Bank Reprint Series: Number 159 Gershon Feder …documents.worldbank.org/curated/en/435141467988885506/...World Bank Reprint Series: Number 159 Gershon Feder and Richard E. Just A

World Bank Reprint Series: Number 159

Gershon Feder and Richard E. Just

A Model for AnalyzinrtgLenders' Perceived Risk

Optimal International Borrowing,Capital Allocation,and Credit Worthiness Control

Reprinted with permission from Applied Economics, vol. 12 (1980), pp. 125-44; andKredit and Kapital, vol. 12, no. 2 (1979), pp. 207-20.

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Applied Econonics. 1980, 12, 125-144

A model for analysing lenders perceivedrisk

GERSHON FEDER and RICtIARD F JUST'

Developmient Researclh Centre, Iorldl Bank, Brigham Young Unihe.sity, and UnirersUiv ofCalifordia. Berkeley, U.S.A.

Banik lending involves, in most cases, the risk that the borrower will not be able or willingto honour his obligations. The existence of default risk is an important factor in explainingthe observed behaviour of lenders as demonstrated in the works of Jaffee and Modigliani(1969), Smith (1972), Azzi and Cox (1976), Jaffee and Russel (1976), and others. Lenders'behaviour in this case depends crucially on their subjective evaluation of the probability ofdefault. Tlhus, to explain lending behaviour, knowledge of lenders' subjective inforimiation isgenerally unobservable, and empirical analysis of lending behaviouir is, therefore, difficult.Furthermore, there is often reason to believe that subjective information may vary con-siderably from lender to lender or from transaction to transaction because of previousexperience, personal relationships, etc.; and, hence, the role of subjective perceptions cannotbe ignored.

The purpose of this paper is to develop a model which facilitates inference about lenders'subjective default probabilities. Several alternative approiches of constructing both pointand interval estimates of subjective default probabilities are proposed under various as-sumptions about competition, loss rate distribution, risk aversion, and relative loan size.

The assumption which makes these approaches possible is that lending takes place with afixed probability of default. That is, lending transactions are assumed to be of sufficiently neg-ligible size relative to the borrower's scale of operations (e.g. lending to sovereign borrowers)so that the probability of default is not influenced by the lender's current decision, i.e. theinterest rate on the loan does not affect default probability. This assumption differs fromthat employed in the theoretical papers cited above and, hence, removes complicationsassociated with endogeni7ing thie probability of default. Nevertheless, the simplificationwhicli leads to the empirical possibilities developed in this paper is plausible in manysituations--particularly in international lending. For example, suppose the state of &aoPaulo, Brazil, borrows i100 inillion in the Eurodollar market with the guarantee of thle

Giannini Fountdation Paper No. 523. T.-e views expressed here are the authors'; they do not representthose of the World Bank. Estimates used in the text are provided froi i Euromarket data and are nieantsolely to be illustrative. There is no presumption that these estimates are the actual probabilities nor thatthe Bank in any way concurs with these perceptions.

0003-6846/80,020 125-20 $04.00/0 © 1980 Chapman and Hall Ltd. 125

126 Gershon Feder and Richard E. Justtederal government of Brazil. The likelihood of default in this case would surely be affectedonly slightly by the lender's decision since the overall debt of Brazil is $32 billion dollars(Newsweek, 1978).

I. TIHE MODEL

Consider a simple one-period model similar to the one presented by Smitlh (1972). Thelender is assumed to have a given amount, W, of loanable funds, all or part of which can belent to tlhe particular borrowver under consideration at interest rate P. The interest rate isexogenously given if the lender operates within a competitive capital market, or it dependson the volume of loans granted (according to the borrower's demand schedule) if the leniderhas some degree of monopoly power. Alternatively, part or all of the lender's funds can belent at interest p which is the opportunity cost of capital and which is considered risk free. 2The borrower's demand schedule is given by the relation,

P = f(L), P' < 0 in the monopolist's caseP' = 0 for the competitive lender, (1)

where L denotes the volume of loans granted. It will be useful to define the risk premium,r, which is the difference between the lender's opportunity cost of capital and the interestrate actually paid by the borrower, namely,

-r-+ p. (2)Using Equation 1 and considering p as a constant, the demand schedule can be defined

for the risk premium r,

r = r(L), r' = < S0, (3)where, as before, r' = 0 applies to the case of the competitive lender.

The borrower is assumed to possess (at the time of loan negotiations) assets totalling Kwhich were financed in part by previous borrowing and in part from the borrower's ownresources. IHis total outstanding debt as of the time of the loan under consideration is givenby D. Following Smith (1972), it is assumed that the liquidity situation of the borrower atthe time of loan maturity is related to borrower resources (including the new loan) by astochastic, linear homogenotus function of the form y x [K + L3, where y is a continuousrandom variable with probability density function fty) defined over the interval (0, mo). Thepaiameters off are influenced by various borrower characteristics as viewed by the lender;and f(y) reflects, therefore, the latter's subjective assessmnent of borrower prosplects.

When the loan under consideration becomes due, the borrower mvst pay (l + F) x L inaddition to other obligations wlhichl will be due on his older outstanding debt D, say p x Dwhere 0 denotes thle rate of interest and amortization payments; 0 is not atTlected by thenew lender.

2As in the earlier works on lending under uncertainty cited above, portfolio considerations are ignored forthe sake of simplicity.

A model for analysing len?ders' perceived risk 127

Obviouisly, if y will be sufficiently high so as to maintain the inequality

'yx (K+L)- (I +f)x L - x D > , (4)

then all creditors will be paid in full and no default takes place. Using Equation 4, one candefine thle critical y value, (say, y*)

* ( +)xL+xD (5)ly K+L

If y < y*, not all obligations can be met, and a partial default takes place. In such a case,it is assumed that creditors are paid according to the proportion of debt they hold. Inparticular, the lendur under consideration will be paid

'y x (K+ L) x L(L + D)

The present value of the new lender's wealth (say, y) is thus given by

W-L + ( x L if y > Y*

Y=

W-L+x (+ p) (L+D) if y < 'y. (6a)

Rearranging the terms in Equation 6, present value y can be rewritten as

w r x x L if y> y*y{ w-- hxL if < y* (6b)

where

Il+p

K+Lh -y x (LT D) x (I + p)

and

r = risk premium as defined in Equation 2.

Note that hi is random since it is a transformation of the random variable -y. From the formu-lation in Equation 7, it follows that, for a givenl value of y, h is the rate of loss incurred bythe lender. The range of values over which Ih is defined is (0, -y*), 'thus, using the definitionof y* as given in Equationi 5, Ih is confinied between upper and lower bounds I and h,respectively, where

h ( ) x D- ((r p] x LI /(L + D).

3 It is implicitly assumed that no priority debt exists. However, one could assume that priority debt is paidfirst and tllat the other creditors share the remaining liquidity according to their share in regular debt. Theappropriate analysis, however, remlains unchanged.

128 Gerslion Feder and Richlard E. Juist

Denote thc lender's utility function by U, where U is defined over prc;'.nt value of wealth.The utility function is assumed to have the properties

v >0, U" < , (7)where U" < 0 implier risk aversion and C' 0 reflects risk neutrality. It is further assumedthat, in the case U" ; relative risk aversion is no greater than one, i.e.,

x ('u)y < 1.

This is a plausible assuzmption as argued by Arrow (1971, p. 98). Tlle lender's objective isto maximize the expected utility of present-value wealth by an optimal choice of loan size,i.e. using Equation 6b

Max f- U(TV + r x 0 x L) x f(y) d7 + U(WV - h x L)f(-) d. (8)L o

Note, however, that

U(W + r x 0 x L)f(y) dy U(J V + r x 0 x L) x [1 -P(y*)],

where

,.P(y*)= ff(y) dy

0

represents the probability of default. Note, also, using the definitions of /z and 11, that

| U(JV - h x L)f('y) dy U(IV - h x L) x p(hl) dli

where

p(h1).-f [(1- mo + p) x (L+ )J'

The objective function in Equation 8 can thus be written as

Alar 7l I [ 1 -P( 7 *)] x U(JV + r x 0 x L) + U(Pt -hL) x p(hi) dlh. (9)L f

.1i the general case, the probability of default P(7*), as well as the specific loss rate densityp(hQ, depends on the size of loan granted by the current lender and on tlle rate of intercstcharged for the loani (as inspections of y* and i verify). I lowever, as will be shown betow, inthe case where the relevant loan size is small, relative to the size of the borrower's assets (i.e.wlhen LIK' is small), the probability of defaullt anid the distribution of loss rates can betreated as given (fromli the point of view of the current lender), This simiiplifies the analysisconsiderably and facilitates several applications as will be demonlstrated in the followinw,sectiolns.

In order to verify the as3ertions regarding PQy*) anid p(h), define the loan to asset ratioi (i.e. = LIK). Rewriting the definitions of y*, h, arnd h by simply dividing numerators and

A model for analysinig lenders 'perceihed risk 129

denominators by K, one obtains

(1 + r) x t+ 4 x (DIK)

1 (Ih I - yx (11

[I + (D/IK(I + p)

and

-( [r( x PO + PA-[!(/n

Taling the limit as 1 e 0 yields

7* = D

= X xDK K'

LI 1-7iL.=o [(1+p)x(D/K)i I

Thus, when I is stufficiently small, y* and hi approach fixed paramiieters from the currentlender's point of view, and the (random) loss rate It is independent of the size of the currentloan (L) and of the risk premium (r) charged oni the loan.

These results imaply that, with the assumption of negligible 1, the probability of defaultP(y*) and the density of the loss rate p(lz) are not appreciably affected by the actions of thecurrent lender and can be treated as fixed parameters in the lender's problem. Rather, thedominant factors underlying these probabilities are the existing economic prospects facingthe borrower.

Before proceeding to characterize the optimal soluition, one fLurthier modification needsto be made-namely, extending the model to a loan duration of N years, Assuming that,at most, one default takes place withiin a loan's duration and that loans are repaid in 'balloon'form (interest payments paid annually and the principal repaid in the last year), the formu-lation of the objective function (Eqtuation 9) can be retained provided that 0 is redefined as

0 - 1 - G + p) ]/P

(note that one can show that

Et= IxLx(1 p)-'+Lx(l +p)-N L rx0xL),and h is reinterpreted as a discowuited rate of loss. This is possible since both p and N areviewed as exogenous by the lender with the former paranieter (leterminii(ed by m1arket oppor-tunities and the latter being dictated to a large extent by the lender's and borrower's overallliquidity projections (Beim, 1977, p. 723).

For notatio..,al convenienice in the following analysis, define

P = P(y*), y X--W + r x 0 x L} and Yh=- H -- It x L.

130 Gershoni Feder t. 'id Richla(rd E. JtstDifferentiation (Equation 9) witlh respect to L, the following first-ordler condJition clharac-

terizes the optimal solution:

aL (I - P) x 0 x U'(y1 ) x (r + Lr') - fh V'(Yh) X l1 x p(hl) dh2 = 0. (10)

Thc second-order condition requires

a2 (1 P) x U"(y1 )02 (r + Lr')2 + f U"1(v,;)h 2p (h) dli

+(1 -- P)x Ox U'(vy,)x (2r'+Lr")<0. (11)

From Equation I 1, it is obvious that risk aversion is a necessary and sufficient conditionfor

a2 H/WL2 < 0

in the case of a competitive lender (r' = r" = 0). In the case of a risk neutral rmonopolisticlender,

(2r' + Lr") < 0

is necessary and sufficient to guarantee concavity of the objective function. With risk aversionand monopoly, the latter condition is not necessary but remains sufficient. Hiencefortlh, itwill be assumed that second-order conditions hold for the lender tinder consideration.

II. COMPARATIVE STATIC RESULTS

Before proceeding to demonstrate the wide range of empirical uses of this nmodel, it is firstuseful to examine the various comparative static properties of the model for the purpose ofshowing that the model is indeed plausible.

Demand elasticity

Considering the nmonopolistic lender, Equation 10 imtlplies tfat the lender operates oni thlenonelastic portioni of thze borrower's demnand schiedule. To see this, note that r + Lr' =r x (1 - -), where T? is the absolutevalue of demand elasticity at the optimal point. Obviously,if7 > 1, Equation 10 cannot hold since the left-hand side is negative.

Default probability

A borrower who is mnore riskv (i.e., who carries a higher )robability of defautlt) will begran1ted less credit whetlier the lenider is competitive or tIot. This can be cojiifirmied byassuming that, foranygiven rate of loss/h, the relative likelihood is at least as great as before,dp(h)> 0 forh•li S 1.

Hence, using the fact that

-PP= f (h) dhi,

A model for analysing lenders' perceived risk 131

one finds that

fh dp(h) > O

if dP > 0. By differentiating Equation 10, one then obtains

dP xa J U; x [Ucx1)x x(r+Lr')+U'(Yh)xhI (12)

where tne sign is established using Equation 11 and the earlier result that r + Lr' > 0 for botha monopolist and a competitive lender. V., result in Equation 1 2 implies, in the case of amonopolist, that riskier borrowers are charged a higlher risk premium. This is also tile casein a competitive market since, with a highler probability of default, a smaller amnount ofloans will be offered by any inidividuLal lender in the market at any given interest rate. Theaggregate supply of loans to the borrower under consideration will thus decline. With anegatively sloped demand for loans and a positively sloped aggregate suLpply, a decline insupply must yield a higher equilibrium level of risk premiunm (withl a lower amount of loansin a competitive market is indeed positively sloped (i.e., that supply is increasing witlh higherinterest rates). This is done by differentiating Equation 10:

dL (3 2 1 1 rU"WV.,(3

dr aL2) (I-P)xvU (y 1 )x [L+UP')'rOLJ (13)

Under the assumption that relative risk aversion is nio greater than one, the termi in squarebrackets on the right-hland side of Equation 13 is positive; and it is thuls conCludLe.d that

dL/dr > 0.

It should be noted in the case where the lender considers the loan to have an impact on theprobability of default, however, that the offer curve by each individual lender is backwardbending irrespective of attitudes toward risk as shown in Jaffee and Modigliani (1969),Smith (1972), and Azzi and Cox (1976). The dli'ferent results in the present analysis aredue to the fact that a higher interest rate increases the marginal expected utility while, inthe earlier models, a higher interest rate may reduce expected utility by causing a higher pro-bability of default.

Risk aversion

Conisiderinig tle. role of risk aversion in the imiodel, it is iiitjuitivcly expected tizat, whenleInders are nmore risk averse, the volume of loans wvill be lower aicd risk premiumzzls will behiiglher. This can be shiown simply by assuming a specific form of the utility function suclthat risk aversion is reflected in a single parameter. The two most commlonl suchi utilityfunctions are the constant relative risk-aversion falmily of functionis and the constaint absoluterisk-aversion fanmily of functions. Thliese are givenI, respectively, by

U(y) = aj, - 0 O < a< I (14a)

U(y) = a -e Y, cr> 0 (14b,i

132 Gerslhon Feder arad Riclhard E. Jlustwhere a is a paraineter of risk aversion such that the hiighler cx is associated with higlher riskaversion. Assuming the case of Equatiori; 14a, differentiation of Elquatioll 10 obtains

d_ [a2 ny'-- X(Wda = -[ -| x( r+r6L)- x i(l -P)x a x (r Lr')

x /h x (W hL)-c x p(z) ln ( - rzL ) ] (I Sa)h W +rOL xl IaSince

In (IV'- hL) < 0,Iu (I± rOL)J

it follows that

dL-<0.duz

Sinmilarly, assuming the case of Equation 14b and differentiating Equation 10 yieldsdL- [a 2 n-I 1 F 1

1 I 1)X -aI Ldi= [L2J xx [| hxp(h+) xea(;v/Il)x(I+ rO) x Lx dhj <0. (15b)

The resuilts in FqLuations 15a and b imply that a higher le,gree of risk aversion causes asmallervoluLme of loars and higher risk premiulms in botlh cominpetitive antd nmoncmlpetitive marklets,

III. SUGGESTED APPLICATIONS

The results in the previoLus section are plausible and intuitive and imply that the equilibriUrmrelation in Equationi 10 can be usefully considered for furtlher e.\positiollal and enmPiricalpurposes, In this section several such enmpirical applrlicatiniis are demniioistrated. For thispurpose, suppose that lenders' inlormiiation and evaluation distingLlislh between the overallprobability that a default will take place and the (conditional) probability that a given rateof loss (h) will be incurred. Hence, for conceptual purposes, and so tlhat subjective defaultprobability can be meaningfully discuissed, suppose that the probability density function ofloss rate li is proportional to the overall default probability suLch that the condlitional pro-bability of h dependls on hi alone and not on the economlic factors wlhiclh explain the overallprobability. This inmplies

p(h)= I (h), ;,Ir(h) d= 1, (16)

where ' depenids only on hi irrespective of the econiomiiic factors whliichi detcrmiinle P. Note,that, wvi th this assumiption, the (LuncConLlditiollal) i.trobability of ally givell loss rate inereaseswhen the overall default probability increases, as onie would expect, since FEqualtion 16unp1lies

p(hZ)= ~(h) X P;

A model for analysing lenders' perceiped risk 133

thus,

dp(h) >O.dP

Applying EquaLion 16 in Equation 10 and rearranging yields

__ P )ir= ) ( -P) R (17)

where

h Jf (h)h dh

and

|f h x 4(h) x U'(yh) dh

hx U'(yl)

For a monopolistic lender, Equation 17 demonstrates that the risk premium includes ademand elasticity factor, (I -- ;_)-I ; the odds of defauilt, P/(1 -- P); a time-effect factor,0 '1; an average loss-rate factor Ii; and a risk-aversion factor, R. The latter factor is identicallyequal to one under risk neutrality in wlhich case

Vyh ) =@ 1 ).

