- Email: [email protected]
Spatial models of the Eurodollar market
Journal of Banking and Finance 8 (1984) 51-65. North-Holland
SPATIAL MODELS OF THE EURODOLLAR MARKET Sten THORE* Unirersity of Texas, Austin, TX 78705, USA Received March 1983, final version received September 1983 The Eurodollar market can be viewed as a logistic system, converting Eurodollars deposited by exporters and other traders into Eurodollar loans raised by the borrowers (importers). This paper traces the spatial distribution of such funds-flow. An extremal principle for the determination of spatial equilibrium, originally due to Samuelson, is here for the first time applied to a problem of financial equilibrium. The behavior of participating international banks is specified by standard portfolio models. At the point of extremum, there is equilibrium in all regional markets, and each international bank maximizes the net return on its Eurodollar portfolio.
1. Motivation and antecedents The functions of banking are to convert low-risk deposits into risk-willing loanable funds, to convert short-term deposits into longer-term loans and to bring savers and borrowers together. The present paper is a study in this last and third function, in the process which redistributes geographically the flowof-funds from a regional vector of deposits into another regional vector of bank loans. The Eurodollar market has since its inception served as a conduit for international redistribution of short-term funds. In the 1970s, there arose a need for recycling the mounting dollar surplus of the OPEC countries back to oil-importing countries which needed loans to finance their trade deficits; the Eurodollar market became a vehicle for such recycling at the short-term end of the maturity spectrum. In the early 1980s, this international financial intermediation appears to be more crucial (and more risky) than ever. As the world has plunged into a global recession, several countries (Mexico, Brazil) have run into international payment difficulties, not to mention the plight of many poor countries of the third world. Several large international banks have reportedly overextended themselves in accommodating the Eurodollar needs of their customers. The Eurodollar market still functions, but its redistributive role is more precarious than ever. Seen against this background, it should be of some interest to develop a *The author is the Gregory A. Kozmetsky centennial fellow at the Institute for Constructive Capitalism, University of Texas.
0378-4266/84/$3.00 9 1984, Elsevier Science Publishers B.V. (North-Holland)
52
S. Thore, Spatial models of the Eurodollar market
model of the spatial and regional distribution of Eurodollar funds. In attempting to accomplish this task, the present paper brings together theoretical developments fetched from two very different fields of economic analysis: the analysis of systems of bank portfolios on the one hand, and spatial and regional equilibrium on the other. In a series of publications [Thore (1969, 1970, 1980), Thore and Kydland (1972)-I, the present author has attempted to evolve a theory of the fundsflow through a linked series of financial portfolios. The micro-behavioral unit in the theory is the portfolio of the individual commercial bank, savings and loan association, or other financial intermediary. The portfolio choice is described in some kind of optimizing framework; in the case of bank portfolios, the optimization may take the form of linear programming [see Chambers and Charnes (1961), Cohen and Hammer (1967)1, chance constrained programming (for references, see section 3), etc. The portfolios are joined together into a network where each separate portfolio can be visualized as a node and assets and liabilities as directed links. The fundsflow is viewed as a logistic process converting primary assets, like deposits, shares in S&L associations, etc., into final assets, like bank loans. In other words, the system of financial intermediation is conceived of as a kind of financial 'production' system, where the financial intermediaries transform the supply of primary assets of the savers into a suitable array of final assets issued by the ultimate borrowers [see Thore (1980, p. 40),1. Using this terminology, we may say that the Eurodollar market constitutes an international financial network, which receives 'primary' Eurodollar deposits from exporters and other traders and re-channels these funds into an optimal supply of 'final' Eurodollar loans raised by importers and other traders. A major tool in regional analysis is the Samuelson and Takayama-Judge model of spatial equilibrium. In the original formulation by Samuelson (1952), one is concerned with explaining the spatial allocation of demand and supply and the pricing of one single consumer good. Samuelson defined in each region an economic function, which he called 'social benefit' [and which Takayama and Judge (1971) later called 'quasi-welfare',1, by calculating the algebraic value of the area between the demand curve and the supply curve. Maximizing total social benefit net of transportation costs, Samuelson showed that, at the optimum, all regional markets are in equilibrium. Takayama and Judge (1964) extended Samuelson's analysis to the case of several consumer goods. They also showed how it is possible to provide an explicit specification of production in the model [Takayama and Judge (1964, 1971, ch. 14)]. They assumed the presence in each region of primary resources (including labor); the resources are converted into finished consumer goods by an activity analysis model. At the point of optimum, all markets are in equilibrium.
S. Thore, Spatial models of the Eurodollar market
53
The Takayama-Judge model has been used extensively in agricultural economics and energy economics, i.e., Uri (1974) and Kennedy (1974). As a result of such applications, the Takayama-Judge model has been extended and developed in several directions, including multi-period applications [Fuchs et al. (1974), Martin and Zwart (1975)], studies of disequilibrium [Taylor and Frohberg (1977)] and the development of computational routines for numerical solution [Woodland (1974)]. Recently, Thore (1982) showed that much more general specifications of the nature of the production process are permitted. Conventional smooth production functions of any suitable form may be used. So much for the theoretical background. What I propose to do in the present paper is to apply these techniques of spatial analysis to the study of the process of financial intermediation. Section 2 starts out with the simple Samuelson model. There is in this model just one single financial 'good': three months Eurodollar funds. There is a given demand function and a given supply function for such funds in each region. The contribution to quasi-welfare in each region is again obtained as the surface between the demand curve and the supply curve. By 'on-lending', often through a sequence of banks operating in the Eurodollar market, the original deposits in a region are re-routed world-wide to the ultimate Eurodollar borrowers. The transactions cost (or 'transportation cost', to use the term of logistics modelling) is small; if so desired, it may be entirely neglected. The analysis in section 3 digs deeper. The Eurodollar portfolio choice of the participating international banks is spelled out explicitly. In other words, we are now turning to a closer examination of each node of the Eurodollar network and providing an account of the 'production' of Eurodollar funds. The financial intermediaries which operate in the Eurodollar markets and which ultimately convert Eurodollar deposits into Eurodollar loans are subsidiaries of large U.S. banks, large Eurobanks and their subsidiaries, or just ordinary local commercial banks with some activity in the Eurodollar markets. These various institutions will be called the 'Eurobanks', for brief. An optimizing model of portfolio choice is specified for each Eurobank (or group of Eurobanks). This portfolio allows for the presence of dollar reserves kept on deposit in New York and for the possibility of U.S. main offices borrowing back Eurodollars from their European subsidiaries. The distinction between non-official and official funds in the Eurodollar market is important. Central banks, the IMF, and other international official bodies hold considerable amounts of Eurodollar deposits as part of their short-term portfolio in dollars and foreign currencies. The analysis both in section 2 and in section 3 will handle the presence of official funds, as long as they are given and known. No attempt will be made to explain the portfolio choice of international official bodies.
54
S. Thore, Spatial models of the Eurodollar market
The present analysis is an instance of partial economic theory. The income of exporters and importers is supposed to be given and known; so are the prices of all commodities and all financial instruments other than Eurodollars. In particular, the quotations on other foreign currencies and the entire foreign exchange portfolio of the Eurobanks is supposed to be given and fixed. Section 4 contains a summing-up and indicates briefly how some of the limitations now mentioned may be removed.
