INFLATION AND DEBT INDEXATION: THE
EQUIVALENCE OF TWO ALTERNA TIVE SCHEMES FOR
THE CASE OF PER/ODIC PA YMENTS
CI01,is de Faro
Outubro l/e /994
INFLATION AND DEJaT I~TJ)EXATION:
THE
E QUIV ALENCE OF TWO
ALTERNATXVE
SCHEMES FOR THE CASE OF PERIODIC
JPAYMENTS
Clovis de Faro 1
Revi~
October, 1994
1 Graduatc School of Economics, Getulio Vargas F01mdation .
INFLATION AND DEBT INDEXATION: Il:m EQUIV AI .ENCE OF TWO
CASE DF PEBIODIC PAYMENTS
ALTERNATIYE SCHEMES FOR
nm
AbstI'act
Thc presencc of inflation has induccd the financial instituticms to
implement proccdun:s deviscd to protcct the real valucs of thcirB loans.
Two of such procedurcs, the floaaing rale scheme and the monetary
corrcction mecbanjsm, tcnd to lcad to VCIY differcnt streams of paymcldS.
Howevcr, whencver the tloating rale scheme foJlows the rule of S1rict
adbercnce to lhe Fmhcr equation, lhe two procedures are financiaDy
c:quivalent.
INFLA.TION AND DEBT INDEXA.TION: 'lHE EQUIVALENCE OF 1WO
ALTERNATIVE SCHEMES FOR TIIE CASE OF PERIODIC PAYMENTS
1. Igtroduction
The moderate to high inflation cnviromncnt tbat has bcen cxpcricnced by
sevcral coUlllrics, inclucting the Unitcd Statc:s in the late 70's and spccially Brazil,
C\'CIl
today,
led thcir rcspcctive financiai instituIions to devisc mcchanisms aimcd to prateet the real valucs
of mcdium to long tcrm 10l1li. In general, such mcchanisms, broadIy known as indcxaJion
procedures, are based on ODe of the following two altcmatives. The first, wbich is usuaIly
adoptcd in the intcmationa1 arena and wbich has its thcorctica1 foundations on the c "Iica1
"
Fisbcr equation, is the 50 caIlcd fJoaUug rue schcme. The sccond ,wbich has bcen oflicially
adoptcd in Bram sincc 1964 and Iw bcen known as the monctary corrcction schcme, is
basecl on the principie of ma.intaining the purchasing powcr of the ple-spccificd paymcms.
As it could bc expcctcd, the two schcmcs tcnd to lcad to very diffcrmt
s1rcams of paymcnts; both in real and in nominal tcnns. Howcver, at least for the case of
constam rcpaymcnt of principal, the spccjfication of the same real rale of intcrcst and the
strict adhcrcnce to the Fishcr cquatiOD make lhe floating rale schcme financiaJly equivalcnt to
the monetary corrcction proccdurc.
It will also be shown that, notwithstanding the financiai equivalcnce, the
tlóating ratc schcme implics initial payments
gI'Cau:r
than the corrresponding ones under
monctary corrcction. 1bis pcculiarity, which amounts to an amortization of the dcbt at a
faster pace, was advanced by Dombusch [ 1]
and can be considcrcd as one of lhe
ingredicnts that contributed to the devcloping coumries debt crisis of 1982.
In fact, 1his
2
author pointcd out that, givcn the real rale of interest. whcn highcr inflation increascs the
nominal intcrest rale the effcct on dcbtors is a shortcDing of the effcctivc manuity of the
dcbt, wbich may rcsuIt in liquidity problcms for the dcbtors. Our finda lend support to the
hypothcsis that, had the cffccts ofinilation in the dcvelopcd counttics, chiefly in the U.S., bc
COIIIidcml tbrough the adoption of monetary correction. ratbcr than via thc ftoating rale
schcme, the dcbt crisis, at lcast for those dcveloping counaics whose loans
wcre takcn
to
financc productivc iDvcs1mCIÚI with some time to mature. could possibly have bccn mildcr.