With risk aversion, it is easy to show that

R > I and aRIaL > 0.

Equation 1 7 offers a great deal of flexibility in empirical inference of subjective defaultprobabilities because of its simplicity. This flexibility is demoinstrated in the following threeapplications.

Threshold defatult probability

A simple application of the model is related to the concept of 'tlhreshold probability' forcompetitive lenders. The tlhreslhold probability is the highest value of default probability(say, P*) for whiclh a loan will be granted to any borrower and depends on r, 0, and Ti.Borrowers witlh probability lower than P* will be given credit with the amount of creditincreasing in the ratio

[P*/(l - P*)]

[PA(1 - P)J

Using EqLuation 17, the threslhold probability is calculated by noting that, at P = P*,it must hold that L = 0 whliich implies that

UV,) U(=ych)= U'(TV).

134 Gershlon Feder and Richtard E Just

This obtains

p$= r(h/0)+ r (18)

By differentiating EqLiation 18, one can verify that thc threshold probability is positivelyrelated to the risk premium and the loan duration (i.e.

aP*ar > °, aP*IaN > 0)

while being negatively related to the average loss rate and to the opportunity cost ofcapital (i.e.

ap*Iah < O, aP*/ap < 0).

Using Equation 18 in 1 7, with T?. = 0, further yields

R(L) = -/P( p;) (19)

The right-hand side of Equation 19 is the ratio of odds evaluated at the tlhrslhold and actualvalues of probability. From EqLuation 19, one can derive thl result that loans (and the riskaversion premium) increaise as the riglht-hand side of Equation 19 increases.

A useful property of the threshold 'probability is that it is indelpendenlt of the degree ofrisk-aversion as is apparent from Equation 18. Since the values of p, N, and Xl are eitlherknown or can be eitinmated without major difficulty, one can calcuiate the threshold pro-bability which applies to different risk premiLlms. This approachi is denmonstrated in Table Iforp = .065 and h =.15, .2. The results are presented in terms of q* (the short-run thireslholdprobability) so as to allow comparison between cases witlh diflerent loan durations. That is,if at most, one default can take place within the duration of the loan, then thc probabilityP of default over the entire period of the loan is

P= 1 -(1 q)N,

provided that q, the probability of default in any given year, is conlstalt . 4 It is reasonable toexpect that, given current data, lenders assumeq to remain essentially constant since no databeyond the period in which the loan is granted are available for projectionl. The q* in Table 1is a threshold or upper boLlu d on q and is related to P* just as q is ielated to P.

Using this approach, one can very easily confine the subjective probabilityq to the interval[0, q*] under competition by simply usillg data on r, p, N and Ti (note that 0 is determineidby p and N; r-oreover, the interv.al [0, P*] is fairly narrow even for small IT. Furthlermore,if Ei is unknown but can be boundecd from below by X*, then P catn be confined to thieiniterval [0,P**] where

p** = r

(Ff *l/) + r

(since P* is decreasing in Ti) or q can be confinedl to an associated interval [0, q**]where P** = I - ( I ^ q**)N I lence, the simniple transactioni data oln r, p. and N is stufficient

4 The reader will note that this definition of Pwas indeed in the descrip)ion of the nrodel.

A model for analysing letnders' perceived risk 135

Table 1. 77hreshold siort-n'n probabilities and risk premnrms

for -5- and I ayear loans'

Loanduration r q(years) (per cent) (h = .15) (h .2)

N= 5 .25 .013 .010N= 10 .25 .011 .009

N 5 .50 .026 .020N= 10 .S0 .021 .016

N= 5 .75 .037 .029N= 10 .75 .03 .024

N= 5 1.00 *048 .037N 10 1.00 .038 .03

N 5 1.25 .058 .045N= 10 1.25 .046 .036

N= 5 1.50 .067 .053N= 10 1.50 .053 .042

N 5 1.75 .076 .060V= 10 1.75 .059 .048

N= 5 2.00 .085 .067N= 10 2.00 .065 .053

aAssuming the opportunity cost of capital p is 6.5 per cent andN = 5, the value of 0 is 4.156, with N = 10, the value of 0 is7.1888.

to develol) some intormation about the miagnIltucs Of subjective dlefault pronabilities which

might exist.

An applicarion in inferring leniders' subjective probabilities: The case of tie Elurodollar

market

Another application of the model in this paper can be made by inferring interval estimlates of

lenders' subjective probabilities of default from a niniiiial amount of observed niark;et data.

To demonstrate this possibility, 20 observations on loanis granted in the Eurodollar nmarket

during the third quiarter of 1973 have been used. Tlese observations (presented in the

Apipendix) refer to loans made dLuring that period to public (or publicly guarantced) Cletities

in developing couuntries for which data on risk premium and loani duLrationi were available. 5

5The short period covered was chosen so as to avoid any significant variations in the opportunity cost or

capital among observations. The estimates are meant solely to be illustrnifive. There is no presuLmption

that these estimates are the actual (objective) probabilities nor that the World Bank in any way concurs

with these perceptions.

136 Gershotn Feder and(i Richawrdt E. Jiust

The procedure employed here uses the assumptions that the nmarket operates competitivelyand that the subjectivc loss-rate distribution is singular in order to establish bounds on thlelenders' perceived default probabilities. rhe first assuLmptiorn imnplies that 77 = 0, and thesecond implies that fI is a constant, say, Th. Assur: ;nr thiat the utility functioni can be specifietdby the constant elasticity formulation (Equiation 14b), the risk-aversion premitum R canthen be written

R= -- (20)

where I L/AI is the share of the loan in total loanable funds anid a is the reljtive risk-aversioni parameter (O < c < 1).

While the 20 observations on Eurodollar loans range in loan duration between 10 andi 15years, the probability concept which is of interest is again one that is free of the loan dLurationeffect. Thus, combining Equations 1 7 and 20 and the relation between q and P the followingformnula is obtained:

q ( 1 (1 p)l /N = 1- I- +rO il- (| -t rO/1- Fz 1 +rOl (1

To make use of Equation 21 in the Eurocurrency market, it can be observled that theaverage rate of loss in international banlkinig is low; in almost all cases of debt-service problems,loans -are renegotiated and resclheduled.6 Hence, for exeniplauiy 1p)Urpo-,e. a valLue of/St= .1 isused.7 As for 1, p, and as, it can be noted from EqUation 21 that

aqlal, aqlap, and aq!lau < 0.

Thus, specifying sets of upper and lower bounds on these coeff-icients leads to lower andupper bounds, respectively, for the short-run probability q. For example, consider uLpperbounds of I = .3, p = .08, and a - .8 and lower bounds of I = 0, p = .06, and Q= 0. Tflesevalues seem to span the range of reality. That is. I = 0 anid a = 0 are the lowest possible vXlLines(assuming no risk-lovinig behaviour), while I- .3 and a = .8 seemii to be above and beyondl alllikely possibilities. No bank seems to loan anytlhine near 30 per cenit of its lunds to oneborrower; also a close to 1, in the case of the constant elasticity utility fllnctioll, implies anextremely high degree of risk aversion. Finally, the opportunity cost of capital seCemls to bebetween 6 and 8 per cent. Using these bounds, the associated interval estimates for subjectivedefault probability (as perceived by Eurodollar lenders) in Table 2 are possible for the Euro-dollar market transactions considered above. hlowever, these estimates do not nkecessarilyimply Judginents on the objective (truLc) probability.

Several interesting observations can be ni- ue onl the basis of these resuLlts. First, the upperand lower bouLnlds span a range for the probability ofeach case withiin .8 of I per cent or less.This is a small interval relative to the overall variation in Table 2. Scond, thle calculated q sior those countries with more than one observation (e.g., Algleria) are t'airlv close for imiost61n this respect, Friednman (1977, p. 55) noted: 'Losses in U.S. banks' overseas olrertions h1aVe been less,both in absolute termis and as a proportion of total risk assets, tlhan in U.S. operations.' A sunil.ir observationhas been made by Beilm (1977, p. 717).'Normitally, one would want to estimate Jl since the estimates are somewhat sensitive to this particular

paramleter.

A model for antalysing lenders' perceived risk 137

Table 2. Perceived slort-nitn probabilities in thze Euromarket third quarter,1973

Obse-- q qvation upper lowernumber Country bounda boundb

1 Algeria (N= 15) .044 .0392 Algeria (N= 12) .047 .0423 Algeria (N= 10) .048 .0444 Brazil (N 10) 052 .0495 Brazil (N= 12) .049 .0446 Colombia .043 .0397 Gabon .079 .0728 Greece .040 .0369 Iran (Qv= 12) .037 .033

10 Iran (N= 10) .037 .03311 Korea (N- 10) .071 .06512 Korea (N 1 1) .059 .05313 Mexico (N= 12) .036 .03214 Mexico (N= 10) .035 .03115 Nicaragua .066 .06016 Peru .075 .06817 Philippines .087 .07918 Senegal .083 .07519 Zaire .079 .07220 Zambia .069 .063

aCa'.,ulated with a 0, I = 0, p = .06, and h .1.bCalculated witho a .8, 1 = .3, p = .08, andh .1.

of these cases (theoretically, the, should be identical). Also, Peru, Zaire, and Zambia, wlhichseem to have been considered as higher risks by Euromnarket lenders as indicated by their qvalue, have inideed dxperienced serious debt-service problems in years after 1973. Theseobservations may serve as an indication of the creditability of the assumptions. In summary,it appears that fairly precise estimates of default probabilities, as perceived by lenders, arepossible under competition using only the terms of the loan if loss rate for the event ofdefault can be determined (estimated).

Possible ecotnometric applications

In addlition to the above exanmples, Equation 17 also has interesting econometric possibilities.For the purpose of .2conoilletric work, it is usefuil to consider the comlnoni logistic specifi-cation for the probability, nanmely,

I + eCo+PaX (22)

138 Gershon Feder andt c Richl(ard E. Just

where g3o is a constant, , is a vector of coefficiefits, and X is a vector of economic indicatorsincluding loan durationl whlich are considered by lenders as relevant risk indicators. Withthis specification, the odds are log linear, i.e.,

In [(l Pp)]

Hence, Eqtuation 1 7 becomiies

In r =-In (I - r)+ In h-In 6+ : 0 + ''X+e, (23)

where

e =In R In [fh h1(h)U'Oh ) d/l In [FhU'(y1,)].

The model in EquLation 23 has several interesting potential simplificationis. First, for risk-neutral lenders, it is clearly the case from above that e = In R = 0; lience,

In r = -In (1 -- 77) + In h -In 0 +P go + P'X. (24)

With the presence of a disturbance term, Equation 23 can be estinmated in a variety ofsituations to determine whichi factors are considered inm portant by lenders in assessing defaultprobabilities. For instance, 77, Ti, and 9 can be calctulated (0 depends only on p and iN whichare often recorded, and Ti is also occasionally recorded for different classes of borrowers)and then included in a regression estimlating go and , wlhere a coellffient of' I or I isimposed for In (1 - 77). In hi, and In 0 as needed according to Equationi 24. Alternatively,one may find data where combinations of 77, Ti, and 0 are held conistant and can tlhus beincluded in a constant tenr. Another approach (Feder and Just, I 977a) is to consider someof the first right-hand terms as randomly distribLuted among borrowers and then uise avariance--components approach in estinmatin-g- P from a time series ot- cross-section data.

Finally, an interesting observation can be made for Equation 23 in the general case of riskpreferences. As it stands, the e term in Equation 23 would be very difficult to treat econo-metrically; besides, it would imply ineluisioni of the dependeiit variable r on the ritght-hatndside. But in a broad range of cases, it turnis out that e is negligible relative to both In r andthe usuial variations in In r amongT observatio.ns. This can be demonstrated asstuming aconstan( elasticity utility ftunction in whiclh case EquLation 20 applies. Using, the sameplausible limits oln parainieters used to generate Table 2, togetlher with additional limits on rof 0 < r < 0.025 and on Nl of 0 < A' 10, it can be slhown throLughl simple calctulatiolns thatI S R < 1.07 and lhence 0 < In R < 0.068. By comparision, as r varies from 0.00575 to0.025 (which seenms to spati the observed range of premiiitums in the Eurodollar miarlket), thedependent variable In r varies from -3.698 to 5.159. ilence, even witlh most conceivablelimits on 1, p, and ca, it is thius clear that e is nlegligible anid can, for mtlost practical pUrposes,be inCluded witlh atly distuLrbalnce in Equation 23.

Considering a conipetitive miarket, EquLation 24 canl becoine f airly simiple once e isincorporated in the error term if comnpetilioil imlplies 77 = 0. The equttationl is tlhen reduLced to

In r = lnT -- In 0+ g +O'X + v (25)

A mnodel for analysing lenders'perceived risk 139

where l} is a randonm disturbance. This formulation is exemplified in the followinig estimation

example which pertains to transactions in the Eurodollar market in the period January,

1975, using a set of 29 observations on public or publicly puaranteed loan transactions(corresponding to 23 developing couintries) as reported in Appelldix 2.' Reliable indications.

in the period covered imply that the cost of capital (p) was quite stable. For instance, the

Eurodollar three-monthl and six-month deposit rates (which are closely related to the cost

of capital) ranged from 6 per cent to 7 per cent ancd 7 per cent to 8 per cent, respectively,

while in the previous year they fluctuated sUbstantially from 8.5 per cent to 14 per cent.

Adopting the assumptioni that p was approximately constant througlhout the period covered

thus simplifies thle treatment of 0 in Equation 25 since one can experillient withl a numlber

of likely values of p (say, .06, .07, and .08) in alternative estimates, The vector X includes,

in addition to loan duration (whichi is expected tobbe positively related to default probability),

three variables wlhich have been suLggested as potentially related to debt-servicing capacity

of developing counltries and, hence, to the risk of default (Frank and Cline, 1971, and Feder

and Just, 1977b). These variables are:(1) The ratio of debt-service payments to exports (also referred to as the debt-service

ratio). Higher ratios reflect a heavier burden of debt service paymnents an!d are thus expected

to be positively related to the probability of default.(2) The ratio of imports to G.N.P., whiclh reflects claims of forei-n exchange which

are competing with debt-service obligations, Higher import to G.N.P. ratios should thus be

positively related to the probability of defaul t.(3) The level of G.N.P. per capita as a proxy for degree of developinent and the ability to

divert resources away from consumption in order to service external debt. Hligier lper capita

G.N.P. is thus negatively relatW,d to default risk,Data for the calculation of these variables were obtained irom various World Bank and

International Monetary Fund publicitions. Having defined the vector X and usingt tlhree

alternative values forp forthe calculations of 0, the corresponding three regression estimates

of Equation 25 are reported in Table 3.The results are practically identical for the different values of p aid thtLs imnily that lack

of information regarding the exact cost of capital is not serious. All pa:irameters are statistically

significant at a 7.5 per cent one-sided sionificance level, andl their signs confirm the a priori

arguments advanced.9 Thus, longer loan maturities, higher debt-service ratios, and higlher

import to G.N.P. ratios increase thie perceived probability of debt-service problerns of

borrowing governments wlile highecr per capita incomes reduce the perceived probability.

To examine one useful nrin'ication of results of this type, note that the constant term

reported in Table 3 is an -,immate of In Ti +& (plus the mean of the error term v if the latter

has a non7ero mean). For given valuies of h (and meani error whlen applicable), one call cal-

culate the corresponding 00 values. Ihese, togetler with the , coerficients reported in TIable

3, can be used in calculating perceived probabilities using EqUation 22, wlhere eachi alternative

value of gO corresponds to a set of probabilities (Feder and Just, 1977a).

Using the parameters in Table 3, one can also establish a dlirect link between terms of

"This period provides a larger rep)resentation of countries in the sample thani other perio(ds, thus contri-buting to tthe quality of econometric results.90ne-sided tests are more appropriate in the prescent case since there are strong a priori arguments relatingto the signs of the coefficients.