2. A spatial equilibrium model of the Eurodollar market Consider a spatial partitioning of the Eurodollar market into several regions (different countries, for instance) h = l , 2 ..... I. The importers and other private traders in each region demanding Eurodollar loans will be treated as one composite unit. Let x h represent the amount of Eurodollar loans they demand. The time period of analysis is three months; assume that all Eurodollar contracts have a maturity of three months and that they are evenly distributed over time. Let the cost of Eurodollar loans (the average cost rate during the period of analysis) be 100 r h percent. As we shall see later, any possible geographical variation in this rate would immediately be wiped out by arbitrage deals. There is, in each region, a demand function for Eurodollar loans, relating the demand x h to the loan cost r h. The interest cost to the borrowers.in other (national) financial markets, and their income, are all supposed to be constant and given. The model that we are about to develop is a partial model for the international Eurodollar market, assuming all other national variables to be fixed. Inverting the demand functions for Eurodollar loans, there will exist in each region a 'demand price '1 function for Eurodollars D(Xh) showing the interest cost at which the local demand for Eurodollar loans would equal x h. The demand price functions D(xh), h=1,2 ..... ! are supposed to be single-valued, positive, decreasing and differentiable. They, too, are defined for given interest costs of all alternative national channels of borrowing and for a given income. The contribution to the 'quasi-welfare' of the importers and other traders in region h demanding Eurodollar loans will now be discussed. The contribution equals the surface below the demand curve for loans (i.e., the total incremental interest payments which the borrowers would be willing to incur if the loans were extended to them incrementally, dollar by dollar), 2' 3 1The correct, but somewhat awkward, term would of course be 'demand interest rate'. 2Forming a concept analogous to 'consumers' surplus', we may say that the integral (1) equals borrowers' interest payments plus 'borrowers' surplus' in region h. aln the common fashion of an indefinite integral, the upper limit of integration of (1) is x h. The lower limit is some arbitrary fixed initial point. The integral is concave, i.e., dD/dx ~ is <0.
S. Thore, Spatial models of the Eurodollar market
S D(x h)dx h.
55
(1)
Turn now to the other side ol the market: the depositors. The exporters and other private traders in each region holding Eurodollar deposits will (like the borrowers) be treated as one composite unit. Let wh represent the amount of Eurodollars they desire to hold (=supply). There will then, in each region, exist a supply function for Eurodollar deposits relating the supply wh to their yield. The yield on other (national) financial investments, and the income of the composite group of depositors, are all supposed to be constant and given. Inverting these supply functions, there will exist in each region a 'supply price' function for Eurodollars S(w h) showing the interest yield at which the local supply of Eurodollar deposits would equa! w h. The supply price functions S(wh), h = l , 2 .... ,l are supposed to be single-valued, positive, increasing and differentiable. They, too, are defined for given yields on alternative national financial instruments and for a given income. The contribution to quasi-welfare of the exporters and other private traders in region h supplying Eurodollar deposits equals the negative of the surface below the supply curve for deposits (i.e., the total incremental interest returns which the depositors would require in order to motivate them to supply the funds, dollar by dollar), 4' 5
- ~ S(~e)dw ~.
(2)
One does well in remembering Samuelson's admonition not to try to read any economic significance into the concept of 'social pay-off' or 'quasiwelfare'. It is formed for operational purposes only. 6 (In particular, it is clearly not a utility function. Increased Eurodollar loans and an increased indebtedness position of the borrowers conventionally reduce their utility rather than adding to it.) Form the consolidated 7 portfolio of the Eurodollar assets and liabilities of all Eurobanks operating in each region h. I shall discuss this consolidated portfolio in some detail in section 3; at the present, it will be assumed that it simply reads:
4The expression (2) equals the negative of depositors' collected interest returns plus 'depositors' surplus' in region h. 5The upper limit of integration is wh. The lower limit is some arbitrary fixed point. The expression (2) is concave, since dS/dwh ~0. 6'An economist looking at these figures would naturally think of some kind of consumer's surplus concept ... However, the name consumer's surplus has all kinds of strange connotations in economics. To avoid these and to underline the completely artificial nature of my procedure I shall simply define a "net social pay-off' function...' [Samuelson (1952, p. 288)]. 7The consolidation mentioned refers to the netting out of all interbank assets and liabilities within each region. J.B.F.-- C
S. Thore, Spatial models of the Eurodollar market
56 Assets
Liabilities
Eurodollar loans to importers and other non-bank borrowers xh
Eurodollar deposits from exporters and other non-bank depositors
Eurodollar loans to Eurobanks in other regions
Eurodollar deposits from Eurobanks in other regions
~----~2thh2
~_~hI t hlh
Net (Eurodollar) worth ah
As illustrated by this balance sheet, the Eurobanks in each region are, of course, in no way constrained to lend locally the Eurodollar deposits and the net worth that originates in each region. The net interregional on-lending between blinks into region h is
~ t h ' h - - ~ t hh2 hI
(3)
h2
(t hlh2 denotes the gross on-lending from region tll to region h2). Consider now the program max ~ S D(xh)dxh -- ~ ~ SCwh)dwh, h
(4)
h
subject to
xh--.wh--(~th'h--~thh'~
h = l , 2 .... ,l,
x,w,t>O.
In words, program (4) instructs us to maximize the total quasi-welfare arising world-wide from interest, payments and interest receipts in the Eurodollar markets, subject to a balance sheet condition in each region (stating that the total out-flow of Eurodollar funds from the region cannot exceed the total in--flow of Eurodollar funds into the region)fl Program (4) is an instance of concave programming over a linear constraint set. Denote the Lagrange multiplier of the constraint by r h. It is non-negative. Let the optimal r h be established as the market interest rate on Eurodollar funds. SAny official Eurodollar funds transacted in each regional market may be absorbed by the entity ah (officialdeposits being added to a~; officialloans deducted from it).
S. Thore, Spatial models of the Eurodollar market
57
The Kuhn-Tucker conditions spell out the following results, holding at the point of optimum: 9 (i)
In each region, the demand price for Eurodollar loans equals the market interest rate,
O(x h)= r
h.
(5)
(ii) In each region, the supply price for Eurodollar deposits equals the market interest rate, S(w~) = r h.
(6)
Taken together, (i) and (ii) simply state that there will be equilibrium in the market for Eurodollar funds in each region. (iii) All regional Eurodollar quotations will be equal. For one gets t~ rht=r h2 for all hi and h 2.
(7)
3. Eurobank portfolio constraints In the simple model developed in section 2, Eurodollar funds are automatically shuffled back and forth between different banks and different regions in order to see to it that the supply of Eurodollar funds at all times matches the demand in each region. The banks serve as passive conduits for such Eurodollar flow without any behavioral relations of their own; On a more realistic level of model-building, it becomes necessary to specify an optimizing portfolio choice for each participating Eurodollar bank. Write the balance sheet of Eurobank k (kh = I, 2..... nh; the superscript is suppressed most of the time) in region h: Assets
Liabilities
Eurodollar loans to importers and other non-bank borrowers
Eurodollar deposits from exporters and other non-bank depositors
9Assuming that positive Eurodollar transactions occur in all markets, relations (5), (6) and (7) have been written as equalities. But if any .~, v/', t h~h: is zero, an inequality sign ( < , > , < , respectively) should be entered in the corresponding relation instead. l~ writing down (4), it was assumed that unit transactions costs, say c hjh~, were all zero. If they are non-negligible, however, the term ~..c*Jh~t ~h2 should be deducted from the maximand. Relation (7) will then come out as - r h ' + r ~ = - c h'h2.
