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The theory of interest rate arbitrage : review and extensions Dolf, Benedikt 1973

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THE THEORY OF INTEREST RATE ARBITRAGE: REVIEW AND EXTENSIONS by BENEDIKT DOLF Licence en Droit, Universite' de Geneve, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION 1 in the Faculty of Commerce and Business Administration We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Commerce The University of British Columbia Vancouver 8, Canada Date A p r i l 9, 1973 i i ABSTRACT The theory of interest rate arbitrage is the subject of this paper. Three objectives have guided the structure of the analysis. The f i r s t concerns the l i t e r a t u r e about interest arbi-trage and the determination of the forward exchange rate. The contributions are numerous and diverse; a general consen-sus has not yet been reached. A c r i t i c a l survey of the l i t e r a t u r e i s therefore undertaken with the aim of putting the various contributions into perspective. It is the con-tention underlying this part of the present paper that, diverse as the many models of interest arbitrage may be, they are a l l s p e c i f i c applications of Keynes' basic theory. In this sense, the evolution of theoretical thought about i n -terest arbitrage can be seen as descending from the general to the s p e c i f i c . An important d i s t i n c t i o n is made, however. Keynes and his followers saw interest arbitrage as the major determinant of the forward rate. Modern economists, on the other hand, recognize that interest arbitrage is only one of many factors instrumental in the determination of the forward rate. Two apparently s i g n i f i c a n t considerations have not received the attention they deserve in the l i t e r a t u r e . The f i r s t concerns the impact of arbitrage on interest rates ; i i i the other the existence of multiple interest d i f f e r e n t i a l s . The relationship between forward and interest rates has not been explored in depth in the l i t e r a t u r e . To investigate the impact of arbitrage on interest rates forms therefore the second objective of this study. It is contended that the relationship between foreign exchange rates and interest rates is of a dif f e r e n t nature under different systems of exchange rate determination. Accordingly, Chapter III examines the impact of interest arbitrage on interest rates under pegged exchange rates; Chapter IV discusses the problem in a setting of f l e x i b l e exchange rates. Under pegged exchange rates, money markets are d i r e c t l y affected by arbitrage. The r e l a t i v e variations of forward and interest rates are found to depend on the r e l a t i v e size of the respective markets. Small markets must absorb a greater amount of variation than do larger ones. Under f l e x i b l e spot rates, the burden of adjustment is borne e n t i r e l y by foreign exchange rates. The r e l a t i v e varia-tion of spot versus forward rates depends again on the r e l a -tive size of the two markets. Chapter IV has been expanded in order to investigate the consequences of an autonomous change in forward rates. The findings are contrasted with the re-sults of an autonomous change in interest rates. This analy-sis solves a puzzling paradox to be found in the l i t e r a t u r e . Most models of interest arbitrage are based on the u n r e a l i s t i c assumption that a unique interest d i f f e r e n t i a l exists between any two markets. To examine the cognitive i v status of such models forms the t h i r d objective in this paper. It is concluded in Chapter V that conventional models apply only under very r e s t r i c t i v e conditions. S p e c i f i c a l l y , i t is required that one p a r t i c u l a r interest d i f f e r e n t i a l dominates a l l others in the sense that arbitragers concentrate their e f f o r t s exclusively on i t . The alternative condition would be that the r i s k structures in two money markets are such that a l l interest d i f f e r e n t i a l s are i d e n t i c a l at a l l times. These conditions are not bound to be realized in the real world. The predictions of s i m p l i f i e d models are therefore not l i k e l y to be accurate both with respect to the level of the forward exchange rate and to the amount of funds transferred by arbitragers. V TABLE OF CONTENTS CHAPTER I. INTRODUCTION 1.1 Objectives 1.11 C r i t i c a l Review of the Literature 1.12 The Influence of Interest Arbitrage under Pegged Exchange Rates 1.13 Interest Arbitrage under Flexible Exchange Rates 1.14 Relevant Pair of Interest-Bearing Securities 2 Methodology . 3 Definitions and Symbols Foreign Exchange Foreign Exchange Market Foreign Exchange Rates Expected Spot Rates 6 8 31 32 33 34 35 36 37 38 39 40 Spot Transactions Forward Outright Transactions Swap Covering Hedging Money Market CHAPTER II. CRITICAL REVIEW OF THE LITERATURE The , 11 , 12 . 13 .14 2.1  Interest Rate Parity Theory The Keynesian Theory of Interest Arbitrage Summary of Keynes' Theory Other Considerations Summary 2.2 Modern Developments: Imperfect Substitutes and Simultaneous Determination of Spot and Forward Rat es 2.21 Neldner's Model 2.22 Other Considerations 2.3 Summary and Appraisal 14 14 CHAPTER III THE INFLUENCE OF INTEREST ARBITRAGE ON INTEREST RATES UNDER PEGGED EXCHANGE RATES Problem . . and Interest 3.1 Outline and Relevance of the 3.2 Short-Term Capital Movements under Pegged Exchange Rates 3.3 Interest Rates and Forward Exchange Rates 3.31 Interactions between Foreign Exchange Markets and the Foreign Money Market Rates 27 59 73 73 75 77 v i 3.32 Interactions between Foreign Exchange Markets and Domestic and Foreign Money Mark ets 3.4 Market Size and Price Variations 82 3.5 Reinterpretation of Results 84 CHAPTER IV. INTEREST ARBITRAGE UNDER FLEXIBLE EXCHANGE RATES 90 4.1 Introduction 90 4.2 Capital Movements under a System of Flexible Exchange Rates 91 4.3 The Process of Adjustment 93 4.31 Autonomous Change in Interest Rates 4.32 Autonomous Change in the Forward Rate 4.4 Reinterpretation 103 CHAPTER V. MULTIPLE INTEREST DIFFERENTIALS AND THE THEORY OF INTEREST RATE ARBITRAGE . . . . . . 109 5.1 Interest Rate D i f f e r e n t i a l s in the Real World . . 109 5.2 Single Interest Rate Models 110 5.3 Multiple Interest Rate Models I l l 5.31 Risk as Discriminating Factor 5.32 Other Discriminating Factors 5.4 Implications for the Theory of Interest Rate Arbitrage 118 5.5 Problems of Empirical Validation 121 5.6 Summary and Conclusion 123 CHAPTER VI. CONCLUSION 129 APPENDIX 137 BIBLIOGRAPHY 144 v i i LIST OF ILLUSTRATIONS Figure Page 1. Arbitrage schedule for spot market (IRPT) 29 2. Supply of forward exchange by individual bank acting as arbitrager 33 3. Supply of forward exchange by individual non-bank acting as arbitrager 34 4. Arbitrage schedule 35 5. Individual supply curves under dif f e r e n t quantities of money supply 36 6. AA schedule for di f f e r e n t quantities of money supply 37 7. Demand and supply of forward exchange by individual speculators 41 8. Speculator's schedule 42 9. Speculators' demand for spot exchange at t+90 . . . 43 10. Demand and supply of forward exchange by merchants 45 11. Demand (supply) for spot exchange by non-arbitragers 49 12. Determination of the spot rate, the forward rate and the foreign interest rate 78 13. Determination of equilibrium in the forward market 94 14. Equilibrium after autonomous change in interest rates 98 15. Equilibrium after autonomous change in the forward rate 102 16. Determination of spot and forward exchange rates after autonomous change in the forward rate . . . . 137 v i i i 17. Flow diagram 139 18. Variation of the forward rate 141 19. Amplifying effect of k 141 20. Amplifying effect of ^ 141 21. Amplifying effect of /3 142 22. Amplifying effect of C) 142 1 CHAPTER I INTRODUCTION The theory of forward exchange has two major objectives. The f i r s t is to explain the behavior of individual economic units operating in the forward market. The second objective is to explain the interaction of aggregate demand and supply of forward exchange stemming from a n a l y t i c a l l y distinguishable groups of market participants. The theory thereby explains the determination of the forward exchange r a t e s . 1 Just as the forward exchange market is a segment of the foreign exchange market, so is the theory of forward exchange a subset and integral part of the theory of foreign exchange. In turn, the theory of foreign exchange deals systematically with problems ar i s i n g from the interface of two or more economic systems. Until quite recently the importance of a theory of forward exchange for the understanding of relations between economic systems has not been recognized. It was not u n t i l 1923 that Keynes proposed the f i r s t systematic set of 2 rules to explain the behavior of forward exchange rates. As late as 1939, a prominent author could claim that the forward contract introduces no real changes in the theory of foreign exchange. 3 Since then, most writers have come to re a l i z e that the presence of forward contract f a c i l i t i e s introduces q u a l i t a t i v e 2 changes into the system of foreign exchange. The workings of a modern foreign exchange market cannot be understood i f the c l a s s i c theory is not supplemented by a theory of forward exchange.^ Were exchange rates fixed once and for a l l , there would be no need for forward f a c i l i t i e s . It is only in times of uncertainty about the future behavior of spot exchange rates that individuals resort to the forward market. Not surpris-ingly, most theoretical contributions to the theory of forward exchange were made in periods of t r a n s i t i o n from one system of exchange rate determination to another.^ There is h i s -t o r i c a l logic to the renewed intensity with which economists have directed t h e i r e f f o r t s towards elaborating and testing forward exchange theory in recent years. The intention in this study is to explore some avenues of the theory which, i t is f e l t , have not received the attention they deserve. Before progressing to the substan-tive part of this paper, i t s objectives w i l l be indicated, the methodological approach discussed and a number of d e f i -nitions and symbols w i l l be presented for l a t e r use. 1.1 Obj ectives This study has four objectives: 1.11 C r i t i c a l Review of the Literature It is an indication of the current state of the theory of forward exchange that there are v i r t u a l l y as many theories as there are authors. Ever since Keynes set forth what has 3 come to be ca l l e d the interest rate parity theory (IRPT), economists have been irked by i t s f a i l u r e to explain r e a l i t y . As a res u l t , variations have been expounded and numerous alter-natives proposed to the point where this student of the l i t e r a t u r e had to spend the best part of his time trying to see order where there appears to be chaos. Nonetheless, the task was undertaken. In Chapter II, the attempt is made to isol a t e the essential features of IRPT and post-Keynesian models. We sh a l l try to demonstrate that modern theories are extensions of Keynes' system; they incorporate additional factors which modern writers believe to be systematically related to the phenomena to be explained. On the basis of that information, furthermore, we hope to indicate which aspects of contem-porary forward exchange theory require further elaboration or substantiation. 1.12 The Influence of Interest Arbitrage on Interest Rates under Pegged Exchange Rates Chapter III contains the f i r s t part of an analysis of the relationship between interest rates and foreign exchange rates as they are affected by interest arbitrage. In a world of pegged spot rates, cap i t a l transfers by arbitragers lead to changes in the internal money supply of an economy. Interest rates are therefore also affected. It is the con-tention in this chapter that interest rates must therefore be included as endogenous variables in the analysis. To this 4 end, Chapter III presents a simple model of interest arbitrage which accommodates interest rates of two international centers as well. This model comprises four markets: the spot, the forward and two money markets. In this context, i t w i l l be attempted to determine whether conclusions derived from simple models are s t i l l V a l i d . The solution depends on the burden which the various markets w i l l have to bear in the process of adjustment. A theory w i l l be set forth, designed to explain the r e l a t i v e impact of arbitrage on the four market prices contained in the model. It says, in essence, that the variation of each price depends on the r e l a t i v e size of the respective market. With the help of this theory i t would be possible to pinpoint situations where the impact of arbitrage is most strongly f e l t in the forward market and others where interest rates are mainly affected by interest arbitrage. 1.13 Interest Arbitrage under Flexible Exchange Rates Chapter IV contains the second part of the analysis of the r e l a t i v e impact of arbitrage on exchange rates and on interest rates. It is argued there that, under a pure system of f l e x i b l e exchange rates, the national supply of money and hence interest rates remain unaffected by ca p i t a l flows. Accordingly, currently available models of interest arbitrage under such systems are exhaustive and need not be expanded. The analysis in Chapter IV centers therefore on the problem of the r e l a t i v e variation of spot versus forward rates. The 5 relevance of the r e l a t i v e size of the two markets i s examined again. Moreover, the analysis attempts to i s o l a t e the con-sequences of autonomous changes in money markets for the spot and forward rates. It contrasts these findings with the consequences of direct intervention in the forward market. This is done in an e f f o r t to resolve the d i f f i c u l t i e s which are encountered when various contributions to the theory of interest arbitrage are compared. 1.14 Relevant Pair of Interest-Bearing Securities Arbitragers base their decisions as to where to invest th e i r funds on a comparison of domestic with foreign interest rates, taking into consideration the loss or gain associated with foreign exchange operations. Given that there are a multitude of interest-bearing s e c u r i t i e s available to inves-tors, the question must be asked which of the many possible combinations arbitragers consider to be relevant. It w i l l be shown la t e r (Section 2.11) that a l l authors, with the exception of Einzig, assume that a unique rate of interest prevails in a l l markets.^ Such an assumption is obviously a gross o v e r s i m p l i f i -cation of r e a l i t y . It is pertinent to ask, however, whether such an abstraction is not j u s t i f i e d in the sense that the conclusions s t i l l conform quite well with observed phenomena. This question is related to another problem, namely that of the cognitive status of models based on such simplifying assumptions. Can empirical work tr u l y reject the underlying 6 theory i f tests are based on a r b i t r a r i l y selected pairs of securities? So far, there exists indeed no theory which would predict what se c u r i t i e s are selected by arbitragers. Chapter V is devoted to an exploration of these problems. 1. 2 Methodology Chapter II assembles major contributions to the theory of forward exchange. The main elements of Keynes 1 IRPT are presented to set the stage for the discussion. In following sections, additional factors that other authors have intro-duced into Keynes' theory w i l l be analysed. In contrast to the foregoing, Neldner's model is presented in order to high-l i g h t the points where he and other representatives of what 7 may be called the modern school depart from IRPT. Neldner's model i s followed by an analysis of factors that other modern authors have stressed. In an attempt to reconcile the many contradictory opinions which have been voiced over the years, reference is made to Nagel's theory g of the structure of science. Chapters III and IV u t i l i z e simple models of interest arbitrage. The analysis in both chapters is carried out in terms of s t a t i c equilibrium theory. Most of the discussion is concerned, furthermore, with s e n s i t i v i t y analysis of various market prices. The s e n s i t i v i t y of these prices is related f i r s t to the slopes of the constituent supply and demand schedules. A further section relates the notions of slopes and of market size. The results are subsequently reinterpreted in terms of market si z e . 