140 Gershon Feder antd Richlard E. Just

Table 3. Regression resuilts with Eurodollar data: January-Julv, 19 7S

Case I Case 2 Case 3Variable (p = .06) (p = .07) (p = .08)

Loan duration (years) .153 .149 .145(6.515)' (6.341) (6.171)

Debt-service ratio .467 .467 .467(1.671) (1.672) (1.673)

Import to G.N.P. ratio .432 .431 .431(2.511) (2.510) (2.510)

Per capita G.N.P. ($1000) -.089 -.088 -.088(1.541) (1.541) (1.541)

Constant -3.496 -3.502 -3.509

R2 .69 .68 .67j~2 .64 .63 .61

'Figures in parentheses are 't' ratios.

credit and economic performance criteria which can be useftul for a borrower in ilnanagingeconomiic resources wisely. Consider, tor example, a five-year loan sotiglht by an economywith a G.N.P. of $1000 per capita. Table 4 presents the risk premiums corresponding tovarious debt-service ratio and import to G.N.P. ratio combinations. Using the results of thisanalysis, a borrowing countrv can thus evaluate the interaction of, say, import policy andborrowing policy with respect to its terms of credit. In a similar way, one can represent therisk premiums corresponding to other combinations of explanatory variables.

Table 4. Risk prendiuins for a five-year loan by a borrowingcouintry with $1000 per capita G.W.P, a

Debt- Import to G.N.P ratioserviceratio .1 .2 .3 .4 .5

.1 .0155 .0162 .0169 .0177 .0184

.2 .0163 .0170 .0177 .0185 .0193

.3 .0170 .0178 .0186 .0194 .0202

'The calculations are based oni Case 2 of Table 3, but almostidentical results are obtained with Cases I and 3.

A model for analysing lenders' perceived risk; 141

IV. CONCLUSION

This paper presents a model of lending under default risk. With an undclerlying assumptionthat the probability of default is determiiined by factors not controlledt by the lelnder (wlhichlis essentially the case when a new loan is small relative to the borrower's existing debt),the effects of risk, risk aversion, and expected loss rate on the equLilibriulm levels of loarsupply and risk premiumLis are analysed for bothl competitive and noncompetitive markets.The model is shown to be useful for assessing subjective default probabilities from miiarketdata, using the Eurodollar market as an examzple. The model also enables the determinationof upper bound threshold probabilities for acceptable customers, given the relevant dturationof loan and interest rate. Further utilization of the model for various econloiiietric purposesis discussed in detail, and it is showIn that in many cases the complicated tern reflecting riskaversion can be ignored since its rnagnitude and range of variation are niegligible. Tllissimplifies greatly the procedures for economnetric application of the model as demonstratedin an example utiliziing additional Eurodollar market data.

Each of these approaches can provide useful informationi depending on data availability.If one simply observes the risk premium, cost of capital, loan duration, and average lossrate in the event of default, then the lender's subjective probability can be confined to aninterval (0, P*). if, in addition, bounds can be established on the share of the indi% id ual loanof the lender's total loanable funds and on thc lender's risk aversion, tlle sUbjective pro-babilities can be confined to a muchl narrowevr interval. Finally, if information can alter-natively be gained on the economic factors whiich influence a lenider's subjective probability,then regression techniquies can be used to produce point estinmates of lender's suLbjective pro-babilities tiider certain conditions. Results of the latter type show particular promise for -borrowcr in managing credit worthiness since they provide information oni hlow the borrower'sother economic decisions affect its cost of capital.

APPENDIX

Table IA. Tranisactionis itn the Etreodollarmarket: third quarter, 1 97 3a

Risk LoanBorrowing premium durationcountry (per cent) (years)

Algeria 1.008 15Algeria .938 12Algeria .875 10Brazil 1.00 12Brazil .955 10Colombia .75 10Gabon 1.75 10Greece .75 12Iran .688 12

142 Gers/lon Feder and Richard E. Just

Table IA. (cont.)


Iran .625 10Korea 1.205 l lKorea 1.5 10Mexico .659 12Mexico .575 10Nicaragua 1.325 10Peru 1.6 10Philippines 2.0 10Senegal 1.875 10Zaire 1.75 10Zambia 1.425 10

Source: World Bank, 1976.aWhen several premiums were charged indifferent stages of the loan duration period,a weighted average was calculated. Also, whenseveral loans with the same duration wereobserved, a weighted average was calculated.

Table 2A. Transactions in the Eurodollarnarket: January-July, 19 75 a

Risk LoanBorrowing preraium durationcountry (per cent) (years)

Algeria 1.375 7Argentina 2.0 5.5Bolivia 2.179 8Bolivia 2.0 5Brazil 1.75 5Brazil 1.875 7China 1.5 6Colombia 1.75 7Costa Rica 2.0 4.75Ecuador 2.0 5Ecuador 1.5 6El Salvador 1.75 5


Table 2A, (cont.)


Guyana 2.5 5Indonesia 1.725 - 5Ivory Coast 1.875 5Jamaica 1.9 5Korea 2.0 5Malaysia 1.625 5Mexjio 1.5 3Mexico 1.5 5Nicaragua 2.07 5Panama 1.75 5Panama 1.875 7Peru 1.788 7Philippines 1.755 6Spain 1.5 5Spain 1.5 7Sudan 1.75 5Zambia 2.0 5

Source: World Bank, 1976.'When several premiums were charged in dif-ferent stages of the loan duration period, aweighted average was calculated. Also, whenseveral loans with the same duration wereobserved, a weighted average was calculated.

REFERENCES

Arrow, K. J. (1971) Essays in the Theory of Risk-Bearing, Markham Publishing Company, Chicago, Ch. 3.Azzi, C. F. and Cox, J. C. (1976) A Theory and Test of Credit Rationing: Conmment, Anzerican Econo-niic

Review, 66 911-17.Beim, D. 0. (1977) Rescuing the LDCs, Foreign Affairs, 55, 717-31.Feder, G. and Just, R. E. (1977a) An Analysis of Credit Terms in The Eurodollar Market, Euzropcan

Econonmic Review, 9, 221-43.Feder, G. and Just, R. E. (1977b) A Study of Debt Servicing Capacity Applying Logit Analysis, Jornoal

of Developnment Economtics, 4, 25-38.Frank, C. R. and Cline, WV. R. (1971) Measurement 'of Debt Scrvicing Capacity: An Application of

Discriminant Analysis, Jouirnzal of Ilternational Economnics, 1, 327--44.Friedman, I. S. (1977) The Enierginig Role oJ'Private Banks in tlhe Developing Wlorld, Citicorp, New York.Jaffee, D. M. and Mlodigliani, F. (1969) A Theory and Test of Credit Rationing, Anmerican Economzic

Review, 59, 850-72.

144 Gerslhon Feder anid Richlard E. Juist

Jaffee, D. M. and Russel, T. (1976) Imperfect Information, Uncertainty, and Credit Rationing, QuarterlyJournal of Econonmics, 90 651-66.

LDC Debt: The Load Eases, Newsweek, February 27, 1978, p. 70.Smith, V. L. (1972) A Theory and Test of Credit Rationing: Somc Generalizations, Anterican Economnic

Review, 62, 477-83.World Bank (1976) Borrowing in Internzational Capital Markets (1973--197S), Report No. EC-181,

Washington, D.C.

Optimal International BorrowingCapital Allocation and Credit-Worthiness Control*

By Gershon Feder, Washington D. C. and Richard E. Just, Berkeley

"Official borrowers like any others, wish to borrowon the best available terms in the private market . ..After all, thley do command the reserves of thecountry as well as access to domestic capital mar-kets and the central bank, Such1 factors enter intheir balance of paymients management and, moreparticularly, their extev-nal debt and reserve man-agement, They know that their external debt man-agement strategies must be played out in the mar-ket place." (Friedman 1977 a, p. 16)

The determination of appropriate international borrowing policy hasattracted the attention of many economists because economic growth,particularly in developing countries, depends heavily on the inflow offoreign funds. Work on this topic dates back to Adamn Smith; but morerecent contributions have been made by Hamada (1966, 1969), Bard-han (1967), Van Long (1974), Manning (1972), and McCabe and Sibley(1976). A common approach in related economic analyses has involveda "small country" assumption that has been taken to imply an infinitelyelastic supply of loans at a given interest rate. It has been arguedrecently by Hanson (1974), however, that the supply of loans may notbe completely elastic even for the small country because, as debts getlarge, the probability of a debt crisis for the individual country in-creases. That is, as debts get large relative to the country (not themarket), it becomes more likely that the borrower will not be able tomake repayments on schedule; hence, lenders will become more reluc-tant to lend as the expected, discounted returns decline. As indicatedby Mohammed and Saccomanni (1973), there is no doubt that this riskplays a major role in the lending practices of commercial banks. Non-profit-oriented lending institutions are also likely to be concerned withthe probability that reschedulings will be required; hence, their lend-

* The views and suggestions expressed in this paper belong to the authorsalone and do not necessarily reflect those of the institutions with which theyare affiliated.

208 Gershon Feder and Richard E. Just

ing decisions may also be influenced to some degree by their borrowers'credit status (Cline and Sargan (1975)). It thus follows that even smallcountries do not face an infinitely elastic supply of foreign funds;rather, the terms of credit are endogenous depending on credit-worth-iness. Credit-worthiness, in turn, is determined by the countries' eco-nomic performance (usually measured by a few accepted economicindicators).

Since economic-performance indicators are affected by economicpolicies, it is important to explore borrowing policies for optimalgrowth, taking into account the relationship between ecnomic perfor-mance and terms of credit'. Although empirical results suggest thatseveral economic variables are related to debt servicing capacity, onemay accept the use of only a few indicators as a useful simplificationfor theoretical analysis (Feder and Just (1977 a, b)). Hanson has sug-gested the use of the debt-equity ratio for this purpose. F-zwever, inthe one-sector model developed by Hanson, this approach implicitlyleads to the assumption that "resources can always be costlessly allo-cated to obtain the necessary foreign exchange for debt servicing"[11, p. 619, footnote 10]. Hence, the only factor limiting the availabilityof foreign exchange is the overall productive capacity of the countrywhich may be represented by the capital stock.

Once installed, however, capital is not a malleable factor than canbe equally productive in any sector. Although some types of capitalare more flexible than others (transportation equipment, power plants,etc.), it is more likely with many exports, particularly in developingcountries where exports are composed mainly of nonmanufacturedgoods, that investments are highly specialized (mines, dams, etc.). Evenwhen reallocation of capital is possible, the time lag involved may betoo long for lenders to consider the productive capacity of other sour-ces as relevant for the evaluation of export potential in the case ofdebt payment difficulty. Since the export sector is the main sourceof foreign exchange earnings in many countries and capital reallocationmay not be possible, it thus seems that the most appropriate case fordeveloping countries may be where the terms of credit depend specif-ically on the size of the export sector (as well as the debt burden). Thisis consistent with the findings of a recent study regarding lenclingbehavior in the Eurodollar market where two variables related to ex-

I Such policies have been suggested by observers of initernational rnarkets;see Irvine et al. (1970), Friedman (1977 b).

Borrowing, Capital Allocation and Credit-Worthiness Control 209

port performance (the ratio of debt service to export earnings and ameasure of export variability) appear to be considered as relevant bylenders, while the ratio of debt to gross national product (which is aproxy for the debt-equity ratio) was found insignificant (Feder andJust 1977 a).

Another balance of payments item which is of importance in credit-worthiness evaluations is the size of foreign exchange reserves, whichis usually viewed in comparison to the import bill (Feder and Just(1977 a), Goodman (1977)). That rational reserve policies are carried outwith reference to credit worthiness (among other arguments) is evidentfrom Friedman (1977 a, p. 18) who states: "Other developing countries,like Brazil, are borrowing both to finance external deficits and to re-build reserves, profiting from the experience that a strong, visible levelof reserves improves creditworthiness as well as providing an additio-nal cushion to meet contingencies."

The purpose of the present paper is to consider a two-sector modelwhere export and import activities are defined explicitly. The modelrecognizes lenders' concern regarding default on international debtthrough a credit-worthiness dependent supply function of foreignfunds. It is assumed that capital, once allocated between the exportor nonexport sector, cannot be reallocated 2 . Accordingly the credit-worthiness assessment carried out by lenders (and which focuses onthe foreign exchange attributes of the economy) is reflected in an in-terest rate which is positively related to the volume of anticipated debtsorvice payment relative to the magnitude of export earnings. Similarly,the interest rate is negatively related to the size of foreign exchangereserves relative to the volume of import expenditures.

The model presents a characterization of optimal investment andborrowing decisions and discussed their policy implications in a mixedeconomy where both governmental and private decision making coexist.

The Model

The model below describes as small, two-sector open economy. Onesector produces an export good, while the other sector produces a goodwhich can be used for either consumption or investment. Investmcnt

2 This assumption has been adopted in several other optimal growth modelswhere terms of credit and optimal borrowing are not tthe major interest; seeworks by Bose (1968), Chakravarty (1969), Dasgtpta (1969) Jo1hansen (1967),and Ryder (1969).

14 Kredit und Kapital 2/1979


can be augmented wr.h imports of capital goods. The economy has accessto international capital markets, but the terms o' credit it obtainsdepend on its credit-worthiness status. The planning authority isassumed to be interested in maximizing the discounted utility streamsof consumption.

The following notation is used in the analysis. All variables shouldcarry a time subscript, which is omitted for the purpose of brevity.A dot over a variable denotes a rate of change over time.

A. Variables(i) Flow variables

F = A composite good which can be used for both consump-tion and/or investment;

G = Export good (not used domestically);C = Consumption;Id = Capital formation from domestic output;I = Total investment;M = Quantity of imports of capit'l goods;L = Gross loans;U = Utility of consumption,

(ii) Stock variablesK1 , Kg = Stock- of capital in F and G sectors, respectively;B = Outstanding external debt;R = International reserves.

(iii) Other variablessx = The proportion of total investment which is directed to

the F sector;r = Interest rate on external debt.

B. Parameters

i = Rate of interest earned on foreign exchange reserves;= The planner's rate of time discount;

15 = Rate of amortization of external debt;a = Rate of capital depreciation;Pg = International price of the export good;Pm = International price of the capital good;r = Average interest rate as perceived by lenders.

C. Model equations

(1) G G (K); G'> ; GI < °(2) F-F (KI); F'> O; Fg < O


The above equations describe the production teclhnologies of the twosectors, which utilize capital only. Labor is assumed a non-bindingconstraint. For simplicity, labor growth is ignored.

(3) F(K/)=C + Id X

Equation (3) describes the allocation of the F good output betweenconsumption and investment.

(4) r=rCP G-(+r)B,R-PM;r1 [P G( )B]<

rr,'<0; r (co,oc)=i

r92- a3 (R - P" MjT I ;r(O }

Equation (4) defines the endogenous interest rate on external debt asa (declining) function of two indicators of debt servicing capacity:(i) The difference between export foreign exchange earnings and theaverage volume of debt service payments (the latter being approxim-ated by some perceived average rate of interest T, which lenders expectto prevail on the average); and, (ii) The difference between internationalreserves held by the economy and the import bill.

These two indicators correspond to two concepts widely used in creditworthiness assessment by lenders in international capital markets:(i) the debt service pressure measure which compares the major sourceof foreign exchange (exports) to the most rigid expenditure item in thebalance of payment (debt service); and, (ii) the import-coveragemeasure which compares the size of reserves to the volume of imports,thereby reflecting the extent to which imports can be financed byreserve depletion if, for some unexpected reason, foreign currency in-flows (exports and new loans) are not available.

While in most cases these measures are expressed as ratios (ratherthan differences), it is more convenient to follow the present formu-lation. The economic essence of the analysis is unaffected3 . As bothcommon sense and empirical evidence suggest, the interest rate will be

3 Evidence on the prevalence of debt service and import coverage measuresin credit worthiness analysis is available in many studies. See for instanceGoodman (1977) who reports the results of a survey conducted amongAmerican banks engaging in international lending. Similar evidence is pro-vided in Feder and Jutst (1977 a) who analysed factors affecting credit termsin the Eurodollar market. See also the articles by Friedman (1977 a), Bracken-ridge (1977), Puz (1977) and van-A gtnwac (1976), all of whom are seniorexecutives in international banks.

14'


lower for borrowers with higher exports (relative to debt service) andhigher reserves (relative to imports), thus the partial derivatives rland r2 are negative. Also, for a risk-free borrower (such that botharguments of the r function tend to infinity), the rate of interest willapproach the going rate on international deposits (i).

(5) U = U (C) ; UJ' > 0 ; TU" < O ; U' (O)-=oo.

The utility of consumption has the standard properties of utilityfunctions, namely, positive but decreasing marginal utility, and infinitemarginal utility at C = 0.

(6) I-Id +M.

Total investment is composed of domestic capital formation augmen-ted by imports of capital goods. While the present model assumes thatimports are composed of capital goods only, one can show that iden-tical results may be derived from a model where imports are composedof consumption goods.

(7) R-(Pg G + L + iR)-[P. M+(r+t9)B].

The difference between foreign exchange inflows (composed of ex-ports, loans and interest earnings on reserves) and foreign exchangeoutflows (composed of imports and debt service payments) equals thechange in international reserve holdings.

(8) K, = es -d K.

MS kg = (I1-a) I - 6 KD

Equations (8) and (9) present the rate of change (over time) of capitalstocks in the F and G sectors (i. e., net investments) as the differencebetween the new investments allocated to each sector and the volume ofcapital depreciation 4 .