S. Thore, Spatial models of the Eurodollar market
58
Assets
Liabilities
Eurodollar loans to Eurobanks in same and other regions
Eurodollar deposits from Eurobanks in same and other regions
Dollar account in New York
Net Eurodollar worth
,hh2 z J'21t
Dollar loans from a London subsidiary of a U.S. bank to head office in New York
The bank's supply of Eurodollar loans to private non-bank customers is written ~k- These customers are all supposed to reside in region h where the bank itself is located. Eurodollar loans made to other Eurobanks in other regions is written ~ ] ~ , where the summation is carried out over all regions h 2 except the home region h where the bank itself is located. Eurodollar loans made to other banks in the same region is ~ . Eurodollar reserves held in New York are written ~k. Any Eurodollar deposit received by a Eurobank must physically be transferred to the Eurobank's correspondent bank in New York and will add to the Eurobank's reserves kept on deposit there; conversely, any Eurodollars lent by a Eurobank will entail a drain of such reserves in New York. If, during some time period, the amount of Eurodollar deposits received by the Eurobank surpasses the amount of Eurodollars lent, there will build up a net accumulation of reserves in New York. Loans to the U.S. main office of a U.S.-owned Eurobank are written yh4k. If a U.S. bank borrows back Eurodollars from its London subsidiary (using the proceeds-of such a loan for domestic U.S. operations), Eurodollar funds which would otherwise have been available for lending in the Eurodollar market will be tied up as loans to New York. Deposits from private non-bank customers is ~k. Eurodollar deposits received from other Eurobanks in other regions are ~ ] h , where the summation is carried out over all regions hx except the home region h. Deposits received from other banks in the same region is ~6h. Now, denote the vector of balance sheet entries of bank k in region h by ~ . Assets are entered with a plus sign, and liabilities are entered with a minus sign. Let
~ r~,
(8)
S. Thore, Spatial models of the Eurodollar market
59
where yh is the set of all feasible portfolio vectors of the bank. Assume that yh is convex, bounded and closed. It What this symbolism means is that I am subsuming the existence of some suitable constraints which are laid down on the portfolio choice of the Eurobanks. At this stage of the development of the argument, it is not necessary to specify precisely what these constraints would look like; in some general fashion they must reflect the opportunities and the risks which confront the banks and the evaluation of the management of those opportunities and risks. 12 Thus prepared, consider the programming problem max ~
I D(xh) dxh -- ~ I S(wh)dwh,
h
(9)
h
subject to x h-x~,~ <0 , /,YlI~
0
k
h,k
k
)~k~ Y~,,
h = l , 2 . . . . . l, k = l , 2 . . . . . n,
x,w>~
y
unrestricted in sign.
The maximand represents total quasi-welfare arising world-wide in the Eurodollar markets. It is calculated as in section 2.13 The first constraint states that the private non-bank customers in region h cannot borrow more than what the Eurobanks collectively are willing to lend them. The second constraint states that the market for inter-bank Eurodollar loans must clear. The third constraint states that the total amount of HThis assumption is by no means always fulfilled in reality but is introduced here in order to simplify the notation. 12More specifically, what I have in mind is some certainty-equivalent formulation of a general bank portfolio management problem. One attractive approach is to lay down a suitable set of chance-constraints (reflecting the need for liquidity, return and solvency in an uncertain environment) for each bank and to convert these constraints into corresponding certaintyequivalents. See Charnes and Thore (1966), Charnes and Littlechild (1968), and Charnes et al. (1982). 13It should perhaps be pointed out that there is an invisible term in the maximand equal to -~0-)~ (the interest paid on the demand accounts in New York is by definition put equal to zero). Geometrically, the demand for reserves by the New York main offices is a horizontal line r 3 = 0 in a price-quantity diagram. The contribution to quasi-welfare equals the surface below this line. There is no net contribution to quasi-welfare from inter-bank Euroloans. Nor is there any contribution from loans to U.S. main offices in New York because the U.S. demand for such loans will be assumed to be exogenously determined. Geometrically, this demand is a vertical line in a price-quantity diagram.
S. Thore, Spatial models of the Eurodollar market
60
Eurodollar funds lent back to domestic U.S. banks must equal the demand for such repatriated funds (the exogenous demand is written b). The fourth constraint states that the Eurobanks in each region collectively have to share the total local supply of Eurodollar deposits. Finally, there is the condition that each bank portfolio vector must be feasible. In general, program (9) is an instance of concave programming over a convex constraint set. Denote the .Lagrange multiplier of the first and fourth constraints by and ~, respectively. Let the optimal ~ be established as the market quotation of Eurodollar loans and the optimal ~ as the market quotation of Eurodollar deposits in the region. Denote the Lagrange multiplier of the second constraint by r2; it will be the market rate of inter-bank funds. Finally, denote the multiplier of the third constraint by r4; it is the market interest rate charged on Eurodollar loans to domestic U.S. banks. The Kuhn-Tucker conditions spell out the following results, holding at the point of optimum: t4 (i)
In each region, the demand price for Eurodollar loans equals the market interest rate,
(1o) (ii) In each region, the supply price for Eurodollar deposits equals the market interest rate, S(wh) = ~ .
(I 1)
There are also the obvious: (iii) in each region, there is equilibrium in the market for Eurodollar loans and in the market for Eurodollar deposits, and (iv) the market for inter-bank funds in Europe and the market in New York for repatriated Eurodollar funds are both in equilibrium. Pausing for a moment at this stage of the development of the ideas, note that if the balance sheet condition hz h2
lh
=
Qh
02)
h1
were the only constraint appearing in the constraint set yh,lS then the 14Assuming that positive Eurodollar transactions occur in all markets, relations (10) and (11) have been written as inequalities. But if any x h, ~r is zero, an inequality sign ( < , > , respectively) should be entered instead. 15Together with some suitable truncation to ensure that the constraint set were bounded and closed, as stipulated.
S. Thore, Spatial models of the Eurodollar market
61
following results could be established. The market quotations on all Eurodollar funds (loans to non-bank customers, inter-bank loans in Europe, loans to domestic U.S. banks, deposits) in all regions would all be equal. The amount of Eurodollar reserves held in New York by each bank would equal zero. However, these are purely speculative statements, since the constraint set Y~ surely contains portfolio constraints in addition to the balance sheet condition. At the very minimum, there must be some constraints laying down the need of each E u r o b a n k to hold some reserves in New York against the contingency that the physical clearing of incoming funds in New York might be delayed. While inter-bank Eurodollar transactions are often considered riskless, there is the lending risk to non-bank customers to be considered. In order to model in a satisfactory fashion how the E u r o b a n k tries to protect itself against such contingencies, it is necessary to spell out its investments in the local national currency as well. A satisfactory explanation of the Eurodollar portfolio choice of the b a n k can only be achieved within the framework of an integrated portfolio model that covers both Eurodollar and other national currency transactions. (v) At the point of optimum of program (9), each Eurobank maximizes the net return on its Eurodollar portfolio. Denote the optimal solution to p r o g r a m (9) and the corresponding Lagrange multiplier by an asterisk. Then ~ = ~ " is an optimal solution to * * m a x ~ *~ k +r2* Eyh2kh 2 -Ji-r4yh4k--d y~sk--r2* E~6kl h , h2
(13)
h1
subject to ~eyh. F o r some indication of the proof, see the footnote. 16
4. Summary In these pages, I have discussed the spatial flow of Eurodollar funds from the original depositors to the ultimate borrowers. The Eurodollar market 16In order to prepare for the proof, one needs first to ascertain that the constraint set in (9) is convex, bounded and closed. The variable x is clearly bounded from above; in addition some suitable upper bounding condition on w should be added to the constraint set in (9). Further, since the world-wide supply of Eurodollar deposits is certainly finite, it must always be possible to find some upper bound for w large enough so that it remains slack at the point of optimum. Then using a version of the Kuhn-Tucker theorem, a saddle point program equivalent to (9) may be written down. In the common fashion, each linear constraint in (9) is moved up into the objective function, multiplied by its Lagrange multiplier. The property (13) then follows from the definition of this saddle point. See Thore (1982).