2. The Monem Correction Sdleme
CODSidcr the 10m of C UDds of capital, at the fixcd real rale of iDtercst i , per
period, during m + n pcriods; the fim m rcquiring onIy the paymcnt of intcrcst. AssumiDg
thc spccification of CODStant rcpaymcnt of principal, it folioM that, as exptcsscd in tcrms of
pricca at the date ofthe 10m (time O), the valuc ofthe k-th paymcnt is cqual to:
iC, k =l, ... ,m
Pto= {
.
c{I + i( n + m+ 1- k)} / n,
(1)
k = m + 1. ...• m + n
On the othcr hand, if ~ denotes lhe value of the relevant pricc indcx at time
k, the adoption of the monetary cOITCCtion schcme implies that the actual (or cUlTCllt) value
ofthe k-lh payment is equal to
(2)
3
If 8 k denotes the infJation ratc rclative to lhe k-th pcriod of the loan, that is
8 t =It /I t_l- 1
I
k=l, ... ,m+n
(3)
it is casily vcrificd that
(4)
In words, rclation (4) rewaIs that the monctary corrcction schcme does not
dcItroy the cquivalcncc, at the real
me
~
betwccn the scqucncc of m+n paymcnts, wbcn
expreuccl at CODStant priccs, and the lCDl principal c.
3. The Fi_tini Rate Scheme
Oncc thc real intcrcst ratc i is fix~ the sttict adhcrcnce to the FJShcr cquation
implics that the currcnt valuc of the nominal ratc of intcrcst that apptics in the k-th pcriod is
cqua1 to :
~=(I+iXI+eIJ-l,
k=l, ... ,m+n
(5)
Under this circunstancc, the adoption of the floating rale schcme, for the 10an
CODSidcrcd on the ~ous scction, imptics that the current valuc ofthe k-th paymcnt is cqual
to:
4
RtC ,k = 1, •.• ,m
P'ltk=
.
li
{ c{ 11 n + ~ [1- (k - m -1) / n ,k = m + 1, ... , m + n
(6)
As a simple numcrical mustration of the difIcrcnccs betwecn the two
proccdures, suppose a loan of 12,000 unities of capital,. at the real ratc of intcrcst of 10% ,
with m = 6 and n
= 12. If the rale of infiation varies as indicated in the seçond column of
Table 1, the corrcsponding valucs ofPu and p'u appcar ín lhe tbird and seçond coJumns,
respective1y. Furthcnnorc, to stress lhe obvious fact tbat they are a1so di1ferent, Table 1
inc1udca, bctwccn parentheais, the corrcsponding real valucs Pk.o and p't.o (dc:fIatcd to time O).
As could be expccted, lhe figures in T able 1 providc evidcncc tbat lhe
llIIIIlCrical di1ferenccs bctwccn the two dcbt indcxation schcmes can bc subtantial.
A point that should be strcIIcd is that, as long as wc do Dot bavc dc:ftation,
we always haw p' 1.1 ~ Pu and p'~
~
plttm,lrta .This CID be easiIy seen, sincc :
(7)
anel
(8)
5
TABLE 1
Monctary C()IRCtion versus Floating Ratc
k
e.r(%)
Pu'Ú\.O>
1
10
2
lS
3
12
4
11
5
9
6
8
7
10
8
13
9
15
10
20
11
26
12
24
13
22
14
21
15
20
16
22
17
23
18
25
1,320.00
(1,200.00)
1,518.00
(1,200.00)
1,700.16
(1,200.00)
1,887.18
(1,200.00)
2,057.02
(1,200.00)
2,221.59
(1,200.00)
4,480.20
(2,200.00)
4, 832,SO
(2,100.00)
S,292.74
(2,000.00)
6,033.73
(1,900.00)
7,202.36
(1,800.00)
8,434.77
(1,700.00)
9,685.10
(1,600.00)
10,986.53
(1,500.00)
12,304.92
(1,400.00)
13,939.71
(1,300.00)
15,826.94
(1,200.00)
18,135.03
(1,100.00)
~
I
i
I
I
pluJ'(PIk.O)
2,S20.00
(2,290.91)
3,180.00
(2,SI3.83)
2,784.00
(1,964.99)
2,652.00
(1,686.33)
2,388.00
(1,393.08)
2,2S6.00
(1,218.S9)
3,520.00
(1,728.S0)
3,673.00
(1,596.13)
3,6S0.00
(1,379.25)
3,880.00
(1,221.80)
4,088.00
(1,021.66)
3,548.00
(71S.09)
3,052.00
(504.20)
2,655.00
(362.49)
2,280.00
(259.41)
2,026.00
(188,94)
1,706.00
(129.35)
1,375.00
(83.40)
6
4. Financiai Equivalence
Dcspitc lhe
diffClenccs
of the rcspcctivc scquenccs of ClDlent payIiICIUI, lhe
fIoatiDg rat.c schcme of dcbt indeDtion is financiaJIy cquiYa1ent to thc monctary corrcction
proccdure. That is, as we are going to show, lhe prcICDt valucs, at the real intercst rat.c i, of
cach
ODe
of the two COlTCIpOIlding
scqUCDCCS
of paymcnts, whcn dct1atcd to time O, are
cqual.
