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Duality, interest rates, and the theory of present value
JOURNAL
OF ECONOMIC
THEORY
30, 98-l
14 (1983)
Duality, Interest Rates, and the Theory of Present Value* PHILIP
H.
DYBVIG
School of Management, Yale University, New Haven, Connecticut 065.20 Received
July
27, 1981
This paper uses duality theory to simplify and extend previous work which characterized technologies which have present value decreasing in interest rates. For “unbounded” interest rate sets, value nonincreasing is equivalent to truncation never being desirable. For bounded interest rate sets, the result is true if truncation is replaced by limited truncation, appropriately defined. Similarly. for equal forward rates, undesirability of geometric truncation characterizes nonincreasing value. Another result shows that if truncation is undesirable at an internal rate of return, then the internal rate of return is well behaved, even if value is not decreasing everywhere. Journal of Economic Literature Classr$cation Numbers: 022, 111, 522.
1.
INTRODUCTION’
Duality theory has yielded insights into the standard theories of consumption, production, and general equilibrium (see, for example, Cass [2], Diewert [3], Jorgenson and Lau [4], Kehoe [5], McKenzie [7,8], and Scarf [ 12, p. 164, ff.]). Duality theory has been particularly fruitful in the theory of production; in particular, the value function or Hamiltonian representation of a technology has proven simpler to work with than the conventional production function or activity set representation for some purposes (see Shell [ 131). This paper represents an application of duality theory to the theory of interest rates and present values. Duality (and the conversion to prices from interest rates) is useful because prices and quantities occur symmetrically in the typical (bilinear) formula for value. More specifically, multiplying any particular quantity by a positive amount has the same effect on value as multiplying the corresponding price by the same positive amount. In addition, the value is a convex function of
* The author thanks William Brainard and John Long for useful comments, and especially Stephen Ross and Chester Spatt for useful comments and related previous joint work.
98 0022-0531/83 Copyright All rights
$3.00
0 1983 by Academic Press, Inc. of reproduction in any form reserved.
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99
prices (although not necessarily a convex function of interest rates), and any convex function is the value function (over prices) for some technology. The introduction of interest rates into the value function follows a simple intuition. Raising a single forward rate decreases all subsequent prices, all in the same proportion, leaving previous prices unchanged. As the interest rate becomes arbitrarily large, this represents a truncation of the price vector, i.e., all prices after the particular interest rate are set to zero. Therefore, raising interest rates truncates prices (partially or fully) which has the same effect on value as truncating flows. This intuition illustrates why there is such an intimate relationship between truncation (the option of costless abandonment of projects) and value decreasing as interest rates rise. This paves the way to give very simple new proofs of old results and to prove new results. The old results which are reproven here are all proven by Ross, Spatt, and Dybvig [lo] (henceforth, RSD). The central result of RSD is that, in a world where all forward rates are free to vary, it is true not only that a truncation technology will not increase in value as interest rates rise, but also, conversely, if value is globally nonincreasing in all interest rates, the technology resembles a truncation technology in two important respects. First, the value of the technology is always the same as the value of the truncation technology it generates. Second, the technology is the same as the truncation technology it generates, up to convexification and free disposal in a value sense. This paper shows that the first form of the result is more robust, since it generalizes easily to other cases. (There exist generalizations of the other form as well, but they seem to be convoluted.) A very simple direct proof of the first form of the central result of RSD is given here. The proof uses simple duality and an observation (formalized in Proposition 1) made jointly with Ross, that the price sets associated with global interest rate sets bounded below but not above (as used in RSD’) are actually truncation sets of a sort themselves. This observation is consistent with the intuition (elaborated above) that raising interest rates is like truncating prices. Another old result which can be reproven by the dual approach is the characterization of k-monotonicity given in RSD. A sketch of the new proof is given in footnote 6. Several new results are proven here. It is shown that in a strong sense, the RSD central result is valid only if we are looking at technologies which are glob&b nonincreasing in interest rates; i.e., the equivalence between truncation and nonincreasing value is not valid if we can bound interest rates in any substantial way. However, a characterization of technologies with nonincreasing value on a class of bounded interest rate sets is derived here. The characterization involves limited truncation, which is a natural generalization
I In RSD.
it is assumed
that (Vr. r’)((r
E T&
r’ > r) *
(r’ E r)).
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PHILIP H. DYBVIG
of truncation. In the same vein, nonincreasing value when forward rates are identical is shown to be equivalent to geometric truncation. This paper also addressesthe problem of determining when the generalized internal rate of return set (IRR) is well behaved. As noted in RSD, value decreasing in interest rates is equivalent to the IRR being well behaved for all levels (positive and negative) of fixed cost. This paper gives conditions under which the IRR is well behaved for a single level of fixed cost, both for the global and bounded interest rates cases. Potential application of this result is also discussed. Section 2 reviews the basic structure of the model (which is basically as in RSD). Section 3 contains the new direct proof of the RSD central result, the proof that the RSD characterization is valid for only the global problem, and the characterization of technologies which have value nonincreasing in interest rates over bounded rate sets. Section 4 characterizes when the IRR will be well behaved for a single value of fixed cost, gives additional results which may be germane for those wishing to use the internal rate of return criteria, and discusses applicability of these results. Section 5 closes the paper.
2. STRUCTURE OF THE MODEL The structure and notation used here will be roughly identical to that used in RSD. The basic structure is the familiar neoclassical activity framework. The technology A c R”+ ’ is a nonempty set representing feasible return streams over the n + 1 periods. The vector (a,, a, ,..., a,) = a E A. denotes a feasible activity with net positive or negative return aj in periodj. The activity set A can represent mutually exclusive or interdependent choices by including all mutually exclusive alternatives. The value function for an arbitrary technology A is defined to be V(r; A) = sup G asA ,yo r-g= ,‘;: + ri) ’ where nfZk+ I (1 + ri) 3 1. We will assume V(r; A) is finite for all feasible interest rates. The feasible rates will be representedby a set T c R ‘. Various choices for T will be used throughout the paper. (T is the set in which interest rates can be restricted to lie a priori.) An equivalent formulation is to associatewith each interest rate vector r > -e = -( 1, l,..., 1) the price vector p E R”+ ’ defined by P(r)=
c l,-,
1 1 + rl
1 (1 + f-,)(1 + r2> ‘-’ nl=,
1 (1 + ri) 1’
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Then the projit function
l7(p;A)-
sup i Pjaj, OEAj=o
with p E P(r), is equivalent to the value function V(r; A), and ZI(p; A) will be finite whenever p E P(T). Working with the profit function is useful becauseprice space is the dual of activity space. Now we formalize the concept of truncation. The kth truncation operator truncates an activity after the first k flows: tr,(a) = (a,, a, ,..., ak- t, 0 ,..., 0). The truncation
of an activity set A is given by tr(A) = {tr,(a) 1a E A, k = 1, 2 ,..., n + 1}.
Note that A G tr(A) since (Va E R”“)a = tr,,+ ,(a). It will also be useful to consider the strict truncation of an activity setA, defined by tr-(A)
= (tr,(a) 1a E A, k = 1, 2,..., n),
which does not normally include A.