62
S. Thore, Spatial models of the Eurodollar market
serves as a medium for the world-wide redistribution of short-term funds from surplus units and surplus regions to deficit units and deficit regions. The institutions which perform this role of financial intermediation have simply been referred to as 'Eurobanks'. In the first analysis, the Eurobanks were simply represented as passive agents, automatically accepting Eurodollar deposits and shuffling them on to the borrowers. The transactions cost was neglected; hence on-lending through a chain of several Eurobanks became in principle costless. The Eurodollar flow was frictionless, like that of water. One uniform Eurodollar interest became established world-wide, being the same for both depositing and lending. Wherever there was a demand for Eurodollar funds, at that interest rate, it would be met. The mathematical format employed has its roots in the pioneering work by Samuelson 1952 on spatial equilibrium. The optimization is carried out by maximizing an economic potential function, called by Samuelson 'net social pay-off and later by Takayama and Judge 'quasi-welfare'. To the knowledge of the present writer, these techniques are used here for the first time in a problem involving financial equilibrium. 'Quasi-welfare' in each region in calculated as the surface between the demand curve for Eurodollar loans and the supply curve for Eurodollar deposits. At the point of optimum, there is equilibrium in each regional market. Thus prepared, the attention was focused at the portfolio behavior of the Eurobanks. In order to understand the intermediating function of the participating banks, it is of course necessary to spell out the portfolio options of the banks and to discuss their portfolio choice. There is in the real world no automatism in the Eurodollar flow 'through' a Eurobank; the bank has full discretionary power to accept or not accept deposits and to invest incoming funds in any fashion it sees fit. In the simple portfolio model discussed in the text, incoming Eurodollar deposits can in principle be invested in any of the following ways: (i) (ii) (iii) (iv)
(v)
Deposits received from an exporter or other non-bank traders in a region can be lent to an importer or private trader in the same region. The funds can be redeposited with another bank and thus be routed to another region. The funds can be kept on deposit in New York, thus building up c/ash reserves. The Eurodollar funds can be lent back to the main office of a U.S. bank. What this means is that the funds enter the U.S. domestic economy as ordinary resident dollar funds. Physically, the funds leave the clearing circuit in New York. Finally, the individual Eurobank may sell the dollar funds outright in the foreign exchange markets rather than lending them. In order to analyze such a transaction, one would need to model the entire foreign
S. Thore, Spatial models of the Eurodollar market
63
exchange portfolio decisions of the bank. This would be a much more demanding task than the one presently envisioned. For the purpose of the present analysis, it suffices to note that the dollar funds would then be acquired by another Eurobank, so that eventually one would end up with one of the outcomes (i)-(iv) anyhow. The choice between these various options is supposed to be made within the context of a portfolio model of the Eurodollar holdings and liabilities for each bank (or group of banks). Considerations of liquidity and risk are reflected in the constraint set. If so desired, the portfolio model may be given a normative content by laying down provisions for lending risk which the banks 'ought to' consider (rather than the provisions for risk which they actually seem to be making). It might be argued that many international banks do not take adequate provisions for Eurodollar lending risk. It has become a generally accepted custom to consider inter-bank Eurodollar redepositing riskless. No collateral is ever asked for, nor is it offered. The name of a bank in good standing is supposed to offer protection enough. Reserves held on deposit in New York are typically kept at a minimum, and they are held for transactions purposes only. Incoming funds in New York are routinely considered available for immediate investment. How is it possible to extend Samuelson's extremal principle so that it will accommodate the presence of individual Eurobanks, each one optimally managing its own Eurodollar portfolio? What one is confronted with here is no longer one single optimization problem but a series of individual optimization problems that have to be solved simultaneously. As demonstrated by the mathematical analysis in section 3, Samuelson's extremal principle is stronger than what has previously been known. Define quasi-welfare as before. Spell out the constraint set of each participating bank portfolio. Maximize total world-wide quasi-welfare subject to the portfolio constraint sets and also subject to an available constraint in each regional market. Although the elements of the portfolio holdings do not appear in the mathematical expression for quasi-welfare, that same maximand will still seek out an allocation and pricing system which yields equilibrium in all regional markets and solves all the participating portfolio problems. In each region, there is equilibrium in the market for Eurodollar deposits and the market for Eurodollar loans. But these two interest rates may now no longer be equal. There will be a typical spread between the depositng rate and the lending rate. Further, at the point of optimum, each bank maximizes the net return on its portfolio constraint set. It must be recognized that the discussion of the balance sheet now displayed can only be the beginning of an analysis of Eurobank portfolio
64
S. Thore, Spatial models of the Eurodollar market
choice. F o r a m o r e detailed investigation, one would have to bring into the picture the entire foreign exchange portfolio of the Eurobanks. The individual E u r o b a n k can at any m o m e n t of time buy dollars outright in the exchange markets rather than borrowing them, and sell dollars rather than lending them. Such decisions are influenced b y the spot and forward quotations in the exchange markets and the range of possible returns on (local) financial investments in national markets. Selected notations
h=l,2 ..... l k = 1,2 . . . . . n h xh wh thlh2
~k,~2k, etc.
yh
Regions Eurobanks D e m a n d for Eurodollar loans Supply of Eurodollar deposits On-lending M a r k e t interest rate E u r o b a n k balance-sheet entries (see section 3.) Set of all feasible portfolio plans of a E u r o b a n k
References Chambers, D. and A. Charnes, 1961, Intertemporal analysis and dptimization of bank portfolios, Management Science, July, 393-410. Charnes, A. and S.C. Littlechild, 1968, Intertemporal bank asset choice with stochastic dependence, Systems Research Report No. 188 (Northwestern University, Evanston, IL). Charnes, A. and S. Thore, 1966, Planning for liquidity in financial institutions: The chanceconstrained method, Journal of Finance, Dec., 649-673. Charnes, A., J. Gallegos and S. Yao, 1982, A chance-constrained approach to bank dynamic balance sheet management, Center for Cybernetic Studies Research Report no. 428 (The University of Texas at Austin, Austin, TX) May. Cohen, K.J. and F.S. Hammer, 1967, Linear programming and optimal bank asset management decisions, The Journal of Finance, May, 147-165. Cohen, KJ. and S. Thore, 1970, Programming bank portfolios under uncertainty, Journal of Bank Research, Spring, 42-61. Fuchs, H.W., O.P. Farrish and R.W. Bohall, 1974, A model of the U.S. apple industry: A quadratic interregional intertemporal activity analysis formulation, American Journal of Agricultural Economics 56, 739-750. Kennedy, M., 1974, An economic model of the world oil market, The Bell Journal of Economics and Management Science 5, 540-577. Martin, LJ. and A.C. Zwart, 1975, A spatial model of the North American pork sector for the evaluation of policy alternatives, American Journal of Agricultural Economics 57, 55-56. Samuelson, P.A., 1952, Spatial price equilibrium and linear programming, The American Economic Review, 283-303. Takayama, T. and G.G. Judge, 1964, An interregional activity analysis model of the agricultural sector, Journal of Farm Economics 46, May, 349-365. Takayama, T. and G.G. Judge, 1964, Equilibrium among spatially separated markets: A reformulation, Econometrica 32, 510-524. Takayama, T. and G.G. Judge, 1971, Spatial and temporal price and allocation models (NorthHolland, Amsterdam).
S. Thore, Spatial models of the Eurodollar market
65
Taylor, C.R. and K.K. Frohberg 1977, The welfare effects of erosion controls, banning pesticides and limiting fertdlzer apphcatlon in the Corn Belt, American Journal of Agricultural Economics 59, 25-35. Thore, S., 1969, Credit networks, Economica, Feb., 42-57. Thore, S., 1970, Programming a credit network under uncertainty, Journal of Money, Credit and Banking, May, 129-246. Thore, S., 1980, Programming the network of financial intermediation (Universitetsforlaget, Oslo). Thore, S., 1982, The Takayama-Judge model of spatial equilibrium extended to convex production sets, Journal of Regional Science 22, no. 2, 203-212. Thore, S. and F. Kydland, 1972, Dynamic flow-of-funds networks, in: S. Eilon, ed., Applications of management science in banking and finance (Gower, London) 259-276. Uri, N.D., 1974, A spatial equilibrium analysis of electrical energy pricing and allocation, American Journal of Agricultural Economics 56, 628--634. Woodland, A.D., 1974, The solution of non-linear separable programs, American Journal of Agricultural Economics 56, 628-634.