To prove lhe abovc asacrtion, it sufticcs to show that:
(9)
That ia, takiDg imo account (6) and clefinjng
(10)
(11)
wc nccd to show that ~ + BD
=1
It mav bc casüv vcrificd that
7
(lO')
Thcrcfore, as prown in the Appendix, we necd oniy to show that
(11')
5. Possible Breakdown
An intcresting question tbat may be poaed is thc ODe relatiw to lhe impacts of
dcYWioDa from de FiIher equation. DaviatioDa may
harPen whenever the nomiDal nae of
intaClt is fixed ex-mtc, iDstcad of ex-polt (after the dctcrmiDation of the infIatjon lc\'cl), aud
1Mre ia a breakdown of raâonal eApOCtatioDI.
A general answer is not poaible. In principie,
ODe
can cxpect tbat lhe financiai
equivalence between the two indexation schemes will not hold However, focusing attcntion
to the
particular case where n
= 2 and
m
=O
, we em show tbat, al1hough for a vcry
fortuitoua lCt of circunUltmcca, the equivalence may hold evcn when there ia a breakdown of
ratioDal ex:pectatioua.
Deno1iDg by 8 t • lhe cxpec;ted valuc of the jnfJation rale reJatiw to the k-tb
period of the loan, it fonows 1hat, in real
ténns,
the ex-post present valuc .of the payments
relative to the tloating rale scheme can be writtcn as
(12)
8
Obviously, we wi1lhave V = C if
IDa)' havc
e1- = 8 1 and e,: = e2 • Howevcr, wc stiIl
V = C, which imptics cquivalcncc with thc monctary COIl"CCUon schcmc, ewn if
et - ~ 9 1 anel
9 2- ~ 8 2 • This wi1l happcn whcnevcr thc fonowiDg rclationsbip holels
(13)
To givc a numcrical examp1e, let i = 8% and supposc that it ia cstimatcd tbat
the first pcriod iDfIation ratc ia e1-
= lI%, while the actual vaIuc ia e1 = lO%.
pcriod inflaôon ratc ia C'rim·tcd at the leYd
bc
If thc sc:cond
e,/ = 7.624% anel the actual valuc tums out to
e% = 10%, the corrcspondiDg cstimation
Cl1'OI"S,
which have diffClcnt sigas, wiIl bc
6. Condusion
.As we haw indicatcd, lhe diffcrcnccs bctwccn the scqucnccs of paymcnts, in
nomiDa1 tcrms, which are rcspcctivcly associatcd with the tloating
rale
schcme anel tbat of
monctary COIl"CCtion, can bc dramatica1ly high. Morcover, thc tloating ratc proccdurc impIics
tbat thc iDitial
payDlCDtS
are gRatcr tban thosc corresponding to the monctary corrcction
schcme, with the opposite occurring with the final 0IlCS.
Thcrcfore, since lhe two proccdurcs are financiaI1y equivalem, if thc 10m is
bcing madc to finance some productivc inws1ment which takes some time to mature, the
borrower should always insista on having the dcbt indcxed in accordancc to the monctary
corrcction scheme. Othcrwise, as experienced with the 1argeJy indebtcd Third World
9
Countrics which were Icei to fali in the floaJing rale trap, the borrowcr may face a iDitial caah
flow jmbaJancc tbat may rcsult in iDsoM:ncy.