(10) B=L-O B

The change in indebtedness is equal to the 0 :Ierence between grossloans and amortization payments.

4 The proportion of investment allocated to each sector should be con-strained between zero and one. It will be assumed that this condition holdsat the relevant range.


The planner's objective is maximization of discounted utility of con-sumption. Assuming an infinite horizon the problem can be formu-lated as

(1)Max fU(C) dt.a, M,L,C f0

subject to initial stock values and equations (1) - (10).

Problem (11) can be solved using the Pontryagin Maximum Prin-ciple. Denote the present value Hamiltonian byH and define the constantvariables A, It, r, and 9, which are dynamic shadow prices of the stockvariables K,, KG,) B and R, respectively. Then, using the relation I=F (K)) - C + M (from (3) and (5)), one can write

(12) H. e-e t = U (C) + {o( [F (K) -C + MI-( K1}

+ u {((1 - ) [F (K!) - C + M] - & K,} + 77 (L - i B)

+ p {[Pg G (Kg) + L + iR] - [Pm M + (r + 2) B]}

It should be noted that the non-negativity requirement for C is notspecified since the assumption U' (0) = oo eliminates such a solution.The non-negativity of M and L, and the requirement 0 < a < 1 areassumed to be non-effective constraints at the range of the optimalpath described in the present paper. It thus follows that the optimalityconditions to be derived below are applicable only at a certain rangeof the optimal path (including the steady state). Other portions of theoptimal path include corner solutions (e. g., zero investment in onesector or the other, zero borrowing, etc.) and seem to us to be less in-teresting from a practical point of view. Similarly, existence and uni-queness of the solution depend on additional constraints which are notspelled out.

The optimality conditions require (in addition to appropriate trans-verality conditions), at the range where all controls are positive;

(13) H U' -o a) U - (1 -a)y z40

(14) aa ^=O eoa~ + (Ie)- Pm [1 -r2 B]-O

(15) H8g = ° t4 (f- M A,s

aH(16) a -0 4-1+9 =°(16) L


(17) 2-1 K = 2 (( + 6 )- [x + (1-a*)4 U ] F' (K1 )

(18) sz - '14 = - a # X G=(' +K ) -q*paG (I - r B)

(19) H (? +O)+p[r+ - r,( +r) B]

(20) i + [i±r-B]

Analysis of the Results

The economic interpretation of the optimality conditions is fairlystraightforward:

Equation (13) implies that the marginal benefit of consumption (i. e.,marginal utility) equals its opportunity cost [a-A + (1 - a,) It], which isthe marginal value of investment foregone. The value of marginal in-vestment is a weighted average of the value of capital in both pro-duction sectors. As will become apparent, the value of a unit of capitalin both sectors is equal, thus the implication is that the marginal utilityof consumption equals the value of a unit of capital whiclh is foregone.

Equation (14) states that the value of a unit of investment [oA +(1 - a) /t] (afforded by one additional unit of imports of capital goods)equals its opportunity cost. The latter is the amount of foreign ex-change (withdrawn from reserve stocks), adjusted for the deteriorationin credit terms caused by the decline in import coverage by reserves.We note that if imports had no effect on the term of credit (i. e., r2 C 0)then the value of investment equals the shadow price of foreign ex-change needed to implement that investment.

Equation (15) implies that value of capital in each of the two pro-duction sectors (that is, the export sector and the investment-consump-tion sector) is equal, (i. e., A =- I) even though capital is not transferablebetween the sectors once investments are implemented (except in thecase F (K) - C + M = 0, which implies no investment in the economy).Given A = Iu, it follows that the value of capital is higher than the valueof the direct foreign exchange cost of a unit of capital imports. Thisis explained (as indicated above) by the additional cost, in terms ofreduced credit worthiness, of imports.

Equation (16) implies that the value of a unit of reserves (or theshadow cost of foreign exchange) equials its opportunity cost (the


reduction of external debt). The value of a unit debt (77) is obviouslynegative, as the value of reserves (p9) is positive.

Using the result A = = p P, (1 - r2 B) (which also implies X- =it

equations (17) and (18) can be combined to yield

(21) Pil F1 -rlB ) -

Equation (21) implies that in general the value of marginal productof capital in the two production sectors (evaluated at internationalprices) will not be identical. Obviously, in the case where no creditworthiness considerations are taken into account (i. e., ri = r2 = 0), themarginal productivity of capital would be maintained equal in allsectors. Indeed. in an economy where most investments are made byprivate (and competitive) investors, one would expect them to operatesuch that Pm F' = Pg G', since each one of them is too small to con-sider the national credit-worthiness implications of imports and exports.The planner (or the government), however, caniiot ignore the impact ofsuch variables as represented by the right hand side of equation (21).Thus, intervention in the sectoral allocation of investment is called forby export promotion and/or import substitution. Export promotion isreflected in the term - rl B which measures (in units of output) theamount of subsidy that should be given to producers of the export good.Similarly, - r2 B measures (in physical units) the subsidy due to pro-ducers of the F good (which can substitute imports of capital goods).Private investors will then equate (at the margin) the actual rates ofreturn received by them, namely

(22) PU G' I [- r, B] = P,1 F. [1 -To B],

which will ensure that the economy maintains the optimal conditiongiven by equation (21). At the steady state, however, the optimal con-

dition for the F sector requires (from equation (17) with 2 = 0),

(23) F'=6+, .

Whether the export sector is (at the margin) more profitable than theimport substituting sector depends on whether rl/r2 _ 1. The size ofri/r 2 is affected by the differential weight attached to the differentcredit-worthiness indicators and is thus an empirical question.

It should be noted that in practice, many governments find it moreconvenient to control the volume of imports (by tariffs or quotas), so as


to avoid foreign exchange crises which will deteriorate their creditstatus. In terms of the present model, the optimal tariff on imports is- r2 B. Imposing such an import tax will eliminate the need to sub-sidize the F sector.

Using the equality i? =-, which implies ~r/99 = /77, equations (19)and (20) can be combined to yield

(24) r-r 1 (O +±)B=i-r 2 B

The left hand side of equation (24) is the marginal cost of borrowingto the economy, which is necessarily higher than the rate of interest r,since it includes the extra cost due to the deterioration in terms ofcredit brought about by higher debt (the latter effect is represented bythe term - r - (2 + r) * B). The right hand side represents the marginalbenefit from reserve holdings, which is composed of the direct rate ofinterest plus the contribution to credit worthiness gained by additionalreserves (given by - r2 B). Equation (24) thus describes the optimalityrelation for reserve and debt stocks. Obviously, in the present model,if reserves had no impact on credit worthiness (ro =( 0), then a zeroreserves level should be maintained. This can be seen by observingthat with r2 =n0, it is impossible for equation (24) to hold (since r -r

(O + r) B > r > i for all values of the arguments of r, as indicated in(4)). To approach the equality, R will be depleted (and B reduced by anequal amount), which verifies the assertion made above.

In an economy where part of the external borrowing is done byprivate corporations, the model points out the necessity of imposing atax on foreign borrowing. The reason for such a tax is that while thecost of borrowing for each individual borrower is r (the rate of interest)the cost to the economy is the marginal borrowing cost which includesthe terms of credit effect. The presence of credit worthiness consider-ations thus implies an externality, which may be corrected by an appro-priate tax. Obviously, direct measures can also be applied. For instance,recently, the government of Turkey banned foreign loans which areobtained at a rate of interest higher than a certain upper limit (Clarke,1977). Other governments imposed administrative restrictions whichamount to an implicit tax on borrowing from foreign sources.

Comparisons between the marginal rate of return and the externalinterest are possible, in the nwesent model, only at the steady state.Thus, setting L = R= 0 and using the results A = ji, - 7= p, one ob-tains from (17) and (19):


(25) F'-6=r-rj (O f)B .

Equation (25) implies that at the steady-state, the marginal produc-tivity of capital in the import-substituting sector (net of capital de-preclation allowance) should be maintained at a level higher then therate of interest of external debt. The difference is equal to the credit-worthiness effect of marginal borrowing. When credit-worthiness con-siderations are negligible (r0m 0), the standard result is obtained,namely, the net marginal productivity of capital equals the interestrate.

The relation between the marginal rate of return of capital in theexport sectvr and the rate of interest is less straightforward. Setting

= 1 = 0, and using the relation - = = [/[Pm, (1 -r2 B)], oneobtains from (18) and (19), at the steady state,

(26) P0 G' (1 -,r1 B)(26) P'm (r - = r - rP (r + () .B.

Since the magnitude of the various terms on the left hand side is notknown a-priori, one can not conclude as to whether the rate of returnto capital should optimally be higher or lower then the direct rate ofinterest. We note, however, that the left hand side describes the fullimpact of the marginal investment in the export sector: the direct con-tribution to export plus the improvement in credit terms induced byhigher export [Pg G' (1 - ri B)] discounted by the total cost of a unitof capital imports (that is, both direct and indirect cost). Equation (26)thus states that the full marginal contribution of investment in the ex-port sector should equal the marginal cost of borrowing.

Setting 2 = 0 also implies (by equation (19)) that at the steady statethe rate of time preference of the economy (e) exceeds the interest rateon external debt, as it needs to equal the marginal cost of borrowing.

From equation (17) with 2 = 0, one obtains the familiar "golden rule"of neo-classical models, asserting that at the steady state capital accu-mulation is maintained at a level such that the net marginal produc-tivity equals the rate of time preference (if labor growth is nil). Thisrule applies, in tle present model, only to the import substitutingsector. For the export sector the optimal steady state accumulation lawis more complicated and depends on credit-worthiness effects.


Conclusion

By way of summary, the model implies the following set of policiesfor an economy with private borrowing, investment and production:

(i) Export promotion through a subsidy (of magnitude - Pg rj * B) tothe export sector;

(ii) Tax on capital goods imports (of magnitude - PI, r2 B), whichamounts to a policy of import substitution;

(iii) Tax on foreign borrowing by private corporations (of magnituderj (7 + r) B).

These recommendations are, of course, of a partial nature, due to thesimplistic structure of the model. But the paper makes the generalargument that development policies should take into account the re-lationship between economic policies and the supply of foreign funds,and it demonstrates that such considerations require central govern-ment intervention in the operation of the market. This conclusion ismost relevant for developing economies who depend on commercialinternational capital markets for a substantial portion of their foreignexchange inflow.

References

[1] Bardhan, P. K.: "Optimum Foreign Borrowing", in K. Shell (ed.), Essayson the Theory of Optimal Economic Growth. Cambridge: MIT Press. 1967.pp. 117 - 128.- [2] Bose, S.: "Optimal Growth and Investment Allocation",Review of Econonmic Studies, Vol. 35. October 1968. pp. 465 - 480. - [3] Brackc-enridge, A. B.: "Evaluating Country Credits", Institutional Investors. June1977. pp. 13 - 16. - [4] Chakravarty, S.: Capital and Development Planning.Cambridge: MIT Press. 1969. - [5] Clarke, P.: "Will the Banks and theFund Make an Example of Turkey?" Euromoney. September 1977. pp. 16 - 23.- [6] Cline, W. R. and N. P. Sargan: "Performance Criteria and MultilateralAid Allocation". World Development. Vol. 3. June 1975. pp. 383 - 391. - [7]Dasgupta, P. S.: "Optimum Growth When Capital Is Nontransferable".Review of Economic Studies. Vol. 36. January 1969. pp. 145 - 154. - [8] Feder,G. and R. E. Just: "An Analysis of Credit Terms in the Eurodollar Market".Europear Economic Review. Vol. 9. May 1977 a. pp. 221 - 243. - [9] Feder, G.and R. E. Just: "'A Study of Debt Servicing Capacity Applying Logit Ana-lysis". Journal of Development Economics. Vol. 4. March 1977 b. pp. 25 - 39. -[10] Feder, G. and U. Regev: "International Loans, Direct Foreign Invest-ment, and Optimal Capital Accumulation". The Economic Record. Vol. 51.September 1975. pp. 320 - 325. - [11] Friedman, I. S.: "Most EurocurrencyBorrowers Remain Good Risks". Eurornoney. March 1977 a. pp. 16 - 21. -


[12] Friedman, I. S.: The Emerging Role of Private Banks in the DevelopingWorld. New York: Citicorp. 1977. - [13] Goodman, S.: "How the Big U. S.Banks Really Evaluate Sovereign Risks". Euromoney. February 1977.- [14]Hamada, K.: "Economic Growth and Long-Term International Capital Move-ments". Yale Economic Essays, Vol. 6. Spring 1966. pp. 49 - 96. - [15] Hamada,K.: "Optimal Capital Accumulation by an Economic Facing an InternationalCapital Market". Journal of Political Economy. Vol. 77. July/August 1969.pp. 684 - 697, - [16] Hanson, J. A.: "Optimal International Borrowing andLending". American Economic Review. Vol. 64. September 1974. pp. 616 - 630.- [17] Irvine, R. J.; Y. Maroni and H. F. Lee: "How to Borrow Successfully".Columbia Journal of World Business. Vol. 5. 1970. pp. 42 - 48. - [18] Johansen,L.: "Some Theoretical Properties of a Two-Sector Model of Optimal Growth".Review of Economic Studies. Vol. 34. January 1967. pp. 125 - 141. - [19] Man-ning, R.: "Optimal Taxation When Borrowing Alters the Foreign SupplyCurve for Capital". The Economic Record. Vol. 48. September 1972. pp. 400 -410.- [20] McCabe, J. L. and D. S. Sibley: "Optimal Foreign Debt Accumu-lation With Export Revenue Uncertainty". International Economic Review.Vol. 17. October 1976. pp. 675 - 686. - [21] Mohammed, A. F. and F. Sacco-manni: "Short Term Banking and Euro Currency Credits to DevelopingCountries". IMF Staff Papers. Vol. 20, November 1973. pp. 612 - 638. - [22]Puz, R.: "How to Find Out When a Sovereign Borrower Slips from A-1to C-3." Euromoney. December 1977. pp. 65 - 68. - [23] Ruder, H. E.: "Opti-mal Accumulation in a Two-Sector Neoclassical Economy with Nonshift-able Capital". Journal of Political Economy. Vol. 77. July/August 1969.pp. 665 - 683. - [24] van Agtmael, A. W.: "Evaluating the Risks of Lend-ing to Developing Countries". Euromoney. April 1976. pp. 16 - 30. - [25] varXLong, N.: "International Borrowing for Resource Extraction". InternationalEconomic Review. Vol. 15. February 1974. pp. 168 - 183.

Zusammenfassung

Optimale internationale SchuldenaufnahmeKapitalallokation und Kreditviirdigkeitskontrolle

In diesem Beitrag wird ein Zweisektorenmodell einer wachsenden Wirt-schaft einschlief3lich des Verhaltnisses zNvischen der Kapazitat des Schulden-dienstes und den Kreditbedingungen eines Landes entwickelt. Diese Bedin-gungen werden durch makro8konomische Variablen beeinflulBt, wie z. B,durch das Ein- und Ausfuhrvolumen, die Hbhe der Devisenreserven, dieHohe der Schuldendienstzahlungen entsprechend der bestehenden Schuld. DieTatsache, daB Importeure, Exporteure und Hersteller oftmals zu unbedeu-tend sein k6nnon, um die Auswirkungen ihrer Handlungsweisen auf dieKreditbedingungen der Wirtschaft in Rechnung zu stellen, schafft Diskre-panzen zwischen den optimalen privaten und offentlichen Kreditl6sungen.Das Modell kommt daher zu dem Ergebnis, daB optimales Wachstum Export-f6rderung verlangt, wahrend gleichzeitig sowohl die private Schuldenauf-nahme von Auslandsgeldern als auch der Import von Kapitalgiutern be-steuert wird.


Summary

Optimal International Borrowing, Capital Allocationand Credit-Worthiness Control

The paper develops a two-sector model of a growing economy, incorporat-ing the relation between debt servicing capacity and the terms of creditfacing the country. These terms are affected by macro-economic variablessuch as the volume of imports and exports, the size of foreign exchangereserves and the magnitude of debt service payments due on outstandingdebt. The fact that importers, exporters and producers may be too small totake account of the impact their actions have on the economy's terms ofcredit creates discrepancies between private and public optimal solutions.The model suggests that optimal growth may require export promotionwhile simultaneously taxing both privr'e borrowing of foreign funds andimportation of capital goods.

Resum6

Endettement international optimal, allocation de capitalet contr6le de la solvabilite

La presente etude developpe un mod6le bisectoriel d'econoanie en expansion,incluant la relation entre la capacite d'endettement et les conditions decredit d'un pays. Ces conditions sont influencees par des variables macro-6conomiques telles que le volume des importations et des exportations, lestock des reserves de devises, le montant des paiements du service de ladette correspondant a l'endettement en cours. Le fait que les importateurs,les exportateurs et les producteurs n'ont frequemment pas la taille requisepour influer par leur maniere d'agir sur les conditions du credit de l'eco-nomie provoque des disparit6s entre les endettements optimaux prive etpublic. Le modele conclut que la croissance optimale requiert la promotiondes exportations et parallelement la taxation des emprunts prives en devises-ftrangeres et de l'importation de biens d'investissement.