7 The analysis in Chapter IV i s supplemented by a mathe-matical appendix. In i t , the dynamic process of adjustment to equilibrium is detailed. This was done in order to ascer-tain the possible range of variations in market prices. The mathematical analysis thus provides a convenient check of the diagrammatical treatment which is of necessity somewhat haphazard when more than one market is involved. Chapter V contrasts the i d e a l i z e d setting of a model containing a single interest d i f f e r e n t i a l with the more complicated s i t u a t i o n where multiple interest d i f f e r e n t i a l s can be formed. Considerable space is devoted to the explora-tion of discriminating factors which serve to distinguish between different types of s e c u r i t i e s . The analysis i s separated into two parts, corres-ponding to different types of discriminating factors. The f i r s t part deals with d i f f e r e n t i a l risk of default and considerable importance is attributed to this factor. It i s then examined under- what circumstances, i . e . what con-fig u r a t i o n of r i s k d i f f e r e n t i a l s , models containing single interest d i f f e r e n t i a l s could be usefully applied to situa-tions where multiple interest d i f f e r e n t i a l s do in fact exist. The second part draws an analogy with problems rel a t i n g to long-term debt instruments of the same risk class. It is an attempt to l i s t several other factors which tend to discriminate between apparently similar s e c u r i t i e s . These factors have been largely overlooked in the l i t e r a -ture because their relevance becomes evident only when 8 multiple interest d i f f e r e n t i a l s are considered. These considerations w i l l help to support the propo-s i t i o n that situations where simple models can be used d i r e c t l y are unlikely to exist in the real world. The consequences of these findings for the theory of interest arbitrage are then examined. 9 1.3 Definitions and SymboIs 1.31 Foreign Exchange The exhaustive d e f i n i t i o n is given byN E i n z i g : ^ Means of payment or instruments of short-term credit of other countries with different monetary units, regarded from the point of view of th e i r purchase or sale against the national currency, or that of t h e i r holdings of reserves. Einzig's term "foreign exchange" is equivalent to the more commonly used "foreign currency." In banking c i r c l e s the expression "forex" i s widely used. 1.32 Foreign Exchange Market Foreign currencies are exchanged against domestic currency on the foreign exchange market. It can be thought to be the place where supply of and demand for foreign currency are met.1''" The foreign exchange market consists of a spot market and a forward market. The d i s t i n c t i o n is based on the length of time elapsed between conclusion of the foreign exchange contract and delivery. 1.33 Foreign Exchange Rates (Exchange Rates) 9 1.331 Spot Rate The price of one unit of foreign currency, expressed in terms of domestic currency. Delivery takes place two days after conclusion of the contract. The spot rate is quoted as the number of units (or fractions thereof) of domestic currency required to buy one unit of foreign currency. Example: the 1 2 price of one Deutsche Mark is $.3600 per DM. Delivery takes 13 place two days after conclusion of the contract. The symbol SX w i l l be used to denote the spot rate. Where i t is necessary to indicate the time at which the contract has been concluded, the subscript t w i l l be used, as in SX t. 1.332 Forward Rate The price of one unit of foreign currency, expressed in terms of domestic currency, for delivery at a future date. Symbol: FX, respectively FX^. Forward rates are usually quoted as a discount from, or a premium on, the spot rate. One way of notation is to indicate the amount by which forward rates deviate from the spot rate. The difference (the forward margin) is expressed in fractions of domestic currency. If the'margin is posi-tive--!.e. the forward rate is higher than the spot rate--the forward rate is said to be at a premium. It is said to be at a discount when i t is lower than the spot rate. It is more common to express premium/discount as a percentage deviation of the forward from the spot rate, usually on a per annum basis. The forward percentage margin is com-puted as fo1lows: 10 Forward margin = (FX-SX) 100 12 t SX Where FX ... forward rate SX ... spot rate t ... duration of forward contract in months A forward margin is calle d premium when po s i t i v e , discount when negative. Note that a premium on the foreign currency implies a discount on the domestic one, and vice versa. This is not a term commonly used by foreign exchange dealers. It i s used for a n a l y t i c a l purposes by theoretical writers. It i s defined as the spot rate expected to pr e v a i l at a certain date in the future. Symbol: E(SX). The subscript t is used when i t is necessary to indicate the time for which these expectations are. held. A purchase or sale of foreign exchange on the spot market. A purchase or sale of foreign exchange on the forward market. A swap consists of a spot transaction and a forward outright transaction in the opposite d i r e c t i o n . They are undertaken simultaneously. 1.34' Expected Spot Rate 1.35 Spot Transactions 1.36 Forward Outright Transaction (Forward Transaction) 1.37 Swap 11 1.38 Cove ring Covering i s defined as an arrangement to safeguard against the exchange ris k on a payment of a de f i n i t e amount to be made or received on a de f i n i t e date with a s e l f -l i q u i d a t i n g commercial or f i n a n c i a l transaction.14 Covering is undertaken by means of a forward outright t rans act ion. 1.39 Hedging Hedging by means of forward transactions i s an arrangement to safeguard against an i n d e f i -nite and indir e c t exchange risk arising from the existence of assets or l i a b i l i t i e s whose value is l i a b l e to be affected by changes in spot rates.15 Hedging i s also implemented by a forward outright transaction. 1.40 Money Market For the purpose of this paper, money market is defined as the thought-of location where demand for and supply of short-term s e c u r i t i e s are met. In a two-country model of interest arbitrage, two money markets are involved. For convenience, they are usually referred to as domestic and foreign money markets. Accordingly, the y i e l d on domestic se c u r i t i e s w i l l be denoted by RD; RF stands for the y i e l d on foreign s e c u r i t i e s . In keeping with a well-established convention in the l i t e r a t u r e , i t i s assumed that s e c u r i t i e s used by interest arbitragers have a uniform maturity of 90 days. 12 CHAPTER I NOTES ^For definit i o n s of the technical terms used in this study see Section 1.3. 2 John Maynard Keynes, A Tract on Monetary Re form (London: Macmillan, 1923), pp. 115-39. Hereafter referred to as Monetary Reform. 3 Charles Poor Kindleberger, "Speculation and Forward Exchange," Journal of P o l i t i c a l Economy, XLVII ( A p r i l , 1939), 163-81. 4 For a thorough discussion of this question see Paul Einzig, A Dynamic Theory of Forward Exchange (London: Macmillan § Co. Ltd.; New York: St. Martin's Press, 1962). Hereafter referred to as Dynamic Theory. 5 F o r a history of forward exchange theory see Einzi g , Dynamic Theory, Chap. X; also Fred R. Glahe, Aii Empirical  Study of the Foreign Exchange Market: Test of a Theory. Princeton Studies in International Finance, No. 20 (Prince-ton, N.J.: International Finance Section, 1967). Hereafter referred to as Empirical Study. ^Einzig, Dynami c Theory, Chap. XI. 7 Manfred Neldner, Die Kursbi1 dung auf dem Devisenter-minmarkt und die Devisenterminpolitik der Zentralbanken (Berlin: Walter de Gruyter § Co., 1970). Hereafter referred to as Devisenterminmarkt. 8 Ernest Nagel, The Structure of Science; Problems in  the Logic of S c i e n t i f i c Explanation (New York: Harcourt, Brace 5 World, 1961) . By the same author: "Assumptions in Economic Theory," Readings in Microeconomics, ed. by William Breit and Harold Hochman (New York: Holt, Rinehart and Winston, Inc., 1968). 9 These definit i o n s are e s s e n t i a l l y based on Einzig, Dynamic Theory, pp. 2-5. A comprehensive l i s t of definit i o n s can be found in Paul Einzig, A Textbook on Foreign Exchange (London: Macmillan, 1966), pp. 2,33-45. ^ E i n z i g , Dynamic Theory, p. 2. ^ F o r an excellent description of a t y p i c a l foreign exchange market see Alan R. Holmes and Francis H. Schott, The New York Foreign Exchange Market (New York: Federal Reserve Bank of New York, 1965). Hereafter referred to as New York Foreign Exchange Market. 13 CHAPTER I NOTES (cont'd) 12 The rate of exchange is always the price of one currency in terms of another. Since there are always two currencies involved, two conventions can be followed. Most countries use what is sometimes called the price notation which is defined above. From a Canadian point of view the price of DM i s thus expressed as $/DM. In London, bankers use the value quotation; they would also quote Canadian dollars as $/^. Thus the exchange rate is quoted in London as the number of foreign units which can be bought with one domestic unit. B r i t i s h writers commonly use the value quotation. This should be kept in mind when the i r contri-butions are compared with Continental or North American l i t e r a t u r e . See Neldner, Devisenterminmarkt, p. 12 and John P. Young, The International Economy (4th ed.; New York: The Ronald Press Company, 1963), p. 55. 13 Grubel erroneously states that the spot rate applies to currency to be delivered the day following the contract. Herbert G. Grubel, Forward Exchange, Speculation, and the  International Flow of Capital (Stanford, Cal.: Stanford University Press, 1966), p. 3. Hereafter referred to as Forward Exchange. Einzig, Dynamic Theory, p. 2, is correct in saying that spot rates apply to currencies "bought and sold two days after conclusion of the deal and paid for on delivery." 14 Einzig, Dynamic Theory, p. 3. 1 5 I b i d . , p. 4. Writers on foreign exchange are not unanimous in t h e i r use of this term. Neldner, Devisenterminmarkt, p. 60, con-curs with Einzig. For a different d e f i n i t i o n see Grubel, Forward Exchange, p. 5. 14 CHAPTER II CRITICAL REVIEW OF THE LITERATURE 2. 1 The Interest Rate Parity Theory The term interest rate parity theory (IRPT) is usually reserved for the Keynesian theory of interest ar-bitrage. In a broader sense, however, i t may be used to describe a l l theories which require interest parity for equilibrium. There i s , according to the proponents of IRPT, a natural tendency for the forward margin to equal the differences in interest rates. Other writers maintain, however, that arbitragers represent only one of many kinds of participants in the foreign exchange markets. Speculators and commercial traders may exert a countervailing power, so that the f i n a l outcome cannot be predicted unless more is known about the behavior and r e l a t i v e importance of the various components. The propositions of these two schools of thought can therefore be conveniently examined in two separate sections. The following presents Keynes1 theory in order to set the stage for the discussion. Other authors of the same school are mostly concerned with adapting this theory to more r e a l i s t i c s ituations. Their contributions are reviewed in the subsequent paragraphs. Section 2.2 reviews the ideas of what might be called 15 the modern school. Neldner's theory is seen as the most comprehensive formulation of the propositions of this school and is therefore presented at some length. Other writers have stressed different aspects of the same problem. Their contributions w i l l be reviewed towards the end of Section 2.2. 2.11 The Keynesian Theory of Interest Arbitrage Keynes' interest rate parity theory f i r s t appeared in print in 1923.''' Although i t has undergone many variations and revisions since that time, i t s t i l l stands as the basic postulate upon which other authors have b u i l t t h e i r own models. We perceive i t as a theory whose realm of applica-2 tion is confined to a "pure case." S p e c i f i c a l l y , the theory is said to hold under the following circumstances: 1. Spot rates are f l e x i b l e . 2. Currencies are freel y convertible. 3. There is only one maturity available for forward contracts; i t is 90 days in the following exposition. 4. Investment (and borrowing) of capit a l in foreign or domestic se c u r i t i e s is limited to the same period of time. : 5. Interest rates in foreign and domestic money markets are represented by a single interest rate, respectively. 6. Only arbitragers operate in the forward market. 7. Interest rates are not affected. 16 The analysis is r e s t r i c t e d to a two-country model. A holder of l i q u i d funds i s free to invest his money at home or in the foreign money market. If he chooses to lend i t to a domestic borrower, one unit w i l l grow in 90 days to A = (1+RD) (2-1) (where RD ... domestic interest rate) expressed in domestic currency. A l t e r n a t i v e l y , he may convert his money into foreign currency. For one unit of domestic currency, he w i l l obtain B = s | (2-2) (where SX ... spot exchange rate) units of foreign currency. At this point, the arbitrager is uncertain what the exchange rate 90 days hence w i l l be. He can eliminate the loss associated with a possible depreciation of the foreign currency by s e l l i n g the same amount of foreign currency forward ( i . e . by buying domestic currency forward). Exchange risks are now eliminated, since the arbitrager knows at what exchange rate he can repatriate the funds. After this swap operation, he holds P Y c - S (2-v (where FX ... forward exchange rate) in terms of domestic currency. This amount, when invested in the foreign money market, w i l l grow i n 90 days to 17 F X D = | i ( i + R F ) (2-4) The two alternatives are equivalent in terms of monetary return when A = D (2-5) or F Y (1+RD) = Ii(l+RF) (2-5') (note that both expressions are in terms of domestic currency). Equation (2-5') i s commonly ca l l e d the equilibrium condition because an arbitrager is said to be in d i f f e r e n t between the two alternatives. There is no incentive to transfer funds from one market to another. From equation (2-5'), the rule governing demand for foreign exchange by arbitragers can be derived: M+(M•RF) = RD-RF (2-6) 3 The expression (M"RF) represents interest earned on invest-ment converted into domestic currency. This factor has been neglected by Keynes because i t is an order of magnitude 4 smaller than M, RD or RF. Therefore, in order to express Keynes' ideas correctly, this factor i s commonly dropped. Equation (2-6) then becomes M = RD-RF (2-7) which i s an approximation of the f i r s t order to (2-6). Equation (2-7) summarizes Keynes' theory that the forward margin (M) on foreign exchange tends to equal the difference between the interest rates prevailing in the two money markets concerned. The forward rate which yields such a margin w i l l 18 be referred to as the parity rate. To i l l u s t r a t e , i f U.S. rates were higher than Canadian interest rates, the theory would predict that the U.S. dol l a r would be at a discount equal to the difference between the U.S. and the Canadian interest rate. It has been said at the beginning of this section that the IRPT can be viewed as a law formulated as the "pure case." A look at the assumptions l i s t e d on page 15 reveals that the setting for Keynes' theory i s indeed highly ide a l i z e d . There i s no need to belabor the point; i t is quite obvious that such conditions never have and probably never w i l l prevai1. The question remains then how a law of this nature can be asserted to be true. The very nature of such a law precludes direct empirical validation because i t is impossible to create a controlled environment. To be empirically testable, the theory needs to be supplemented by a number of assumptions to take care of the influence of other factors r e l a t i n g to the "impurity" of the environment. There are a great many d i f f i c u l t i e s associated with such an approach. However, there are other grounds for accepting a theory without the benefit of empiri-cal v a l i d a t i o n . F i r s t , a theory is acceptable so long as no better one i s available. More important, our confidence in the v a l i d i t y is increased i f i t conforms with other generally accepted p r i n c i p l e s of economics. Third, a theory is accept-able when meaningful empirical models can be derived from i t . 