sup u(tI )
vH(t )"
0
tI |P(t0 )
which, by de"nition, exists, is unique and is the sup in P (the NPV once the optimal sequence of maturities
has been chosen).
It can be proved (Theorems 4.2}4.5 of Stokey and Lucas [10]) that vH(t ) satis"es (5.1) and, conversely, that
0
if there is a function v which satis"es (5.1) and is such that
lim v(t )"0
n
n?`=
for every admissible sequence tI , then v is the sup in problem (P): v"vH(t ). In both cases the corresponding
0
optimal timings, i.e. the ones that attain the supremum in (P), and which we denote with MtHN, satisfy
s
v(tH )"H(tH!tH )/(tH )#v(tH)
(B.1)
s~1
s
s~1
s~1
s
for every s. Conversely, every time sequence tI which satis"es (B.1) and has the property
lim sup v(tH))0
s
s?`=
is optimal.

E. Luciano, L. Peccati / Int. J. Production Economics 69 (2001) 65}83

83

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