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World Bank Reprint Series: Nunrber 159

Gershon Feder and Richard E. Just

A Model for AnalyzingLenders' Perceived Risk

Optimal International Borrowing,Capital Allocation,and Credit Worthiness Control

Reprinted with permission from Applied Econiomics, vol. 12 (1980), pp. 125-44; andKredit and Kapital, vol. 12, no. 2 (1979), pp. 207-20.

Applied Ecoionomics. 1980, 12, 125--144

A model for analysing lenders' perceivedrisk

GERSHON FEDER and RICHIARD E. JUST'

Development Researchi Centtre, W18orld Bank, Brighlam Yotunig University, and University ofCaliforniia, Berkeley, U.S.A.

Bank lending involves, in most cases, the risk that the borrower will not be able or williligto honour his obligations. The existence of default risk is an important factor in explainingthe observed behaviour of lenders as demonstrated in the works of Jaffee and Modi-liani(1969), Smith (1972), Azzi and Cox (1976), Jaffee and Rus7el (1976), and others. Lenders'behaviour in this case depends crucially on their subjective evaluation of the probability ofdefault. Tlhus, to explain lending behaviour, knowledge of lenders' subjective informiiation isgenerally unobservable; and empirical analysis of lending behaviotur is, therefore, difficult.Furthennore, there is often reason to believe that subjective information may vary con-siderably from lender to lender or from transaction to transaction because of previousexperience, personal relationships, etc.; and, hence, the role of subjective perceptions cannotbe ignored.

The purpose of this paper is to develop a model which facilitates inference about lenders'subjective default probabilities. Several alternative approaches of constructing both pointand interval estimates of subjective default probabilities are proposed under various as-sumptions about competition, loss rate distribution, risk aversion, and relative loan size.

The assumption which makes these approaches possible is that lending takes place w,ith afixed probability of default. That is, lending transactions are assumed to be of sufficienltly neg-ligible size relative to the borrower's scale of operations (e.g. lending to sovereign borrowers)so that the probability of defaolt is not influenced by the lender's current decision, i.e. theinterest rate on the loan does not affect default probability. This assumption differs fromthat employed in the theoretical papers cited above and, hence, renmoves complicationsassociated with endogeni7ing tde probability of default. Nevertlieless, the simpliFicatioiwhiclh leads to the empirical possibilities developed in this paper is plausible in manysituations -particularly in international lending. For example, suippose the state of SdoPaulo, Brazil, borrows $100 million in the Eurodlollar market with the guarantee of the

1 Giannini Foundation Paper No. 523. The views expressed here are the authors'; they do not representthose of the World Bark. Estimates used in the text are provided from Euromarket data and are nmeantsolely to be ilustrative, There is no presumption that these estimates are the actual probabilities nor thatthe Bank in any way concurs with these perceptions.

0003-6846/U//020125-20 $04.00/0 i 1980 Chapman and Hall Ltd. 125

126 Gerslhon Feder and Richard E. Justtederal government of Brazil. The likelithood of default in this case would surely be affectedonly slightly by the lender's decision since the overall debt of Brazil is $32 billion dollars(Newswcek, 1978).

I. THE MODEL

Consider a simple one-period model similar to the one presented by Smithi (1972). Thelender is assumed to have a given amount, W4, of loanable funds, all or part of which can belent to the particular borrower under consideration at interest rate P. The interest rate isexogenously given if the lender operates within a competitive capital market, or it dependson the volumec of loans granted (according to the borrower's demand schedule) if the lenderhas some degree of monopoly power. Alternatively, part or all of the lender's funds can belent at interest p which is the opportunity cost of capital and which is considered risk free. 2The borrower's demand schedule is given by the relation,

f = f(L), P' < 0 in the monopolist's casef' = 0 for the competitive lender, (1)

where L denotes the volume of loans granted. It will be useful to define the risk premium,r, which is the difference between the lender's opportLnity cost of capital and the interestrate actually paid by the borrower, namely,

P-r+p. (2)Using Equation 1 and considering p as a constant, the demand schedule can be defined

for the risk premium r,

r = r(L), r' P'f < 0, (3)where, as before, r' = 0 applies to the case of the coompet7"ve lender.

The borrower is assumed to possess (at the time of loan negotiations) assets totalling Kwhich were financed in part by previous borrowing and in part from the borrower's ownresources. IHis total outstanding debt as of the time of the loan under consideration is givenby D. Following Smuith (1972), it is assumed that the liquidity situation of the borrower atthe time of loan maturity is related to borrower resources (including the new loan) by astochastic, linear homnogenous function of the form y x [K + L], where y is a continuousrandom variable with probability density function f(y) defined over the interval (0, m). Theparameters of f are inflLenced by various borrower characteristics as viewed by the lender;and f(-y) reflects, therefore, the latter's subjective assessment of borrower prospects.

Whenl the loan under consideration becomes due, the borrower ImlUst pay (I + f) x L inadditioni to other obligations which will be due on his older oLutstanding debt D, say 0 x L)where 0 denotes the rate of interest and ainortizatioin paymnents; p is not affrected by thenew lender.

"As in the earlier works on lending under uncertainty cited above, portfolio considerations are ignored forthe sake of simplicity.

A model for analysJng lencders' perceived risk 127

Obvioxisly, if -y will be sufflcicntl,;i .Vxgh so as to maintain the inequality

y x (K+L)-(I+fr)x L-OxD>0, (4)

then all creditors will be paid in full and no default takes place. Using Equation 4, one candefine the critical y value, (say, y*)

(1 O+ f) x L + 0 x D(5K +K L (+)

If y < y*, not all obligations can be met, and a partial default takes place. In such a case,it is assumed that creditors are paid according to the proportion of debt they hold.3 Inparticular, the lender under consi deration will be paid

7yx (K+L)x (L+D)

The presc.nt value of the new lender's wealth (say, y) is thus given by

W-L + (+ x L if Y > 'y*1 + pY=I

W-L+ (l+ )K (L+D) ifY <-Y. (6a)

Rearranging the terms in Equation 6, present value y can be rewritten as

fw +rxOxL ifY >Y*' ww-hxL if <'Y* (6b)

where

01+ p

K+Lh- 1 - x (LTD)x (1+p)

and

r= risk premium as defined in Equation 2.

Note that hi is random since it is a transformation of the random variable -Y. From the formu-lation in Equation 7, it follows that, for a giveni value of Y, h is the rate of loss incurred bythe lenlder. The range of values over which Ih is defined is (0, *). Thus, using the definitionof y* as giveni in Equation 5, It is confined between upper and lower bounds I and h,respectively, where

h=- +p) x D -(+) x L l(L +D)

31t is implicitly assumed that no priority debt exists. Howcever, one could assume that priority debt is paidfirst and that the other creditors share the remaining liquidity according to their shiare in regular debt. Theappropriatc analysis, lhowever, remains unchanged.

128 Gershon Feder and Riclatrd E. JuistDenote the lender's Lutility function by U, where U is def'ined over present value of wealth.

The uttility function is assumed to have the properties

(f> o, VI< o, (7)where U" < 0 implies risk aversion and U" = 0 reflects risk neutrality. It is furthler assumedthat, in the case U1' < 0, relative risk aversion .s no greater than one, i.e.,

-("/U) x y < I .

This is a plausible assumption as argued by Arrow (1971, p. 98). The lcender's objective isto maximize the expected utility of present-value wealth by an optimal choice of loan size,i.e. using Equation 6b

Max 11f U(JV+ rx 0 x L) x f(y) d7+ | U(W- h x L)fty) d-y. (8)L fa

Note, however, that

f U(W+ r x O x L)f(y) dy = U(W/+ r x 0 x L) x [ 1-P(y*)],

where

P(y*)- =f f(y) d-yfo

represents the probability of default. Note, also, using the definitions of hz and h1, that

f0 U(W - h x L)f(yf) dy =f U(1' - h x L) x p(hl) dli

where

p(h).=-f [( - h)(1 + p) x (+ L) ]

The objective function in Equation 8 can thus be written as

Max a = [1 P(,y*)I x U(JV + r x O x L) + U(Ji'- hL) x p(li) dhl (9)

In the general case, the probability of default P(y*), as well as the specific loss rate densityp(h), depends on the size of loan granted by the current lenider and oln the rate of interestcharged for tlle loani (as inspections of y* and hi veriFY). lowever, as will be shownv belowv, inthe case where the relevant loan size is small, relafive to the size of the borrower's assets (i.e.wlhen LIK is small), tlhe probability of default and the distribution of loss rates can betreated as given (from) the point of view of tlle cLirrenit lender). This simplynlifies the analysisconsiderably and facilitates several applications as, will be demonstrated in thie followig,sectionis.

In order to verify the assertions regarding P(y*) and p(hl), define the loan to asset ratioi (i.e. i LIK). Rewriting the definitionis of y*, h, and h by simply dividing numerators and

A model for analysinig lentders' percei'Led risk 1 29

denominators by K, one obtains

- (I + ;) x 1+ x (DIK)1+1

h-- - x - (I i)+( 1 + p)

and

k-frl¢ D\ [r x 1/(1 + p)]L tl +p)j U ( K) [1+ (D/K)] |

Taking the limit as i - 0 yields

D7*t= 0 x DfX= K'

h=i=o [(1 +p) x (D/K)]

I 1-¢h =

Thus, when I is sufficiently sm-nall, y* and h approach fixed parameters from the currentlender's point of view, and the (random) loss rate /l is independenit of the size of the cuLrrentloan (L) and of the risk premium (r) charged on the loan.

These results imply that, with the assumption of negligible 1, the probability of defaultP(y*) and the density of the loss rate p(h) are not appreciably affected by the actions of thecurrent lender and can be treated as fixed parameters in the lender's problem. Rather, thedominant factors underlying these probabilities are the existing economic prospects facingthe borrower.

Before proceeding to characterize the optimal solution, one fturther modification needsto be made-namely, extending the model to a loan duration of N years. Assullming that.at most, one default takes place within a loan's duration and that loans are repaid in 'balloon'form (interest payments paid annually and the principal repaid in the last year), the formu-lation of the objective function (Equation 9) can be retained provided that 0 is redefined as

0 I I -( O+p) , I/P

(note that one can show that

I=IxL x(l +p)-f +L x(1+ p)-N-L rx0 xL),and h is reinterpreted as a discountited rate of loss. This is possible since botht p and N areviewed as exogenous by the lender wvitlh the former paranmeter dletermiineed by mlarket oppor-tunities and the latter being dictated to a large extent by the lcnder's anld borrower's overallliquidity projections (Beim, 1977, p. 723).

For notationial convenience in the following analysis, define

P=P(y*),yI=W+rx0 xL, andyh=IV--I/t x L.

130 Gershon Feder andt Riclhardt E. JuistDifferentiation (E!quation 9) with respect to L, the following first-order condition clharac-

terizes the optimal solution:

a(I -P) x O x U'(Cy ) x (r + Lr') -I U'('h) x I X p(hl) d = 0. (10)

The second-order condition requires

aL2 11 (I P) x U"(y 1 )02 (r + Lr')2 +f UJ( 2 (hd

+ (I - P) x 0 x U'(v) x (2r'+ Lr") < 0. (11)

From Equation 11, it is obvious that risk aversion is a necessary and sufficient conditionfor

a21n/aL2 <0

in the case of a competitive lender (r' = r" = 0). In the case of a risk neutral monopolisticlender,

(2r' + Lr") < 9)

is necessary and sufficient to guarantee concavity of the objective function. With risk aversionand monopoly, the latter condition is not necessary but remains sufficient. fer.-eforth, itwill be assumed that second-order conditionis hold for the lender uinder consideration.

11. COMPARATIVE STATIC RESULTS

Before proceeding to demonstrate the wide range of empirical uses of this model, it is firstuseful to exanmine the various comiiparative static properties of the model for the purpose ofshowing that the model is indeed plausible.

Demand elasticity

Considerinig the monopolistic lender, Equation 10 imtiplies that thze lender operates onl thenoneclastic portionz of tlze borrower's demand schtedule. To see this, note that r + Lr' =r x (1 - t), where 17 is the absolute value of demand elasticity at the optimal point. Obviously,if 17 > 1, Equiation 10 cannot hold since the left-hand side is negative.

Default probability

A borrower who is more risky (i.e., who carries a hiighier prolbability of defauilt) wvill begranzted less credit whlethier the lender is comn petiti'e or tiot. This can be colnirmled byassuming tlhat, forany given rate of loss l, the relative likelihood is at least as great as before,d p(h) > 0 for I <SI < 1.

Hence, uising the fact that

-Pe = Ih p(h ) dhi,


one finds that

fh dp(h)>0

if dP > 0. By differentiating Equiation 10, one then obtains

x x ] O f[u( )x(r+L) +Ut(yh)xl] d p(h)- x(r+ r' +U'yj dp 1 < 0, (12)

where tne sign is established using Equation I I and the earlier result that r + Lr' > 0 for botha r. .r-polist and a competitive lender. The result in Equatioin 12 implies, in the case of amonopolist, that riskier borrowers are clharged a higlher risk premium, This is also the casein a competitive nmarket since, with a highler probability of default, a smaller amount ofloans will be offered by any individual lender in the market at any given interest rate. Theaggregate supply of loans to the borrower under consideration will thus declinie. With anegatively slo,ped demand for loans and a positively sloped aggregate supply, a decline insupply must yield a higher equilibrium level of risk premium (witlh a lower amount of loansin a competitive market is indeed positively sloped (i.e., that sutpply is increasing with higherinterest rates). This is done by differentiating Equation 10:

dL _ ____ -

dr (2 ) (- P) X 0 X U (Y 1) x [1 + U()rOL] (13)dr L I V ) I

Under the assumption that relative risk aversion is nlo greater thani one, the terili in s(quarebrackets on the ri-lt-hand side of Equation 13 is positive; and it is thtls coniluded that

dL/dr > 0.

It should be noted in the case where the lender considers the loan to have an impact on theprobability of default, lhowever, that the offer curve by each individual lender is backwardbending irrespective of attitudes toward risk as shIowII in Jaffee and Modi-liani (1969),Smith (1972), and Azzi and Cox (1976). The different results in the present anialysis aredue to the fact that a higlher interest rate increases the marginal expected utility while, inthe earlier models, a higher interest rate may reduce expected utility by causing a highler pro-bability of default.

Risk aversion

Coiisideribig the role of risk aversion in the model, it is intuitively expecte(d thiat, whenleniders are more risk averse, the volumte of loanis will be lowver anid risk premiums will behiighler. This cani be shown simply by assuming a specific formii of the utility function sucIthat risk aversion is reflected in a single paramieter. The two most comtmlon sLuchi utilityftunctions are the constant Iclat ive risk-aversion family of functions anad the constant absoluterisk-aversion family of f unctionis. Thliese are given, respectively, by

U(y) =ay1 11, 0 <o < I (14a)

U(y) = a - e Y,c> 0 (14b)

132 Gershl&e -eder antd Riclhairld E. Justwhere a is a parameter of risk aversion such that the higher a is associated witlh higher riskaversion. Assuming the case of Equation 14a, differentiation of Eq luation 10 obtains

dL [a 2 f 1 ( I ' )1x [ - P)x x(r+Lr')

x fhx (IV -- hL)W+IOL 'I x dli (I5a)

Since

n [ (IV- hL) 1< 0,L(W + rOL)J

it follows that

dcL

Similarly, assuming the case of Equation 14b and differentiating Eq(uLlation 10 yieldsdl, [ 2 f 1 1~E 11 1 (5b

= - aLL2 ] x x p(l) x e (rY-hL) X (h + rO) x L x dlzj < O. (I5b)The resuLlts in Equationis I5a and b imply thlat a highier clegree of risk aversioll causes a smallervolume of loans and higher risk premiums in botlh cornpetitive anxd noncom petitikc mlaikets.

III. SUGGESTED APPLICATIONS

The results in the previotus section are plausible and intuitive and imply tliat the equilibriuimrelation in Equationi 10 can be usefully considered for fulrther ex positional anid empiricalpurposes, In this section several SuIci empirical applications are demiionistrated. For thispurpose, suppose that lenders' intormation and eUalu1tion distillgluislh between the overallprobabdity that a default will take place and the (conditionial) probability that a given rateof loss (h) will be incurred. Hence, for conceptual purposes, and so thlat subjective defaultprobability can be nieoningfully' discLsSLd, suppose that the&probability density Function ofloss rate hi is proportional to the overall default probability suici that the condttitionzal pro-bability of h delpenids oni h alone and not on the econoniic factors whiich explain the overallprobability. This implies

p(h) ) (h) dh = 1, (16)

wlhere 4' depends Only on II irrespective of the ec(uion 1ic factois which determiln /'. Notc,that, withi tllis assinwftioli, the (un11conditional) probability of any given loss rate increaiseswhen the overall default probability increases, as onei would expect, since [quation 16imlplies

p(h) = i("l) x P;


thus,

dP(') = i(h) > 0.dP

Applying Equation 16 in Equation 10 and rearranging yields

P hr= 1 ) (I -P) -R (17)

where

f- i t(h)h dh

and

fhx i(h) x U'(yh) dh

hx U'(y,)

For a monopolistic lender, Equation 17 demonstrates that the risk premium includes ademand elasticity factor, (1 - 77)- ; the odds of default, P/(1 - P); a time-effect factor,0 -1; an average loss-rate factor Ti; and a risk-aversion factor, R. The latter factor is idcrnticallyequal to one tinder risk neutrality in which case

U(Yh ) = U' 1 )-With risk aversion, it is easy to show that

R > I and aRIaL > 0.