3. SOME FUNDAMENTAL
RESULTS
The approach taken in this paper was motivated by the observation, made jointly with Ross, that the price setsgenerated by the interest rate sets T (for T satisfying the unbounded interest rate assumption used in RSD’) were truncation sets of a sort themselves. Loosely speaking, this is because sending the kth rate to infinity is like truncating all subsequentprices, which has the sameeffect on value as truncating all subsequentflows. This observation is illustrated for n = 2 in Fig. 1. The formal statement is: PROPOSITION
Proof:
1. Let F > -e. Then cl(P( (r 1r > ?})) = Hull(tr(P(F))).
We will show that
P( (r I r > r”}) = Hull(tr(P(r^))) n ( p 1p > O} from which the result follows since Hull(tr(P(i))) convex and contains P(f) > 0. First we prove that
g Rt+’
P( (r / r > t}) s Hull(tr(P(?))) n {p / p > 0).
is closed and
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PHILIP
H. DYBVIG
0
b
FIG. 1. The shaded area represents the set cl[P({r 1r > F})) = Hull(tr(P(i))). c = P(t). b = tr2(P(i)), and 0 = tr,(P(i)). (Note that p0 is the numeraire and always 1, so it is not shown.) Higher interest rates imply lower prices, i.e., ?, > r: > r(l for i = I. 2.
Suppose p* E I’( { I ( r > P}). Then there exists r > r” such that pi* =
1 rIzl
Define A* = (AT,..., AZ+ ,) inductively
(1 + Ti) . (in decreasing order) as
Then let A = (A,,..., An+ ,) be defined by
Then CAj=
1, L>O,
A,+, > 0, and2 p* = \‘ k=il
=(l+?j+,)*=l+rj,,. ,+ I
A, trJP(f))
> 0.
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RATES
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Therefore,
Now we show that P({r / r > F)) 3 Hull(tr(P(i)))
n (p 1p > O}.
Now supposep* E Hull(tr(P(r))) n {p / p > O}. Then there exists Lr, AZ,..., Antl such that C Lj = 1, ,l> 0, and p* = \‘
A, tr#(P)).
kTI
Since p* > 0, it follows that A,+ 1 > 0. Define r = (ri , rz ,..., I~) as
Since A>0 and A,,, > 0 the expression is well defined and r > L Furthermore, it is readily verified that p* = P(r). Therefore, P(P I r 2 ~1) E HWtr(W-1))
n {P / P > 01,
and by the comment at the start of the proof, we are done.
Q.E.D.
Now we will use Proposition 1 to give the simple proof of the value characterization of nonincreasing value given in RSD (Theorem 1 in both RSD and this paper). The only regularity required on the activity set A is that its value V(r; A) be finite for all feasible rates (i.e., for all r E 7). This is in contrast with RSD, where the proof would necessarily be complicated considerably by dropping the assumption that A is compact. Note that the present proof is valid even if V(r; tr(A)) is not necessarily finite for all r E T. In other words, if V(r; tr(A)) = co, then by Proposition 1 the value becomes unbounded as some rk gets large, so that V(r; tr(A)) = co, and the value must increase in rk somewhere. THEOREM 1. Suppose that the feasible rate set T satisfies (r E T & r’ > r) S- (r’ E T). Then V(r; A) is nonincreasing in r (r’ > r => V(r’; A) < V(r; A)) over T if and only if V(r; A) = V(r; tr(A)) for all r E T. (Equivalently, V(r; A) > V(r; tr - (A)).)
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ProoJ: The definition of V nonincreasing is that for r, r’ E T, r’ > r * V(r’; A) < V(r; A). This is equivalent to V(r; A) = sup V(r’; A) r’>r for all r E T. If we define V(T, A) to be the supremum of present value over activities in A and rates in T (and 17(S;A) to be the corresponding profit function), then monotonicity is equivalent to (Vr E T) V(r;A)
= V({r’ 1r’ > r};A)
so that it suffices to show that V({r’ 1r’ > r};A)
= V(r; tr(A)).
V((r’Ir’>r};A)=ZI(P((r’Ir’>r});A) = sup $ Pj(r) aj, SEA
k = l,..., n + 1
jzo
= ZZ(P(r); tr(A)) = V(r; tr(A)) which is the required result. (The second equality follows from Proposition 1 and the fact that the value of a linear objective depends only on the closed convex hull of the constraint set.) Q.E.D. This approach is also useful in providing a simple proof of the characterization of technologies with value decreasing in interest rates. This characterization was alluded to in RSD in a footnote. THEOREM l*. Suppose that the feasible rate set T satisfies (r E T & r’ > r) => (r’ E T). Then V(r; A) is decreasing in r (i.e., r’ > r 3 V(r’; A) < V(r; A)) over T if and onZy if (Vr E r)( V(r; A) > V(r; tr(A))).
Proof: Proposition 1 and the bilinearity decreasing in r is equivalent to V(r; A) > n(tr-(P(r)); and the rest of the proof is as in Theorem 1.
of IZ tells us that value A) Q.E.D.
To this point, we have assumed that the set T of relevant interest rate vectors is not significantly bounded above. Therefore, our characterizations have been for global monotonicity of value as interest rates change. Perhaps more relevant is the question of monotonicity over a bounded range of interest rates. The proof of the RSD is valid only if each interest rate can be sent to infinity independently. Since sending an interest rate to infinity has the effect on value as truncating the activity, intuition tells us that if there is
DUALITY
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FIG. 2. The value of A = (b, c) is decreasing e = tr,(c) and d = trt(c; ?, F).
105
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in r for r < F and increasing
in r for r > F.
a bound on an interest rate this corresponds to partial or limited truncation of the activity. Limited truncation as it applies to the 2-flow case is illustrated in the Fisher diagram in Fig. 2. The technology A = {b, c} clearly has value decreasing in the single rate I for r> r >,r, since, in this range, b is supported and since V(r; {b}) is decreasing in r for all rates r > -1. At the lower bound 1, it is possible to truncate c entirely without increasing the value. At the upper bound r it is not possible to truncate c at all without increasing value. Therefore, we can expect the degree of limited truncation to depend on r. Limited truncation is defined formally as: DEFINITION.
The kth limited truncation aO,al ,..., akP,,-a
operator is defined to be
1 + rk 1 + rk -a 1 + r;, k’ 1 t-r;,
Note that trk(a; r, F) = aa + (1 - a) tr,(a), where a = (1 + rJ/( 1 + Tk), and that trf;,, (a; r, F) E a. The limited truncation of a technology A is defined to be trL(A; r, F) E {tri, o tri2 0 ... o trf;l(a) 1a E A, k,‘s = l,..., n + 1 distinct}, where for brevity, the parameters r and f of the limited truncation operators trii have been omitted. The strict limited truncation set is defined to be tP(A;
r, F) E {trkl 0 trk2 0 . . . 0 @$a) / a E A, kj’s = l,..., n, distinct with (Vj) rk, # Fk,}.
Note that since the limited truncation operators commute, trL(a; r, f) will
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H. DYBVIG
FIG. 3. The shaded area represents the set P({r 1 r’ > r > r”}) = Hull(tr’ (P(r”): r”. r’)). Note that c = P(r”). d = P(ri. r;) = tr:(P(r”); r”, r’). e = P(r;. ry) = tri(P(r”): r”. r’). and f= P(r’) = trf(tr$(P(r”); r”, r’); r”, r’). (As in Fig. I, p,, = 1.)
have up to 2” elements and trL-(a; I, f) will have up to 2” - 1 elements. When r = i=, the strict limited truncation set will be empty and its value (by convention) will be --oo. It should be clear that this definition is a generalization of normal truncation. The (unlimited) truncation operators also commute. Additionally, (VJ k)
Hence, in generating (unlimited) truncation sets we need not consider multiple truncations. The importance of limited truncation as defined above is that for all ?, P({r 1r> r > ?)) is the same as Hull(trL(P(?): ?, f)). This observation is illustrated for n = 2 in Fig. 3. The formal statement is PROPOSITION
Hull(trL(P(r”); Proof.
2. For r”, r’)).
all
r’ > r” > -e.
P((rIr’>r>r”})=
The proof is similar to Proposition 1; use the samefunctions. Q.E.D.