REFERENCE
[ 1] Dombusch, Ructigcr, "Dcbt Problcms anel the Worid Macroeconomy", in DcM:Ioging
Cmgmy Dcbt and Economic Pctiormancc , Voll, cd. ]cffi'cy D. Sachs, The UuMrsity of
Chicago Prcu, (1989), pp. 335-338.
lO
APPENQIX
Aaswniog now 1hat (9) ia 1nJc for h > 1, we haw:
1 1D~+1 [1+ Rt(h+l+m+ 1- k)~I+itt
B.s+l :: -
~
h + 1 k ••+l
k(
1t 1 + e.
;'1
J
)
11
r--
(l+i)-m-l
(1 + i
!(l+S)
!1(1+6j )
II
••
ffiOGRAFmCAL sKEICHE
CLOVIS DE FARO is Profeaor of Quantitativc Metbods at thc Gradulte School of
Economics, Getulio Vargas Fowdarion, Rio de Janeiro.
He rcccM:d a B.Sc. in Civil
Editor of Revista Brasikira de Economia, thc audlOr of 5 boob and of sewral artidcs tbat
havc becn publishcd in Portugucsc, bis papcn havc aJso appcarcd in joumaJs such as: The
Engineering Economist, JOU17IIli of Financiai and Qw;mtitative A1IDiysis. Joumal of
/nI617UltionaJ Busi1UlSs Sbubes anti OR Spektrum (in GemJan).
,..
ENSAIOS ECONOMICOS DA EPGE
200.
A VISÃO TEÓRICA SOBRE MODELOS PREVIDENCIÁRIOS: O CASO BRASILEIROLuiz Guilhenne Schymura de Oliveira - Outubro de 1992 (esgotado)
201.
IDPERINFLAÇÃO: CÂMBIO, MOEDA E ÂNCORAS NOMINAIS - Fernando de Holanda
Barbosa - Novembro de 1992 - (esgotado)
202.
PREVIDÊNCIA SOCIAL: CIDADANIA E PROVISÃO - Clovis de Faro - Novembro de
1992
203.
OS BANCOS ESTADUAIS E O DESCONTROLE FISCAL: ALGUNS ASPECTOS Sérgio Ribeiro da Costa Werlang e Armínio Fraga Neto - Novembro de 1992 - (esgotado)
204.
TEORIAS ECONÔMICAS: A MEIA-VERDADE TEMPORÁRIA - Antonio Maria da
Silveira - Dezembro de 1992
205.
THE RICARDIAN VICE AND THE INDETERMINATION OF SENIOR - Antonio Maria
da Silveira - Dezembro de 1992
206.
HIPERINFLAÇÃO E A FORMA FUNCIONAL DA EQUAÇÃO DE DEMANDA DE
MOEDA - Fernando de Holanda Barbosa - Janeiro de 1993 (esgotado)
207
REFORMA FINANCEIRA - ASPECTOS GERAIS E ANÁLISE DO PROJETO DA LEI
COMPLEMENT AR - Rubens Penha Cysne - fevereiro de 1993.
208.
ABUSO ECONÔMICO E O CASO DA LEI 8.002 - Luiz Guilhenne Schymura de Oliveira e
Sérgio Ribeiro da Costa Werlang - fevereiro de 1993 (esgotado)
209.
ELEMENTOS
DE
UMA
ESTRATÉGIA
PARA
O
DESENVOLVIMENTO
DA
AGRICULTURA BRASILEIRA - Antonio Salazar Pessoa Brandão e Eliseu Alves Fevereiro de 1993
210.
PREVIDÊNCIA
SOCIAL
PÚBLICA:
A EXPERIÊNCIA
BRASILEIRA
-
Hélio
Ponocarrer? de Castro, Luiz Guilhenne Schymura de Oliveira, Renato Fragelli Cardoso e
Uriel de Magalhães - Março de 1993.
211.
OS SISTEMAS PREVIDENCIÁRIOS E UMA PROPOSTA PARA A REFORMULACAO
DO MODELO BRASILEIRO - Helio Ponocarrero de Castro, Luiz Guilhenne Schymura de
Oliveira, Renato Fragelli Cardoso e Uriel de Magalhães - Março de 1993. (esgotado)
212.
