Is Purity Necessary and Sufficient for Truncatability ?
Quaderno n. 40 Istituto di Matematica
Facoltà di Scienze Economiche e Bancarie Università degli Studi
Siena
SUMMARY
In order to ensure the uniqueness of the Internal Rate of Return I.R.R. of a givenÐ Ñ project, two different proceedings can be found in the economic-financial literature: one based on the uniqueness of the I.R.R., later performed by the concept of purity, and another referring to the so-called Truncation Theorem.
The aim of this paper is to explain the connections between the purity of the I.R.R.
of a given project and the truncatability of the project itself.
We will clarify a very important relation between purity and truncatability: if a given investment project has a “pure" I.R.R. , then its Present Value is maximum3!
compared with the Present Values of each of its shorter lives, when the rate of interest 3 belongs to an interval of the type Ó à 31 7+BÒ, whith 3 Ÿ 3! 7+B, and, as a main consequence, we will show that purity is necessary and sufficient for truncatability.
1PRELIMINARY DEFINITIONS
In this paper we will adopt the following definitions.
Definition 1 : We define as an investment project a vector of expected net outputs
œ + ß + ß ÞÞÞß +! " 8 such that:
. ,
,
, for at least one couple Ú
ÛÜ
+ ! + Á !
+ † + ! 4ß 5
! 8
4 5
For every project we consider its 8 1 Present Value at time functions:!
•: 5
5œ!
:
3 œ + † " 3 5 œ
...œ + + + + + ,
" 3 " 3 " 3 " 3
! " # :" :
# :" :
a : À ! Ÿ : Ÿ 8 ,
and its 8 1 Cumulated Value at time functions::
’: 5
5œ!
:
3 œ + † " 3 :5 œ
...œ + † " 3! : + † " 3" :" +:"† " 3 +:, .
a : À ! Ÿ : Ÿ 8 Obviously:
Ð Ñ1 , ’: 3 œ " 3 : †•: 3 a : À ! Ÿ : Ÿ 8. Using these functions we have first of all:
Definition 2 : We define as an I.R.R. attached to project œ + ß + ß ÞÞÞß +! " 8 an interest rate such that:3!
’8 3! œ•8 3! œ ! .
Following 14 and 15 , we give then:Ò Ó Ò Ó
Definition 3 : We define a project œ + ß + ß ÞÞÞß +! " 8 as a “pure investment project" for a given rate of interest if the following inequalities are satisfied:3!
Ð Ñ2 ’: 3! œ + † " 35 ! Ÿ !, a : À ! Ÿ : Ÿ 8 " .
5œ!
:
:5
or, by 1 , ifÐ Ñ
•: ! 5 ! , .
5œ!
:
3 œ + † " 3 5 Ÿ ! a : À ! Ÿ : Ÿ 8 "
2PROPRIETIES OF PURE PROJECTS
As lim , 1 , referring to 14 , we can easily conclu-
3Ä∞’ 3 œ ∞ a : À: Ÿ : Ÿ 8 Ò Ó de:
Proposition 1 : Given an investment project œ + ß + ß ÞÞÞß +! " 8 then there exists an interest rate 3738 such that:
, ’: 3 Ÿ ! a 3 − Ò3738; ∞Ò, a : À ! Ÿ : Ÿ 8 ".
Clearly 3738 1 , and we can easily see that 3738œ 1 if and only if is the+8 only positive cash-flow of the project.
In Appendix 1 we demonstrate the following:
Proposition 2 : Given an investment project œ + ß + ß ÞÞÞß +! " 8 , ’: 3 is a non-positive decreasing function of , 3 a 3 − Ò3738; ∞Ò a : À, 1Ÿ : Ÿ 8 1 . Moreover, ’ 3 is a de- creasing function in the same interval.
In addition, as demonstrated in reference 13 and 5 , the following holds:Ò Ó Ò Ó
Proposition 3 : An I.R.R. attached to the project 3! œ + ß + ß ÞÞÞß +! " 8 is unique if is a pure investment project at that rate of interest .3!