This leads to the characterization of nonincreasing value for T= (r / ?> r >r}. It should be noted that for activity sets A containing only a single activity, the condition here is equivalent to the condition of RSD, namely, (Vr E T)( V(r; A) = V(r; tr(A))) for r > r.j
’ By Proposition 1, the RSD (Hull(tr(P(r))) c cl(P(77)).
condition
(used
in Theorem
I) can be rewritten
as (Vr E T)
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THEOREM 3. Let T = {r ) F > r > T}. Then V(r; A) is nonincreasing in r over T if and only if (Vr E T)( V(r; A) = V(r; trL(A; r, F))).4
ProoJ
V nonincreasing is equivalent to (Vr E T)
V(r:A)=
;;c
V(r’;A)=
V((r’IF>r’>r};A)
T’ET
= ZZ(P((r’ 1?> r’ > r)); A) = n(tr”(P(r);
r, f)); A)
= sup{Z7(tri, 0 trk-I 0 ..- 0 trij(P(r)); a) 1a E A, k, ,..., k,i distinct \ =
sup
(II(P(r);
tr:, 0 t& o . . +0 t&(a)) (a E A, k, ,..., k,i distinct }
= ZZ(P(r); trL(A; r, F)) = V(r; tr’-(A; r, f)).
Q.E.D.
COROLLARY. Let T = (r / F> r > r}. Then V(r; A) is nonincreasing in r over T if V(r; A) = V(r; trL(A; r, F)) for all r E T.
ProoJ: Immediate, since (Vr E 7) trL(A; r, F) s Hull(tr’.(A; r, F)). Q.E.D. The corollary has the virtue that the truncation set used is independent of r. Although its condition is not necessary (in contrast to the condition of Theorem 3), it is interesting becauseof its relative simplicity. It should be apparent that technologies with value decreasing in r over T = (r 1F> r > T} can be characterized by resorting to the strict limited truncation set trL-(A; r, F): THEOREM 3*. Let T= (r 1F> r > r). Then V(r; A) is decreasing in r over T if and only if V(r; A) > V(r; trLp(A; r, f)) for all r E T.
ProoJ
Similar to proofs of Theorems 3 and 1*.
Q.E.D.
COROLLARY. Let T = {r I 7 > r > r}. Then V(r; A) is decreasing in r over T if V(r; A) > V(r; trL-(A;r, F)) for all r E T.
Proof: Immediate by convexity of L’ and the fact that tr’*-(A;!. F) represents at least as much truncation for each k as tr’.- (A; r, F) for all rE T. Q.E.D. ’ It should
be apparent
(1)
(VrE
(2)
(Vr E T)(Tn
that this generalizes
to any
T for which
(3F)
T)(r
(r’ 1r’ > rJ = (r’ 1F> r’ > r)).
The slightly less general form is used in the text for expositional apparent that it is easy to characterize value decreasing for bounded and others unbounded.
simplicity. It should also be hybrid cases, i.e.. some rd’s
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PHILIP H. DYBVIG
It should be noted that the geometric characterization (A = tr(A) up to convexification and a form of free disposal) of globally nonincreasing value given in RSD does not appear to be appropriate to the bounded case presented here, since the value comparison here is between the technology A and a d@kent set trL(A ; r, i;> at each rate r E T. It is, however, possible to give a geometric characterization (A = trL(A; r, F) up to convexification and free disposal in value terms for all r E Z’) of the sufficient condition given in the corollary to Theorem 3 (also possible for 3*). This is because in these cases the set whose value must be the same as the original technology is independent of the rate r, and P(T) is convex. It is interesting to note the features of the sets T = (r / r > ?) and T = (r 1r> r >T} that make it possible to construct characterizations of decreasing value as done in this paper. The key properties seemto be that WV) n P({r I r > r* I)) can be generated as the set-valued image of some linear operator L(p; r*) such that
(VS, A)(lqL(S;
r*); A)) = fl(R JqA; r*))*
In mathematical terms, this is equivalent to saying that the linear operator L(.; r*) is self-adjoint (or Hermitian) under the bilinear form ZZ(.; .). Another case in which the price set can be generated by a self-adjoint operator is the familiar identical forward rates case. The price set generated is the geometric truncation set generated by the price corresponding to the lowest rate involved: tr’(A)=
((a,,a,x,a,x*,...,a,x”)JaEA,O,
Tr WVN
11;
{rel r>r};
= tr”P@>).
The corresponding characterization of nonincreasing value, illustrated for n = 2 in Fig. 4, is THEOREM 4. Let T= (re 1r >r} (i.e., unbounded identical rate case). Then V(r; A) is nonincreasing in r if and only if V(r; A) = V(r; tr’(A)) for all r E T.
Proof
The proof is straightforward.
Q.E.D.
Theorem 4 is simple to prove even without the special approach used here, but it illustrated nicely why it has been so difficult to find a simple and useful characterization of nonincreasing value for this case: unlike the other cases,the characterization is no simplification. It should be apparent that a
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0 FIG. 4. The entire arc represents the set cl(P({ re 1r > r^))) = tr”(P(?e)). P(k) and P(r’e) represents the set P((re 1r’ > r > i)) = tr”‘(P(?e); rl r’). and is left implicit.
The arc between As before. pO= 1
similar characterization, also illustrated in Fig. 4, is valid for the bounded case:5 THEOREM 4*. Let T = (re ] f> r >I} (i.e., the bounded identical rate case). Then V(r;A) is nonincreasing over T if and only if V(r: A) = V(r; trGL(A ; r, q), where the limited geometric truncation of A is defined to be
trGL(A;r,f)ProofI
(a,,a,x,a,x’,..., 1
The proof is straightforward.
a,x”)laEA,
l+r ---r
I
. Q.E.D.
It should be noted that another case in which value nonincreasing can be characterized using methods of this paper is the k-monotonicity case treated in RSD (Proposition 2).6 Also, there is a related characterization of kmonotonicity on bounded sets which is entirely analogous to Theorem 3.
’ The characterizations for decreasing value can be derived, but is necessarily messy because of closure problems. Also, as with Theorems 3 and 3*. Corollaries to 4 and 4* having truncation sets independent of r can be derived. However, such corollaries are theme selves of limited use since geometric truncation (and limited geometric truncation) involves an infinite number of activities, even when the initial technology contains only one activity. Of course. Theorems 1, 3, etc. give sufficient conditions for nonincreasing value for the geometric case. 6 The proof relies on the fact, analogous to Propositions 1 and 2, that cl(P(( r / rk > i,, (Vj# k)r,= ?,})) = Hull(tr,(i)), with the details of the proof similiar to the details of Theorem 1.
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PHILIP H. DYBVIG
4. THE GENERALIZED
INTERNAL
RATE OF RETURN SET
The generalized internal rate of return set (IRR) introduced implicitly by Bailey [I] and explicitly by Weingartner [ 141 is defined to be the set of interest rate vectors r E T such that V(r; A) = 0. For the IRR to be useful in evaluating whether or not to undertake a project corresponding to the activity set A, it is convenient if (Vr E IRR) (Vr’ E T) (r’ > r)*
V(r’;A)
<0
(r’ ,< r) =s-V(r’; A) > 0.
(4 (b)
Either (a) or (b) alone is sufficient to guarantee that the internal rate of return set is “skinny.” Condition (a) assuresthat at all rates above the IRR (i.e., above any r E IRR), the activity set has negative value and should be rejected. Condition (b) assuresthat at all rates below the IRR, the activity set has positive value and should be used. Both conditions together imply that the IRR will classify the good and bad projects. (Note that we have implicitly been assuming some regularity on T, i.e., that T is connected. Also, if IRR = 4, we know that the technology is unambiguously useful or useless,independent of r. If we know this is the case, we need only evaluate V(r;A) at a single rate r E T.) As noted in RSD, value decreasing is necessary and sufficient for these properties to hold across all levels (positive and negative) offixed cost (i.e., for all translations acting identically on the first flow of every activity in A, and leaving the other flows unchanged). The contribution here is to give necessary and sufficient conditions for (a) and (b) to hold for a given level of fixed cost. The necessary and sufficient condition for (a) and (b) holding (for bounded T’), is given by THEOREM 5. Suppose T={rlf>r>r} and 3r^ V(r; trL- (A; r, F)).