19 With these remarks in mind, let us re-examine the content of Keynes' theory. The IRPT says, in essence, that the forward rate w i l l assume a level such that the rate of return on domestic and foreign assets i s the same. But this i s the equivalent of the statement "that two units of the same asset should y i e l d , in a perfect market, exactly the same return."*' It appears that Keynes' theory i s no more than a reformulation of a generally accepted p r i n c i p l e of economics, as applied to the foreign exchange market. This consistency with the body of economic thought should give the IRPT high c r e d i b i l i t y . Furthermore, the theory is indeed amenable to l o g i c a l derivations which are empirically useful. The models examined in the remainder of this chapter bear this out. Keynes, however, went further in his analysis than is generally recognized. He perceived a number of other factors which might influence the forward rate and prevent the establishment of a parity rate in the real world. A l l these factors were subsumed by Keynes under an even more compre-hensive theory: The difference between the spot and the forward rates i s , . ., precisely and exactly the measure of the preference of the money and exchange market for holding funds in one international centre rather than in another, the exchange risk apart, that is to say under conditions in which the exchange risk is covered.6 In this more comprehensive theory, interest d i f f e r e n t i a l s remain the fundamental and dominant determinants of forward 20 rates. But other factors must also be considered in the context of concrete situations. These factors are examined below; i t should become apparent in the process, that these factors are "impurities" with respect to the pure case of the IRPT. 1. Even i f exchange ri s k is covered, fear of an imminent imposition of exchange regulations, the p o s s i b i l i t y of f i n a n c i a l or p o l i t i c a l trouble may deter investors from committing large sums of c a p i t a l to foreign markets. Under such circumstances, investment decisions cannot s o l e l y be based on numerical data. 2. Subjects other than arbitragers may operate i n the forward market. In such cases, purchases (sales) by arbitragers may be matched by sales (purchases) by merchants 7 or speculators. Conversely, i t i s conceivable that, say, speculators exert such a strong pressure on the forward rate that i t becomes pr o f i t a b l e for arbitragers to lend funds in the market with the lower interest rates. As a re s u l t , a so-called " u p - h i l l " flow of funds may be observed. Other authors have stressed this aspect as w i l l be seen g presently. 3. In many instances, transactions in forward exchange are controlled by a small number of banks. Tacit agreements between themselves to maintain abnormal p r o f i t s may prevent arbitragers from exploiting differences in interest rate to the f u l l . 4. There are l i m i t s to the c a p i t a l available for 21 arbitrage. Keynes attributes this to a general lack of 9 understanding on the part of most businessmen. 2.12 Summary of Keynes' Theory A b r i e f summary and c r i t i c a l appraisal of the argument developed so far is in order. Keynes holds that the d i f f e r -ence between spot and forward rates i s a measure of the preference of investors--he does not mention borrowers--for holding assets in one money market rather than another. Formulated in these general terms the theory can hardly be disputed. However, Keynes goes on to say that the fundamental reason why one market is preferable to another i s the covered interest d i f f e r e n t i a l . This p a r t i c u l a r aspect of his theory has been singled out by later writers who dubbed i t the interest rate parity theory (IRPT). On the surface, i t i s easy to c r i t i c i z e the IRPT because of glaring deviations observed when i t is applied to real world conditions in a straightforward manner. We have t r i e d to point out that such a procedure i s uncalled for, since the theory i s postu-lated as an "ideal type" or "pure case." Keynes himself has recognized this fact; he therefore proceeded to l i s t a number of additional factors of varying importance which prevent parity from being attained in r e a l i t y . There are a number of writers who have b a s i c a l l y espoused Keynes' theory; their work is distinguished by the emphasis they lay on such additional factors. The most 22 i n f l u e n t i a l contributions are discussed in the following section. 2.13 Other Considerations 2.131 Limited Funds The IRPT assumes a world where funds are unlimited. It requires that funds continue flowing from one center to another u n t i l a l l incentives are eliminated. Hawtrey, on the other hand, undertakes to demonstrate that such an assumption may not be warranted.^ According to him, funds available for arbitrage consist of 1. Balances forming part of commercial traders' working capi t a l and not immediately required for their business and 2. Liquid assets of banks. Only big enterprises w i l l be able to keep up with foreign developments. Moreover, investments must be made in large quantities in order to absorb the fixed costs associated with the transfer of funds. Banks, for their part, may incur an opportunity loss by withdrawing funds from domestic operations, mostly because deposits abroad are less l i q u i d than domestic b i l l s . Such considerations lead Hawtrey to expect that the flow of funds from one country to another w i l l stop abruptly once the available balances are drawn down. Furthermore, the migration of funds is almost instantaneous; as soon as an incentive for arbitrage is created (through extraneous shock), 2 3 capi t a l w i l l flow in large quantities in a short period of time and then subside completely. The implication of this modification of assumptions underlying the theory should be clear. The flow of funds between money markets may not be s u f f i c i e n t to influence the forward rate to the point where parity is restored. Indeed, incentives for arbitrage may persist and yet no capital i s transferred, simply because balances available for arbitrage are exhausted. 1 1 In other words, interest arbitrage may not be of a s u f f i c i e n t magnitude to bring the forward rate to parity l e v e l . The usefulness of Hawtrey's theory is somewhat doubtful, however, since i t holds only up to the point where funds available for arbitrage run out; beyond that point the theory is unable to explain or predict the behavior of the forward rate. This point i s well made by modern theor i s t s ; i t i s taken up in Section 2.2. 2.132 Market Imperfections To convert, say, s t e r l i n g into dollars and back in order to buy and cover foreign s e c u r i t i e s i s obviously more troublesome than purchasing instruments denominated in domestic currency, even i f commissions and other incremental expenses are l e f t out. Arbitragers w i l l require some compen-sation for this inconvenience. Hence they may stop trans-f e r r i n g c a p i t a l before numerical equality between domestic and foreign s e c u r i t i e s is restored. For the purpose of 24 exposition, Keynes situated this compensation required by 12 arbitragers at around %% p.a. This and similar arguments pertaining to cost of information and other expenses are in themselves i n s u f f i c i e n t to f u l l y explain lasti n g deviations from p a r i t y . For one thing--as w i l l be pointed out later--banks often engage in arbitrage even when monetary returns on foreign investments are lower. Second, one would have to argue that compensa-tions for inconvenience w i l l vary from about 1/32% p.a. to approximately 2% p.a. to account for observed deviations 13 over a period of time, which i s a p r i o r i an untenable position. 2.133 Moral Suasion A number of writers have argued that monetary authorities can and do e f f e c t i v e l y c u r t a i l the volume of 14 arbitrage through moral suasion. The effect of such an intervention would be equivalent to a case of limited funds. While moral suasion may contribute to deviations from parity in some cases, i t is nevertheless true that such deviations exist even in the absence of o f f i c i a l intervention. 2.134 Speculation Introduction of speculation in the model requires the removal of the assumption that only arbitragers operate in the forward market. Although Keynes had recognized the p o s s i b i l i t y that speculation may at times overpower the influence of arbitrage, 25 i t was Spraos who incorporated speculation (and trade) as an 15 integral part of the theory. How does speculation prevent arbitrage from establishing parity? A b r i e f i l l u s t r a t i o n may make this relationship clear. Suppose there is no speculation. Arbitragers s e l l forward exchange to banks who make the market. Banks, in turn, w i l l cover themselves by s e l l i n g an equivalent amount of spot exchange. 1 6 This s e l l i n g pressure on both spot and forward markets w i l l eventually lead to a weakening of primarily the forward rate (because arbitragers buy spot exchange simul-taneously). Thus, either the cost of covering becomes too high or the pickup resulting from conversion too small so that arbitrage i s no longer p r o f i t a b l e . Suppose now that speculators are also in the market. The banks are now able to match sales of forward exchange by arbitragers with purchases by speculators and the exchange rate need not move. The more active are speculators, the more funds arbitragers can transfer without affecting the forward rate. In such cases there i s l i t t l e or no tendency for parity to be attained. However--and this i s a c r u c i a l p o i n t - - i f funds available for arbitrage are v i r t u a l l y un-limited, arbitragers continue s e l l i n g even after speculators 17 run out of funds. Under such circumstances IRPT may hold. 2.135 Ignorance 1 8 Sohmen is another supporter of IRPT. According to him, forward rates should not digress from their parity-value by more than h% p.a. The fact that in r e a l i t y those rates 26 appear to deviate by considerably more than %% p.a. i s attributable to a l i m i t a t i o n on funds available for arbitrage. This shortage of funds, in turn, i s due to ignorance on the part of potential arbitragers. Ignorance, however, is i r r e l e -vant to the o r e t i c a l arguments, according to Sohmen. Therefore IRPT holds! 2.14 Summary The exposition of Keynes' theory was followed by a discussion of other factors that might influence the determi-nation of forward rates. Most of these had already been considered by Keynes, but other writers stressed t h e i r im-portance for a r e a l i s t i c model. In r e l a t i o n to IRPT, these factors constitute "impurities" which must be introduced i f a highly " p u r i f i e d " theory such as IRPT is to be applied to real world conditions. Even so, the theory and i t s variations are open to c r i t i c i s m on theoretical grounds. However, i t i s best to postpone a c r i t i c a l examination of IRPT u n t i l an account of some recent developments in the theory of forward exchange is given. In the meantime, let us b r i e f l y summarize the basic tenets of the proponents of IRPT. 1. Funds available for arbitrage are for a l l p r a c t i c a l purposes unlimited. The migration of funds from one i n t e r -national center to another takes place within a very short period of time. 27 2. Arbitragers require a minimum incentive, (most commonly estimated at %% p.a.) before they are inclined to transfer c a p i t a l . 3. Equilibrium prevails only when the forward margin expressed in percentage equals the difference between 19 national interest rates. 2.2 Modern Developments: Imperfect Substitutes and  Simultaneous Determination of Spot  and Forward Rates C r i t i c s reject IRPT on two counts: 1. Domestic and foreign s e c u r i t i e s are not perfect substitutes for each other. There are various advantages and risks associated with foreign s e c u r i t i e s not to be found in domestic ones. 2. IRPT stipulates that the forward rate w i l l assume a value such as to establish equilibrium. In re a l i t y - - s a y modern writers--equilibrium must be attained simultaneously in the spot and forward markets for the two are intimately related through the a c t i v i t y of arbitragers and speculators. Modern theorists have developed various models designed to cope with these two problems. It is the intent in this section to examine some of the unique features of important recent contributions. 20 Neldner's model w i l l be presented at some length. Considerable space is devoted to i t because i t represents an eff o r t to synthesize most of the work done previously. Where other authors rely almost exclusively on, say, ris k of default to j u s t i f y their model, Neldner treats the same factor in a 28 much more comprehensive manner. His work is probably the most rigorous and c a r e f u l l y formulated contribution to the theory of foreign exchange available today. It has not received the attention i t deserves, which i s probably due to the fact that i t appeared in German only. 2.21 Neldner's Model A d i s t i n c t i v e feature of Neldner's and other modern models i s the integration of demand for (and supply of) both spot and forward foreign exchange by arbitragers, speculators and merchants. Accordingly, separate demand and supply 2 1 functions w i l l be presented successively. At the end of the of the discussion, these demand curves w i l l be used to deter-mine simultaneous equilibrium on the spot and forward markets. 2.211 The Arbitrage Schedule The arbitrage schedule depicts the demand for funds by arbitragers as a function of a price, a l l other factors held constant. Depending on the p a r t i c u l a r market under investigation, the AA schedule may represent demand for spot exchange as a function of the spot rate, or the demand for forward exchange as a function of the forward rate. It may also relate demand for domestic (foreign) s e c u r i t i e s to the domestic (foreign) interest rate. IRPT i m p l i c i t l y assumes a perf e c t l y e l a s t i c AA schedule for values of the pa r i t y forward rate ± p.a. (Fig- ! ) • 29 SX DEMAND where AA PP SX SX' ft.?. SX' SUPPLY arbitrage schedule parity l i n e . It cuts the SX axis at B, where parity p r e v a i l s . For s i m p l i c i t y , i t i s assumed that at B, FX = SX and therefore RD = RF. spot rate Fig. 1. Arbitrage schedule for spot market (IRPT) Suppose the price of spot exchange increases from B to a l i t t l e over SX', with FX and RD-RF = 0 held constant. Then i t pays a foreign arbitrager to s e l l foreign spot exchange, invest the proceeds in the domestic money market and to buy back the same amount on the forward market at a lower rate (B). This process is often called inward arbitrage The spot rate cannot remain at a level beyond SX' for any pro-longed period of time, since funds continue to flow u n t i l the rate returns to SX'. The converse would be true i f SX f e l l below SX" and outward arbitrage would occur. If on the other hand, another parameter changed (ceteris paribus), the AA schedule would s h i f t p a r a l l e l to i t s e l f up or down along the SX axis. S p e c i f i c a l l y , i f RD increases, the curve s h i f t s down (the curve is specified in foreign exchange.'). If RF increases, the curve s h i f t s up. 30 In constrast to the schedule shown in Fig. 1, Neldner develops a demand function which rises with increasing i n -centives for arbitrage. According to this author, the amount of capital that arbitragers are w i l l i n g to transfer i s determined by two basic considerations: return on investment and l i q u i d i t y . 