As a consequence, following reference 5 , we also have:Ò Ó
Proposition 4 : If an investment project œ + ß + ß ÞÞÞß +! " 8 is pure in its I.R.R. , then3!
there exists a set:
: ; : 1
˜ˆJ33" J33‰ 3 œ ß ÞÞÞß 8™
of consecutive investment operations such that:8
J"! œ +! ;
, : 1 ; J33 œ " 3! †J33" a 3 Ÿ 3 Ÿ 8
, : 1 1 ; J33 J3"3 œ +3 a 3 Ÿ 3 Ÿ 8
J88 œ +8 ;
which formally are see 4 :Ð Ò ÓÑ
, : 1 ; J33" œ’3" 3! Ÿ ! a 3 Ÿ 3 Ÿ 8
, : 1 . J33 œ " 3! †’3" 3! ! a 3 Ÿ 3 Ÿ 8
So the I.R.R. is an interest rate uniformly applied to consecutive investment3! 8 operations into which a project, cumulated from time to time , can be uniquely! 8 decomposed (see figure below).
J"! ÊJ""
Ì
† J#" ÊJ##
† Ì
† † J$# ÊJ$$
† † Ì
...
† † † J%$ Ê
...
† † † Ì
...
† † † † Ê J8#8#
† † † †
† † † † J8"8# ÊJ8"8"
† † † † Ì
† † † † † J88" ÊJ88
† † † † † Ì Ì
...
+! +" +# +$ +8# +8" +8
It is demonstrable, by Proposition 2 see also 9 , that the following hold:Ð Ò ÓÑ
Proposition 5 : An investment project œ + ß + ß ÞÞÞß +! " 8 satisfies conditions 2 if andÐ Ñ only if ’ 3738 !.
Proposition 6 : A necessary condition for a project œ + ß + ß ÞÞÞß +! " 8 to have a pure I.R.R. is that 3! + † + !! 8 .
3ANOTHER ECONOMIC INTERPRETATION OF PURITY Let us now introduce the following functions of the rate of interest :3
È: 85
5œ!
:
3 œ + † " 3 5: œ
...œ + + + + ,
" 3 " 3 " 3
8: 8:" 8" 8
:" :
, a : À ! Ÿ : Ÿ 8 "
Present Value functions of the cash-flows +8:ß +8:"ß ÞÞÞß +8 from time to time 8 8 :, and
Æ: 85
5œ!
:
3 œ + † " 3 5 œ
...œ +8:† " 3 : +8:"† " 3 :" +8"† " 3 +8, ,
a : À ! Ÿ : Ÿ 8 "
Cumulated Value functions of the cash-flows +8:ß +8:"ß ÞÞÞß +8 from time 8 : to time 8.
We clearly have ’8 3 œ Æ8 3 and •8 3 œÈ8 3 . After easy calculations we get:
1 ,
• 3 œ •:" 3 " 3 :†È8: 3 a : À Ÿ : Ÿ 8
and:
1 ,
’ 3 œ " 3 8:"†’:" 3 Æ8: 3 a : À Ÿ : Ÿ 8 from which, as Æ8: 3 œ " 3 8:†È8: 3 , we get also:
,
’ 3 œ " 3 8:†È8: 3 " 3 8:"†’:" 3 and finally:
Ð Ñ3 " 3 8:†È8: 3 œ’ 3 " 3 8:"†’:" 3 . Let us now consider the equation • 3 œ !, in which we put C œ " 3
1 , to have:
...+ † C +8 8 8"† C8" + † C + œ !" ! .
If the equation • 3 œ ! has a solution , we put 3! " 3! " œ C! to obtain:
...+ † C +8 8 8"† C8" + † C + œ" ! C † C C! .
Under the hypotesis + !8 , if the coefficients of the polynomial C are all non- negative like the first, , then, for Descartes' rule of signs applied to polynomial +8 C , the root is unique.3!