SPATIAL MODELS OF THE EURODOLLAR MARKET Sten THORE* Unirersity of Texas, Austin, TX 78705, USA Received March 1983, final version received September 1983 The Eurodollar market can be viewed as a logistic system, converting Eurodollars deposited by exporters and other traders into Eurodollar loans raised by the borrowers (importers). This paper traces the spatial distribution of such funds-flow. An extremal principle for the determination of spatial equilibrium, originally due to Samuelson, is here for the first time applied to a problem of financial equilibrium. The behavior of participating international banks is specified by standard portfolio models. At the point of extremum, there is equilibrium in all regional markets, and each international bank maximizes the net return on its Eurodollar portfolio.
1. Motivation and antecedents The functions of banking are to convert low-risk deposits into risk-willing loanable funds, to convert short-term deposits into longer-term loans and to bring savers and borrowers together. The present paper is a study in this last and third function, in the process which redistributes geographically the flowof-funds from a regional vector of deposits into another regional vector of bank loans. The Eurodollar market has since its inception served as a conduit for international redistribution of short-term funds. In the 1970s, there arose a need for recycling the mounting dollar surplus of the OPEC countries back to oil-importing countries which needed loans to finance their trade deficits; the Eurodollar market became a vehicle for such recycling at the short-term end of the maturity spectrum. In the early 1980s, this international financial intermediation appears to be more crucial (and more risky) than ever. As the world has plunged into a global recession, several countries (Mexico, Brazil) have run into international payment difficulties, not to mention the plight of many poor countries of the third world. Several large international banks have reportedly overextended themselves in accommodating the Eurodollar needs of their customers. The Eurodollar market still functions, but its redistributive role is more precarious than ever. Seen against this background, it should be of some interest to develop a *The author is the Gregory A. Kozmetsky centennial fellow at the Institute for Constructive Capitalism, University of Texas.
0378-4266/84/$3.00 9 1984, Elsevier Science Publishers B.V. (North-Holland)
52
S. Thore, Spatial models of the Eurodollar market
model of the spatial and regional distribution of Eurodollar funds. In attempting to accomplish this task, the present paper brings together theoretical developments fetched from two very different fields of economic analysis: the analysis of systems of bank portfolios on the one hand, and spatial and regional equilibrium on the other. In a series of publications [Thore (1969, 1970, 1980), Thore and Kydland (1972)-I, the present author has attempted to evolve a theory of the fundsflow through a linked series of financial portfolios. The micro-behavioral unit in the theory is the portfolio of the individual commercial bank, savings and loan association, or other financial intermediary. The portfolio choice is described in some kind of optimizing framework; in the case of bank portfolios, the optimization may take the form of linear programming [see Chambers and Charnes (1961), Cohen and Hammer (1967)1, chance constrained programming (for references, see section 3), etc. The portfolios are joined together into a network where each separate portfolio can be visualized as a node and assets and liabilities as directed links. The fundsflow is viewed as a logistic process converting primary assets, like deposits, shares in S&L associations, etc., into final assets, like bank loans. In other words, the system of financial intermediation is conceived of as a kind of financial 'production' system, where the financial intermediaries transform the supply of primary assets of the savers into a suitable array of final assets issued by the ultimate borrowers [see Thore (1980, p. 40),1. Using this terminology, we may say that the Eurodollar market constitutes an international financial network, which receives 'primary' Eurodollar deposits from exporters and other traders and re-channels these funds into an optimal supply of 'final' Eurodollar loans raised by importers and other traders. A major tool in regional analysis is the Samuelson and Takayama-Judge model of spatial equilibrium. In the original formulation by Samuelson (1952), one is concerned with explaining the spatial allocation of demand and supply and the pricing of one single consumer good. Samuelson defined in each region an economic function, which he called 'social benefit' [and which Takayama and Judge (1971) later called 'quasi-welfare',1, by calculating the algebraic value of the area between the demand curve and the supply curve. Maximizing total social benefit net of transportation costs, Samuelson showed that, at the optimum, all regional markets are in equilibrium. Takayama and Judge (1964) extended Samuelson's analysis to the case of several consumer goods. They also showed how it is possible to provide an explicit specification of production in the model [Takayama and Judge (1964, 1971, ch. 14)]. They assumed the presence in each region of primary resources (including labor); the resources are converted into finished consumer goods by an activity analysis model. At the point of optimum, all markets are in equilibrium.
S. Thore, Spatial models of the Eurodollar market
53
The Takayama-Judge model has been used extensively in agricultural economics and energy economics, i.e., Uri (1974) and Kennedy (1974). As a result of such applications, the Takayama-Judge model has been extended and developed in several directions, including multi-period applications [Fuchs et al. (1974), Martin and Zwart (1975)], studies of disequilibrium [Taylor and Frohberg (1977)] and the development of computational routines for numerical solution [Woodland (1974)]. Recently, Thore (1982) showed that much more general specifications of the nature of the production process are permitted. Conventional smooth production functions of any suitable form may be used. So much for the theoretical background. What I propose to do in the present paper is to apply these techniques of spatial analysis to the study of the process of financial intermediation. Section 2 starts out with the simple Samuelson model. There is in this model just one single financial 'good': three months Eurodollar funds. There is a given demand function and a given supply function for such funds in each region. The contribution to quasi-welfare in each region is again obtained as the surface between the demand curve and the supply curve. By 'on-lending', often through a sequence of banks operating in the Eurodollar market, the original deposits in a region are re-routed world-wide to the ultimate Eurodollar borrowers. The transactions cost (or 'transportation cost', to use the term of logistics modelling) is small; if so desired, it may be entirely neglected. The analysis in section 3 digs deeper. The Eurodollar portfolio choice of the participating international banks is spelled out explicitly. In other words, we are now turning to a closer examination of each node of the Eurodollar network and providing an account of the 'production' of Eurodollar funds. The financial intermediaries which operate in the Eurodollar markets and which ultimately convert Eurodollar deposits into Eurodollar loans are subsidiaries of large U.S. banks, large Eurobanks and their subsidiaries, or just ordinary local commercial banks with some activity in the Eurodollar markets. These various institutions will be called the 'Eurobanks', for brief. An optimizing model of portfolio choice is specified for each Eurobank (or group of Eurobanks). This portfolio allows for the presence of dollar reserves kept on deposit in New York and for the possibility of U.S. main offices borrowing back Eurodollars from their European subsidiaries. The distinction between non-official and official funds in the Eurodollar market is important. Central banks, the IMF, and other international official bodies hold considerable amounts of Eurodollar deposits as part of their short-term portfolio in dollars and foreign currencies. The analysis both in section 2 and in section 3 will handle the presence of official funds, as long as they are given and known. No attempt will be made to explain the portfolio choice of international official bodies.
54
S. Thore, Spatial models of the Eurodollar market
The present analysis is an instance of partial economic theory. The income of exporters and importers is supposed to be given and known; so are the prices of all commodities and all financial instruments other than Eurodollars. In particular, the quotations on other foreign currencies and the entire foreign exchange portfolio of the Eurobanks is supposed to be given and fixed. Section 4 contains a summing-up and indicates briefly how some of the limitations now mentioned may be removed.