ProoJ Similar to the proof of Theorem 3 *, but relying on convexity of Q.E.D. the profit function to show that (a) G- (b). The drawback to Theorem 5 is that the test is impossible to use unlessthe whole IRR is known in advance. A more useful result is THEOREM 6. Let T= (r / J>r>y}, and suppose iE IRR, with ?< 7. Then V(r;A)i (with r-ET) if and only if V(i; trL-(A; ?, F)) < 0. Furthermore, if this condition holds, then V(r; A) > 0 for all r < F.
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DUALITY AND INTERESTRATES
Proof. Suppose V(r; A) < 0 for all r E T with I > ?. Then, in particular, V(r; A) < 0 for r having each forward rate the same as the corresponding forward rate of either r^or f, so long as r,, = rk for somek. The various such rate vectors r correspond to prices p(r) which are the elements of trL-(P(P); f, f), so that by duality, valuing A at these prices corresponds to valuing trL- (A; ?, F) at P(f). Therefore, V(i, trL- (A ; ?, F)) < 0 since it is the maximum of a finite number of negative numbers and therefore negative. Supposethen that V(e trL-(A; ?, f)) < 0. Then by duality V(r*;A) < 0 for all r* having each forward rate the same as the corresponding rate of either r or i (and the same as r as least once), and V(i; A) = 0 since r^E IRR. Now choose any r E T with r > r. Then P(r) is a convex combination of P(F) and the P(r*)‘s (see Fig. 5) so by convexity of the profit function V(r; A) < 0. (V(r; A) # 0 since P(r) f P(q). Finally, suppose V(r; A) < 0 for all r E T with r > ?. Choose any r* E T with r* ,< ?. Then there is some r E T such that P(F) is a convex combination of P(r) and P(r*), where r < ? since this represents a direct reversal of the linear inequalities (on prices) relative r* > F. But V(r;A) < 0 and V(r^:A) = 0. Therefore, by convexity of the profit function, V(r*: A) > 0. Q.E.D. For unbounded interest rates, the following result is appropriate: THEOREM 7. Let T= {r/r>-e) and suppose FE IRR. Then V(r; A) < 0 for all r > i if and only if either (a) V(r^; tr-(A)) < 0 or (b) V(?Jtr(A)) = 0 and V(< trL-(A; p, f)) < 0, where f is any rate chosen so that f > 6 Furthermore, if this condition holds, then V(r; A) > 0 for all r ,< F.
ProoJ
The proof is similar to that of Theorem 6. Cl
p2
'2
0 FIG.
642!30/1-8
PI
5. For T= (r [ r > r > r} and i E IRR, value is negative
and positive
Q.E.D.
for r < i (shaded
area “+“)
if and only
if tr’
for r > i (shaded (A ; r? F) has a negative
area “-“) value.
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PHILIP H. DYBVIG
Theorems 6 and 7 may be useful for determining in what regions V(r; A) is positive, negative, and zero, even if the internal rate of return criterion is not well behaved. Also, these theorems have useful interpretations for rate sets which are subsets of the rate sets in the theorems. For example, the following result is a direct consequenceof Theorem 7: COROLLARY. Let T = {re 1r > - 1) (equal forward rates), and suppose FE IRR. Then if V(re; trr(A)) < 0, the internal rate of return criterion is well behaved in the sensethat the technology should be rejected (has negative value) for higher interest rates, and should be accepted (has positive value) for lower interest rates.
Proof
The proof is immediate from Theorem 7.
Q.E.D.
This result is illustrated in Fig. 6. For those who are interested in using internal rates of return for capital budgeting with the identical forward rates approximation, this result provides a criterion which is a sufficient condition for nicely behaved internal rate of return. The criterion is simple, more general than the possibility of costless abandonment, and less cumbersome than the roots of polynomials tests. Only experience will tell whether this approach is more useful in practice than the asymptotically exact method developed by Pratt and Hammond [9]. (It should go without saying, but may not, that in many practical situations the internal rate of return criterion has no justification, since the present value criterion is more reliable and simpler to apply than the internal rate of return criterion. On the other hand, the internal rate of return criterion may be useful if the planners in a firm wish to know whether interest rate changeswill affect the decision whether to adopt a technology. The internal rate of return may also be useful to economists as an intuitive notion, both for research and for teaching.)
FIG. 6. The internal rate of return i is unique even if the value does not always decrease in r.
whenever
V(F; tr-(A))
< 0. This may occur
DUALITY
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5. CONCLUSION
We have found that the use of duality allows us to construct simple proofs of both old and new results in the theory of production over time, which could apply to the valuation of bonds and other cash flows as well. Some results of this paper have potential for application. There are indications in Section 4 that the results will have application in validating the internal rate of return criterion in specific cash flow problems. Future theoretical research could extend in several directions. Alternative bounded sets with either mathematically simple or economically interesting characterizations of decreasing value might be found. Or, the class of rate sets generated by appropriate self-adjoint operators (and hence with characterizations of nonincreasing value) might be characterized. More importantly, extending the results to nontrivial cases of uncertainty might be fruitful. The ultimate hope. as in Ross et al. IlO] is to enhance understanding of project selection criteria and their impact on investment and economic growth.
REFERENCES
1. M. J. BAILEY, Formal criteria for investment decisions. J. Political Econ. 67 (1959). 476488. 2. D. CASS. Duality: A symmetric approach from the economist’s vantage point. J. Econ. Theory 7 (1974), 272-295. 3. W. DIEWERT. Applications of duality theory, in “Frontiers of Quantitative Economics” (D. A. Kendricks and M. D. Intriligator, Eds.), Vol. II, pp. 106-171, 1974. 4. D. W. JORGENSON AND L. J. LAW, Duality of technology and economic behavior, Ret). Econ. Studies 41 (1974). 181-200. 5. T. KEHOE. An index theorem for general equilibrium models with production, Econometrica 48 (1980), 1211-1232. 6. J. M. KEYNES, “The General Theory of Employment, Interest, and Money.” MacMillan. London, 1936. 7. L. MCKENZIE, On the existence of general equilibrium for a competitive market. Econometrica 27 (1959), 54-7 1. 8. L. MCKENZIE. On the existence of general equilibrium for a competitive market: some corrections, Economefrica 29 (196 1). 9. J. W. PRATT AND J. S. HAMMOND III, “Evaluating and comparing projects: a simple detection of false alarms. J. Finance 34 (1979), 123 1-1242. 10. S. A. Ross, C. S. SPATT, AND P. H. DYBVIG, Present values and internal rates of return, J. Econ. Theory (1980), 66-81. I 1. P. A. SAMUELSON, “Foundations of Economic Analysis,” Harvard Univ. Press. Cambridge, Mass., 1947. 12. H. SCARF, (with the collaboration of Terje Hansen), “The Computation of Economic Equilibria,” Yale Univ. Press, New Haven and London, 1973.
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13. K. SHELL, The theory of Hamiltonian dynamical systems, and an application to economics, in “The Theory and Application of Differential Games” (J. D. Grote, Ed.). Reidel, Dordrecht/Holland, 1975. 14. H. M. WEINGARTNER. The generalized rate of return, J. Financial and Quantitatice Anal. 1 (3) (1966). I-29.