But it is easy to see that the coefficients of the polynomial C are given by the È: 3! , a : À ! Ÿ : Ÿ 8 ", so the following holds:
Proposition 7 : Given a project œ + ß + ß ÞÞÞß +! " 8 with an I.R.R. , if the following3! inequalities are satisfied:
Ð Ñ4 È: 3! œ +85† " 3! !, a : À ! Ÿ : Ÿ 8 ",
5œ!
:
5:
then the I.R.R. is unique.3!
As Æ: : È: , 4 is equivalent to 3 œ " 3 † 3 Ð Ñ
Æ: ! 85 ! , .
5œ!
:
3 œ + † " 3 5 ! a : À ! Ÿ : Ÿ 8 "
Using 3 , as Ð Ñ ’ 3! œ !, we have immediately:
Proposition 8 : Conditions 2 are satisfied if and only if conditions 4 are satisfied.Ð Ñ Ð Ñ We can get another economic interpretation of purity, because we can demonstrate, as in Proposition 4, that the following holds:
Proposition 9 : Given a project œ + ß + ß ÞÞÞß +! " 8 , which is pure in its I.R.R. , there3! exists a set
: ; : 1
ƒ ˜ˆK33" K33‰ 3 œ ß ÞÞÞß 8™
of consecutive financing operations such that:8
K!" œ +8 ;
, : 1 ; K33 œ " 3! "†K3"3 a 3 Ÿ 3 Ÿ 8
, : 1 1 ; K33K3"3 œ +83 a 3 Ÿ 3 Ÿ 8
; K88œ +! which formally are:
, : 1 ; K3"3 œÈ3" 3! ! a 3 Ÿ 3 Ÿ 8
, : 1 .
K33 1 È
! 3" !
œ † 3 Ÿ ! a 3 Ÿ 3 Ÿ 8
" 3
So, the I.R.R. is an interest rate uniformly applied to consecutive financing3! 8 operations into which a project, discounted from time to time 8 8 : : À ! Ÿ : Ÿ 8, , can be uniquely decomposed.
It is easy to see, by the definitions and as ’ 3! œ !, that the following relation holds:
. K8383"œ " 3! †J33"
If + !8 , we have : 3 œ ∞ a : À, 1Ÿ : Ÿ 8, so it is easily deducible
3Älim
1 È that:
Proposition 10 : If in the project œ + ß + ß ÞÞÞß +! " 8 is + !8 , then an interest rate 37+B exists such that:
, È: 3 ! a 3 − Ó à 31 7+BÒ, a : À ! Ÿ : Ÿ 8 1 .
Proposition 11 : If in the project œ + ß + ß ÞÞÞß +! " 8 is + !8 , then È: 3 is a non- negative decreasing function of , 3 a 3 − Ó à1 7+BÒ a : À, 1Ÿ : Ÿ 8 1 . Moreover, • 3 is a decreasing function in the same interval.
The Proof is given in Appendix 2.
It is also possible to demonstrate see 9 that:Ð Ò ÓÑ
Proposition 12 : A project œ + ß + ß ÞÞÞß +! " 8 satisfies conditions 4 if and only ifÐ Ñ È8 37+B œ• 37+B Ÿ ! .
Finally, see 9 , the following holds:Ò Ó
Proposition 13 : A project œ + ß + ß ÞÞÞß +! " 8 has a pure I.R.R. if and only if3!
3738 Ÿ 37+B. In this case we also have that : 3738 Ÿ 3 Ÿ 3! 7+B.
On the contrary, when a project has a unique I.R.R. which is not pure or it has more than one I.R.R., 3 ß 3 ß ÞÞÞß 3" # 5, in this case it is easily deducible that 3738 37+B and that
37+B Ÿ 3 ß 3 ß ÞÞÞß 3 Ÿ 3" # 5 738.