Eq'uation 17 offers a great deal of flexibility in empirical inference of subjective defaultprobabilities because of its simplicity. This flexibility is demnonstrated in tlle following threeapplications.

Threshold defatult probability

A simple application of the model is related to the concept of 'threshold probability' forcompetitive lenders. The threshold probability is the highest value of default probability(say, P*) for which a loan will be granted to any borrower and depends on r, 0, and Iti.Borrowers with probability lower than P* will be given credit with the amount of creditincreasing in the ratio

[P*/(l - P*)][P/(l -P)]

Using Equation 17, the threslhold probability is calculated by noting thiat, at P = P*it must hold tllat L = 0 whiclh inmplies that

Ut lI ) = U (Yh ) = U'(IV).

134 Gershlon Feder and Riclard E. Just

This obtains

P*= r(hO) + r (18)

By differentiating Equation 18, one can verify that the threslhold probability is positivelyrelated to the risk premium and the loan duration (i.e.

aP*a>0, fap*/a3pl> )

while being negatively related to the average loss rate and to the opportunity cost ofcapital (i.e.

aP*/Iw < o, ap*/ap < o).

Using Equation 18 in 17, with 77.= 0, further yields

R(L)P= P*(1 - P) (I9).P/( - P) (9

The right-lhand side of Equation 19 is the ratio of odds evaluated at the threslhold and actualvalues of probability. From Equation 19, one can derive the resllt that loans (and the riskaversion premium) increase as the right-hand side of FLuatiorl 19 increases.

A useful property of the threshold 'probability is that it is indelenrdient of the degree ofrisk-aversion as is apparent from Equation 18. Since the values of p, A', and T are eitherknown or can be estimatedI without major difficulty, one can calcullate the threslhold pro-bability whiich applies to different risk premiums. This approach is demonistrated in Table 1fcr p = .065 and h =.15, .2. The results are presented in terms of q* (the short-run threslholdpvobability) so as to allow comiiparison between cases witlh ditterent loan duLrations. That is,i'; at most, one default can take place within the duration of the loan, then the probabilityP of default over the entire period of the loan is

P = I -. (I - q)N,

provided that q, the probability of default in any given year, is constant.4 It is reasonable toexpect thlat, given current data, lenders assumeq to remain essentially constant since no databeyond the period in wlhich the loan is granted are available for projectioni. The q* in Table Iis a threshold or upper bound on q and is related to P* just as q is related to P

Using this approach, one can very easily confine the sLubjectiv e probahility q to the interval[0, q*] under competition by simply uSinlg data on r, p, V and h (note that 0 is determlliniedby p and A); moreover, the interval [0, P*J is fairly narrow even for small i. FLurtlhermnore,if h is unknown btut can be bounded from below by h*, then P can be Confined to theinterval [0,P**] where

P** = ,,

(F*I0) + r

(since P* is decreasing in f) or q can be conlfinecd to an associated interval [0, q **]where P** = 1 - (1 q- q**)N. lience, the simlple transaction data on r, p. anld N -is suifm-cielnt

4 The reader will note that this definition ofPwvas indeed in the description of the nodel.

A model for analysing letnders' perceived risk 135

Table 1. 7hreshold short-nrim probabilities and risk premiums

for S- and I ayear loan?s

Loanduration r q* q*(years) (per cent) (h 15) (h - .2)

N= 5 .25 .013 .010N= 10 .25 .011 .009

N= 5 .50 .026 .020N= 10 .50 .021 .016

N= 5 .75 .037 .029N= 10 .75 .03 .024

N= 5 1.0 .048 .037NV= 10 1.00 .038 .03

N= 5 1.25 .058 .045N= 10 1.25 .046 .036

N= 5 1.50 .067 .053

NI= 10 1.50 .053 .042

N= 5 1.75 .076 .060NJ= 10 1.75 .059 .048

NV= 5 2.00 .085 .067N= 10 2.00 .065 .053

'Assuming thle opportunity cost of capital p is 6.5 per cent andN = 5, the value of 0 is 4.156; with N = 10, the value of 0 is7.1888.

to develolp some inlormation about the magnitLudes of subjective default probabilities which

might exist.

An application in inferring leniders' suibjectfie probabilities: Thze case of the Euirodollar

market

Anothler applicationl of the model in this paper can be made by inferring initerval estimlates of

lenders' subjective probabilities of default fronm a minimal amount of obser-ved iniarket data.

To demonstrate this possibility, 20 observations oni loans granted in the Eurodollar nmarket

during the tlhird quarter of 1973 have been tised. These observations (presented in the

Appendix) refer to loans made during that period to public (or publicly guiaranted(l) entities

in deVeloping countrics for which data on risk pr-emiiLnm and loani dUration wvere available.5

5The short period covered was chosen so as to avoid any signiflicant variations in the opportuinity cost of

capital among observations. The cstimuates are nmeant solely to be illustrative. There is no pcesiniption

that these estimates are the actual (objcetive) probabilities nor that the World Bank in any way concurs

with these perceptions,

136 Gershotn Feer eand(1 Riclhard( E. JuistThe procedure employed here uses the assumfitions that the market operates compilretitively

and that the subjective loss-rate distributionl is singular in order to establish bouLnds on thlelenders' perceived default probabilities. The first assumlnption implies that 77 = 0, and thesecond if pies that / is a constant, say, Ti. Assuming that thle utility functioni can be specifiedby the constant elasticity formulationi (Eqqationi 14b), the risk-aversion premiLum R ca;nthen be written

R 1 c, (20)

where I L/AM is the share of the loan in total loanable funds and Q' is the relative risk-aversion parameter (O < a < 1).

While the 20 observations on Eurodollar loans range in loan duration betwceli 10 and 15years, the probability concept which is of interest is again one that is free oftthe loan durationeffect. Thus, combininig Equations 17 and 20 and the relation between q and P the followingformula is obtained:

( 1 '( p)l IN = /1 -I+ (l +2 1.)

To make use of Equation 2 1 in the Eurocurrency market, it can be observed that theaverage rate of loss in international bankiing is low; in almliost all cases of debt-service problems,loans are renep-otiited and rescheduled.6 hlence, for exempl:iry purpn,es, a value of l .1 isused.7 As forl, p, and a, it can be noted from E(qLua1tioni 21 that

aq/al, aqlap, and aq/ao < 0.

Thus, specifying sets of upper and lower bounds on these coeFMicients leads to lower andlupper bounds, respectively, for the shiort-run probability q. For exaniple, consitder upperbounds of ! = .3, p = .08, and a - .8 and lower botunds of ! = 0, p = .06, and a 0=. Tlesevalues seem to span the range of reality. That is, I = 0 an1d a = 0 are the lowest lpossible vilues(asstunming no risk-loving behaviour), while / = .3 and at= .8 seemii to be above and beyond alllikely possibilities. No bank seems to loan anything near 30 per cenit of its fuinids to oneborrower; also a close to 1, in the case of the constant elasticity utility fLunctioln, implies anextremely highl degree of risk aversion. Finally, the opportunity cost of capital seems to bebetween 6 and 8 percent. Using these bounds, the aissociated interval e.stimates lor subjectivedefault probability (as perceiVed by Eurodollar lenders) in Table 2 are possible for the Euro-dollar imiarket transactions considelred above. However, these estimates do not necessarilyinmply judgmilents on the nbjective (truLc) probabilitytv.

Several interestinig observations can be made oni the basis of these resuLlts. First, thei upperand lower boLundics span a range for the probability ofeach case within . of Iper )cent or less,This is a small interval relative to the overall variatioi in Table 2. SeCond, thIe ca;C1alcuLhd (I sfor those countries witlh more than one observatiOn (e.g., Algeria) are lairlv close for mliost6 In this respect, Friedtman (1977, p. 55) noted: 'Losses in U.S. banks' overseas operations have been less,both in absolute terimis aind as a p1ropo)rtion of total risk asse r. than in U.S. opertlions.' A sin llhi obscivltionhas been nmade by Beimii (1977, p. 717). _'Nornrally, one would want to estiniatc Ih since the esLiiliates are soniewhat sensitive tO this particularparameter.

A model for anialysing lenders' perceived risk 137

Table 2. Perceived short-nn probabilities in the Euromarket third quiarter,1973

Obser. q qvation upper lowernumber Country bounda boundb

1 Algeria (-= 15) .044 .0392 Algeria (N= 12) .047 .0423 Algeria (N= 10) .048 .0444 Brazil (N 10) .052 .0495 Brazil (N= 12) .049 .0446 Colombia .043 .0397 Gabon .079 .0728 Greece .040 .0369 Iran (N= 12) .037 .033

10 Iran (N= 10) .037 .03311 Korea (N 10) .071 .06512 Korea (N= 1 1) .059 .05313 Mexico (N= 12) .036 .03214 Mexico (N= 10) .035 .03115 Nicaragua .066 .06016 Peru .075 .06817 Philippines .087 .07918 Senegal .083 .07519 Zaire .079 .07220 Zanibia .069 .063

'Calculated with a= 0, 1= 0. p = .06, and h= .1.bCalculated with a = .8,1 = .3, p = .08, andh= .1.

of these cases (theoretically, they should be identical). Also, Pertu, ZaYre, and Zambia, whichseem to have been considered as higher risks by Euromarket lenders as indicated by their qvalue, have inideed experienced serious debt-service problems in years after 1973. Theseobservations may serve as an indication of the creditability of the assumptions. In summary,it appears that fairly precise estimates of default probabilities, as perceived by lenders, arepossible under competition using only the terms of the loan if loss rate for the event ofdefault can be determined (estinmated).

Possible econometric applications

In addlition to the above examples, Equation 17 also has interesting economletric possibilities.For tlle purpose of econonmetric work, it is useftul to consider the commnloni logistic specifi-cation for the probability, namely,

P= + ePo+'X (22)

138 Gershon Fedcr and;t Riclhard E. Just

where go is a constant, ,B is a vector of coefficienits, and X is a vector of econiomic indicatorsincluding loan dLuration whlich are considered by lenders as relevant risk indicators. Witlthis specification, the odds are log linear, i.e.,

In [1 p) =B+X

Hence, EqLuation 17 becomiies

In r =-In (1 i?) + In IFT- ln 0 + g + P'X+ e, (23)

where

-=InR R In hI hV'(h)U'(Yj, ) dli] -In [f1tU( 1)]

The model in EquLation 23 has several interesting potential simplificationls. First, for risk-neutral lenders, it is clearly the case from above that e = In R 0 O; lhence,

In r = -In (1I - )+InT -In 0 + go +,B'X. (24)

With the presence of a disturbance term, Equationi 23 can be estinmated in a variety ofsituations to determine which factors are considered imiportant by leII(Iels in assessing defaultprobabilities. For instanice, n, Ti, and 0 can be calcutlated (0 decends only on p anld iN whichare often recordel, and Ti is also occasionally recordled for dtiflerent clisses of borrowers)and theni included in a regression estimnating f,l and B where a coefficient of I or -I isimposed for In (1 77). In Ti, and In 0 as needed according to EqLaation 24. Alternatively,one may find data wlhere combinationls of ?7, Ti, anid 0 are held constant and can tlhus beincluded in a conistant term. Another approach (Feder and Just, I 977a) is to consider someof the first right-hand terms as randomly distributed among borrowvers and then uise avariance-components approaclh in estinmating B from1 a timiie series of cross-sectioni data.

Finally, an interestin- observation can be made for Eqtuation 23 in the general case of riskpreferences. As it stands, the e term in Equation 23 would be ,,ery difficult to treat econo-metrically: besides, it wotild imply incilusioni of the dependent variable r on the right-lhandside. But in a broad range of cases, it turns out that e is negligible relative to both In r andthe usuial variatioIns in ln r among- observationis. This can be demonstrated assuming aconstant clasticitv titility funiction in wlhichi case Equation 20 applies. Using the samcplausible limits on paramneters used to generate Table 2, togethler with additional liniits on rof 0 < r < 0.025 and on N of 0 < N < 10, it can be slhown tirouglh simple caleLclatiolns tlhatI < R < 1.07 and hcence 0 < In R < 0.068. By coniparision, as r varies fromi 0.00575 to0.025 (which seemis to span the observed range of premiums in the 'ur,d)dollar niiarket), thedelpendent variable Ifi r varies from1 - 3.698 to - 5.159. Hlelnce, evenl witli Imlost conceivablelimlits on 1, p, an(d a, it is thuts clear that E is negligible and can, for most practical p)Urposcs,be inicluided witlh any diStUrbance in Eiquation 23.

Considelin- a conmpetitive nm:mrket, Equtation 24 can becomiie fairly sinmple once e isincorporated in the error tenrm if comipetition imiiplies 77 = 0. Thle equiationi is tlenl reduced to

In r = ln T- In O + go + g'X + v (25)


where v is a randonm disturbance. This formulation is exemplifiedl in the following estimationexample whichi pertains to tratnsactions in the Eurodollar rnarket in the period January,1975, using a set of 29 observations on public or publicly gUarainteel loan transactions(corresponding to 23 developing countries) as reported in Appendix 2.' Reliable indlications

in the period covered imply that the cost of capital (p) was quite stable. For instance, theEurodollar three-month and six-month deposit rates (which are closely related to the costof capital) ranged from 6 per cent to 7 per cent an(d 7 per cent to 8 per cent, respcctively,while in the previous year they flucttuatedl substantially from 8.$ per cent to 14 per cent.Adopting the assumption that p was approximately constant throLIghoLut thle period coveredthus simplifies the treatment of 0 in Equation 25 since one can experimnent witlh a numberof likely values of p (say, .06, .07, and .08) in alternative estimates. The vector X includes,in addition to loan duration (which isexpected tobe positively related to default probability),three variables which have been suggested as potenitially related to debt-servicing capacityof developing countries and, hence, to the risk of default (Frank and Cline, 1.97 1, and Federand Just, 1977b). These variables are:

(I) The ratio of debt-service payments to exports (also referred to as the debt-serviceratio). IHigher ratios reflect a heavier burden of debt service payments and are thus expectedto be positively related to the prohability of default.

(2) The ratio of imports to G.N.P., whiclh, reflects claims of foreign exchange whichare competing with debt-service obligations. Higher import to G.N.P. ratios shouldI titus bepositively related to the probability of default.

(3) The level of G.N'.P. per capita as a proxy for degree of devel)pnment and the ability todivert resources away fromn consump11tion in order to service external debt. hfigler per capitaG.N.P. is thus negatively related to default risk.

Data for the calculation of these variables were obtained trom various World Bank andInternational Monetary Fund pUblications. Having defined the vector X and sing tlhreealternative values forp for the calculations of 0. the corresponlding tlhree regression cstimlates

of Equation 25 are reported in Table 3.The results are practically identical for the differenit values of p and tlhtus inply that lack

of information regarding the exact cost of capital is not serious. All pLaranmcters are statisticallysignificant at a 7.5 per cent nne-sided significance level, andl their sig ns confirm the a prioriarguments advanced.9 Thus, longer loan miaturities, higher debt-service ratios, and higherimport to G.N.P. ratios increase the perceived probability of debt-service problems ofborrowing govemmenits whlie higher per capita incomes reduce the perceived probabilitV.

To exaniine one useftul application of results of this type, note that the constalnt term

reported in Table 3 is an estimate of In h + (plus the mean of the error term v if the latterhas a nonzero mean). For given Values Of 11 (and meani error whenl applicable), one canl cal-culate the correspondingl-, Po valtLes. Ihese, togethier withi the cocfficients reported in Table3, can be used in calculating perc:ived probabilities uisintg (quLtion 22, whtere eaclh alternativevalue of go corresponds to a set of probabilities (Feder and Just, 19 77a).

Using the parameters in Table 3, one caTn also establish a (lirect link betweeln terms of

8This period provides a larger representation of countries in the sanmple than other periods, thuls contri-buting to the quality' of econonietric results.90ne-sided tests are more appropriate in the present case since there are stronga priori arguments relatingto the signs of the coefficients.