2.2111 Return on Investment The return on investment accruing to arbitragers con-s i s t s of the interest earned plus the pickup on foreign exchange operations (or minus the loss on these transactions, as the case may be). However, there are a number of addi-tional factors which may affect return on investment. They are: 2.21111 Cost. Costs to arbitragers consist of fixed and variable components. The f i r s t category comprises cost of obtaining information and overhead, the second commission and taxes. Although the incidence of costs varies from one investor to another (a bank w i l l have to bear minimal costs only), costs are not l i k e l y to affect an investor's decision s i g n i f i c a n t l y . Bear in mind that most fixed costs must be borne regardless of the f i n a l decision. 2.21112 Convenience Yield. Writers who have come in close contact with foreign exchange markets emphasize that a good deal of money is placed abroad under arbitrage arrangements which are not necessarily advantageous from a monetary point 31 of view. To quote Ein z i g : Nowadays certain banks operate from time to time even without any p r o f i t , which, in view of the heavy overhead of Foreign Exchange Departments, means in practice that they operate at a loss. They do this partly for considerations of prestige, to impress the market with the frequency and volume of their operations, and partly to attract more business . . .22 These and other intangible returns associated with investing c a p t i a l abroad are subsumed under "convenience y i e l d . " The other side of the coin is the convenience y i e l d forsaken i f capi t a l i s withdrawn from the domestic market. Neldner stipulates that the foreign marginal convenience y i e l d decreases with increasing demand for arbitrage, while the domestic counterpart of convenience y i e l d forgone i s l i k e l y to increase. Net marginal convenience y i e l d is there-fore l i k e l y to approach zero as transferring of funds for purposes of arbitrage continues. 2.21113 Risk. The t h i r d factor affecting return on invest-ment is r i s k . Since the objective of the analysis is to determine under what circumstances investors are prepared to transfer funds to foreign markets, only incremental risk 2 3 associated with foreign investments is considered. Con-ventionally, two types of risks that the arbitrager must take into account are distinguished: 1. That banks default on the forward contract, 2. That foreign exchange controls are imposed in the foreign country. 32 The f i r s t type of risk i s not of a serious kind. Barring bankruptcy, a contractual partner can only default i f the spot rate has turned strongly in his disfavor, other-wise he stands to make a gain. The second type of r i s k is decidedly more serious. It most l i k e l y accounts for the l i m i t that banks in p a r t i c u l a r set to the amount of funds to be committed to a p a r t i c u l a r 24 market. The question remains, however, whether the demand for spot exchange by arbitragers is perfectly e l a s t i c up to the self-imposed l i m i t or whether i t changes with increasing incentives. Neldner refers to Kalecki to s e t t l e this issue: "The rate of r i s k of every investment i s greater the larger is the investment." Further on Kalecki argues that the notion "that the rate of risk is independent of the amount in -vested . . . has to be dropped . . . in order to obtain a r e a l i s t i c solution to the problem of 'limited investment.'" For, as Kalecki continues ". . . the greater . . . the investment of an entrepreneur, the more is his wealth position 25 endangered in the event of an unsuccessful business." The inference must be drawn that since investors require higher returns as r i s k increases, they w i l l continue trans-fer r i n g funds only as returns i n c r e a s e . ^ Borrowing-arbitrage can be represented by one and the same AA schedule. The risk in this case is borne by the non-arbitrager lender. No doubt, the impact of borrowing-arbitrage on foreign exchange markets is the same as that of 1 ending-arbitrage. This leads a l l contemporary writers to treat borrowing-arbitrage simply as the dual of lending-AA schedule can be derived geometrically. Arbitragers w i l l transfer capital only i f return on investment in the broad sense ( i . e . including convenience) equals at least return on domestic investment plus cost. If they dispose of unlimited funds, the condition for e q u i l i b -rium can be written as: (Marginal Cost + Marginal Risk) - Net Marginal U t i l i t y of 2 8 Investing Abroad = Net Interest D i f f e r e n t i a l . The magnitudes on the l e f t hand side of the equation vary from one investor to another. Consequently, the micro-economic functions must be derived f i r s t . Summation of individual functions yields the macro-economic AA schedule. Fig. 2 i l l u s t r a t e s an arbitrage function thought to be t y p i c a l for a bank. Costs of transfer and of obtaining information are r e l a t i v e l y low while intangible returns 29 (convenience yield) are considered to be high at f i r s t . arb itrage. 27 On the basis of the above considerations, the Net interest d i f f e r e n t i a l Individual supply Marginal u t i 1 i t y Marginal r i s k Q forward exchange/t Marginal cost Fig. 2. Supply of forward exchange by individual bank acting as arbitrager 34 Neldner obtains the supply function by adding the marginal cost to the marginal risk curve and subtracting the marginal . . „ . 30 u t i l i t y function. Fig. 3 shows a supply schedule said to apply to a non-bank acting as arbitrager. Marginal u t i l i t y is lower while costs are considerably higher. Hence, non-banks w i l l s t a r t arbitraging only once the net interest d i f f e r e n t i a l has reached a certain level which w i l l be call e d the "threshold." It is quite l i k e l y that this threshold is different for each arb i t rager. Net interest d i f f e r e n t i a l Ind. supply function Marginal u t i l i t y Marginal risk Marginal cost Q forward exchange/t Fig. 3. Supply of forward exchange by individual non-bank acting as arbitrager Horizontal summation of a l l individual functions yields the aggregate AA schedule (Fig. 4). It relates arbitragers' demand for and supply of forward exchange to the net interest d i f f e r e n t i a l . It starts at the o r i g i n because some arbitragers w i l l transfer funds even in the absence of pecuniary incentives 35 Net interest d i f f e r e n t i a l favoring foreign country A Q forward exchange/t Q forward exchange/t + A Net interest d i f f e r e n t i a l favoring domestic market DEMAND SUPPLY Fig. 4. Arbitrage schedule Fig. 4 incorporates demand for forward exchange as well. Demand is the symmetric counterpart of supply of forward attributable to the two factors mentioned above: 1. The " p r i n c i p l e of increasing risk" 2. The "threshold-effect" which i s different for each individual arbitrager. This completes the f i r s t part of our analysis of the AA schedule. It has been shown how cost, convenience and ri s k affect the willingness of arbitragers to invest funds abroad. The second part of the analysis removes the assumption that arbitragers dispose of unlimited amounts of fund which they could--if they so desired--invest abroad. Questions pertaining to the volume of funds available are discussed under the heading " l i q u i d i t y . " exchange. 31 The AA schedule is upward sloping. This i s 36 2.2112 Li q u i d i t y Monetary authorities can influence the money supply by a variety of means. If they chose, for example, to reduce the money supply, the volume of domestic l i q u i d c a p i t a l amenable to arbitrage is l i k e l y to decrease. S i m i l a r l y , foreign borrowers w i l l find that funds are less easily available. Consequently, i t is possible that arbitragers run out of funds before they reach t h e i r self-imposed l i m i t s . This means that arbitragers cannot transfer as much capital as they wish. Therefore, the amount of ca p i t a l transferred by arbitragers under conditions of r e s t r i c t i v e monetary policy i s l i k e l y to be smaller than i t would be under conditions of easy money. Fig. 5 i l l u s t r a t e s this relationship for individual a r b i t r a -gers; Fig. 5a relates to a bank, Fig. 5b to a non-bank. 5a 5b where Mi, M2 , M3 ... domestic money supply A-p A 2, A 3 ... corresponding supply of forward exchange by arbitragers A^Ai ... supply function for i n d i -vidual arbitrager Fig. 5. Individual supply curves under different quantities of money supply32 37 The supply function for an individual arbitrager becomes completely i n e l a s t i c at OA^. Subsequent tight monetary policy reduces l i q u i d i t y even more; the i n e l a s t i c portion of the A^Ai function is shifted to A2 and then A3. Aggregation of these micro-economic functions y i e l d the AA schedule for different magnitudes of M as i l l u s t r a t e d in Fig. 6. Net interest d i f f . favoring for. market l A 3 < A 2 < A i A ML M3 M2 M l + Q forward exchange / t SUPPLY Net interest d i f f . favoring dom. market where AA o r i g i n a l arbitrage schedule AA^_3 ... arbitrage schedule under Mj_g Ml-3 ... domestic money supply M1 ... foreign money supply Fig. 6. AA schedule for dif f e r e n t quantities of money supply - 3 - 3 The discussion thus far has centered on demand for forward foreign exchange. It should be borne in mind, how-ever, that for every unit of forward exchange demanded (supplied) by arbitragers there is an equivalent amount of M' Q forward exchange/t spot exchange supplied (demanded). Thus, i t is redundant to analyse demand for or supply of spot exchange by arbi-tragers. The derivation of supply and demand curves for spot exchange would be by straightforward analogy. 2.212 Speculation The second component of t o t a l demand for foreign exchange i s demand originating from speculators. Let us assume for the moment that speculators conduct their business in the forward market exclusively. We had defined FX as the price of foreign exchange to be delivered at t+90 quoted at time t. S i m i l a r l y , E(SX) t +gg is the spot rate expected to pre v a i l at time t+90. Thus speculators buy forward exchange i f E(SX) t + 9 ( ) > FX t (2-8) and they w i l l s e l l forward exchange i f E ( S X ) t + 9 0 < FX t (2-9) Once E ( S X ) t + 9 g and FX t are determined, the volume of funds offered or demanded by speculators depends upon costs and r i s k . Costs incurred by speculators are similar to those faced by arbitragers and need not be investigated again. The risk a speculator has to bear i s of a different nature, however. A n a l y t i c a l l y , two types of risks pertaining to speculation can be distinguished: 1. Risk of default on the part of the contractual partner in the forward transaction. This type of risk is almost n e g l i g i b l e in this case. Since the speculator did not 39 put up any funds at time t, at worst he loses an opportunity to make a p r o f i t . 2. Exchange r i s k : exchange risk i s an essential ingre-dient of speculation. It manifests i t s e l f in the p o s s i b i l i t y that the spot rate may either move in a sense contrary to the one predicted; or i t may move in the predicted d i r e c t i o n , but not s u f f i c i e n t l y so. A speculator w i l l therefore have to include a risk premium in his calculations. Neldner suggests that the magnitude of this premium is determined by three factors : 3 5 a. The degree of certainty about the level of the future spot rate, b. The subjective evaluation of risk by speculators (attitude towards r i s k ) , c. The magnitude of past commitments, so-called "open pos i t i o n s . " Given this lack of perfect foresight, the best specu-lators can do i s to assign p r o b a b i l i t i e s to a number of 3 6 possible outcomes. E(SX) is then taken to be the weighted average of such a pr o b a b i l i t y d i s t r i b u t i o n . The degree of certainty with which expectations are held i s reflected in the dispersion of probable outcomes around E(SX). The lower the certainty about the future spot rate the more pronounced this dispersion. Accordingly, the standard deviation (c») can be employed as a measure of r i s k . The higher d, the more pronounced the risk associated with speculation for given value of E(SX) and FX. 40 The subjective evaluation of risk determines at which level of FX a speculator is prepared to enter into a forward contract. A risk-averter might require a substantially higher level of FX than a r i s k - l o v e r for the same E(SX). The t h i r d factor--open positions--determining the r i s k premium required by speculators is associated with Kalecki's "principle of increasing r i s k " outlined above. Presumably, the higher past commitments, the less a speculator is w i l l i n g to buy or s e l l additional forward exchange. In other words, the higher the quantities invested at t-k, the higher the ri s k premium required by speculators at time t. The p r i n c i p l e s outlined above can now be u t i l i z e d to derive the speculator's schedule (SS schedule). Incorporating the additional data required for decision-making by specula-tors, the equilibrium condition can be stated to be: a) Marginal Cost + Marginal Risk = E(SX) - FX for E(SX) >FX, i. e . purchase of foreign exchange, respectively: b) Marginal Cost + Marginal Risk = FX - E(SX) for E(SX) <FX, i.e. sale of forward exchange.^ Considering that FX is the only objective datum available to a l l speculators, the micro-economic demand functions must be 3 8 derived f i r s t . Fig. 7 i s drawn under the assumption that E(SX) and O are given and constant. The S^S^ function is arrived at by adding marginal cost and marginal risk curves. It r e f l e c t s how supply and demand for forward exchange by an individual speculator changes with the forward rate. Fig. 7. Demand and supply of forward exchange by i n d i v i d u a l speculators 42 Fig. 7a represents the demand (supply) function of a r i s k -averter. The ri s k aversion of the speculator in Fig. 7b is substantially less pronounced. At FXj, for example, he i s prepared to supply B units of forward exchange while the risk-averter supplies only A units. Horizontal summation of a l l individual demand and supply curves yields the aggregate SS schedule as shown in Fig. 8. FX FX 1 S i ( C 2 ) ^ E (SX) S f s' 0 A B + Q forward exchange/t DEMAND SUPPLY 40 Fig. 8. Speculators' schedule The SS schedules intersect the FX-axis at the point where E(SX) = FX. If E(SX) is revised (cet. par.), the SS schedule w i l l move p a r a l l e l to i t s e l f up or down along the FX-axis. SS and S-^ S-i have been drawn to i l l u s t r a t e how the degree of uncertainty about E(SX) affects speculators' demand. ^ S ^ is associated with higher uncertainty (d^ > o^) • For any given value of FX above or below E(SX), say FX', and i f d = d 1, speculators w i l l supply A units of forward exchange as compared to B i f d = c / „ . 4 3 At time t+90, speculators must unwind their deal. To do so, they resort to the spot market, so that to each forward contract concluded at t, there i s a corresponding spot con-tract in the opposite direction at t+90. Fig. 9 complements Fig. 8 in this sense. S Xt+90 s 2 S l S2 Si - B A 0 + Q spot exchange/t DEMAND SUPPLY Fig. 9. Speculators' demand for spot exchange at t+9041 Assuming speculators sold amounts A, respectively B, of forward exchange at t, they would need to buy the same amount, respectively B, of spot exchange at t+90 to r e a l i z e gains or losses. This completes our evaluation of speculation in the forward market. It is the prevalent form of foreign exchange speculation because i t does not require an immediate outlay of cap i t a l which would involve opportunity cost. Nor i s there 42 any cost of borrowing involved. Nonetheless, speculation does occur in the spot market. However, i t can be shown that speculation in the spot market involves i m p l i c i t arbitrage combined with forward speculation. 44 (The assumption must be made, of course, that spot speculators invest their proceeds in foreign currency in the foreign money market.) Therefore, speculation in the spot market can be absorbed--in this model--by the AA schedule and SS schedule outlined above.