4THE TRUNCATION THEOREM
Starting from the behavioural hypothesis that the investor might choose the duration of a given project, neglecting scrap values, two different procedures for choosing the optimal duration can be found in the economic literature.
The first, due to P. H. Karmel (see 6 and 5 ), suggests to choose that particular du-Ò Ó Ò Ó ration to which the maximum I.R.R. is attached, and in article 6 it has been proved that ifÒ Ó an investor chooses as the optimum duration of a project that particular duration to which the maximum I.R.R. is attached, then the I.R.R. connected to this duration is unique.
S. Gronchi demonstrates in 5 that, doing so, such an I.R.R. is pure too.Ò Ó
Many Authors instead, see 1 , 3 , 16 and 17 , object that the best truncation cri-Ò Ó Ò Ó Ò Ó Ò Ó terion is to pursue the maximisation of the Present Value.
In fact, K. J. Arrow and D. Levhari in 1 state that: “...choosing a truncation periodÒ Ó so as to maximise the I.R.R. is not using the proper criterion for the selection of a truncation period ... one should choose that investment which maximises Present Value, using the given rates of discount ... the aim in the choice of a truncation period should be that of maximising the Present Value of the investment project."
J. F. Wright has demonstrated in 16 that if, at a given rate of discount, from theÒ Ó possible lenghts of life of the project the enterpreneur selects that with the greatest Present Value, then the Present Value function is decreasing at that rate.
Incidentally he also demonstrates that if at a rate the Present Value of the project is3 maximum compared to every shorter life of its, then at this rate the project is also pure.3
This result has been generalizes in two ways, the first in 3 , and furthermore in 11Ò Ó Ò Ó and 12 , where it is supposed that the rate can vary during the periods, and the second inÒ Ó Ò Ó1 , where K. J. Arrow and D. Levhari say to have proved that if, with a given constant rate of discount, we choose the truncation period so as to maximise the Present Value of the project, then the I.R.R. of the truncated project is unique.
However it has been proved in 10 that Karmel's procedure and Arrow andÒ Ó Levhari's one lead to the same numerical result.
To be more precise, Arrow and Levhari prove that if the life of the project is optimally chosen, then the maximised Present Value of the project is a monotonic decreasing function of the rate of interest.
But they also say that “... < 3 œmax • 3 , the Present Value of the best
: :
truncated project, is a decreasing monotonic function of ."3
This is not exact, as < 3 is not, generally, the Present Value of the best truncated project; it is the maximum, varying , among the 3 •: 3 , and we shall demonstrate that this coincidence is true if and only if the project has a pure I.R.R. .
This incorrectness is also found in 17 , where J. F. Wright says:Ò Ó
“if truncatability costless extricability holds, then the Present Value of a project for itsÐ Ñ whole life will be greater than its Present Value for some shorter life..." and “... thus the simple time-profile ensured by truncatability is identical with that implied by purity. But a project might happen to have such a profile even though it cannot be truncated. Thus truncatability is a sufficient rather than a necessary condition; and consequently less fundamental than purity."
5TRUNCATABILITY AND PURITY
The following Proposition could clarify the above-mentioned ambiguities.
In fact, we can prove that:
Proposition 14 : If the project œ + ß + ß ÞÞÞß +! " 8 has a pure I.R.R. , then 3! • 3 is maximum compared to all •: 3 , ! Ÿ : Ÿ 8 1 , a 3 − Ó à 31 7+BÒ.
Proof : Let us say that • 3 •8:" 3 ! a : À ! Ÿ : Ÿ 8 ", .
In fact we have:
...• 3 • 3 œ + + + œ
" 3 " 3 " 3
8:" 8: 8" 8
8: 8" 8
1 ...
œ † + œ
" 3 " 3
+ +
" 3
– —
8: 8: 8" 8 :
:"
1 .
œ † 3
" 3 8: È:
But the I.R.R. of the project is pure, and so, by Proposition 11 we see that, a : À ! Ÿ : Ÿ 8 1 , È: 3 is a non-negative function, a 3 3 − Ó à 3: 1 7+BÒ and so we have the Proof.