2. A spatial equilibrium model of the Eurodollar market Consider a spatial partitioning of the Eurodollar market into several regions (different countries, for instance) h = l , 2 ..... I. The importers and other private traders in each region demanding Eurodollar loans will be treated as one composite unit. Let x h represent the amount of Eurodollar loans they demand. The time period of analysis is three months; assume that all Eurodollar contracts have a maturity of three months and that they are evenly distributed over time. Let the cost of Eurodollar loans (the average cost rate during the period of analysis) be 100 r h percent. As we shall see later, any possible geographical variation in this rate would immediately be wiped out by arbitrage deals. There is, in each region, a demand function for Eurodollar loans, relating the demand x h to the loan cost r h. The interest cost to the borrowers.in other (national) financial markets, and their income, are all supposed to be constant and given. The model that we are about to develop is a partial model for the international Eurodollar market, assuming all other national variables to be fixed. Inverting the demand functions for Eurodollar loans, there will exist in each region a 'demand price '1 function for Eurodollars D(Xh) showing the interest cost at which the local demand for Eurodollar loans would equal x h. The demand price functions D(xh), h=1,2 ..... ! are supposed to be single-valued, positive, decreasing and differentiable. They, too, are defined for given interest costs of all alternative national channels of borrowing and for a given income. The contribution to the 'quasi-welfare' of the importers and other traders in region h demanding Eurodollar loans will now be discussed. The contribution equals the surface below the demand curve for loans (i.e., the total incremental interest payments which the borrowers would be willing to incur if the loans were extended to them incrementally, dollar by dollar), 2' 3 1The correct, but somewhat awkward, term would of course be 'demand interest rate'. 2Forming a concept analogous to 'consumers' surplus', we may say that the integral (1) equals borrowers' interest payments plus 'borrowers' surplus' in region h. aln the common fashion of an indefinite integral, the upper limit of integration of (1) is x h. The lower limit is some arbitrary fixed initial point. The integral is concave, i.e., dD/dx ~ is <0.
S. Thore, Spatial models of the Eurodollar market
S D(x h)dx h.
55
(1)
Turn now to the other side ol the market: the depositors. The exporters and other private traders in each region holding Eurodollar deposits will (like the borrowers) be treated as one composite unit. Let wh represent the amount of Eurodollars they desire to hold (=supply). There will then, in each region, exist a supply function for Eurodollar deposits relating the supply wh to their yield. The yield on other (national) financial investments, and the income of the composite group of depositors, are all supposed to be constant and given. Inverting these supply functions, there will exist in each region a 'supply price' function for Eurodollars S(w h) showing the interest yield at which the local supply of Eurodollar deposits would equa! w h. The supply price functions S(wh), h = l , 2 .... ,l are supposed to be single-valued, positive, increasing and differentiable. They, too, are defined for given yields on alternative national financial instruments and for a given income. The contribution to quasi-welfare of the exporters and other private traders in region h supplying Eurodollar deposits equals the negative of the surface below the supply curve for deposits (i.e., the total incremental interest returns which the depositors would require in order to motivate them to supply the funds, dollar by dollar), 4' 5
- ~ S(~e)dw ~.
(2)
One does well in remembering Samuelson's admonition not to try to read any economic significance into the concept of 'social pay-off' or 'quasiwelfare'. It is formed for operational purposes only. 6 (In particular, it is clearly not a utility function. Increased Eurodollar loans and an increased indebtedness position of the borrowers conventionally reduce their utility rather than adding to it.) Form the consolidated 7 portfolio of the Eurodollar assets and liabilities of all Eurobanks operating in each region h. I shall discuss this consolidated portfolio in some detail in section 3; at the present, it will be assumed that it simply reads:
4The expression (2) equals the negative of depositors' collected interest returns plus 'depositors' surplus' in region h. 5The upper limit of integration is wh. The lower limit is some arbitrary fixed point. The expression (2) is concave, since dS/dwh ~0. 6'An economist looking at these figures would naturally think of some kind of consumer's surplus concept ... However, the name consumer's surplus has all kinds of strange connotations in economics. To avoid these and to underline the completely artificial nature of my procedure I shall simply define a "net social pay-off' function...' [Samuelson (1952, p. 288)]. 7The consolidation mentioned refers to the netting out of all interbank assets and liabilities within each region. J.B.F.-- C
S. Thore, Spatial models of the Eurodollar market
56 Assets
Liabilities
Eurodollar loans to importers and other non-bank borrowers xh
Eurodollar deposits from exporters and other non-bank depositors
Eurodollar loans to Eurobanks in other regions
Eurodollar deposits from Eurobanks in other regions
~----~2thh2
~_~hI t hlh
Net (Eurodollar) worth ah
As illustrated by this balance sheet, the Eurobanks in each region are, of course, in no way constrained to lend locally the Eurodollar deposits and the net worth that originates in each region. The net interregional on-lending between blinks into region h is
~ t h ' h - - ~ t hh2 hI
(3)
h2
(t hlh2 denotes the gross on-lending from region tll to region h2). Consider now the program max ~ S D(xh)dxh -- ~ ~ SCwh)dwh, h
(4)
h
subject to
xh--.wh--(~th'h--~thh'~
h = l , 2 .... ,l,
x,w,t>O.
In words, program (4) instructs us to maximize the total quasi-welfare arising world-wide from interest, payments and interest receipts in the Eurodollar markets, subject to a balance sheet condition in each region (stating that the total out-flow of Eurodollar funds from the region cannot exceed the total in--flow of Eurodollar funds into the region)fl Program (4) is an instance of concave programming over a linear constraint set. Denote the Lagrange multiplier of the constraint by r h. It is non-negative. Let the optimal r h be established as the market interest rate on Eurodollar funds. SAny official Eurodollar funds transacted in each regional market may be absorbed by the entity ah (officialdeposits being added to a~; officialloans deducted from it).
S. Thore, Spatial models of the Eurodollar market
57
The Kuhn-Tucker conditions spell out the following results, holding at the point of optimum: 9 (i)
In each region, the demand price for Eurodollar loans equals the market interest rate,
O(x h)= r
h.
(5)
(ii) In each region, the supply price for Eurodollar deposits equals the market interest rate, S(w~) = r h.
(6)
Taken together, (i) and (ii) simply state that there will be equilibrium in the market for Eurodollar funds in each region. (iii) All regional Eurodollar quotations will be equal. For one gets t~ rht=r h2 for all hi and h 2.
(7)
3. Eurobank portfolio constraints In the simple model developed in section 2, Eurodollar funds are automatically shuffled back and forth between different banks and different regions in order to see to it that the supply of Eurodollar funds at all times matches the demand in each region. The banks serve as passive conduits for such Eurodollar flow without any behavioral relations of their own; On a more realistic level of model-building, it becomes necessary to specify an optimizing portfolio choice for each participating Eurodollar bank. Write the balance sheet of Eurobank k (kh = I, 2..... nh; the superscript is suppressed most of the time) in region h: Assets
Liabilities
Eurodollar loans to importers and other non-bank borrowers
Eurodollar deposits from exporters and other non-bank depositors
9Assuming that positive Eurodollar transactions occur in all markets, relations (5), (6) and (7) have been written as equalities. But if any .~, v/', t h~h: is zero, an inequality sign ( < , > , < , respectively) should be entered in the corresponding relation instead. l~ writing down (4), it was assumed that unit transactions costs, say c hjh~, were all zero. If they are non-negligible, however, the term ~..c*Jh~t ~h2 should be deducted from the maximand. Relation (7) will then come out as - r h ' + r ~ = - c h'h2.