OF ECONOMIC
THEORY
30, 98-l
14 (1983)
Duality, Interest Rates, and the Theory of Present Value* PHILIP
H.
DYBVIG
School of Management, Yale University, New Haven, Connecticut 065.20 Received
July
27, 1981
This paper uses duality theory to simplify and extend previous work which characterized technologies which have present value decreasing in interest rates. For “unbounded” interest rate sets, value nonincreasing is equivalent to truncation never being desirable. For bounded interest rate sets, the result is true if truncation is replaced by limited truncation, appropriately defined. Similarly. for equal forward rates, undesirability of geometric truncation characterizes nonincreasing value. Another result shows that if truncation is undesirable at an internal rate of return, then the internal rate of return is well behaved, even if value is not decreasing everywhere. Journal of Economic Literature Classr$cation Numbers: 022, 111, 522.
1.
INTRODUCTION’
Duality theory has yielded insights into the standard theories of consumption, production, and general equilibrium (see, for example, Cass [2], Diewert [3], Jorgenson and Lau [4], Kehoe [5], McKenzie [7,8], and Scarf [ 12, p. 164, ff.]). Duality theory has been particularly fruitful in the theory of production; in particular, the value function or Hamiltonian representation of a technology has proven simpler to work with than the conventional production function or activity set representation for some purposes (see Shell [ 131). This paper represents an application of duality theory to the theory of interest rates and present values. Duality (and the conversion to prices from interest rates) is useful because prices and quantities occur symmetrically in the typical (bilinear) formula for value. More specifically, multiplying any particular quantity by a positive amount has the same effect on value as multiplying the corresponding price by the same positive amount. In addition, the value is a convex function of
* The author thanks William Brainard and John Long for useful comments, and especially Stephen Ross and Chester Spatt for useful comments and related previous joint work.
98 0022-0531/83 Copyright All rights
$3.00
0 1983 by Academic Press, Inc. of reproduction in any form reserved.
DUALITY
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99
prices (although not necessarily a convex function of interest rates), and any convex function is the value function (over prices) for some technology. The introduction of interest rates into the value function follows a simple intuition. Raising a single forward rate decreases all subsequent prices, all in the same proportion, leaving previous prices unchanged. As the interest rate becomes arbitrarily large, this represents a truncation of the price vector, i.e., all prices after the particular interest rate are set to zero. Therefore, raising interest rates truncates prices (partially or fully) which has the same effect on value as truncating flows. This intuition illustrates why there is such an intimate relationship between truncation (the option of costless abandonment of projects) and value decreasing as interest rates rise. This paves the way to give very simple new proofs of old results and to prove new results. The old results which are reproven here are all proven by Ross, Spatt, and Dybvig [lo] (henceforth, RSD). The central result of RSD is that, in a world where all forward rates are free to vary, it is true not only that a truncation technology will not increase in value as interest rates rise, but also, conversely, if value is globally nonincreasing in all interest rates, the technology resembles a truncation technology in two important respects. First, the value of the technology is always the same as the value of the truncation technology it generates. Second, the technology is the same as the truncation technology it generates, up to convexification and free disposal in a value sense. This paper shows that the first form of the result is more robust, since it generalizes easily to other cases. (There exist generalizations of the other form as well, but they seem to be convoluted.) A very simple direct proof of the first form of the central result of RSD is given here. The proof uses simple duality and an observation (formalized in Proposition 1) made jointly with Ross, that the price sets associated with global interest rate sets bounded below but not above (as used in RSD’) are actually truncation sets of a sort themselves. This observation is consistent with the intuition (elaborated above) that raising interest rates is like truncating prices. Another old result which can be reproven by the dual approach is the characterization of k-monotonicity given in RSD. A sketch of the new proof is given in footnote 6. Several new results are proven here. It is shown that in a strong sense, the RSD central result is valid only if we are looking at technologies which are glob&b nonincreasing in interest rates; i.e., the equivalence between truncation and nonincreasing value is not valid if we can bound interest rates in any substantial way. However, a characterization of technologies with nonincreasing value on a class of bounded interest rate sets is derived here. The characterization involves limited truncation, which is a natural generalization
I In RSD.
it is assumed
that (Vr. r’)((r
E T&
r’ > r) *
(r’ E r)).
100
PHILIP H. DYBVIG
of truncation. In the same vein, nonincreasing value when forward rates are identical is shown to be equivalent to geometric truncation. This paper also addressesthe problem of determining when the generalized internal rate of return set (IRR) is well behaved. As noted in RSD, value decreasing in interest rates is equivalent to the IRR being well behaved for all levels (positive and negative) of fixed cost. This paper gives conditions under which the IRR is well behaved for a single level of fixed cost, both for the global and bounded interest rates cases. Potential application of this result is also discussed. Section 2 reviews the basic structure of the model (which is basically as in RSD). Section 3 contains the new direct proof of the RSD central result, the proof that the RSD characterization is valid for only the global problem, and the characterization of technologies which have value nonincreasing in interest rates over bounded rate sets. Section 4 characterizes when the IRR will be well behaved for a single value of fixed cost, gives additional results which may be germane for those wishing to use the internal rate of return criteria, and discusses applicability of these results. Section 5 closes the paper.
2. STRUCTURE OF THE MODEL The structure and notation used here will be roughly identical to that used in RSD. The basic structure is the familiar neoclassical activity framework. The technology A c R”+ ’ is a nonempty set representing feasible return streams over the n + 1 periods. The vector (a,, a, ,..., a,) = a E A. denotes a feasible activity with net positive or negative return aj in periodj. The activity set A can represent mutually exclusive or interdependent choices by including all mutually exclusive alternatives. The value function for an arbitrary technology A is defined to be V(r; A) = sup G asA ,yo r-g= ,‘;: + ri) ’ where nfZk+ I (1 + ri) 3 1. We will assume V(r; A) is finite for all feasible interest rates. The feasible rates will be representedby a set T c R ‘. Various choices for T will be used throughout the paper. (T is the set in which interest rates can be restricted to lie a priori.) An equivalent formulation is to associatewith each interest rate vector r > -e = -( 1, l,..., 1) the price vector p E R”+ ’ defined by P(r)=
c l,-,
1 1 + rl
1 (1 + f-,)(1 + r2> ‘-’ nl=,
1 (1 + ri) 1’
DUALITY ANDINTEREST
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Then the projit function
l7(p;A)-
sup i Pjaj, OEAj=o
with p E P(r), is equivalent to the value function V(r; A), and ZI(p; A) will be finite whenever p E P(T). Working with the profit function is useful becauseprice space is the dual of activity space. Now we formalize the concept of truncation. The kth truncation operator truncates an activity after the first k flows: tr,(a) = (a,, a, ,..., ak- t, 0 ,..., 0). The truncation
of an activity set A is given by tr(A) = {tr,(a) 1a E A, k = 1, 2 ,..., n + 1}.
Note that A G tr(A) since (Va E R”“)a = tr,,+ ,(a). It will also be useful to consider the strict truncation of an activity setA, defined by tr-(A)
= (tr,(a) 1a E A, k = 1, 2,..., n),
which does not normally include A.