If a project is pure in its I.R.R., then its Present Value is maximum compared with the Present Values of all its truncations, when the rate of interest varies in an interval into which its I.R.R. lies; so the Present Value • 3 is the same as the maximised Present Value and Arrow and Levhari's assertions are true.
The best truncation, if the project is pure in its I.R.R., is the whole project so we can conclude that pureness is necessary and sufficient for truncatability and that, using Karmel's procedure, from every project we can draw out a sub-project, which can be the project itself, the I.R.R. of which is pure.
To conclude, we also have the
Proposition 15 : If the project œ + ß + ß ÞÞÞß +! " 8 has a pure I.R.R. , then 3! Æ 3 is minimum compared with all Æ: 3 , ! Ÿ : Ÿ 8 1 , a 3 − Ò3738à ∞Ò.
Proof : Let us say that Æ 3 Æ8:" 3 Ÿ ! a : À ! Ÿ : Ÿ 8 ", . In fact we have:
Æ 3 Æ:" 3 œ
...œ +8:† " 3 : + † " 3" 8" + † " 3! 8 œ œ " 3 :† +c 8: +8:" " 3 ... + † " 3! 8:dœ
œ " 3 :†’8: 3 .
But, by the hypoteses and as 2 holds, we have the Proof.Ð Ñ
However it is possible for a project to have a Present Value maximum for all acce- ptable rates 3 À 1 3 ∞.
It is easy to prove that it is possible if and only if is the only negative cash-flow.+! In this case we put 37+B œ ∞.
So for the uniperiodal project œ + ß +! " , Ð+ ! + !Ñ! , " we have 3738œ 1 and 37+B œ ∞, and this propriety extends only to the projects of the type
œ + ß !ß ÞÞÞß !ß +! 8 .
APPENDIX 1 Proof of Proposition 2 : If 3 − Ò3738à ∞Ò, we have:
’! 3 œ + !! ;
,
’" 3 œ + † " 3 +! "
for which ’" 3738 Ÿ ! and ’w" 3 œ + !! , and so ’" 3 is a non-positive decreasing function a 3 − Ò3738à ∞Ò .
By induction, suppose ’:" 3 is a non-positive decreasing function a 3 − Ò3738à ∞Ò.
Then:
, ’: 3 œ’:" 3 † " 3 +: with
, ’: 3738 Ÿ ! and
’w: 3 œ’:" 3 " 3 †’w:" 3 ,
so ’: 3 is decreasing because ’:" 3 is non-positive and decreasing.
By induction, it follows Proposition 2, a : À1Ÿ : Ÿ 8 1 . But , ’ 3 œ’8" 3 † " 3 +8 so
, ; .
’w 3 œ’8" 3 " 3 †’w8" 3 Ÿ ! a 3 − Ò3738 ∞Ò
APPENDIX 2 Proof of Proposition 11 : If 3 − Ó à 31 7+BÒ, we have:
È! 3 œ + !8 ;
, È" 3 œ + † " 38 " +8"
for which È" 7+B and Èw 8 , and so È" is a non-negative
" #
3 ! 3 œ + † " 3 ! 3
decreasing function a 3 − Ó à 31 7+BÒ.
By induction, suppose È:" 3 is a non-negative decreasing function a 3 − Ó à 31 7+BÒ.
Then :
, È: 3 œÈ:" 3 † " 3 " +8: with
, È: 37+B ! and
, È:w 3 œ " 3 #†È:" 3 " 3 "†È:"w 3
so È: 3 is decreasing because È:" 3 is non-negative and decreasing . By induction, Proposition 11 holds a : À1Ÿ : Ÿ 8 1 .
But , • 3 œÈ 3 œÈ8" 3 † " 3 " +! so
, •w 3 œ " 3 #†È8" 3 " 3 "†È8"w 3 Ÿ ! a 3 − Ó à 31 7+BÒ.
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