140 Gershon Feder atnd Ricliard E. Just

Table 3. Regressiotn results with Eurodollar data: Jantuary-Julv, 1975

Case 1 Case 2 Case 3Variable (p = .06) (p = .07) (p = .08)

Loan duration (years) .153 .149 .145(6,.515)2 (6.341) (6.171)

Debt-service ratio .467 .467 .467(1.671) (1.672) (1.673)

Import to G.N.P. ratio .432 .431 .431(2.511) (2.510) (2.510)

Per capita G.N.P. ($1000) -.089 -.088 -.088(1.541) (1.541) (1.541)

Constant -3.496 -3.502 -3.509

R2 .69 .68 .67R2 .64 .63 .61

'Figures in parentheses are 't' ratios.

credit and economic perfonnance criteria wliich can be usefuLl for a borrower in managingeconomic resources wisely. Consider, for example, a five-year loan soulglht by an economywith a G.N.P. of $1000 per capita. Table 4 presents the risk premiums corresponding tovarious debt-service ratio and import to G.N.P. ratio combinations. Using the results of thisanalysis, a borrowing countrv can thus evaluate the interactioni of, say, import policy andborrowing policy with respect to its terms of credit. In a similar way, onie c2n represent thierisk premiums corresponding to other combinations of explanatory variables.

Table 4. Risk preL'iumzs for a five-year loan by a borrowingcounttry with $1000 per capita G.N.P.a

Debt- Import to G.N.P. ratioserviceratio .1 .2 .3 .4 .5

.1 .0155 .0162 .0169 .0177 .0184

.2 .0163 .0170 .0177 .0185 .0193

.3 .0170 .0178 .0186 .0194 .0202

aThe calculations are based on Case 2 of Table 3, but almostidentical results are obtained with Cases 1 and 3,


IV. CONCLUSION

This paper presents a model of lending under default risk. With an underlying assumption

that the probability of default is deternmined by factors not controlledl by tlle leneder (wllich

is essentially the case when a new loan is small relative to the borrower's existing debt),

the effects of risk, risk aversion, and expected loss rate oni the equiilibriulll levels of loar.

supply and risk premiums are analysed for both conmpetitive and noncomlpetitive markets.

The model is shown to be useful for assessing subjective dlefault probabilities from imarket

data, using the Eurodollar market as an example. Thle model also enabtes the determination

of upper bound threshold probabilities for acceptable customers, given tlle relevant dturation

of loan and interest rate. Further utilization of the model for various econometric purposes

is discussed in detail, and it is shown that in many cases the complicated term reflecting risk

aversion can be ignored zince its magnitude aind range of variation are negligible. This

simplifies greatly the procedures for econometric application of the nmodel as demonstrated

in an example utilizing additional Eurodollar market data.Each of these approaches can provide useful information depending on data availability.

If one simply observes the risk premium, cost of capital, loan duration, and average losF

rate in the event of default, then the lender's subjective probability can be confined to an

interval (0, P*). If, in addition, bounds can be established on the share of the individual loan

of the lender's total loanable funds and on the lender's risk aversion, the stlbjective pro-

babilities can be confincd to a muclh narrower interval. Filnally, if information can1 alter-

natively be gained on the economic factors which influence a len,'er's subjective probability,

then regression techniques can be used to produce point estimates of lender's subjective pro-

babilities tunder certain conditions. Results of the latter type show particula, ,. rimise for a

borrower in managing credit worthiness since they provide information on hliow the borrower's

other economic decisions affect its cost of capital.

APPENDIX

Table IA. Tratnsactionis in the Eurodollarmarket: third quarter, 1973 a


Algeria 1.008 15Algeria .938 12Algeria .875 10Brazil 1.00 12Brazil .955 10Colombia .75 10Gabon 1.75 10Greece .75 12Iran .688 12

142 Gershion Feder and Richard E. Just

Table 1A. (cont.)


Iran .625 10Korea 1.205 11Korea 1.5 10Mexico .659 12Mexico .575 10Nicaragua 1.325 10Peru 1.6 10Philippines 2.0 10Senegal 1.875 10Zaire 1.75 10Zambia 1.425 10

Source: World Bank, 1976.aWhen several premiums were charged indifferent stages of the loan duration period,a weighted average was calculated. Also, whenseveral loans with the same duration wereobserved, a weighted average was calculated.

Table 2A. Transactions in the Eurodollarmarket: January-July, 19 75a


Algeria 1.375 7Argentina 2.0 5.5Bolivia 2.179 8Bolivia 2.0 5Brazil 1.75 5Brazil 1.875 7China 1.5 6Colonmbia 1.75 7Costa Rica 2.0 4.75Ecuador 2.0 5Ecuador 1.5 6El Salvador 1.75 5


Table 2A. (cont.)


Guyana 2.5 5Indonesia 1.725 5Ivory Coast 1[875 5Jamaica 1.9 5Korea 2.0 5Malaysia 1.625 5Mexico 1.5 3Mexico 1.5 5Nicaragua 2,07 5Panama 1.75 5Panama 1.875 7Peru 1.788 7Philippines 1.755 6Spain 1.5 5Spain 1.5 7Sudan 1.75 5Zambia 2.0 5

Source: World Bank, 1976.aWhen several premiums were charged in dif-ferent stages of the loan duration period, aweighted average was calculated. Also, whenseveral loans with the same duration wereobserved, a weighted average was calculated.

REFERENCES

Arrow, K. J. (1971) Essays inz the Thteory ofRisk-Bearing, Markham Publishing Company, Chicago, Ch. 3.Azzi, C. F. and Cox, J. C. (1976) A Theory and Test of Credit Rationing: Comment, Anmericant Economic

Review, 66. 911-17.Beim, D. 0. (1977) Rescuing the LDCs, Foreign Affairs, 55, 717-3 1.Feder, G. and Just, R. E. (1977a) An Analysis of Credit Terms in The Eurodollar Market, European

EcononmicReview, 9, 221-43.Feder, G. and Just, R. E. (1977b) A Study of Debt Servicing Capacity Applying Logit Analysis, Jotrnial

of Dcvelopmnenr Econzotmiics, 4, 25-38.Frank, C. R, and Cline, W. R. (1971) Measurement of Debt Scrvicing Capacity: An Application of

Discriminant Analysis, Journal of International Economics, 1 327-44.Friedman, [s S. (1977) The Etmiergitg Role oJPrivate Banks in the Developing W'orld, Citicorp, New York.Jaffee, D. M. and Mlodigliani, F. (1969) A TIheory and Test of Credit Rationing, Anmeican Economic

Review, 59, 850-72.

144 Gershton Feder anld Richard E. Juist

Jaffee, D. M. and Russel, T. (1976) Imperfect Information, Uncertainty, and Credit Rationing, QuarterlyJournal of Economzics, 90 651-66.

LDC Debt: The Load Eases,Newsweek, February 27, 1978, p. 70.Smith, V. L. (1972) A Theory and Test of Credit Rationing: Some Gcneralizations, Anterican Ecotnomiiic

Review, 62, 477-83.World Bank (1976) Borrowinig in Iiternational Capital Markets (1973--197SJ, Report No. EC-181,

Washington, D.C.

Optimal International BorrowingCapital Allocation and Credit-Worthiness Control

By Gershon Feder, Washington D. C. and Richard E. Just, Berkeley

"Official borrowers like anV others, wish to borrowon the best available terms in the private irzarkct ...After all, they do command the reserves of thecountry as well as access to domestic capital mar-kets and the central bank. Such factors enter intheir balance of panyments management and, moreparticularly, their external debt and reserve man-agement. They know that their external debt man-agement strategies must be play/ed out in the mar-ket place." (Friedman 1977 a, p. 16)

The determination of appropriate international borrowing policy hasattracted the attention of many economists because economic growth,particularly in developing countries, depends heavily on the inflow offoreign funds. Work on this topic dates back to Adam Smith; but morerecent contributions have been made by Hamada (1966, 1969), Bard-han (1967), Van Long (1974), Manning (1972), and McCabe and Sibley(1976). A common approach in related economic analyses has involveda "small country" assumption that has been taken to imply an infinitelyelastic supply of loans at a given interest rate. It has been arguedrecently by Hanson (1974), however, that the supply of loans may notbe completely elastic even for the small country because, as debts getlarge, the probability of a debt crisis for the individual country in-creases. That is, as debts get large relative to the country (not themarket), it becomes more likely that the borrower will not be able tomake repayments on schedule; hence, lenders will become more reluc-tant to lend as the expected, discounted returns decline. As indicatedby Mohammed and Saccomanni (1973), there is no doubt that this riskplays a major role in the lending practices of commercial banks. Non-profit-oriented lending institutions are also likely to be concerned withthe probability that reschedulings will be required; hence, their lend-

* The views and suggestions expressed in this paper belong to the authorsalone and do not necessarily reflect those of the institutions with which theyare affiliated.


ing decisions may also be influenced to some degree by their borrowers'credit status (Cli?. and Sargan (1975)). It thus follows that even smallcountries do not face an infinitely elastic supply of foreign funds;rather, the terms of credit are endogenous depending on credit-worth-iness. Credit-worthiness, in turn, is determined by the countries' eco-nomic performance (usually measured by a few accepted economicindicators).

Since economic-performance indicators are affected by economicpolicies, it is important to explore borrowing policies for optimalgrowth, taking into account the relationship between ecnomic perfor-mance and terms of creditt . Although emi irical results suggest thatseveral economic variables are related to debt servicing capacity, onemay accept the use of only a few indicators as a useful simplificationfor theoretical analysis (Feder and Just (1977 a, b)). Hanson has sug-gested the use of the lebt-equity ratio for this purpose. However, inthe one-sector model developed by Hanson, this approach implicitlyleads to the assumption that "resources can always be costlessly allo-cated to obtain the necessary foreign exchange for debt servicing"[11, p. 619, footnote 10]. Hence, the only factor limiting the availabilityof foreign exchange is the overall productive capacity of the countrywhich may be represented by the capital stock.

Once installed, however, capital is not a malleable factor than canbe equally productive in any sector. Although some types of capitalare more flexible than others (transportation equipment, power plants,etc.), it is more likely with many exports, particularly in developingcountries where exports are composed mainly of nonmanufacturedgoods, that investments are highly specialized (mines, dams, etc.). Evenwhen reallocation of capital is possible, the time lag involved may betoo long for lenders to consider the productive capacity of other sour-ces as relevant for the evaluation of export potential in the case ofdebt payment difficulty. Since the export sector is the main sourceof foreign exchange earnings in many countries and capital rieallocationmay not be possible, it thus seems that the most appropriate case fordeveloping countries may be where the terms of credit depend specif-ically on the size of the export sector (as well as the debt burdeni). Thisis consistent with the findings of a recent study regarding lendingbehavior in the Eurodollar market where two variables related to ex-

1 Such policies have been suggested by observers of international markets;see Irvine et al. (1970), Friedman (1977 b).


port performance (the ratio of debt service to export earnings and ameasure of export variability) appear to be considered as relevant bylenders, while the ratio of debt to gross national product (which is aproxy for the debt-equity ratio) was found insignificant (Feder andJust 1977 a).

Another balance of payments item which is of importance in credit-worthiness evaluations is the size of foreign exchange reserves, whichis usually viewed in comparison to the import bill (Feder and Just(1977 a), Goodman (1977)). That rational reserve policies are carried outwith reference to credit worthiness (among other arguments) is evidentfrom Friedman (1977 a, p. 18) who states: "Other developing countries,like Brazil, are borrowing both to finance external deficits and to re-build reserves, profiting from the experience that a strong, visible levelof reserves improves creditworthiness as well as providing an additio-nal cushion to meet contingencies."

The purpose of the present paper is to consider a two-sector modelwhere export and import activities are defined explicitly. The modelrecognizes lenders' concern regarding default on international debtthrough a credit-worthiness dependent supply function of foreignfunds. It is assumed that capital, once allocated between the exportor nonexport sector, cannot be reallocated2 . Accordingly the credit-worthiness assessment carried out by lenders (and which focuses onthe foreign exchange attributes of the economy) is reflected in an in-terest rate which is positively related to the volume of anticipated debtservice payment relative to the magnitude of export earnings. Similarly,the interest rate is negatively related to the size of foreign exchangereserves relative to the volume of import expeinditures.

The model presents a characterization of optimal investment andborrowing decisions and discussed their policy implications in a mixedeconomy where both governmental and private decision making coexist.

The Model

The model below describes as small, two-sector opern economy. Onesector produces an export good, while the othier sector produces a goodwhich can be used for either consumption or investment. Investmenit

2 This assumption has been adopted in several other optimal growth modelswhere terms of credit and optimal borrowing are not the major interest; seeworks by Bose (1968), Cliakravarty (1969), Dasgutpta (1969) Johansen (1967),and Ryder (1969).

14 Kredlt und Kapital 211979


can be augmented with imports of capital goods. The economy has accessto international capital markets, but the terms of credit it obtainsdepend on its credit-worthiness status, The planning authority isassumed to be interested in maximizing the discounted utility streamsof consumption.

The following notation is used in the analysis. All variables shouldcarry a time subscript, which is omitted for the purpose of brevity.A dot over a variable denotes a rate of change over time.

A. Variables(i) Flow variables

F = A composite good which can be used for both consump-tion and/or investment;

G -Export good (not used domestically);C = Consumption;id = Capital folrmation from domestic output;I = Total investment;M = Quantity of imports of capital goods;L = Gross loans;U = Utility of consumption.

(ii) Stock variablesK/, Kg = Stock- of capital in F and G sectors, respectively;B = Outstanding external debt;R = International reserves.

(iii) Other variablesa = The proportion of total investment w%rhich is directed to

the F sector;r - Interest rate on external debt.

B. Parameters

= Rate of interest earned on foreign exchange reserves;= The planner's rate of time discount;= Rate of amortization of external debt;

d = Rate of capital depreciation;Pg = International price of the export good;Pm = International price of the capital good;

= Average interest rate as perceived by lenders.

C. Model equations

(1) G =G(K,,); G'>O ; G"<O .(2) F=F(K/); F'>O; Fg<0°


The above equations describe the production technologies of the twosectors, which utilize capital only. Labor is assumed a non-bindingconstraint. For simplicity, labor growth is ignored.

(3) F (K) = C + Id

Equation (3) describes the allocation of the F good output betweenconsumption and investment.

(4) r =r[PO G-(O+ f)B, R-P,n M]; r1- a P,G-D+r)B

-<0; r (c'O,cc)ia(R-P,mM)

Equation (4) defines the endogenous interest rate on external debt asa (declining) function of two indicators of debt servicing capacity:(i) The difference between export foreign exchange earnings and theaverage volume of debt service payments (the latter being approxim-ated by some perceived average rate of interest f, which lenders expectto prevail on the average); and, (ii) The difference between internationalreserves held by the economy and the import bill.

These two indicators correspond to two concepts widely used in creditworthiness assessment by lenders in international capital markets:(i) the debt service pressure measure which compares the major sourceof foreign exchange (exports) to the most rigid expenditure item in thebalance of payment (debt service); and, (ii) the import-coveragemeasure which compares the size of reserves to the volume of imports,thereby reflecting the extent to which imports can be financed byreserve depletion if, for some unexpected reason, foreign currency in-flows (exports and new loans) are not available.

While in most cases these measures are expressed as ratios (ratherthan differences), it is rnore cornvenient to follow the present formu-lation. The economic essence of the analysis is unaffected3. As bothcommon sense and empirical evidence suggest, the interest rate will be

3 Evidence on the prevalence of debt selvice and import coverage measuresin credit worthiness analysis is availrbble in many studies. See for instanceGoodman (1977) who reports the results of a survey conducted amongAmerican banks engaging in international lending, Similar evidence is pro-vided in Feder and Just (1977 a) who analysed factors affecting credit termsin the Eurodollar market. See also the articles by Friedlman (1977 a), Braclcen-ridge (1977), Puz (1977) and vann-Aght7.1el (1976), all of whom are seniorexecutives in international banks.

14*


lower for borrowers with higher exports (relative to debt service) andhigher reserves (relative to imports), thus the partial derivatives riand r2 are negative. Also, for a risk-free borrower (such that botharguments of the r function tend to infinity), the rate of interest willapproach the going rate on international deposits (i).

(5) U = U (C) ; U' > O; U" <0 ; J' (O) - .

The utility of consumption has the standard properties of utilityfunctions, namely, positive but decreasing marginal utility, and infinitemarginal utility at C = 0.

(6) I=Id+M.

Total investment is composed of domestic capital formation augmen-ted by imports of capital goods. While the present model assumes thatimports are composed of capital goods only, one can show that iden-tical results may be derived from a model where imports are composedof consumption goods.

(7) Rk(PgG+L+iR)-[PM,MM+(r+ )Bl .

The difference between foreign exchange inflows (composed of ex-ports, loans and interest earnings on reserves) and foreign exchiangeoutflows (composed of imports and debt service paymnents) equals thechange in international reserve holdings.

(8) aI - K, .

(9) Kg-(1-a)1-6Kg .

Equations (8) and (9) present the rate of change (over time) of capitalstocks in the F and G sectors (i. e., net investments) as the differencebetween the new investments allocated to each sector and the volume ofcapital depreciation4 .

(10) B=L-6B.

The chang'. in indebtedness is equal to the difference between grossloans and amortization payments.

4 The proportion of investment allocated to each sector should be con-strained between zero and one. It will be assumed that this condition holdsat the relevant range.