THE
INDETERMINATION OF SENIOR (OR THE INDETERMINATION OF
WAGNER) AND SCHMOLLER AS A SOCIAL ECONOMIST - Antonio Maria da Silveira
213.
- Março de 1993.
NAS H EQUILlBRIUM UNDER KNIGHTIAN C;-';CERTAINTY BREAKING DOWN
BACKW ARD INDUCTION (Extensively Revised Version) - James Dow e Sérgio Ribeiro da
Costa Werlang - Abril de 1993 .
214.
ON THE DIFFERENTIABILlTY OF THE CONSUMER DEMAND FUNCTION - Paulo
Klinger Monteiro, Mário Rui Páscoa e Sérgio Ribeiro da Costa Werlang - Maio de 1993
(esgotado)
215.
DETERMINAÇÃO DE PREÇOS DE ATIVOS, ARBITRAGEM, MERCADO A TERMO E
MERCADO FUTURO - Sérgio Ribeiro da Costa Werlang e Flávio Auler - Agosto de 1993
(esgotado ).
216.
SISTEMA MONETÁRIO VERSÃO REVISADA - Mario Henrique Simonsen e Rubens
Penha Cysne - Agosto de 1993 (esgotado).
217.
CAIXAS DE CONVERSÃO - Fernando Antônio Hadba - Agosto de 1993.
218.
A ECONOMIA BRASILEIRA NO PERÍODO MILITAR - Rubens Penha Cysne - Agosto de
1993 (esgotado).
219.
IMPÔSTO INFLACIONÁRIO E TRANSFERÊNCIAS INFLACIONÁRIAS
- Rubens
Penha
Cysne - Agosto de 1993 (esgotado).
220.
PREVISÕES DE M1 COM DADOS MENSAIS - Rubens Penha Cysne e João Victor IsslerSetembro de 1993.
221.
TOPOLOGIA E CÁLCULO NO
Rn - Rubens Penha Cysne e Humberto Moreira
-
Setembro de 1993.
222.
EMPRÉSTIMOS DE MÉDIO E LONGO PRAZOS E INFLAÇÃO: A QUESTÃO DA
INDEXAÇÃO - Clovis de Faro - Outubro de 1993.
223.
ESTUDOS SOBRE A INDETERMINAÇÃO DE SENIOR vol. 1 - Nelson H. Barbosa,
Fábio N.P. Freitas, Carlos F.L.R. Lopes, Marcos B. Monteiro, Antonio Maria da Silveira
(Coordenador) e Matias Vernengo - Outubro de 1993. (esgotado)
224.
A SUBSTITUIÇÃO DE MOEDA NO BRASIL: A MOEDA INDEXADA - Fernando de
Holanda Barbosa e Pedro Luiz VaUs Pereira - Novembro de 1993.
225.
FINANCIAL INTEGRATION AND PUBLIC FINANCIAL INSTITUTIONS - Walter
Novaes e Sérgio Ribeiro da Costa Werlang - Novembro de 1993.
226.
LAWS OF LARGE 1\ruMBERS FOR NON-ADDITIVE PROBABILITIES - James Dow e
Sérgio Ribeiro da Costa Werlang - Dezembro de 1993.
227.
A ECONOMIA BRASILEIRA NO PERÍODO MILITAR - VERSÃO REVISADA - Rubens
Penha Cysne - Janeiro de 1994. (esgotado)
228.
THE IMPACT OF PUBLIC CAPITAL AND PUBLIC INVESTMENT ON ECONOMIC
GROWTH: AN EMPIRICAL INVESTIGATION - Pedro Cavalcanti Ferreira - Fevereiro de
1994 (esgotado)
229.
FROM
THE
BRAZILIAN
PAY
AS
VOU
GO
PENSION
SYSTEM
TO
CAPITALIZATION: BAILING OUI THE GOVERNMENT - José Luiz de Carvalho e
Clóvis de Faro - Fevereiro de 1994.
230.
ESTUDOS SOBRE A INDETERMINAÇÃO DE SENIOR - vol. 11 - Brena Paula Magno
Fernandez. Maria Tereza Garcia Duane, Sergio Grumbach, Antonio Maria da Silveira
(C oordenador) - Fevereiro de ·1994. (esgotado)
23 I.
EST ABILIZAÇÃO
DE
PREÇOS
AGRÍCOLAS
NO
BRASIL:
AVALIAÇÃO
E
PERSPECTIVAS - Clovis de Faro e José Luiz Carvalho - Março de 1994.