3. SOME FUNDAMENTAL
RESULTS
The approach taken in this paper was motivated by the observation, made jointly with Ross, that the price setsgenerated by the interest rate sets T (for T satisfying the unbounded interest rate assumption used in RSD’) were truncation sets of a sort themselves. Loosely speaking, this is because sending the kth rate to infinity is like truncating all subsequentprices, which has the sameeffect on value as truncating all subsequentflows. This observation is illustrated for n = 2 in Fig. 1. The formal statement is: PROPOSITION
Proof:
1. Let F > -e. Then cl(P( (r 1r > ?})) = Hull(tr(P(F))).
We will show that
P( (r I r > r”}) = Hull(tr(P(r^))) n ( p 1p > O} from which the result follows since Hull(tr(P(i))) convex and contains P(f) > 0. First we prove that
g Rt+’
P( (r / r > t}) s Hull(tr(P(?))) n {p / p > 0).
is closed and
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PHILIP
H. DYBVIG
0
b
FIG. 1. The shaded area represents the set cl[P({r 1r > F})) = Hull(tr(P(i))). c = P(t). b = tr2(P(i)), and 0 = tr,(P(i)). (Note that p0 is the numeraire and always 1, so it is not shown.) Higher interest rates imply lower prices, i.e., ?, > r: > r(l for i = I. 2.
Suppose p* E I’( { I ( r > P}). Then there exists r > r” such that pi* =
1 rIzl
Define A* = (AT,..., AZ+ ,) inductively
(1 + Ti) . (in decreasing order) as
Then let A = (A,,..., An+ ,) be defined by
Then CAj=
1, L>O,
A,+, > 0, and2 p* = \‘ k=il
=(l+?j+,)*=l+rj,,. ,+ I
A, trJP(f))
> 0.
DUALITY ANDINTEREST
RATES
103
Therefore,
Now we show that P({r / r > F)) 3 Hull(tr(P(i)))
n (p 1p > O}.
Now supposep* E Hull(tr(P(r))) n {p / p > O}. Then there exists Lr, AZ,..., Antl such that C Lj = 1, ,l> 0, and p* = \‘
A, tr#(P)).
kTI
Since p* > 0, it follows that A,+ 1 > 0. Define r = (ri , rz ,..., I~) as
Since A>0 and A,,, > 0 the expression is well defined and r > L Furthermore, it is readily verified that p* = P(r). Therefore, P(P I r 2 ~1) E HWtr(W-1))
n {P / P > 01,
and by the comment at the start of the proof, we are done.
Q.E.D.
Now we will use Proposition 1 to give the simple proof of the value characterization of nonincreasing value given in RSD (Theorem 1 in both RSD and this paper). The only regularity required on the activity set A is that its value V(r; A) be finite for all feasible rates (i.e., for all r E 7). This is in contrast with RSD, where the proof would necessarily be complicated considerably by dropping the assumption that A is compact. Note that the present proof is valid even if V(r; tr(A)) is not necessarily finite for all r E T. In other words, if V(r; tr(A)) = co, then by Proposition 1 the value becomes unbounded as some rk gets large, so that V(r; tr(A)) = co, and the value must increase in rk somewhere. THEOREM 1. Suppose that the feasible rate set T satisfies (r E T & r’ > r) S- (r’ E T). Then V(r; A) is nonincreasing in r (r’ > r => V(r’; A) < V(r; A)) over T if and only if V(r; A) = V(r; tr(A)) for all r E T. (Equivalently, V(r; A) > V(r; tr - (A)).)
104
PHILIP H. DYBVIG
ProoJ: The definition of V nonincreasing is that for r, r’ E T, r’ > r * V(r’; A) < V(r; A). This is equivalent to V(r; A) = sup V(r’; A) r’>r for all r E T. If we define V(T, A) to be the supremum of present value over activities in A and rates in T (and 17(S;A) to be the corresponding profit function), then monotonicity is equivalent to (Vr E T) V(r;A)
= V({r’ 1r’ > r};A)
so that it suffices to show that V({r’ 1r’ > r};A)
= V(r; tr(A)).
V((r’Ir’>r};A)=ZI(P((r’Ir’>r});A) = sup $ Pj(r) aj, SEA
k = l,..., n + 1
jzo
= ZZ(P(r); tr(A)) = V(r; tr(A)) which is the required result. (The second equality follows from Proposition 1 and the fact that the value of a linear objective depends only on the closed convex hull of the constraint set.) Q.E.D. This approach is also useful in providing a simple proof of the characterization of technologies with value decreasing in interest rates. This characterization was alluded to in RSD in a footnote. THEOREM l*. Suppose that the feasible rate set T satisfies (r E T & r’ > r) => (r’ E T). Then V(r; A) is decreasing in r (i.e., r’ > r 3 V(r’; A) < V(r; A)) over T if and onZy if (Vr E r)( V(r; A) > V(r; tr(A))).
Proof: Proposition 1 and the bilinearity decreasing in r is equivalent to V(r; A) > n(tr-(P(r)); and the rest of the proof is as in Theorem 1.
of IZ tells us that value A) Q.E.D.
To this point, we have assumed that the set T of relevant interest rate vectors is not significantly bounded above. Therefore, our characterizations have been for global monotonicity of value as interest rates change. Perhaps more relevant is the question of monotonicity over a bounded range of interest rates. The proof of the RSD is valid only if each interest rate can be sent to infinity independently. Since sending an interest rate to infinity has the effect on value as truncating the activity, intuition tells us that if there is
DUALITY
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FIG. 2. The value of A = (b, c) is decreasing e = tr,(c) and d = trt(c; ?, F).
105
RATES
in r for r < F and increasing
in r for r > F.
a bound on an interest rate this corresponds to partial or limited truncation of the activity. Limited truncation as it applies to the 2-flow case is illustrated in the Fisher diagram in Fig. 2. The technology A = {b, c} clearly has value decreasing in the single rate I for r> r >,r, since, in this range, b is supported and since V(r; {b}) is decreasing in r for all rates r > -1. At the lower bound 1, it is possible to truncate c entirely without increasing the value. At the upper bound r it is not possible to truncate c at all without increasing value. Therefore, we can expect the degree of limited truncation to depend on r. Limited truncation is defined formally as: DEFINITION.
The kth limited truncation aO,al ,..., akP,,-a
operator is defined to be
1 + rk 1 + rk -a 1 + r;, k’ 1 t-r;,
Note that trk(a; r, F) = aa + (1 - a) tr,(a), where a = (1 + rJ/( 1 + Tk), and that trf;,, (a; r, F) E a. The limited truncation of a technology A is defined to be trL(A; r, F) E {tri, o tri2 0 ... o trf;l(a) 1a E A, k,‘s = l,..., n + 1 distinct}, where for brevity, the parameters r and f of the limited truncation operators trii have been omitted. The strict limited truncation set is defined to be tP(A;
r, F) E {trkl 0 trk2 0 . . . 0 @$a) / a E A, kj’s = l,..., n, distinct with (Vj) rk, # Fk,}.
Note that since the limited truncation operators commute, trL(a; r, f) will
106
PHILIP
H. DYBVIG
FIG. 3. The shaded area represents the set P({r 1 r’ > r > r”}) = Hull(tr’ (P(r”): r”. r’)). Note that c = P(r”). d = P(ri. r;) = tr:(P(r”); r”, r’). e = P(r;. ry) = tri(P(r”): r”. r’). and f= P(r’) = trf(tr$(P(r”); r”, r’); r”, r’). (As in Fig. I, p,, = 1.)
have up to 2” elements and trL-(a; I, f) will have up to 2” - 1 elements. When r = i=, the strict limited truncation set will be empty and its value (by convention) will be --oo. It should be clear that this definition is a generalization of normal truncation. The (unlimited) truncation operators also commute. Additionally, (VJ k)
Hence, in generating (unlimited) truncation sets we need not consider multiple truncations. The importance of limited truncation as defined above is that for all ?, P({r 1r> r > ?)) is the same as Hull(trL(P(?): ?, f)). This observation is illustrated for n = 2 in Fig. 3. The formal statement is PROPOSITION
Hull(trL(P(r”); Proof.
2. For r”, r’)).
all
r’ > r” > -e.