^ 3 So far, two components of Neldner's model have been presented, namely demand for foreign exchange by arbitragers and by speculators. But t o t a l demand consists of arbitragers', speculators' and merchants' demand. It is the behavior of merchants that remains to be analysed. 2.213 Foreign Trade The analysis of demand for foreign exchange by merchants is complicated by the number of alternative modes of financing foreign trade. A few di s t i n c t i o n s w i l l help to c l a r i f y the matter. The f i r s t d i s t i n c t i o n refers to the chronological sequence of contract, delivery and payment for the commodities traded. If a l l these events occur simultaneously, the mer-chant w i l l buy foreign exchange on the spot market. On the other hand, i f these events do not coincide, further d i s t i n c t i o n s must be made. The merchant may, as a matter of p r i n c i p l e , eliminate risk by buying foreign currency on the forward market. In such cases, derivation of demand and supply curves for for-ward exchange is straightforward (Fig. 10). Importers (M) demand forward exchange, exporters (X) supply i t . 45 FX M X X M 0 Q forward exchange/t Fig. 10. Demand and supply of forward exchange by merchants44 The e l a s t i c i t i e s of MM and XX depend on p r i c e - e l a s t i c i t i e s of supply of and demand for the commodities traded. In r e a l i t y , the merchant may decide to turn to the spot market. He may do so at three points in time: at the time of contract, at the time of delivery or at the time payment is due. According to this d i s t i n c t i o n , three cases can be is o l a t e d . Case No. 1: An importer decides to buy spot exchange when he signs a contract for delivery of certain commodities. With the proceeds (denominated in foreign currency) he can do one of two things. He may deposit his funds with a foreign bank or he can make a prepayment to the foreign exporter in order to take advantage of a discount. In either case, he has made an investment which w i l l grow to in 90 days. Compare this with the amount he could have earned in the domestic money market: E = J - ( 1 + R F ) (2-10) F = (1+RD) (2-11) 46 At the same time, he could s e l l F on the forward market. His proceeds in foreign currency would then be G = pyCl+RD) (2-12) An importer who wishes to buy spot exchange at the time the contract is signed w i l l do so only i f S7(1 + R F ) > 'p^Cl + RD) (2-13) Conversely, he would buy forward exchange i f ^ L ( 1 + R F ) < J_(1+RD) (2-13') He w i l l be i n d i f f e r e n t between the two alternatives i f ^ ( 1 + RF) = i^U + RD) (2-13' ') But equation (2-13") reduces to ||(1+RF) = (1+RD) (2-13- ' •) 45 which i s i d e n t i c a l with equation ( 2 - 5 ' ) , said to be the equilibrium condition for arbitrage. It appears that, a n a l y t i c a l l y , two separate transactions can be distinguished whenever an importer buys spot exchange at the time the contract for commodities is signed: 1. The importer eliminates exchange risk by entering the forward market; 2. He performs an arbitrage operation by buying spot exchange and s e l l i n g forward exchange simultaneously. Only the purchase of spot exchange is actually carried out. The purchase of forward exchange (covering) and the sale of forward exchange (arbitrage) are imputed to him for a n a l y t i c a l reasons! But the two f i c t i o n a l transactions on the forward market cancel each other; as a result there remains only a demand for spot exchange, which corresponds to r e a l i t y . Case No. 2: The second case involves the merchant who buys spot exchange at the time of delivery. He has thus omitted to cover the exchange risk when his obligation became known to him. He can be said to be a passive speculator because his behavior can only be explained r a t i o n a l l y i f certain expectations about the future behavior of the spot rate are imputed to him. Accordingly, he buys spot exchange at the time of delivery (t+90) i f E(SX) t + 9 Q <FX t (2-14) at time t. If the opposite is true at time t, namely E CSX) t + 9 Q > FX t (2-14') the rat i o n a l importer buys forward exchange. Apparently, his decisions are based on the same considerations as those of an active s p e c u l a t o r . ^ As in the previous case, a number of transactions can be a n a l y t i c a l l y distinguished: 1. Purchase of forward exchange in order to eliminate exchange risk (time t ) . This is a f i c t i o n a l transaction. 2. Sale of forward exchange as a speculative trans-action (time t ) . F i c t i o n a l transaction. 3. Purchase of spot exchange (time t+90). Actual transaction. These two f i c t i o n a l transactions cancel each other. There 48 remains the actual purchase of spot exchange at t+90. In this interpretation, i t is carried out in order to r e a l i z e speculative p r o f i t s . Case No. 3: This leaves the last case where spot exchange i s bought after delivery. This s i t u a t i o n involves the use of a trade credit by the importer. He pays his dues only at t+180 (assuming payment is due 90 days after d e l i v e r y ) . By analogy to the foregoing cases, a number of i m p l i c i t transactions can be imputed to the importer. 1. He buys forward exchange to eliminate exchange ris k (time t ) . 2. He s e l l s forward exchange, acting as a speculator (time t) . 3. At t+90, he undertakes an arbitrage transaction. 4. He s e l l s forward exchange, acting as a speculator (t+90). Transactions 1 and 2 cancel each other for t. Transaction 2 requires a purchase of spot exchange at t+90, so that transactions 3 and 4 are also cancelled as far as t+90 is concerned. There remains the real purchase of spot exchange at t+180, which is undertaken to r e a l i z e speculative p r o f i t s . What is the relevance of this t h e o r e t i c a l construction? It is an attempt to explain why some merchants resort to the forward market to finance their trade while others make use of the spot market, thereby incurring an exchange r i s k . This model permits the assertion that a l l merchants base their 49 decisions regarding price and quantities of the commodities to be traded on the forward rate. The fact that some merchants conduct their actual business in the spot market merely indicates--under this approach--that in addition to financing their trade they also enter the market as arbitragers, speculators or both. Since these spot transactions are absorbed by the AA and the SS schedules r e l a t i n g to the spot market, there must be a demand (supply) for forward exchange corresponding to each commercial transaction. Fig. 10 (p. 45) can now be interpreted to represent t o t a l demand and supply of forward exchange by merchants.^ It was noted at the beginning of this section that whenever contract, payment and delivery coincide, the mer-chant w i l l resort to the spot market. Other a c t i v i t i e s leading to transactions in the spot market include trades financed by long-term c r e d i t , u n i l a t e r a l transfers and movements of long-term c a p i t a l . These transactions are used to construct supply and demand curves for spot exchange, exclusive of arbitrage, as i l l u s t r a t e d in Fig. 11. SX 0 Q spot exchange/t where D SD S ... demand for spot exchange S SS S ... supply of spot exchange Fig. 11. Demand (supply) for spot exchange by non-arbitragers49 50 This completes the discussion of the major components of Neldner's model. 5^ With the help of the various demand and supply schedules, equilibrium in spot and forward markets can now be determined. 2.214 Petermination of Equilibrium Spot and forward markets are in equilibrium when spot and forward rates assume values such that quantities demanded and supplied are equal in both markets respectively. Let us define: Ds ... excess demand for spot exchange by non-arbitragers 5''" A s A f D c excess demand for spot exchange by arbitragers excess demand for forward exchange by arbitragers excess demand for forward exchange by speculators excess demand for forward exchange by non-arbitragers. The spot market is in equilibrium i f -D = A (2-15) s s The equilibrium condition for the forward market is S £+D £ = -Af (2-16) Furthermore, for every unit of foreign exchange supplied by arbitragers on the spot market, there i s a corresponding de-mand for forward exchange (omitting interest earned, as usual) Therefore A = -A, 5 2 (2-17) s f so that, under conditions of equilibrium, the following must also be true: Sr+D. = A (2-18) f f s 51 but A s = -Ds (2-15) and S £ + D f = " D s ( 2 _ 1 9 ) must also hold. In other words, there are three conditions for simul-taneous equilibrium in the spot and forward markets. Equa-tion (2-15) states that the spot market is in equilibrium when excess demand by non-arbitragers equals excess supply by arbitragers (or vice versa). Equation (2-16) expresses the analogous requirement for the forward market. The link between the spot and the forward markets is established by equation (2-18), respectively (2-19). Equation (2-17) is 5 3 merely a d e f i n i t i o n . There are some s t r i k i n g differences between this system and Keynes' theory. F i r s t , spot and forward rates are determined simul-taneously in Neldner's model while the IRPT is only concerned with- the forward rate. Second, speculation and trade are an integral part of modern models. Whereas proponents of IRPT consider speculation and trade to be disruptive factors in the determination of the forward parity rate, modern writers treat these forces as essential and permanent components of a theory of forward exchange. F i n a l l y , Neldner's equilibrium condition for the forward market (2-16) makes no reference to the parity rate. The equilibrium rate depends upon the shapes of the S£, D^, and curves; i t is only under certain 52 circumstances that the rate of equilibrium turns out to be the parity r a t e . 5 4 A l l that can be said without reference to a p a r t i c u l a r case is that there may be a tendency for forward rates to move in the dire c t i o n indicated by IRPT owing to the influence of arbitragers. The more important is arbitrage compared to speculation and trade, the greater the tendency for the forward rate to conform to IRPT. This completes our exposition and examination of Neldner's model as the most advanced and comprehensive treat-ment of contemporary theory of forward exchange. The remainder of this chapter is devoted to a b r i e f analysis of contributions made by other modern writers. It w i l l become apparent that the various authors stress one or the other aspect of the foreign exchange market where Neldner has attempted to integrate most of them in his model. 2.22 Other Considerations 2.221 P o r t f o l i o Theory The f i r s t attempt to apply modern p o r t f o l i o theory to the theory of forward exchange was made by Gru b e l . 5 5 He suggests that whenever arbitragers decide on the proportion of their p o r t f o l i o they wish to a l l o t to covered placements abroad, they are uncertain about the returns on these.assets. This uncertainty relates to the possible need to cancel contracts p r i o r to maturity, to payment moratoria or to exchange controls. These uncertainties j u s t i f y in Grubel's mind the application of p o r t f o l i o theory as developed by 53 Markowitz and Tobin by straightforward analogy. His approach is open to c r i t i c i s m on two grounds. F i r s t , Tobin e x p l i c i t l y disregards the p o s s i b i l i t y of a t o t a l loss due to default, which is the equivalent of exchange controls and payment moratoria in our context. Tobin s p e c i f i c a l l y refers to ris k as re s u l t i n g from possible future 5 7 variations of the price of the asset. But this r i s k is eliminated in the case of arbitrage where assets are t y p i c a l l y held to maturity. It is Grubel's contention, however, that some arbitragers may have to repatriate funds p r i o r to matur-i t y . As a r e s u l t , these arbitragers may or may not incur a p a r t i a l loss, depending on the movement of the spot rate in the meantime. However, S t o l l has demonstrated that whenever premature re p a t r i a t i o n involves an exchange loss, an arbi-trager could offset this loss by re-entering the forward market.^ ^  It is interesting to note that in most instances Grubel arrives at the same conclusions as does Neldner. This coincidence of findings is due to the fact that the two authors employ the same theory; but their models d i f f e r . S p e c i f i c a l l y , t h e i r terms of correspondence are not i d e n t i -c a l . Where Grubel interprets (J to represent the threat of moratoria, exchange controls or premature repatriation, Neldner associates O with the p r i n c i p l e of increasing r i s k and other types of risks mentioned above. Although the main conclusions regarding the behavior of forward rates are i d e n t i c a l , there must, by necessity, be 54 instances where one model applies but not the other. Ultimately, therefore, the question of which model interprets the theory more adequately has to be s e t t l e d empirically. 5 9 The same observation applies to Branson's model. He emphasizes a different aspect of p o r t f o l i o theory. For a given set of rates of return and attendant risk considera-tions, investors w i l l a l l o t proportions of their p o r t f o l i o in such a way as to maximize their expected u t i l i t y functions. There w i l l be an equilibrium level of domestic claims on foreigners. The d i s t r i b u t i o n of assets w i l l be changed only i f rates of return change. Accordingly, i t is changes in interest rates, not their r e l a t i v e l e v e l s , which lead to stock-adjustments and thus determine international flows of t5 0 short-term c a p i t a l . This contention sparked a major controversy over flow- versus stock-models. It w i l l be discussed a l i t t l e further down (Section 2.3). For the moment, s u f f i c e i s to say that Branson's model is seriously impaired by the assumption that trade flows are insensitive to changes in exchange r a t e s . 6 1 2.222 Convenience Tsiang's paper on the theory of forward exchange has been j u s t l y acclaimed for i t s l u c i d exposition of the i n t e r -actions between arbitrage, speculation and commercial 6 2 covering. It also explores in depth the relationship between spot and forward rates. Together with Spraos' 6 3 contribution i t forms the foundation for most modern models, including Neldner's. 55 The present discussion w i l l be confined to an appraisal of Tsiang's derivation of the arbitrage schedule. He takes exception to the proposition embedded in IRPT that arbitrage funds should run out abruptly. What r e a l l y happens is that arbitragers " w i l l generally, after a certain point, become increasingly reluctant to transfer spot l i q u i d resources 64 from the domestic center to any p a r t i c u l a r foreign center." In the same vein, there i s an increasing reluctance on the part of arbitragers to repatriate spot l i q u i d resources held in foreign centers. The reason for this behavior of ar b i t r a g e r s - - t y p i c a l l y banks and other f i n a n c i a l i n s t i t u t i o n s -is that they require balances of spot funds to conduct th e i r day-to-day business. Accordingly, "spot l i q u i d assets y i e l d some intangible returns of convenience or l i q u i d i t y in addition to the i r interest y i e l d . " ^ Consequently, demand for arbitrage funds i s not a function of net interest d i f f e r e n t i a l alone, but also of the "subjective marginal convenience (or l i q u i d i t y ) y i e l d . Assuming that marginal convenience decreases with increasing capi t a l transfers and that there is a corresponding loss of marginal convenience on domestic assets, Tsiang derives an AA schedule which increases with increasing net interest d i f f e r e n t i a l . In t h i s , Tsiang arrives at much the same conclusions as Neldner. However, there is a certain inconsistency in Tsiang's argument. His proposition must be interpreted to mean that capi t a l placed abroad in the course of arbitrage carries a 56 convenience y i e l d in addition to the monetary return. However, i f convenience is equivalent to liquidity--and Tsiang suggests that i t is--then arbitrage is associated with a loss of convenience, not a gain. That is so because capital is t i e d up for 90 days (in our example); i t can therefore not be used for daily business operations. It would seem to be more sensible to interpret Tsiang's position in the sense that l i q u i d assets y i e l d a convenience return. Thus, conversion of l i q u i d assets into short-term investments would en t a i l a loss of convenience y i e l d . Con-sequently, banks would be increasingly reluctant to perform such transformations as t h e i r balances of l i q u i d assets are drawn down. As was the case with Grubel's model, Tsiang's model i s based on the same theory as Neldner's, and the conclusions are the same in i t s main applications. The model is d i f f e r -ent because theoretical terms are interpreted d i f f e r e n t l y . There must be cases, however, where the two models y i e l d d i f f e r e n t r e s u l t s . It is the task of empirical research to is o l a t e such instances and to determine the v a l i d i t y of these models. 