S. Thore, Spatial models of the Eurodollar market
58
Assets
Liabilities
Eurodollar loans to Eurobanks in same and other regions
Eurodollar deposits from Eurobanks in same and other regions
Dollar account in New York
Net Eurodollar worth
,hh2 z J'21t
Dollar loans from a London subsidiary of a U.S. bank to head office in New York
The bank's supply of Eurodollar loans to private non-bank customers is written ~k- These customers are all supposed to reside in region h where the bank itself is located. Eurodollar loans made to other Eurobanks in other regions is written ~ ] ~ , where the summation is carried out over all regions h 2 except the home region h where the bank itself is located. Eurodollar loans made to other banks in the same region is ~ . Eurodollar reserves held in New York are written ~k. Any Eurodollar deposit received by a Eurobank must physically be transferred to the Eurobank's correspondent bank in New York and will add to the Eurobank's reserves kept on deposit there; conversely, any Eurodollars lent by a Eurobank will entail a drain of such reserves in New York. If, during some time period, the amount of Eurodollar deposits received by the Eurobank surpasses the amount of Eurodollars lent, there will build up a net accumulation of reserves in New York. Loans to the U.S. main office of a U.S.-owned Eurobank are written yh4k. If a U.S. bank borrows back Eurodollars from its London subsidiary (using the proceeds-of such a loan for domestic U.S. operations), Eurodollar funds which would otherwise have been available for lending in the Eurodollar market will be tied up as loans to New York. Deposits from private non-bank customers is ~k. Eurodollar deposits received from other Eurobanks in other regions are ~ ] h , where the summation is carried out over all regions hx except the home region h. Deposits received from other banks in the same region is ~6h. Now, denote the vector of balance sheet entries of bank k in region h by ~ . Assets are entered with a plus sign, and liabilities are entered with a minus sign. Let
~ r~,
(8)
S. Thore, Spatial models of the Eurodollar market
59
where yh is the set of all feasible portfolio vectors of the bank. Assume that yh is convex, bounded and closed. It What this symbolism means is that I am subsuming the existence of some suitable constraints which are laid down on the portfolio choice of the Eurobanks. At this stage of the development of the argument, it is not necessary to specify precisely what these constraints would look like; in some general fashion they must reflect the opportunities and the risks which confront the banks and the evaluation of the management of those opportunities and risks. 12 Thus prepared, consider the programming problem max ~
I D(xh) dxh -- ~ I S(wh)dwh,
h
(9)
h
subject to x h-x~,~ <0 , /,YlI~
0
k
h,k
k
)~k~ Y~,,
h = l , 2 . . . . . l, k = l , 2 . . . . . n,
x,w>~
y
unrestricted in sign.
The maximand represents total quasi-welfare arising world-wide in the Eurodollar markets. It is calculated as in section 2.13 The first constraint states that the private non-bank customers in region h cannot borrow more than what the Eurobanks collectively are willing to lend them. The second constraint states that the market for inter-bank Eurodollar loans must clear. The third constraint states that the total amount of HThis assumption is by no means always fulfilled in reality but is introduced here in order to simplify the notation. 12More specifically, what I have in mind is some certainty-equivalent formulation of a general bank portfolio management problem. One attractive approach is to lay down a suitable set of chance-constraints (reflecting the need for liquidity, return and solvency in an uncertain environment) for each bank and to convert these constraints into corresponding certaintyequivalents. See Charnes and Thore (1966), Charnes and Littlechild (1968), and Charnes et al. (1982). 13It should perhaps be pointed out that there is an invisible term in the maximand equal to -~0-)~ (the interest paid on the demand accounts in New York is by definition put equal to zero). Geometrically, the demand for reserves by the New York main offices is a horizontal line r 3 = 0 in a price-quantity diagram. The contribution to quasi-welfare equals the surface below this line. There is no net contribution to quasi-welfare from inter-bank Euroloans. Nor is there any contribution from loans to U.S. main offices in New York because the U.S. demand for such loans will be assumed to be exogenously determined. Geometrically, this demand is a vertical line in a price-quantity diagram.
S. Thore, Spatial models of the Eurodollar market
60
Eurodollar funds lent back to domestic U.S. banks must equal the demand for such repatriated funds (the exogenous demand is written b). The fourth constraint states that the Eurobanks in each region collectively have to share the total local supply of Eurodollar deposits. Finally, there is the condition that each bank portfolio vector must be feasible. In general, program (9) is an instance of concave programming over a convex constraint set. Denote the .Lagrange multiplier of the first and fourth constraints by and ~, respectively. Let the optimal ~ be established as the market quotation of Eurodollar loans and the optimal ~ as the market quotation of Eurodollar deposits in the region. Denote the Lagrange multiplier of the second constraint by r2; it will be the market rate of inter-bank funds. Finally, denote the multiplier of the third constraint by r4; it is the market interest rate charged on Eurodollar loans to domestic U.S. banks. The Kuhn-Tucker conditions spell out the following results, holding at the point of optimum: t4 (i)
In each region, the demand price for Eurodollar loans equals the market interest rate,
(1o) (ii) In each region, the supply price for Eurodollar deposits equals the market interest rate, S(wh) = ~ .
(I 1)
There are also the obvious: (iii) in each region, there is equilibrium in the market for Eurodollar loans and in the market for Eurodollar deposits, and (iv) the market for inter-bank funds in Europe and the market in New York for repatriated Eurodollar funds are both in equilibrium. Pausing for a moment at this stage of the development of the ideas, note that if the balance sheet condition hz h2
lh
=
Qh
02)
h1
were the only constraint appearing in the constraint set yh,lS then the 14Assuming that positive Eurodollar transactions occur in all markets, relations (10) and (11) have been written as inequalities. But if any x h, ~r is zero, an inequality sign ( < , > , respectively) should be entered instead. 15Together with some suitable truncation to ensure that the constraint set were bounded and closed, as stipulated.
S. Thore, Spatial models of the Eurodollar market
61
following results could be established. The market quotations on all Eurodollar funds (loans to non-bank customers, inter-bank loans in Europe, loans to domestic U.S. banks, deposits) in all regions would all be equal. The amount of Eurodollar reserves held in New York by each bank would equal zero. However, these are purely speculative statements, since the constraint set Y~ surely contains portfolio constraints in addition to the balance sheet condition. At the very minimum, there must be some constraints laying down the need of each E u r o b a n k to hold some reserves in New York against the contingency that the physical clearing of incoming funds in New York might be delayed. While inter-bank Eurodollar transactions are often considered riskless, there is the lending risk to non-bank customers to be considered. In order to model in a satisfactory fashion how the E u r o b a n k tries to protect itself against such contingencies, it is necessary to spell out its investments in the local national currency as well. A satisfactory explanation of the Eurodollar portfolio choice of the b a n k can only be achieved within the framework of an integrated portfolio model that covers both Eurodollar and other national currency transactions. (v) At the point of optimum of program (9), each Eurobank maximizes the net return on its Eurodollar portfolio. Denote the optimal solution to p r o g r a m (9) and the corresponding Lagrange multiplier by an asterisk. Then ~ = ~ " is an optimal solution to * * m a x ~ *~ k +r2* Eyh2kh 2 -Ji-r4yh4k--d y~sk--r2* E~6kl h , h2
(13)
h1
subject to ~eyh. F o r some indication of the proof, see the footnote. 16
4. Summary In these pages, I have discussed the spatial flow of Eurodollar funds from the original depositors to the ultimate borrowers. The Eurodollar market 16In order to prepare for the proof, one needs first to ascertain that the constraint set in (9) is convex, bounded and closed. The variable x is clearly bounded from above; in addition some suitable upper bounding condition on w should be added to the constraint set in (9). Further, since the world-wide supply of Eurodollar deposits is certainly finite, it must always be possible to find some upper bound for w large enough so that it remains slack at the point of optimum. Then using a version of the Kuhn-Tucker theorem, a saddle point program equivalent to (9) may be written down. In the common fashion, each linear constraint in (9) is moved up into the objective function, multiplied by its Lagrange multiplier. The property (13) then follows from the definition of this saddle point. See Thore (1982).