The planner's objective is maximization of discounted utility of con-sumption. Assuming an infinite horizon the problem can be formu-lated as

(11) ~,Max Jf U (C) dt

subject to initlial stock values and equations (1) - (10).

Problem (11) can be solved using the Pontryagin Maximum Prin-ciple. Denote the present value Hamiltonian by H and define the constantvariables 2, ,t, ?7 and q, which are dynamic shadow prices of the stockvariables K1 , Kg, B and R, respectively. Then, using the relation I=F (K1 ) - C + M (from (3) and (5)), one can write

(12) H te-t =U(C) +{A.a[F (K1 ) -C+M] -5K 1 }

+ (1-)[F (K,) - C + M] - a K,} + 77 (L - t B)

+ {[Pg G (Kg) + L + iR] - [P, M + (r +± ) B]}

It should be noted that the non-negativity requirement for C is notspecified since the assumption U' (0) = oo eliminates such a solution.The non-negativity of M and L, and the requirement 0 <oe < 1 areassumed to be non-effective constraints at the range of the optimalpath described in the present paper. It thus follows that the optimalityconditions to be derived below are applicable only at a certain rangeof the optimal path (including the steady state). Other portions of theoptimal path include corner solutions (e. g., zero investment in onesector or the other, zero borrowing, etc.) and seem to us to be less in-teresting from a practical point of view. Similarly, existence and uni-queness of the solution depend on additional constraints which are notspelled out.

The optimality conditions require (in addition to appropriate trans-verality conditions), at the range where all controls are positive:

(13) -UC- =0 o; ) , = o

(14) a M ° Xoa~ + (I1-a), #-( Pm [I1-r2 B] =

(15) a 0 XF (K 1)-C + MJ tA-)=0

(16) 0 > =7 + X 0


(17) A a X2--) - E6-A + (1-) 1 t3F'(Kt)

y 18 ) HQ6 -Ir it Q0 #( + 6) - Pg. G' (KD) * (-r B)

(19) H * 7-( tO+Cr+ r,(+F) B]

(20) aq R-X-9p [ - ± r2 * B]

Analysis of the Results

The economic interpretation of the optimality conditions is fairlystraightforward:

Equation (13) implies that the marginal benefit of consumnption (i. e.,marginal utility) equals its opportunity cost [ar + (1 - c) [l], which isthe marginal value of investment foregone. The value of marginal in-vestment is a weighted average of the value of capital in both pro-duction sectors. As will become apparent, the value of a unit of capitalin both sectors is equal, thus the implication is that the marginal utilityof consumption equals the value of a unit of capital which is foregone.

Equation (14) states that the value of a unit of investment [cd +(1 - o) u] (afforded by one additional unit of imports of capital goods)equals its opportunity cost. The latter is the amount of foreign ex-change (withdrawn from reserve stocks), adjusted for the deteriorationin credit terms caused by the decline in import coverage by reserves.We note that if imports had no effect on the term of credit (i. e., r2 = 0)then the value of investment equals the shadow price of foreign ex-change needed to implement that investment.

Equation (15) implies that value of capital in each of the two pro-duction sectors (that is, the export sector and the investment-consump-tion sector) is equal, (i. e., A = It) even though capital is not transferablebetween the sectors once investments are implemented (except in thecase F (K) - C + M = 0, which implies no investment in the economy).Given A = ,u, it follows that the value of capital is higher than the valueof the direct foreign exchange cost of a unit of capital imports. Thisis explained (as indicated above) by the additional cost, in terms ofreduced credit wvorthiness, of imports.

Equation (16) implies that the value of a unit of reserves (or theshadow cost of foreign exchange) equals its opportunity cost (the


reduction of external debt). The value of a unit debt (X) is obviouslynegative, as the value of reserves (q') is positive.

Using the result 2 = p P,, (1 - rL, B) (which also implies A = A),

equations (17) and (18) can be combined to yi-a-ld

(21) PUG F' ( - )

Equation (21) implies that in general the value of marginal productof capital in the two production sectors (evaluated at internationalprices) will not be identical. Obviously, in the case where no creditworthiness considerations are taken into account (i. e., ri = r2 = 0), themarginal productivity of capital would be maintained equal in allsectors. Indeed, in an economy where most investments are made byprivate (and competitive) investors, one would expect them to operatesuch that Pmn F' = Pg G', since each one of them is too small to con-sider the national credit-worthiness implications of imports and exports.The planner (or the government), however, cannot ignore the impact ofsuch variables as represented by the right hand side of equation (21).Thus, intervention in the sectoral allocation of investment is called forby export promotion and/or import substitution. Export promotion isreflected in the term - rj B which measures (in units of output) theamount of subsidy that should be given to producers of the export good.Similarly, - r2 B measures (in physical units) the subsidy due to pro-ducers of the F good (which can substitute imports of capital goods).Private investors will then equate (at the margin) the actual rates ofreturn received by them, namely

(22) P G'G [1- r B] = Pm F [1- r2 B]

which will ensure that the economy maintains the optimal conditiongiven by equation (21). At the steady state, however, the optimal con-

dition for the F sector requires (from equation (17) with 2 0),

(23) F'=d+ Q .

Whether the export sector is (at the marginj more profitable than theimport substituting sector depends on whether rl/r2 1. The size ofrl/r2 is affected by the differential weight attached to the differentcredit-worthiness indicators and is thus an empirical question.

It should be noted that in practice, many governments find it moreconvenient to control the volume of imports (by tariffs or quotas), so as


to avoid foreign exchange crises which will deteriorate their creditstatus. In terms of the present model, the optimal tariff on imports is- r2 B. Imposing such an import tax will eliminate the need to sub-sidize the F sector.

Using the equality ?I =-, which implies j1p 7 /1, equations (19)and (20) can be combined to yield

(24) r-rj(t9 +J)B=i-r 2 B

The left hand side of equation (24) is the marginal cost of borrowingto the economy, which is necessarily higher than the rate of interest r,since it includes the extra cost due to the deterioration in terms ofcredit brought about by higher debt (the latter effect is represented bythe term - *j (2 + r) * B). The right hand side represents the marginalbenefit from reserve holdings, which is composed of the direct rate ofinterest plus the contribution to credit worthiness gained by additionalreserves (given by - r2 B). Equation (24) thus describes the optimalityrelation for reserve and debt stocks. Obviously, in the present model,if reserves had no impact on credit worthiness (r> = 0), then a zeroreserves level should be maintained. This can be seen by observingthat with rs =0, it is impossible for equation (24) to hold (since r -r(§ + r) B > r > i for all values of the arguments of r, as indicated in(4)). To approach the equality, R will be depleted (and B reduced by anequal amount), which verifies the assertion made above.

In an economy where part of the external borrowing is done byprivate corporations, the model points out the necessity of imposing atax on foreign borrowing. The reason for such a tax is that while thecost of borrowing for each individual borrower is r (the rate of interest)the cost to the economy is the marginal borrowing cost which includesthe terms of credit effect. The presence of credit worthiness consider-ations thus implies an externality, which may be corrected by an appro-priate tax. Obviously, direct measures can also be applied. For instance,recently, the government of Turkey banned foreign loans which areobtained at a rate of interest higher than a certain upper limit (Clarke,1977). Other governments imposed administrative restrictions whichamrount to an implicit tax on borrowing from foreign sources.

Comparisons between the marginal rate of return and the externalinterest are possible, in the present model, only at the steady state.Thus, setting A = X = 0 and using the results A- U, -X-u one ob-tains from (17) and (19):


(25) F'-6=r-r 1 -(0+7)B .

Equation (25) implies that at the steady-state, the marginal produc-tivity of capital in the import-substituting sector (net of capital de-preciation allowance) should be maintained at a level higher then therate of interest of external debt. The difference is equal to the credit-worthiness effect of marginal borrowing. When credit-worthiness con-siderations are negligible (ri! 0), the standard result is obtained,namely, the net marginal productivity of capital equals the interestrate.

The relation between the marginal rate of return of capital in theexport sector and the rate of interest is less straightforward. Settingu =i1 = 0, and using the relation - = ( u = Il[Pn,* (1 - r2 B)], oneobtains from (18) and (19), at the steady state,

Pg G' (1 - r1 B)(26)

Pm (1 - r2 B)

Since the magnitude of the various terms on the left hand side is notknown a-priori, one can not conclude as to whether the rate of returnto capital should optimally be higher or lower then the direct rate ofinterest. We note, however, that the left hand side describes the fullimpact of the marginal investment in the export sector: the direct con-tribution to export plus the improvement in credit terms induced byhigher export [Pg G' (1 - ri B)] discounted by the total cost of a unitof capital imports (that is, both direct and indirect cost). Equation (26)thus states that the full marginal contribution of investment in the ex-port sector should equal the marginal cost of borrowing.

Setting I = 0 also implies (by equation (19)) that at the steady statethe rate of time preference of the economy (C) exceeds the interest rateon external debt, as it needs to equal the marginal cost of borrowing.

From equation (17) with -= 0, one obtains the familiar "golden rule"of neo-classical models, asserting that at the steady state capital accu-mulation is maintained at a level such that the net marginal produc-tivity equals the rate of time preference (if labor growth is nil). Thisrule applies, in tlA present model, only to the import substitutingsector. For the export sector the optimal steady state accumulation lawis more complicated and depends on credit-worthiness effects.

218 Gershon Feder and Richiard E. Just

Conclusion

By way of summary, the model implies the following set of policiesfor an economy with private borrowing, investment and production:

(i) Export promotion through a subsidy (of magnitude - Pg 'ri B) tothe export sector;

(ii) Tax on capital goods imports (of magnitude - P,. r2 B), which

amounts to a policy of import substitution;

(iii) Tax on foreign borrowing by private corporations (of magnituderi (0 + r) B)-

These recommendations are, of course, of a partial nature, due to thesimplistic structure of the model. :3ut the paper makes the generalargument that development policies should take into account the re-lationship between economic policies and the supply of foreign funds,and it demonstrates that such considerations require central govern-ment intervention in the operation of the market. This conclusion ismost relevant for developing economies who depend on commercialinternational capital markets for a substantial portion of their foreignexchange inflow.

References

[1] Bardhan, P. K.: "Optimum Foreign Borrowing", in K. Shell (ed.), Essayson the Theory of Optimal Economic Growth. Cambridge: MIT Press. 1967.pp. 117 - 128. - [2] Bose, S.: "Optimal Growth and Investment Allocation",Review of Economic Studies, Vol. 35. October 1968. pp. 465 - 480. - [3] Brackc-enridge, A. B.: "Evaluating Country Credits", In7stitittioizal Investors. June1977. pp. 13 - 16. - [4] Chakravarty, S., Capital and Development Planning.Cambridge: MIT Press. 1969. - [5] Clarke, P.: "Will the Banks and theFund Make an Example of Turkey?" Euromoney. September 1977. pp. 16 - 23.- [6] Cline, W. R. and N. P. Sargan: "Performance Criteria and MultilateralAid Allocation". World Development. Vol. 3. June 1975. pp. 383 - 391. - [7]Dasgupta, P. S.: "Optimum Growth When Capital Is Nontransferable".Review of Economic Studies. Vol. 36. January 1969. pp. 145 - 154. - [8] Feder,G. and R, E. Just: "An Analysis of Credit Terms in the Eurodollar Market".European Economic Review. Vol. 9. May 1977 a. pp. 221 - 243.- [9] Feder, G.and R. E. Just: "A Study of Debt Servicing Capacity Applying Logit Ana-lysis". Journal of Development Economics. Vol. 4. March 1977 b. pp. 25 - 39.-[10] Feder, G. and U. Regev: "International Loans, Direct Foreign Invest-ment, and Optimal Capital Accumulation". The Economic Record. Vol. 51.September 1975. pp. 320 - 325. - [11] Friedman, I. S.: "Most EurocurrencyBorrowers Remain Good Risks". Euromoney. March 1977 a. pp. 16 - 21. -

BorroNving, Capital Allocation and Credit-Worthiness Control 219

[12] Friedman, I. S.: The Emerging Role of Private Banks in the DevelopingWorld. New York: Citicorp. 1977. - [13] Goodman, S.: "How the Big U. S.Banks Really Evaluate Sovereign Risks". Euromoney. February 1977. - [14]Hamada, K.: "Economic Growth and Long-Term International Capital Move-ments". Yale Economic Essays. Vol. 6. Spring 1966. pp. 49 - 96, - [15] Hamada,K.: "Optimal Capital Accumulation by an Economic Facing an InternationalCapital Market". Journal of Political Economy. Vol. 77. July/August 1969.pp. 684 - 697. - [16] Hanson, J. A.: "Optimal International Borrowing andLending". American Economic Review. Vol. 64. September 1974, pp, 616 - 630.- [17] Irvine, R. J.; Y. Maroni and H. F. Lee: "How to Borrow Successfully".Columbia Journal of World Butsiness. Vol. 5. 1970. pp. 42 - 48. - [18] Johansen,L.: 'Some Theoretical Properties of a Two-Sector Model of Optimal Growth".Review of Economic Studies. Vcol. 34. January 1967. pp. 125 - 141. - [19] Man-ning, R.: "Optimal Taxation When Borrowing Alters the Foreign SupplyCurve for Capital". The Economic Record. Vol. 48. September 1972. pp. 400 -410.- [20] McCabe, J. L. and D. S. Sibley: "Optimal Foreign Debt Accumu-lation With Export Revenue Uncertainty". International Economic Review.Vol. 17. October 1976. pp, 675 - 686. - [21] Mohammed, A. F. and F. Sacco-manni: "Short Term Banking and Euro Currency Credits to DevelopingCountries". IMF Staff Papers. Vol. 20. November 1973. pp. 612 - 638. - [22]Puz, R.: "How to Find Out When a Sovereign Borrower Slips from A-1to C-3." Euromoney. December 1977. pp, 65 - 68. - [23] Ryder, H. E.: "Opti-mal Accumulation in a Two-Sector Neoclassical Economy with Nonshift-able Capital". Journal of Political Economy. Vol. 77. July/August 1969.pp. 665 - 683. - [24] van Agtmael, A. W.: "Evaluating the Risks of Lend-ing to Developing Countries". Euromoney. April 1976 pp. 16 - 30. - [25] vanLong, N.: "International Borrowing for Resource Extraction". InternationalEconomic Review, Vol. 15. February 1974. pp. 168 - 183.

Zusammenfassung

Optimale internationale SchuldenaufnahmeKapitalallokation und Kreditwiirdigkeitskontrolle

In diesem Beitrag wird ein Zweisektorenmodell einer wachsenden Wirt-schaft einschlieBflich des Verhaltnisses zwischen der Kapazitat des Schulden-dienstes und den Kreditbedingungen eines Landes entwickelt, Diese Bedin-gungen werden durch makrookonomische Variablen beeinfluBt, wie z. B,durch das Ein- und Ausfuhrvolumen, die Hohe der Devisenreserven, dieHohe der Schuldendienstzahlungen entsprechend der besteheniden Schuld, DieTatsache, daI3 Importeure, Exporteure und Hersteller oftmals zu unbedeu-tend sein konnen, um die Auswirkungen ihrer Handlungsweisen auf dieXreditbedingungen der Wirtschaft in Rechnung zu stellen, schafft Diskre-panzen zwischen den opti-nalen privaten und offentlichen Kreditlosungen.Das Modell kommt daher zu dem Ergebnis, dalI optimales Wachstum Export-firderung verlangt, w*vhrend gleichzeitig sowohl die private Schuldenauf-nahme von Auslandsgeldern als auch der Import von Kapitalguitern be-steuert wird.


Summary

Optimal International Borrowing, Capital Allocationand Credit-Worthiness Control

The paper develops a two-sector model of a growing economy, incorporat-ing the relation between debt servicing capacity and the terms of credit

facing the country. These terms are affected by macro-economic variablessuch as the volume of imports and exports, the size of foreign exchangereserves and the magnitude of debt service payments due on outstandingdebt. The fact that importers, exporters and producers may be too small totake account of the impact their actions have on the economy's terms ofcredit creates discrepancies between private and public optimal solutions.The model suggests that optimal growth may require export promotionwhile simultaneously taxing both private borrowing of foreign funds andimportation of capital goods.

Resume

Endettement international optimal, allocation de capitalet controle de la solvabilit6

La presenbe etude d6veloppe un modele bisectoriel d'kconomie en expansion,incluant la relation entre la capacitO, d'endettement et les conditions decredit d'un pays. Ces conditions sont influencees par des variables macro-6conomiques telles que le volume des importations et des exportations, lestock des reserves de devises, le montant des paiements du service de ladette correspondant a l'endettement en cours, Le f alt que les importateurs,les exportateurs et les producteurs n'ont frequemment pas la taille requisepour influer par leur maniere d'agir sur les conditions du credit de l'eco-nomie provoque des disparites entre les endettements optimaux prive etpublic. Le modele conclut que la croissance optimale requiert la promotiondes exportations et parallelement la taxation des emprunts prives ell devisesetrangeres et de l'importation de biens d'investissement.

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