2.223 Risk of Default At least one author claims that incremental risk of 6 7 default is the sole reason for an upward sloping AA schedule. Foreign investments carry a higher ri s k of default c h i e f l y because of the uncertainty about the foreign government's a b i l i t y to maintain the international flow of ca p i t a l without interference. If that was the only reason, one would expect interest parity to prevail in markets which are not subject to control by any government, such as the Eurodollar market. At least one empirical study rejects this hypothesis, , 6 8 however. 2.224 Lack of Liq u i d i t y According to Stein, foreign assets are not perfect substitutes for domestic ones because they are less l i q u i d . Premature repatriation is more l i k e l y to enta i l a greater 69 loss than disposal of assets on the domestic market. S t o l l ' argument that losses occuring under such circumstances could 70 be offset by additional forward contracts applies again. Stein has also f a i l e d to observe that merchants who resort to the spot market may act as speculators or arbitragers. These omissions may partly account for the fact that 71 Stein's model has not proved to possess predictive power. 2.225 Underdeveloped Forward Markets 72 Frevert's position is quite unique. His is the only known attempt to apply dynamic programming to the theory of forward exchange. According to Frevert, deviations from parity are mainly due to a lack of f u l l y developed forward market for a large number of maturities. For example, i f an arbitrager invests funds abroad for the shortest period available, he cannot reverse forward contracts in the forward market, should the 58 need arise; he must resort to the spot market, thereby possibly incurring a loss. Frevert's argument is a sophisticated extention of Stein's position; i t also counters S t o l l ' s argument. Black has pointed out, however, that Frevert's demand function is c r u c i a l l y dependent on the choice of an arbitrary terminal 73 date for forward contracts. 2.226 Dynamic Theory Einzig's work on the theory of forward exchange i s 74 perhaps the most far-reaching analysis of this topic. In his view, arbitrage must be seen as an ongoing process sub-ject to continual change. The preoccupation of other modern writers with the mathematical formulation of well-known re l a -tionships adds l i t t l e to our understanding of these processes. It also accounts for the widening gap between theory and practice "which has been largely responsible for the lack of realism that characterizes a large part of post-war ,,75 l i t e r a t u r e . " It must be recognized, however, that i f a theory is to achieve a certain degree of generality, i t must be formulated in terms of i t s essential c h a r a c t e r i s t i c s , without regard to circumstantial factors that may be operative at some moment but not at others. But such a theory is by necessity u n r e a l i s t i c . Moreover, while i t is true that foreign exchange markets are never at a s t a n d s t i l l , there are nevertheless relationships 5 9 which are perf e c t l y adducible to comparative s t a t i c analysis. Arbitrage is such a case. For i t is essential to arbitragers that they can base their decisions on ex ante data. Einzig is ri g h t , however, when he i n s i s t s that the analysis should not end with the determination of equilibrium under s t a t i c conditions. It is indeed necessary to i n v e s t i -gate how the system behaves over time. But before this can be done, a proper description of the system in s t a t i c e q u i l i b -rium i s an absolute prerequisite. Dynamic analysis can only proceed where the relationships under s t a t i c conditions are a s c e r t a i n e d . ^ 2.3 Summary and Appraisal Keynes' interest rate parity theory states that the percentage premium or discount of the forward rate over the spot rate tends to equal the difference between interest rates prevailing in two countries. We have attempted to demon-strate that his theory holds under highly idealized circum-stances. In fact, IRPT is a s p e c i f i c formulation of an a l l pervasive economic p r i n c i p l e , namely that two assets of the same tkind tend to y i e l d the same rate of return. It is i n -appropriate to apply such a general theory to p a r t i c u l a r instances. Keynes has recognized this error; he supplies a number of additional factors that have to be taken into con-sideration when actual phenomena are to be explained. Post-Keynesian writers can be divided into two groups. On one hand, the proponents have adopted the basic tenets of 60 IRPT; they attempt to explain deviations from parity by r e f e r r i n g to special circumstances occurring at certain times, but not at others. Such factors include limitations on available funds, market imperfections, moral suasion and speculation. The second group of post-Keynesian authors carry the analysis of foreign exchange a b i t further. They depart from IRPT in three fundamental ways. F i r s t , they stress that spot and forward rates must be determined simultaneously since arbitrage by d e f i n i t i o n involves simultaneous transactions in both markets. Second, they reject the notion that domestic and foreign assets are equivalent. Even i f monetary returns are the same, there are differences in r i s k - - r e a l or perceived--and i n -tangible advantages associated with one type of asset but not with the other. Third, forward exchange, rates are not determined by arbitrage alone. Modern theorists include speculation and foreign trade in a systematic manner in their models. Con-sequently, whether forward rates w i l l tend to parity levels depends on the r e l a t i v e magnitude of speculation, arbitrage and commercial covering in the forward market. Accordingly, parity is only one of many possible outcomes, unless the 7 7 AA schedule is completely e l a s t i c . However, a number of factors influencing the shape of the AA schedule such as the p r i n c i p l e of increasing r i s k , decreasing u t i l i t y of investing funds abroad, limited l i q u i d i t y 61 and exchange risks suggest that the arbitrage schedule is in fact less than pe r f e c t l y e l a s t i c . Compared to IRPT these models are much more " r e a l i s t i c " and empirically meaningful. They are also much more r e s t r i c t e d in their scope of application. Before we conclude this chapter, some of the questions that s t i l l remain unanswered and that require further research are b r i e f l y considered. F i r s t , there is the controversy over stock- versus flow-models. Although stock-concepts are said to be superior on 7 8 lo g i c a l grounds, models employing flow-concepts y i e l d consistently better results when empirically tested. This paradox arises because increases in stocks and lags in adjust-ments to changed rates may be s u f f i c i e n t to remove any 79 measurable effect of p o r t f o l i o realignment. There are some theoretical reasons to believe that stock-adjustments are-not instantaneous. Assuming that p o r t f o l i o -size is held constant and assuming a uniform investment ho r i -zon of 90 days, there is 1/90 of a l l short-term assets available for arbitrage each day. The rest of the funds are tie d up u n t i l maturity. It would require a very substantial covered interest d i f f e r e n t i a l before i t would pay an arbi -trager to cancel existing contracts in order to invest the proceeds at the new rates. Now, i t is perf e c t l y conceivable that 1/90 of a l l short-term assets is not s u f f i c i e n t to nudge the foreign exchange rates s u f f i c i e n t l y to restore equilibrium. Hence, 62 1/90 of a l l short-term assets could conceivably be transferred day after day u n t i l either investors reach th e i r self-imposed l i m i t s or u n t i l rates are pushed to equilibrium levels. The second observation refers to the predictive capacity of contemporary models. It is common to distinguish between arbitrage, speculation and foreign trade. But none of the models can explain or predict the r e l a t i v e magnitude of these components. Modern theory can state, for example, that parity rates w i l l not be attained when there is a certain amount of speculation (provided AA schedule is not perf e c t l y e l a s t i c ) . But i t cannot predict when and to what extent speculation occurs and when i t won't. Attempts have been made to overcome this problem by 8 0 distinguishing periods of " c r i s i s " from normal ones. It is doubtful whether anything can be gained from such an approach since the problem i s simply couched in other terms. The question remains why some people are arbitragers and other speculators. What is the proportion of arbitragers to speculators? There appears to be an elegant way out of this dilemma: "Speculation and covered-interest arbitrage should be viewed as methods of d i v e r s i f y i n g investments and trans-81 action p o r t f o l i o s . " It is in this d irection that future research should be undertaken. Another unanswered question relates to the symmetry of contemporary models. Borrowing-arbitrage is equated with 1 ending-arbitrage, active speculation with passive speculation. It i s quite l i k e l y , however, that lending to a foreign 6 3 borrower is much easier than actively seeking to borrow funds from a foreigner. S i m i l a r l y , we suggest that a subject who acquires foreign exchange in the course of some other trans-action is not l i k e l y to have the same attitude towards exchange risk as the person who acquires or s e l l s foreign exchange with the s p e c i f i c intent to speculate. If this i s true, then contemporary models could be improved by developing separate demand functions for borrowing and lending arbitrage, as well as for passive and active speculation. We have already alluded to another problem that merits elaboration. In the real world, there are a number of d i f -ferent interest rate p a r i t i e s corresponding to pairs of different types of interest-bearing s e c u r i t i e s . Chapter V is devoted to a preliminary exploration of d i f f i c u l t i e s associated with applying a s i m p l i f i e d model to real world conditions. F i n a l l y , the theory of forward exchange is incomplete in that i t treats interest rates as exogenous variables. Hence, the theory appears to imply that forward rates are always determined by interest d i f f e r e n t i a l s , but negates the possi-b i l i t y that forward rates may in turn affect interest rates. In Chapters III and IV we examine how a s i m p l i f i e d version of interest-arbitrage model could be expanded to accommodate this reciprocal relationship between forward rates and rates of interest. 64 CHAPTER II NOTES ^Keynes, Monetary Reform, pp. 115-39. 2 The notion of "pure case" or "ideal type" i s often resorted to in economics. What is meant is that relations so formulated are specified to hold under " p u r i f i e d " conditions between highly " i d e a l i z e d " objects or processes. Laws formu-lated with reference to pure case serve an important function. "A law so formulated states how phenomena are related when they are unaffected by numerous factors whose influence may never be completely eliminable but whose effects generally vary in magnitude with differences in the attendant c i r -cumstances under which the phenomena actually re cur." (Nagel, "Assumptions in Economic Theory," p. 64.) Much of the discussion in this chapter is inspired by Nagel's philosophy of science. His d i s t i n c t i o n between funda-mental assumptions, rules of correspondence and models are p a r t i c u l a r l y useful in this regard. It can indeed be said that much of the controversy around Keynes' theory of i n t e r -est arbitrage arises because the relationship between these notions is not always f u l l y understood. F r i t z Machlup discusses a similar problem in "The Problem of Economic V e r i f i c a t i o n , " Southern E conomi c Journal, XXII (July, 1955), pp. 1-21. See also Milton Friedman, "The Methodology of Positive Economics," Readings in Microeconomics, ed. by William Breit and Harold Hochman (New York: Holt, Rinehart and Winston, Inc., 1968). 3 M i s defined as M = F X ^ ^ o A 4 See John Spraos, "The Theory of Forward Exchange and Recent Practices," Manchester Schoo1 o f E conomi cs and Social  Studies, XXI (May, 1953), 88, hereafter referred to as "Theory of Forward Exchange and Recent Practices"; Sho Chie Tsiang, "The Theory of Forward Exchange and Effects of Government Intervention on the Forward Market," International Monetary  Fund Staff Papers, Vol. VII ( A p r i l , 1959), p. 80, n. 6, here-after referred to as "Theory of Forward Exchange," for a controversy over the correct mathematical treatment of this expression. 5Tagmir Amon, "The Forward Exchange Market: A Capital Market Approach," Unpublished Manuscript (University of Chicago, 1969) , p. 7. ^Keynes, Monetary Reform, p. 124. 65 CHAPTER II NOTES (cont'd) 7 In geometrical terms, supply and demand curves s h i f t to the right simultaneously and by the same amount, so that price remains unchanged, g For a discussion of the interactions between arbitrage, speculation and commercial trade see also Section 2.214. 9 How the presence of limited funds may interfere with the establishment of a parity rate is explored in Section 2.2112. 1 0R. G. Hawtrey, The Art of Central Banking (2nd ed.; London: Frank Cass § Co. Ltd., 1962), pp. 406-11. 11I_bi_d. , p. 408. Similar views are held by the following authors: Spraos, "Theory of Forward Exchange and Recent Practices," p. 88 f f . ; Anthony E. Jasay, "Bank Rate or Forward Exchange Policy," Banca Nazionale de1 Lavoro Quarterly Review (March, 1958), pp. 56-73, hereafter referred to as "Bank Rate"; B. Reading, "The Forward Pound 1951-59 ," Economic Journal, LXX (June, 1960), 304-19, hereafter referred to as "Forward Pound." 12 Keynes, Monetary Re form, p. 128. This figure has acquired something li k e the status of a myth. Hardly any textbook or even theoretical contribution f a i l s to mention that the forward rate may deviate from i t s prescribed level by as much as p.a. However, there is absolutely no empirical evidence or theoretical foundation given for such assertions, other than a reference to Keynes. For a flagrant example see Reading, "Forward Pound," p. 306. Also Egon Sohmen, Flexible Exchange Rates (rev. ed.; Chicago and London: University of Chicago Press, 1969), p. 89, n. 11. Hereafter referred to as Flexible Exchange Rates. 13 Compare the data compiled by Grubel, Forward Exchange, pp. 69-80. 14 Sohmen, Flexible Exchange Rates, p. 89; Spraos, "Theory of Forward Exchange and Recent Practices," p. 92. 1 5Spraos, "Theory of Forward Exchange and Recent Practices," pp. 97-101. ^ F o r the mechanics involved see Holmes and Schott, New York Foreign Exchange Market, pp. 51-52; and Report of  the Royal Commission on Banking and Finance (Ottawa: Queen's Printer, 1965), pp. 291-300, hereafter referred to as Report. 66 CHAPTER II NOTES (cont'd) 1 7 It s t i l l does not necessarily hold. Spot rates or interest rates may change. See John H. Auten, "Forward Exchange Rates and Interest-Rate D i f f e r e n t i a l s , " Journal of  Finance, XVIII (March, 1963), 11-19, hereafter referred to as "Forward Exchange Rates"; Hans R. S t o l l , "An Empirical Study of the Forward Exchange Market under Fixed and Flexible Exchange Rate Systems," Canadi an Journal of E conomi cs, II (Feb., 1968), 55-78, hereafter referred to as "Empirical Study." 