62
S. Thore, Spatial models of the Eurodollar market
serves as a medium for the world-wide redistribution of short-term funds from surplus units and surplus regions to deficit units and deficit regions. The institutions which perform this role of financial intermediation have simply been referred to as 'Eurobanks'. In the first analysis, the Eurobanks were simply represented as passive agents, automatically accepting Eurodollar deposits and shuffling them on to the borrowers. The transactions cost was neglected; hence on-lending through a chain of several Eurobanks became in principle costless. The Eurodollar flow was frictionless, like that of water. One uniform Eurodollar interest became established world-wide, being the same for both depositing and lending. Wherever there was a demand for Eurodollar funds, at that interest rate, it would be met. The mathematical format employed has its roots in the pioneering work by Samuelson 1952 on spatial equilibrium. The optimization is carried out by maximizing an economic potential function, called by Samuelson 'net social pay-off and later by Takayama and Judge 'quasi-welfare'. To the knowledge of the present writer, these techniques are used here for the first time in a problem involving financial equilibrium. 'Quasi-welfare' in each region in calculated as the surface between the demand curve for Eurodollar loans and the supply curve for Eurodollar deposits. At the point of optimum, there is equilibrium in each regional market. Thus prepared, the attention was focused at the portfolio behavior of the Eurobanks. In order to understand the intermediating function of the participating banks, it is of course necessary to spell out the portfolio options of the banks and to discuss their portfolio choice. There is in the real world no automatism in the Eurodollar flow 'through' a Eurobank; the bank has full discretionary power to accept or not accept deposits and to invest incoming funds in any fashion it sees fit. In the simple portfolio model discussed in the text, incoming Eurodollar deposits can in principle be invested in any of the following ways: (i) (ii) (iii) (iv)
(v)
Deposits received from an exporter or other non-bank traders in a region can be lent to an importer or private trader in the same region. The funds can be redeposited with another bank and thus be routed to another region. The funds can be kept on deposit in New York, thus building up c/ash reserves. The Eurodollar funds can be lent back to the main office of a U.S. bank. What this means is that the funds enter the U.S. domestic economy as ordinary resident dollar funds. Physically, the funds leave the clearing circuit in New York. Finally, the individual Eurobank may sell the dollar funds outright in the foreign exchange markets rather than lending them. In order to analyze such a transaction, one would need to model the entire foreign
S. Thore, Spatial models of the Eurodollar market
63
exchange portfolio decisions of the bank. This would be a much more demanding task than the one presently envisioned. For the purpose of the present analysis, it suffices to note that the dollar funds would then be acquired by another Eurobank, so that eventually one would end up with one of the outcomes (i)-(iv) anyhow. The choice between these various options is supposed to be made within the context of a portfolio model of the Eurodollar holdings and liabilities for each bank (or group of banks). Considerations of liquidity and risk are reflected in the constraint set. If so desired, the portfolio model may be given a normative content by laying down provisions for lending risk which the banks 'ought to' consider (rather than the provisions for risk which they actually seem to be making). It might be argued that many international banks do not take adequate provisions for Eurodollar lending risk. It has become a generally accepted custom to consider inter-bank Eurodollar redepositing riskless. No collateral is ever asked for, nor is it offered. The name of a bank in good standing is supposed to offer protection enough. Reserves held on deposit in New York are typically kept at a minimum, and they are held for transactions purposes only. Incoming funds in New York are routinely considered available for immediate investment. How is it possible to extend Samuelson's extremal principle so that it will accommodate the presence of individual Eurobanks, each one optimally managing its own Eurodollar portfolio? What one is confronted with here is no longer one single optimization problem but a series of individual optimization problems that have to be solved simultaneously. As demonstrated by the mathematical analysis in section 3, Samuelson's extremal principle is stronger than what has previously been known. Define quasi-welfare as before. Spell out the constraint set of each participating bank portfolio. Maximize total world-wide quasi-welfare subject to the portfolio constraint sets and also subject to an available constraint in each regional market. Although the elements of the portfolio holdings do not appear in the mathematical expression for quasi-welfare, that same maximand will still seek out an allocation and pricing system which yields equilibrium in all regional markets and solves all the participating portfolio problems. In each region, there is equilibrium in the market for Eurodollar deposits and the market for Eurodollar loans. But these two interest rates may now no longer be equal. There will be a typical spread between the depositng rate and the lending rate. Further, at the point of optimum, each bank maximizes the net return on its portfolio constraint set. It must be recognized that the discussion of the balance sheet now displayed can only be the beginning of an analysis of Eurobank portfolio
64
S. Thore, Spatial models of the Eurodollar market
choice. F o r a m o r e detailed investigation, one would have to bring into the picture the entire foreign exchange portfolio of the Eurobanks. The individual E u r o b a n k can at any m o m e n t of time buy dollars outright in the exchange markets rather than borrowing them, and sell dollars rather than lending them. Such decisions are influenced b y the spot and forward quotations in the exchange markets and the range of possible returns on (local) financial investments in national markets. Selected notations
h=l,2 ..... l k = 1,2 . . . . . n h xh wh thlh2
~k,~2k, etc.
yh
Regions Eurobanks D e m a n d for Eurodollar loans Supply of Eurodollar deposits On-lending M a r k e t interest rate E u r o b a n k balance-sheet entries (see section 3.) Set of all feasible portfolio plans of a E u r o b a n k
References Chambers, D. and A. Charnes, 1961, Intertemporal analysis and dptimization of bank portfolios, Management Science, July, 393-410. Charnes, A. and S.C. Littlechild, 1968, Intertemporal bank asset choice with stochastic dependence, Systems Research Report No. 188 (Northwestern University, Evanston, IL). Charnes, A. and S. Thore, 1966, Planning for liquidity in financial institutions: The chanceconstrained method, Journal of Finance, Dec., 649-673. Charnes, A., J. Gallegos and S. Yao, 1982, A chance-constrained approach to bank dynamic balance sheet management, Center for Cybernetic Studies Research Report no. 428 (The University of Texas at Austin, Austin, TX) May. Cohen, K.J. and F.S. Hammer, 1967, Linear programming and optimal bank asset management decisions, The Journal of Finance, May, 147-165. Cohen, KJ. and S. Thore, 1970, Programming bank portfolios under uncertainty, Journal of Bank Research, Spring, 42-61. Fuchs, H.W., O.P. Farrish and R.W. Bohall, 1974, A model of the U.S. apple industry: A quadratic interregional intertemporal activity analysis formulation, American Journal of Agricultural Economics 56, 739-750. Kennedy, M., 1974, An economic model of the world oil market, The Bell Journal of Economics and Management Science 5, 540-577. Martin, LJ. and A.C. Zwart, 1975, A spatial model of the North American pork sector for the evaluation of policy alternatives, American Journal of Agricultural Economics 57, 55-56. Samuelson, P.A., 1952, Spatial price equilibrium and linear programming, The American Economic Review, 283-303. Takayama, T. and G.G. Judge, 1964, An interregional activity analysis model of the agricultural sector, Journal of Farm Economics 46, May, 349-365. Takayama, T. and G.G. Judge, 1964, Equilibrium among spatially separated markets: A reformulation, Econometrica 32, 510-524. Takayama, T. and G.G. Judge, 1971, Spatial and temporal price and allocation models (NorthHolland, Amsterdam).
S. Thore, Spatial models of the Eurodollar market
65
Taylor, C.R. and K.K. Frohberg 1977, The welfare effects of erosion controls, banning pesticides and limiting fertdlzer apphcatlon in the Corn Belt, American Journal of Agricultural Economics 59, 25-35. Thore, S., 1969, Credit networks, Economica, Feb., 42-57. Thore, S., 1970, Programming a credit network under uncertainty, Journal of Money, Credit and Banking, May, 129-246. Thore, S., 1980, Programming the network of financial intermediation (Universitetsforlaget, Oslo). Thore, S., 1982, The Takayama-Judge model of spatial equilibrium extended to convex production sets, Journal of Regional Science 22, no. 2, 203-212. Thore, S. and F. Kydland, 1972, Dynamic flow-of-funds networks, in: S. Eilon, ed., Applications of management science in banking and finance (Gower, London) 259-276. Uri, N.D., 1974, A spatial equilibrium analysis of electrical energy pricing and allocation, American Journal of Agricultural Economics 56, 628--634. Woodland, A.D., 1974, The solution of non-linear separable programs, American Journal of Agricultural Economics 56, 628-634.