P((rIr’>r>r”})=
The proof is similar to Proposition 1; use the samefunctions. Q.E.D.
This leads to the characterization of nonincreasing value for T= (r / ?> r >r}. It should be noted that for activity sets A containing only a single activity, the condition here is equivalent to the condition of RSD, namely, (Vr E T)( V(r; A) = V(r; tr(A))) for r > r.j
’ By Proposition 1, the RSD (Hull(tr(P(r))) c cl(P(77)).
condition
(used
in Theorem
I) can be rewritten
as (Vr E T)
107
DUALITY AND INTEREST RATES
THEOREM 3. Let T = {r ) F > r > T}. Then V(r; A) is nonincreasing in r over T if and only if (Vr E T)( V(r; A) = V(r; trL(A; r, F))).4
ProoJ
V nonincreasing is equivalent to (Vr E T)
V(r:A)=
;;c
V(r’;A)=
V((r’IF>r’>r};A)
T’ET
= ZZ(P((r’ 1?> r’ > r)); A) = n(tr”(P(r);
r, f)); A)
= sup{Z7(tri, 0 trk-I 0 ..- 0 trij(P(r)); a) 1a E A, k, ,..., k,i distinct \ =
sup
(II(P(r);
tr:, 0 t& o . . +0 t&(a)) (a E A, k, ,..., k,i distinct }
= ZZ(P(r); trL(A; r, F)) = V(r; tr’-(A; r, f)).
Q.E.D.
COROLLARY. Let T = (r / F> r > r}. Then V(r; A) is nonincreasing in r over T if V(r; A) = V(r; trL(A; r, F)) for all r E T.
ProoJ: Immediate, since (Vr E 7) trL(A; r, F) s Hull(tr’.(A; r, F)). Q.E.D. The corollary has the virtue that the truncation set used is independent of r. Although its condition is not necessary (in contrast to the condition of Theorem 3), it is interesting becauseof its relative simplicity. It should be apparent that technologies with value decreasing in r over T = (r 1F> r > T} can be characterized by resorting to the strict limited truncation set trL-(A; r, F): THEOREM 3*. Let T= (r 1F> r > r). Then V(r; A) is decreasing in r over T if and only if V(r; A) > V(r; trLp(A; r, f)) for all r E T.
ProoJ
Similar to proofs of Theorems 3 and 1*.
Q.E.D.
COROLLARY. Let T = {r I 7 > r > r}. Then V(r; A) is decreasing in r over T if V(r; A) > V(r; trL-(A;r, F)) for all r E T.
Proof: Immediate by convexity of L’ and the fact that tr’*-(A;!. F) represents at least as much truncation for each k as tr’.- (A; r, F) for all rE T. Q.E.D. ’ It should
be apparent
(1)
(VrE
(2)
(Vr E T)(Tn
that this generalizes
to any
T for which
(3F)
T)(r
(r’ 1r’ > rJ = (r’ 1F> r’ > r)).
The slightly less general form is used in the text for expositional apparent that it is easy to characterize value decreasing for bounded and others unbounded.
simplicity. It should also be hybrid cases, i.e.. some rd’s
108
PHILIP H. DYBVIG
It should be noted that the geometric characterization (A = tr(A) up to convexification and a form of free disposal) of globally nonincreasing value given in RSD does not appear to be appropriate to the bounded case presented here, since the value comparison here is between the technology A and a d@kent set trL(A ; r, i;> at each rate r E T. It is, however, possible to give a geometric characterization (A = trL(A; r, F) up to convexification and free disposal in value terms for all r E Z’) of the sufficient condition given in the corollary to Theorem 3 (also possible for 3*). This is because in these cases the set whose value must be the same as the original technology is independent of the rate r, and P(T) is convex. It is interesting to note the features of the sets T = (r / r > ?) and T = (r 1r> r >T} that make it possible to construct characterizations of decreasing value as done in this paper. The key properties seemto be that WV) n P({r I r > r* I)) can be generated as the set-valued image of some linear operator L(p; r*) such that
(VS, A)(lqL(S;
r*); A)) = fl(R JqA; r*))*
In mathematical terms, this is equivalent to saying that the linear operator L(.; r*) is self-adjoint (or Hermitian) under the bilinear form ZZ(.; .). Another case in which the price set can be generated by a self-adjoint operator is the familiar identical forward rates case. The price set generated is the geometric truncation set generated by the price corresponding to the lowest rate involved: tr’(A)=
((a,,a,x,a,x*,...,a,x”)JaEA,O,
Tr WVN
11;
{rel r>r};
= tr”P@>).
The corresponding characterization of nonincreasing value, illustrated for n = 2 in Fig. 4, is THEOREM 4. Let T= (re 1r >r} (i.e., unbounded identical rate case). Then V(r; A) is nonincreasing in r if and only if V(r; A) = V(r; tr’(A)) for all r E T.
Proof
The proof is straightforward.
Q.E.D.
Theorem 4 is simple to prove even without the special approach used here, but it illustrated nicely why it has been so difficult to find a simple and useful characterization of nonincreasing value for this case: unlike the other cases,the characterization is no simplification. It should be apparent that a
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0 FIG. 4. The entire arc represents the set cl(P({ re 1r > r^))) = tr”(P(?e)). P(k) and P(r’e) represents the set P((re 1r’ > r > i)) = tr”‘(P(?e); rl r’). and is left implicit.
The arc between As before. pO= 1
similar characterization, also illustrated in Fig. 4, is valid for the bounded case:5 THEOREM 4*. Let T = (re ] f> r >I} (i.e., the bounded identical rate case). Then V(r;A) is nonincreasing over T if and only if V(r: A) = V(r; trGL(A ; r, q), where the limited geometric truncation of A is defined to be
trGL(A;r,f)ProofI
(a,,a,x,a,x’,..., 1
The proof is straightforward.
a,x”)laEA,
l+r ---r
I
. Q.E.D.
It should be noted that another case in which value nonincreasing can be characterized using methods of this paper is the k-monotonicity case treated in RSD (Proposition 2).6 Also, there is a related characterization of kmonotonicity on bounded sets which is entirely analogous to Theorem 3.
’ The characterizations for decreasing value can be derived, but is necessarily messy because of closure problems. Also, as with Theorems 3 and 3*. Corollaries to 4 and 4* having truncation sets independent of r can be derived. However, such corollaries are theme selves of limited use since geometric truncation (and limited geometric truncation) involves an infinite number of activities, even when the initial technology contains only one activity. Of course. Theorems 1, 3, etc. give sufficient conditions for nonincreasing value for the geometric case. 6 The proof relies on the fact, analogous to Propositions 1 and 2, that cl(P(( r / rk > i,, (Vj# k)r,= ?,})) = Hull(tr,(i)), with the details of the proof similiar to the details of Theorem 1.
110
PHILIP H. DYBVIG
4. THE GENERALIZED
INTERNAL
RATE OF RETURN SET
The generalized internal rate of return set (IRR) introduced implicitly by Bailey [I] and explicitly by Weingartner [ 141 is defined to be the set of interest rate vectors r E T such that V(r; A) = 0. For the IRR to be useful in evaluating whether or not to undertake a project corresponding to the activity set A, it is convenient if (Vr E IRR) (Vr’ E T) (r’ > r)*
V(r’;A)
<0
(r’ ,< r) =s-V(r’; A) > 0.