18 Sohmen, Flexible Exchange Rates, pp. 88-90. 19 For a c r i t i c a l appraisal of IRPT see also Neldner, Devisenterminmarkt, pp. 17-24; Tsiang, "Theory of.Forward Exchange," pp. 80-81. 2 0 Neldner, Devisenterminmarkt, pp. 29-69. 2 1 In the following, reference to supply schedules (or demand schedules, as the case may be) w i l l be omitted. When-ever a person demands foreign exchange, he i s prepared to supply some other currency in exchange. Supply curves for one currency can therefore be derived from demand curves for another in the case of a two-country model. For technical details see Gottfried Haberler, "The Market for Foreign Exchange and the S t a b i l i t y of the Balance of Payments," Kyklos, III (1949), 193-218; reprinted in International  Finance: Selected Readings, ed. by Richard N. Cooper (Middle-sex and Baltimore: Penguin Library, 1969), hereafter referred to as "Market for Foreign Exchange." 2 2 Einzig , Dynamic Theory, p. 167. See also Oscar L. Altman, "Foreign Markets for Dollar, S t e r l i n g and Other Currencies," International Monetary Fund  Staff Papers, VIII (1960/1961), 322, hereafter referred to as "Foreign Markets"; Tsiang, "Theory of Forward Exchange," p . 82 . 2 3 For these reasons risk of default can be eliminated, provided such risks are not perceived to be s i g n i f i c a n t l y larger by a majority of arbitragers. Neldner, Devisentermin-markt, p. 34, n. 71. 24 A good description of i n s t i t u t i o n a l constraints in the foreign exchange markets can be found in Oscar L. Altman, "Recent Developments in Foreign Markets for Dollars and other Currencies," International Monetary Fund Staff Papers, X (1963), 53, hereafter referred to as "Recent Developments." 6 7 CHAPTER II NOTES (cont'd) 2 5 Michal Kalecki, "The Pri n c i p l e of Economic Risk," Economica New Series, IV (Nov., 1937), 442, hereafter referred to as "Economic Risk"; as quoted in Neldner, Devisentermin-markt, p. 3 7. 2 6 Neldner does not e x p l i c i t l y draw this conclusion. However, this appears to be the only possible explanation consistent with his derivation of the AA schedule. Devisen-terminmarkt, pp. 37-39. 2 7 See for example: Neldner, Devisenterminmarkt, pp. 37-38; Grubel, Forward Exchange, pp. 9-10; Lawrence H. O f f i c e r and Thomas D. W i l l e t t , "The Covered-Arbitrage Schedule: A C r i t i c a l Survey of Recent Developments," Journal of Money, Credit, and Banking, Vol. II, No. 2 (1970), pp. 250, here-after referred to as "Covered-Arbitrage Schedule." Space does not permit further discussion of this interesting question. It w i l l be b r i e f l y alluded to in our summary, Section 2.3. 2 8 Net interest d i f f e r e n t i a l is defined as the difference between the domestic and foreign interest rates adjusted for the cost or pickup of covering. 2 9 Neldner defines the AA schedule as a function r e l a t i n g demand for forward exchange to net interest d i f f e r e n t i a l . To reconcile Fig. 2 with Fig. 1, the labels "demand and "supply" must be interchanged. The reader should also be warned that B r i t i s h writers employ schedules which are rotated around the ordinate compared to standard diagrams. This p e c u l i a r i t y is due, of course, to the way prices of foreign exchange are quoted in London. 30 Neldner, Devisenterminmarkt, p. 39. 31 Assuming domestic and foreign interest rates remain unchanged, the term net interest rate d i f f e r e n t i a l can be replaced with forward margin. The AA schedule intersects the ordinate at the point where the forward margin is consistent with interest rate parity. As in the case of IRPT, the assump-tion was made that interest earned is not covered in the forward market. The AA schedule could represent supply of and demand for spot exchange i f the labels "supply" and "demand" were interchanged. If the foreign (domestic) interest rate increases (cet• par.), the schedule w i l l s h i f t down (up) because parity is attained at a higher discount, respectively lower premium (lower discount, respectively higher premium). Adapted from Neldner, Devisenterminmarkt, p. 41. 68 CHAPTER II NOTES (cont'd) 33 Neldner, Devisenterminmarkt, p. 42. In r e a l i t y , a change in supply of forward foreign exchange (and the corollary demand for domestic spot exchange) w i l l not only reduce the foreign money supply; i t w i l l at the same time increase the money supply in the domestic center, thereby influencing domestic and foreign interest rates. Neldner assumes that these subsidiary effects are offset by intervention of monetary authorities. But this is an aspect that we plan to investigate in Chapters III and IV. 34 We continue to disregard interest accrual. 35 Neldner, Devisenterminmarkt, p. 42. 3 6 The subscript is dropped when the meaning is clear. 3 7 Neldner, Devisenterminmarkt, p. 46. 5 8 I b i d . , p. 47. 39 Adapted from Neldner, idem. 40 Adapted from Neldner, idem. 41 Adapted from Neldner, i b i d . , p. 49. 4 2 See Holmes and Schott, New York Foreign Exchange  Mark et, pp. 49-50, for a more extensive discussion. 43 Compare Neldner, Devisenterminmarkt, pp. 48-49. An analogous argument w i l l be made with regard to certain spot transactions by merchants (Section 2.213), so that the analysis can be omitted here. 44 Neldner does not e x p l i c i t l y derive these two functions. Devisenterminmarkt, p. 58, contains a reference to this sort of diagram, however. These supply and demand curves w i l l be amalgamated with the schedules to be derived in the following paragraphs for the determination of equilibrium. ^See pp. 16-17. By analogy, exporters may s e l l t h e i r future proceeds on the forward market. A l t e r n a t i v e l y , exporters may borrow foreign exchange, s e l l i t on the spot market and invest the proceeds in the domestic market. The payment made by his partner at t+90 would then be used to pay off the loan to the exporter, which was denominated in foreign currency. See Grubel, Forward Exchange, p. 56, and Tsiang, "Theory of Forward Exchange," p. 93. 69 CHAPTER II NOTES (cont'd) 46 Tsiang, "Theory of Forward Exchange," p. 94, and it Spraos, "Theory of Forward Exchange and Recent Practices pp. 92-95 for a more exhaustive exposition. Tsiang also demonstrates that the same argument applies to exporters by analogy. 47 E l a s t i c i t i e s of supply and demand depend on the price e l a s t i c i t i e s of the commodities traded. 4 8 This is due to our assumption that only 90-day forward contracts are available. 49 Neldner, Devisenterminmarkt, p. 58. This diagram as well as Fig. 10 are highly s i m p l i f i e d . A more detailed analysis can be found in Haberler, "Market for Foreign Exchange," pp. 107-34. "^ We have omitted hedging. Neldner, Devisentermin-markt , pp. 59-62, demonstrates that hedging can be viewed as a special type of speculation. Therefore, hedging is absorbed by the SS schedule. ^Non-arbitragers are in this instance, a l l persons or i n s t i t u t i o n s other than arbitragers and speculators. Excess-demand curves are derived by defining supply as negative demand. 5 2 Disregarding interest accrual. 5 3 For a geometrical derivation of equilibrium rates see Neldner, Devisenterminmarkt, pp. 63-67. 54 Chapter III provides an i l l u s t r a t i o n to this point, "^Grubel, Forward Exchange, pp. 12-21 and references to other a r t i c l e s contained therein. ^ H a r r y M. Markowitz, P o r t f o l i o Selection: Ef f i ci ent  D i v e r s i f i c a t i o n of Investments (New York: John Wiley, 1959); and James Tobin, " L i q u i d i t y Preference as Behavior Towards Risk," Review of Economic Studies , XXV (Feb., 1958), pp. 65-86. Hereafter referred to as " L i q u i d i t y Preference I ! 5 7 Tobin, " L i q u i d i t y Preference," p. 66. 5 8 S t o l l , "Empirical Study," p. 59. It should be noted that S t o l l ' s reasoning requires the assumption that the need for premature repatriation be the only factor which could cause interest parity to be violated. This means that such 70 CHAPTER II NOTES (cont'd) factors as increasing risk or lack of l i q u i d i t y are s p e c i f i -c a l l y eliminated. If Grubel's prime reason for an i n e l a s t i c arbitrage curve is not correct, then the subsidiary considera-r e p a t r i a t i o n - - i s not v a l i d either. The argument hinges, however, on the existence of forward markets for the term concerned. Com-"Causes of Deviation from Interest-Rate J o u rn a1 of Money, Credit, and Banking, IV (Feb. , 115, n. 9., hereafter referred to as "Causes of tion--premature v a l i d i t y of our f u l l y developed pare Hans S t o l l P arity," 1972), p Deviation." See also O f f i c e r and W i l l e t t , "Covered Arbitrage Schedule," pp. 253-54. 59 William H. Branson, Financial Capital Flows in the H-§L' Balance of Payments (Amsterdam: North-Holland Publishing Company, 1968). Hereafter referred to as Financial Capital  Flows. ^Presumably, changes in r i s k levels or in investor's attitudes towards risk can also result in a r e d i s t r i b u t i o n of assets. Branson does not mention this p o s s i b i l i t y . 61, 62 Branson, Financial Capital Flows, p. 3 and Chap. II Tsiang, "Theory of Forward Exchange." 6 3 Spraos, "Theory of Forward Exchange and Recent Practices." 64 65 Tsiang, "Theory of Forward Exchange," p. 81. Idem. 66 67 Ibid., p. 82. S t o l l , "Empirical Study." 6 8 John H e l l i w e l l , "A Structural Model of the Foreign Exchange Market," Canadian Journal of Economics, II (Feb., 1969), 90-105. Hereafter referred to as "Structural Model.' 6 9 Jerome L. Stein, The Nature and E f f i c i e n c y of the  Foreign Exchange Mark et, Princeton Studies in International Finance, No. 40 (Princeton, N. J.: International Finance Section, 1962). Hereafter referred to as Nature and  E f f i c i e n c y . 70 S t o l l , "Empirical Study," p. 59 .71 CHAPTER II NOTES (cont'd) 71 For a good appraisal of Stein's i n f l u e n t i a l c o n t r i -bution consult Glahe, Empirical Study, who bases a number of empirical tests on Stein's model. Compare also Jay H. Levin, Forward Exchange and Internal-External Equilibrium (Ann Arbor, Mich.: University of Michigan, 1970), p. 23. Here-after so quoted. 72 * Peter Frevert, "A Theoretical Model of the Forward Exchange: Part I," International Economic Review, VIII (June, 1967), 153-67. Part II is contained in the same volume and appears on pp. 307-26. Hereafter so quoted. 73 Stanley S. Black, "Theory and Policy Analysis of Short-Term Movements in the Balance of Payments," Yale  Economic Essays, VIII (Spring, 1968), p. 9, n. 14. 74 Einzig, Dynamic Theory• 7 5 I b i d . , p. 143. 7 6 I b i d . , p. 265. 7 7 Chapter IV, pp. 95-6, provides a more detailed analysis of these problems. See also Auten, "Forward Exchange Rates," pp. 14-15. 7 8 Off i c e r and W i l l e t t , "Covered Arbitrage Schedule," p. 249. Compare for example: U.S. Congress, Joint Economic Committee, "Private Capital Movements and the U.S. Balance-of-Payments Position," by P h i l i p W. B e l l , Factors Affecting  the United States Balance of Payments, Compilation of Studies Prepared for the Subcommittee on International Exchange and Payments (Washington, D.C.: Government Printing Office, 1962), pp. 395-482; U.S. Congress, Joint Economic Committee, "Short-Term Capital Movements and the U.S. Balance of Pay-ments," by Peter B. Kenen, Part I_: Current Prob lems and  P o l i c i e s , Hearings before the Joint Economic Committee, 88th Cong., 1st sess., 1963, pp. 153-90; U.S. Congress, Joint Economic Committee, "A Survey of Capital Movements and Findings Regarding Their Interest S e n s i t i v i t y , " by Benjamin J. Cohen, Part I_: Current Problems and P o l i c i e s , Hearings before the Joint Economic Committee, 88th Cong., 1st sess., 1963, pp. 192-207; for a c r i t i c a l 'appraisal of these studies see Branson, Financial Capital Flows, p. 68 f f . ; Jerome L. Stein, "International Short-Term Capital Movements," American  Economic Review, LV (March, 1965), 40-66. Hereafter so quot ed. CHAPTER II NOTES (cont'd) On this question see H e l l i w e l l , "Structural Model," p. 101; O f f i c e r and W i l l e t t , "Covered Arbitrage Schedule," p. 249; Robert A. Dunn, J r . , Canada's Experience with Fixed  and Flexible Exchange Rates in a_ North American Capital  Market. Montreal: Private Planning Association of Canada, 1971, p. 25. Hereafter referred to as Canada's Experience. 8 0 See, for example, Stein, Nature and E f f i c i e n c y ; Frevert, "Theoretical Model of the Forward Exchange." ^ L e v i n continues: "This i s to be contrasted with the conventional approach, which assumes that these a c t i v i t i e s are carried on as unassociated transactions. The former view e x p l i c i t l y recognizes the stock nature of the a c t i v i t i e s and the substitutions between money, domestic s e c u r i t i e s , foreign s e c u r i t i e s , and speculation which occur as the result of changes in the i r rates of return." Forward Exchange and Internal-External Equi1ibrium, p. 7. 73 CHAPTER III THE INFLUENCE OF INTEREST ARBITRAGE ON INTEREST RATES UNDER PEGGED EXCHANGE RATES 3.1 Out li n e and Re 1evan ce o f the Problem The theory of interest arbitrage was o r i g i n a l l y devised as a means of explaining the behavior of the forward exchange rate. By r e l a t i n g demand for forward exchange to the d i f f e r -ence between short-term interest rates in two international centers, Keynes was able to show how the forward rate could systematically deviate from the spot r a t e . 1 This deviation gives r i s e to the forward margin. Much of the theoretical work since Keynes has been concerned with the conditions under which the forward margin apparently f a i l s to adjust to the gross interest d i f f e r e n t i a l . The evaluation of the e l a s t i c i t y of arbitrage schedules and the examination of the effect of speculation on the forward rate have been particu-2 l a r l y s i g n i f i c a n t in this regard. The h i s t o r i c a l development of the theory of interest arbitrage may thus explain why most of the analyses were carried out within the framework of models confined to the foreign exchange market. In order to concentrate the analysis on the behavior of the forward rate, i t is generally assumed that interest rates in both money markets can be treated as data, i . e . exogenous variables. A large number of such 74 models are also based on the further assumption that spot rates are fixed. Yet both assumptions cannot hold at the same time. It w i l l be demonstrated in the next section that under a system of r i g i d l y pegged exchange rates the money supply of an economy must expand and contract as cap i t a l movements take place. One could argue, however, that while there is a d e f i n i t e connection between arbitrage and changes in interest rates, the quantitative effect is so small as to be negligible when 3 compared to the changes in the forward rate. The argument can be s e t t l e d by examining the interactions between the domestic and foreign interest rates and the forward rate. In this context we propose to examine the following hypothesis: The impact of interest arbitrage on interest rates under r i g i d l y pegged exchange rates depends on the r e l a t i v e size and e f f i c i e n c y of the various markets used by arbitragers. The pressure which arbitragers bring to bear on interest rates, e.g., is stronger in small money markets than in 1arge ones. The v a l i d i t y of this hypothesis has not only implica-tions for the usefulness of other models of interest arbitrage and for the design of empirical tests; i t also helps to explain the degree of independence enjoyed by monetary authorities in various countries. Consider, for example, the following statement: In a world where large volumes of private capital flow among nations in quick response to r e l a t i v e l y modest variations in interest rates and where the exchange rate of the Canadian dollar is fixed--as 75 i t was between 1962 and June 1970--the Bank of Canada has been almost compelled to approximate the monetary p o l i c i e s followed by the U.S. Federal Reserve System.4 A p r i o r i , unless certain assumptions about the size and perhaps the e f f i c i e n c y of the two markets are made, there is no reason why the statement could not be reversed, saying that the United States had to approximate Canadian monetary policy. If our hypothesis i s v a l i d , however, the merit of Dunn's statement can be easily judged. 