232.
,., ....
--' -' .
ESTIMATING SECTORAL CYCLES USING COINTEGRATION ANO COMMON
FEATURES - Roben F. Engle e João Victor Issler - Março de 1994
COMMON CYCLES IN MACROECONOMIC AGGREGATES - João Victor Issler e
Farshid Vahid - Abril de 1994
234.
BANDAS DE CÂMBIO: TEORIA, EVIDÊNCIA EMPÍRICA E SUA POSSÍVEL
APLICAÇÃO NO BRASIL - Aloisio Pessoa de Araújo e Cypriano Lopes Feijó Filho - Abril
235.
de 1994 (esgotado)
O HEDGE DA DÍVIDA EXTERNA BRASILEIRA - Aloisio Pessoa de Araújo, Túlio Luz
Barbosa., Amélia de Fátima F. Semblano e Maria Haydée Morales - Abril de 1994.
236.
TESTING THE EXTERNALITIES HYPOTHESIS OF ENDOGENOUS GROWTH
USING COINTEGRATION - Pedro Cavalcanti Ferreira e João Victor Issler - Abril de 1994
- 37 pág. (esgotado)
237.
THE BRAZILIAN SOCIAL SECURITY PROGRAM: DIAGNOSIS AND PROPOSAL
FOR REFORM - Renato Fragelli; Uriel de Magalhães; Helio Ponocarrero e Luiz Guilherme
Schymura - Maio de 1994 - 32 pág.
238.
REGIMES COMPLEMENTARES DE PREVIDÊNCIA - Hélio de Oliveira Ponocarrero de
Castro, Luiz Guilherme Schymura de Oliveira., Renato Fragelli Cardoso, Sérgio Ribeiro da
Costa Werlang e Uriel de Magalhães - Maio de 1994 - 106 pág.
239.
PUBLIC EXPENDITURES, TAXATION AND WELFARE MEASUREMENT - Pedro
Cavalcanti Ferreira - Maio de 1994 - 36 pág.
240.
A NOTE ON POLICY, THE COMPOSITION OF PUBLIC EXPENDITURES AND
ECONOMIC GROWTH - Pedro Cavalcanti Ferreira - Maio de 1994 - 40 pág.
241.
INFLAÇÃO E O PLANO FHC - Rubens Penha Cysne - Maio de 1994 - 26 pág. (esgotado)
242.
INFLATIONARY BIAS AND STATE OWNED FINANCIAL INSTITUTIONS - Walter
Novaes Filho e Sérgio Ribeiro da Costa Werlang - Junho de 1994 -35 pág.
243.
INTRODUÇÃO À INTEGRAÇÃO ESTOCÁSTICA - Paulo Klinger Monteiro - Junho de
1994 - 38 pág.
244.
PURE ECONOMIC THEORIES: THE TEMPORARY HALF-TRUTH - Antonio M.
Silveira - Junho de 1994 - 23 pág.
245.
WELFARE COSTS OF INFLATION - THE CASE FOR INTEREST-BEARING MONEY
AND EMPIRICAL ESTIMATES FOR BRAZIL - Mario Henrique Simonsen e Rubens
Penha Cysne - Julho de 1994 - (esgotado)
246.
INFRAESTRUTURA PÚBLICA,
PRODUTIVIDADE
E CRESCIMENTO - Pedro
Cavalcanti Ferreira - Setembro de 1994 - 25 pág.
247.
MACROECONOMIC POLICY AND CREDIBILITY: A COMPARATIVE STUDY OF
THE F ACTORS AFFECTING BRAZILIAN AND IT ALIAN INFLATION AFTER 1970 Giuseppe Tullio e Mareio Ronci - Outubro de 1994 - 61 pág.
248.
INFLATION
AND
DEBT
INDEXATION:
THE
EQUIVALENCE
OF
TWO
AL TERNAtIVE SCHEMES FOR THE CASE OF PERIODIC PAYMENTS - Clovis de
Faro - Outubro de 1994 -18 pág.
_,
I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
000086015
"li111111 111 111111111111 111111111111