(4 (b)
Either (a) or (b) alone is sufficient to guarantee that the internal rate of return set is “skinny.” Condition (a) assuresthat at all rates above the IRR (i.e., above any r E IRR), the activity set has negative value and should be rejected. Condition (b) assuresthat at all rates below the IRR, the activity set has positive value and should be used. Both conditions together imply that the IRR will classify the good and bad projects. (Note that we have implicitly been assuming some regularity on T, i.e., that T is connected. Also, if IRR = 4, we know that the technology is unambiguously useful or useless,independent of r. If we know this is the case, we need only evaluate V(r;A) at a single rate r E T.) As noted in RSD, value decreasing is necessary and sufficient for these properties to hold across all levels (positive and negative) offixed cost (i.e., for all translations acting identically on the first flow of every activity in A, and leaving the other flows unchanged). The contribution here is to give necessary and sufficient conditions for (a) and (b) to hold for a given level of fixed cost. The necessary and sufficient condition for (a) and (b) holding (for bounded T’), is given by THEOREM 5. Suppose T={rlf>r>r} and 3r^
ProoJ Similar to the proof of Theorem 3 *, but relying on convexity of Q.E.D. the profit function to show that (a) G- (b). The drawback to Theorem 5 is that the test is impossible to use unlessthe whole IRR is known in advance. A more useful result is THEOREM 6. Let T= (r / J>r>y}, and suppose iE IRR, with ?< 7. Then V(r;A)
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Proof. Suppose V(r; A) < 0 for all r E T with I > ?. Then, in particular, V(r; A) < 0 for r having each forward rate the same as the corresponding forward rate of either r^or f, so long as r,, = rk for somek. The various such rate vectors r correspond to prices p(r) which are the elements of trL-(P(P); f, f), so that by duality, valuing A at these prices corresponds to valuing trL- (A; ?, F) at P(f). Therefore, V(i, trL- (A ; ?, F)) < 0 since it is the maximum of a finite number of negative numbers and therefore negative. Supposethen that V(e trL-(A; ?, f)) < 0. Then by duality V(r*;A) < 0 for all r* having each forward rate the same as the corresponding rate of either r or i (and the same as r as least once), and V(i; A) = 0 since r^E IRR. Now choose any r E T with r > r. Then P(r) is a convex combination of P(F) and the P(r*)‘s (see Fig. 5) so by convexity of the profit function V(r; A) < 0. (V(r; A) # 0 since P(r) f P(q). Finally, suppose V(r; A) < 0 for all r E T with r > ?. Choose any r* E T with r* ,< ?. Then there is some r E T such that P(F) is a convex combination of P(r) and P(r*), where r < ? since this represents a direct reversal of the linear inequalities (on prices) relative r* > F. But V(r;A) < 0 and V(r^:A) = 0. Therefore, by convexity of the profit function, V(r*: A) > 0. Q.E.D. For unbounded interest rates, the following result is appropriate: THEOREM 7. Let T= {r/r>-e) and suppose FE IRR. Then V(r; A) < 0 for all r > i if and only if either (a) V(r^; tr-(A)) < 0 or (b) V(?Jtr(A)) = 0 and V(< trL-(A; p, f)) < 0, where f is any rate chosen so that f > 6 Furthermore, if this condition holds, then V(r; A) > 0 for all r ,< F.
ProoJ
The proof is similar to that of Theorem 6. Cl
p2
'2
0 FIG.
642!30/1-8
PI
5. For T= (r [ r > r > r} and i E IRR, value is negative
and positive
Q.E.D.
for r < i (shaded
area “+“)
if and only
if tr’
for r > i (shaded (A ; r? F) has a negative
area “-“) value.
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PHILIP H. DYBVIG
Theorems 6 and 7 may be useful for determining in what regions V(r; A) is positive, negative, and zero, even if the internal rate of return criterion is not well behaved. Also, these theorems have useful interpretations for rate sets which are subsets of the rate sets in the theorems. For example, the following result is a direct consequenceof Theorem 7: COROLLARY. Let T = {re 1r > - 1) (equal forward rates), and suppose FE IRR. Then if V(re; trr(A)) < 0, the internal rate of return criterion is well behaved in the sensethat the technology should be rejected (has negative value) for higher interest rates, and should be accepted (has positive value) for lower interest rates.
Proof
The proof is immediate from Theorem 7.
Q.E.D.
This result is illustrated in Fig. 6. For those who are interested in using internal rates of return for capital budgeting with the identical forward rates approximation, this result provides a criterion which is a sufficient condition for nicely behaved internal rate of return. The criterion is simple, more general than the possibility of costless abandonment, and less cumbersome than the roots of polynomials tests. Only experience will tell whether this approach is more useful in practice than the asymptotically exact method developed by Pratt and Hammond [9]. (It should go without saying, but may not, that in many practical situations the internal rate of return criterion has no justification, since the present value criterion is more reliable and simpler to apply than the internal rate of return criterion. On the other hand, the internal rate of return criterion may be useful if the planners in a firm wish to know whether interest rate changeswill affect the decision whether to adopt a technology. The internal rate of return may also be useful to economists as an intuitive notion, both for research and for teaching.)
FIG. 6. The internal rate of return i is unique even if the value does not always decrease in r.
whenever
V(F; tr-(A))
< 0. This may occur
DUALITY
AND
INTEREST
RATES
113
5. CONCLUSION
We have found that the use of duality allows us to construct simple proofs of both old and new results in the theory of production over time, which could apply to the valuation of bonds and other cash flows as well. Some results of this paper have potential for application. There are indications in Section 4 that the results will have application in validating the internal rate of return criterion in specific cash flow problems. Future theoretical research could extend in several directions. Alternative bounded sets with either mathematically simple or economically interesting characterizations of decreasing value might be found. Or, the class of rate sets generated by appropriate self-adjoint operators (and hence with characterizations of nonincreasing value) might be characterized. More importantly, extending the results to nontrivial cases of uncertainty might be fruitful. The ultimate hope. as in Ross et al. IlO] is to enhance understanding of project selection criteria and their impact on investment and economic growth.
REFERENCES
1. M. J. BAILEY, Formal criteria for investment decisions. J. Political Econ. 67 (1959). 476488. 2. D. CASS. Duality: A symmetric approach from the economist’s vantage point. J. Econ. Theory 7 (1974), 272-295. 3. W. DIEWERT. Applications of duality theory, in “Frontiers of Quantitative Economics” (D. A. Kendricks and M. D. Intriligator, Eds.), Vol. II, pp. 106-171, 1974. 4. D. W. JORGENSON AND L. J. LAW, Duality of technology and economic behavior, Ret). Econ. Studies 41 (1974). 181-200. 5. T. KEHOE. An index theorem for general equilibrium models with production, Econometrica 48 (1980), 1211-1232. 6. J. M. KEYNES, “The General Theory of Employment, Interest, and Money.” MacMillan. London, 1936. 7. L. MCKENZIE, On the existence of general equilibrium for a competitive market. Econometrica 27 (1959), 54-7 1. 8. L. MCKENZIE. On the existence of general equilibrium for a competitive market: some corrections, Economefrica 29 (196 1). 9. J. W. PRATT AND J. S. HAMMOND III, “Evaluating and comparing projects: a simple detection of false alarms. J. Finance 34 (1979), 123 1-1242. 10. S. A. Ross, C. S. SPATT, AND P. H. DYBVIG, Present values and internal rates of return, J. Econ. Theory (1980), 66-81. I 1. P. A. SAMUELSON, “Foundations of Economic Analysis,” Harvard Univ. Press. Cambridge, Mass., 1947. 12. H. SCARF, (with the collaboration of Terje Hansen), “The Computation of Economic Equilibria,” Yale Univ. Press, New Haven and London, 1973.
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13. K. SHELL, The theory of Hamiltonian dynamical systems, and an application to economics, in “The Theory and Application of Differential Games” (J. D. Grote, Ed.). Reidel, Dordrecht/Holland, 1975. 14. H. M. WEINGARTNER. The generalized rate of return, J. Financial and Quantitatice Anal. 1 (3) (1966). I-29.