3.2 Short-Term Capital Movements and Interest  Rates under Pegged Exchange Rates In order to examine the polar case of pegged exchange rates we assume that the spot exchange rate i s r i g i d l y fixed. There is no spread between upper and lower intervention points. The central bank is therefore committed to purchase or s e l l - f o r e i g n exchange at the given rate. A net capital inflow which is not met, e.g., by a simultaneous change in commodity imports must therefore lead to o f f i c i a l intervention in the spot market. Reserves of commercial banks are increased by the equivalent of the capi t a l inflow. Credit conditions tend to ease and interest rates tend to f a l l . Net ca p i t a l outflows lead to a contrac-tion of the money supply and to a s t i f f e n i n g of interest 5 rates. In a model incorporating only two countries, one nation's gain is the other's loss. A ca p i t a l transfer would therefore lead to lower interest rates in the gaining center and to higher interest rates in the losing economy. The movement of arbitrage funds from, say, the domestic to the foreign economy tends therefore to reduce the net incentive for arbitrage in two ways. The purchase of spot funds and the simultaneous sale of forward currency tends to reduce the premium (increase the discount) on the foreign currency. At the same time interest rates tend to r i s e in the domestic economy while those in the foreign economy tend to f a l l . Arbitrage tends therefore to restore equilibrium by reducing the foreign exchange component on one hand and by reducing the gross interest d i f f e r e n t i a l on the other hand.*' This mechanism d i f f e r s from the one incorporated in the models which were examined in Chapter II. Since interest rates were held constant, equilibrium could only be attained through variation in the forward rate under a system of 7 pegged spot rates.. The following sections explore in some d e t a i l the burden that interest rates must bear in the adjustment process Section 3.31 examines the interactions between the foreign exchange markets and one money market. In Section 3.32 the analysis is expanded to accommodate the second money market. The analysis w i l l be carried out in terms of the slopes of the constituent demand and supply schedules. Section 3.4 attempts to provide a link between the notions of market size and e f f i c i e n c y on one hand and the slopes of market schedules on the other. Section 3.5, f i n a l l y , reinterprets and sum-marizes the results obtained in Sections 3.31 and 3.32 in terms of market size. 77 ^•^ Interest Rat es and Forward Exchange Rates 3.31 Interactions between Foreign Exchange Markets and the Foreign Money Market For the ensuing discussion, reference w i l l be made to Fig. 12 (page 78). Note that a l l markets are represented by excess-demand curves. The arbitrage schedule i s thought to be less than perfectly e l a s t i c . For s i m p l i c i t y , a l l schedules are assumed to be l i n e a r . Note that a l l quantities are measured in terms of the same foreign currency. The i n i t i a l equilibrium position implies that the foreign interest rate g (RF^) i s higher than the domestic interest rate. A f i n a l remark concerns the spot market. Only the supply and demand curve of monetary authorities is shown. It i s i n f i n i t e l y e l a s t i c because the central bank is com-mitted to buying or s e l l i n g that which is supplied or demanded at the fixed price SX. By assumption, transfers of short-term cap i t a l leave exports and imports of commodities unchanged. This requires that transfers of cap i t a l by arbi-tragers must be exactly matched by central bank purchases or 9 sales in the spot market. Let us now assume that the foreign interest rate is increased. This is represented in the diagram by a s h i f t to the right of the D r schedule to D^. A new, higher interest rate RF 2 i s established. Arbitragers w i l l therefore tend to buy more spot exchange, shown by the s h i f t of A g to A^. This leads to an expansion of the money supply in the foreign money market by the amount A. Consequently RF f a l l s from FX RF Q spot exchange/t A £ A1 v v x f \N\ c F X i FX 2-^\ \ \ ^ D f - + Q forward exchange/t -A A Q loanable funds/t SX. FX, RF, are i n i t i a l 'I' *"1 where SX ... spot exchange rate FX ... forward exchange rate RF ... rate of interest in the foreign money market D ... excess demand for spot exchange by monetary authorities (arbi-tragers transactions are ex-actly matched by central bank sales or purchases) equilibrium values D „ D excess demand for forward exchange by non-arbitragers excess demand for loanable funds by non-arbitragers arbitragers' excess demand for spot exchange arbitragers' excess demand for forward exchange arbitragers' excess demand for loanable funds 10 Fig. 12. Determination of the spot rate, the forward rate and the foreign interest rate 0 0 79 RF2 to RF3 (and the A' schedule s h i f t s back down somewhat to A"). It can be seen immediately that the steeper D£, the larger the o f f s e t t i n g effect of arbitrage on RF^-But arbitragers must also s e l l forward exchange. Hence Af s h i f t s down as a result of the o r i g i n a l increase in RF. We obtain A£ and a new, lower forward rate FX 2 . But the i n -creased discount reduces the net incentive for arbitrage. Consequently A'^  s h i f t s down to A^ ' and A r - s h i f t s up to A.J. yiel d i n g RF^. The system o s c i l l a t e s in this manner u n t i l the following conditions are f u l f i l l e d : 1. Quantities demanded or supplied by arbitragers are equal in a l l markets, 2. Quantities demanded or supplied by non-arbitragers equal quantities supplied or demanded by arbitragers in a l l markets. Though the equilibrium solution requires a number of ite r a t i o n s and cannot be obtained by inspection, the following observations can be derived from Fig. 12: the steeper the D £ schedules, the larger the var i a t i o n of the forward rate with a given s h i f t of the arbitrage schedule. But the larger the variation of FX, the longer the distance by which the arbitrage schedule w i l l s h i f t back up in the foreign money market. This backward s h i f t can be interpreted as a reduction of arbitragers' impact on the foreign money market. It can be said, therefore, that the stronger the impact of a change in arbitragers' demand on the forward rate, the smaller the impact of arbi -trage on the money market. In other words, the money market gains in independence as the v a r i a b i l i t y of the forward rate increases . We also know that i f the foreign money market is characterized by a steep D r schedule, the impact of arbitrage is stronger than i t would be i f D r was more e l a s t i c . The effect of arbitrage would therefore tend to offset a change in the interest rate induced by monetary authorities more completely in a market with a steep D r curve. But part of the v a r i a t i o n w i l l be absorbed by the forward rate. It appears, therefore, that the slopes of excess demand by non-arbitragers in both forward market and money market are of cr u c i a l importance for the assignment of the variation of forward rates and interest rates. The greater the varia-b i l i t y of the forward rate, the greater the independence of monetary authorities in their pursuit of an optimal monetary policy . On the other hand i t appears that i f the interest rate is highly v o l a t i l e , the forward rate does not vary as much as would be predicted by more r e s t r i c t i v e models of interest arbitrage. At this juncture i t can be said, therefore, that models which treat interest rates as data apply best when demand and supply curves of non-arbitragers in the money market are f l a t compared with the schedules in the forward market. 3.32 Interactions between Foreign Exchange Markets and Domestic and Foreign Money Markets It can now be undertaken to include the domestic money market in the analysis. This integration is not straight-forward, however. Demand for and supply of domestic loanable funds is expressed in a different unit, and the conversion from domestic into foreign currency i s non-1inear. 1 1 But so long as we undertake to keep the spot rate constant this problem is of no concern. Starting from equilibrium, l e t the domestic interest rate increase. This provides a net incentive for inward arbitrage. The inflow of funds leads to an expansion of the domestic money supply. The domestic interest rate w i l l tend to f a l l , thus o f f s e t t i n g the i n i t i a l increase. This off-setting effect is t o t a l i f arbitragers' supply is perfectly e l a s t i c . With an imperfectly e l a s t i c arbitrage schedule the of f s e t t i n g effect of arbitrage i s greater the steeper the demand and supply schedules of non-arbitragers in the domestic money market. Meanwhile the supply of money in the foreign center diminishes and foreign interest rates tend to r i s e . They ri s e more sharply i f that market is characterized by steep curves for non-arbitragers. But this r i s e in the foreign interest rate reduces the incentive for arbitrage. Hence fewer funds w i l l be transferred into the domestic economy and the o f f s e t t i n g effect of arbitrage on the domestic i n -terest rate is also smaller. It follows that the larger the variation of the foreign interest rate, the smaller the off-setting effect of arbitrage on the domestic interest rate. On the contrary we find that the market which i s charac-terized by steep supply and demand schedules of non-arbitragers 82 is strongly dependent on the market where these schedules are f l a t . The effects of variations in the forward rate have been omitted so far. The presence of a v o l a t i l e forward rate tends to disrupt the strong interdependence between the two money markets. Suppose the domestic interest rate is increased. There w i l l be an inflow of arbitrage funds. This inflow tends to lower the domestic interest rate somewhat, thereby p a r t i a l l y o f f s e t t i n g the o r i g i n a l increase. The foreign interest rate tends to r i s e and the o f f s e t t i n g effect of arbitrage is smaller in the domestic market. But arbitragers must also offer funds on the forward market, i.e . demand forward exchange. The for-ward rate increases, thereby reducing the net incentive for arbitrage. Hence less funds w i l l be transferred and the off-setting effect of arbitrage i s smaller in the domestic market. The increase in the foreign interest rate is also smaller. We conclude, therefore, that the greater the v a r i a b i l i t y of the forward rate (the steeper the D £ schedule) the smaller the interdependence and the greater the independence of the two money markets. 3.4 Market Size and Price Variations The foregoing sections explored the v a r i a b i l i t y of interest rates and foreign exchange rates under various assumptions about the slopes of the constituent curves for each market. The use of slopes rather than e l a s t i c i t i e s is j u s t i f i e d so long as the analysis is carried out in terms of 83 a common unit of measurement which in our case was the foreign 12 currency. The results of the preceding analysis can now be re-interpreted in terms of market size instead of slopes. The two notions are stipulated to be linked as follows: large markets (measured by the volume of transactions in terms of foreign currency per unit of time) must be represented graphi-c a l l y by r e l a t i v e l y f l a t schedules while small markets are characterized by r e l a t i v e l y steep demand and supply curves. Even i f the e l a s t i c i t y of demand i s the same in a small and in a large market, the demand schedule must be drawn more steeply in a smaller market. But demand and supply schedules are also l i k e l y to be more e l a s t i c in large markets because such markets tend to be more e f f i c i e n t . This i s so because the cost of buying or s e l l i n g commodities tends to be smaller in a large market. Smith distinguishes three types of costs that a buyer 13 or s e l l e r may incur: 1. The spread between what the s e l l e r receives and what the buyer pays, 2. The difference between the amount the s e l l e r gets and what he would have obtained in a better market. In a sense this type of cost is a measure of the devia-tion of the actual market price from the hypothetical, "true" value of the asset, 3. Costs associated with random fluctuations. 84 A l l three types of costs are negatively related with the volume of a c t i v i t y . If i t is true, therefore, that "the supply of sec u r i t i e s [and presumably the demand] for a secondary market w i l l be highly e l a s t i c with respect to 14 costs" then i t f o i l ows that large markets are characterized by more e l a s t i c supply and demand curves than small ones. A similar view was upheld in a study commissioned by the OECD where the authors contend that national boundaries and small national incomes place certain p r a c t i c a l l i m i t a -tions on the size of European ca p i t a l markets. They go on to say: "Thus, inasmuch as market e f f i c i e n c y is a function of economics of scale, the degree of e f f i c i e n c y which can 15 be achieved is bound to be r e s t r i c t e d . " 3.5 Reinterpretation of Results The conclusions of the foregoing sections can now be b r i e f l y summarized in terms of market size. F i r s t consider the interactions between a single money market and the for-ward exchange market. This corresponds to the analysis in Section 3.31. The smaller the forward market, the larger the variation of the forward rate in response to a given change in demand or supply. The smaller the money market, the stronger the impact of arbitrage on that interest rate. This means that a desired change in the interest rate i s p a r t i a l l y offset by arbitragers; this o f f s e t t i n g effect is larger the more responsive arbitragers are to changes in interest rates. But the impact of arbitrage on the forward 85 rate has the effect of reducing the net incentive for arbitrage. Hence less funds w i l l be transferred and the off-setting effect on the interest rate i s smaller. We find therefore that i f the forward market is large r e l a t i v e to the money market, the impact of arbitrage w i l l be most strongly f e l t in the money market. The reverse is also true. If the forward market is small r e l a t i v e to the money market, then the forward rate must absorb most of the variation and the interest rate is less strongly affected. Hence the money market gains in independence as i t s size increases r e l a t i v e to that of the forward market- and vice versa. The second money market can now be included in our considerations, following the developments in Section 3.32. The interactions between two money markets in the absence of a forward market are exactly analogous to those between the forward and one money market. The results need not be re-stated, therefore. The strong link i s broken with the intro-duction of a forward market. The smaller the forward market r e l a t i v e to the two money markets, the larger the variation of the forward rate and the greater the independence of the two money markets. If one money market is small r e l a t i v e to the other two markets, i t s interest rate has to absorb most of the variat i o n . This i s true no matter where the autonomous change occurs. If the monetary authorities wish to change their domestic interest rate, t h e i r goals are largely frustrated 86 by the effects of interest arbitrage. If the autonomous change occurs in the large market, on the other hand, the a c t i v i t y of arbitragers w i l l do l i t t l e to offset i t . Instead, the smaller market i s forced to approximate the developments abroad. The accuracy of models which treat spot and interest rates as data can now be appreciated. The larger the money markets r e l a t i v e to the forward market, the greater the con-t r i b u t i o n of the forward rate towards the restoration of equilibrium. In the case of i n f i n i t e l y large money markets ( i . e . p e r f e c t l y e l a s t i c demand and supply curves of non-arbitragers), the predictions of such models are perfectly accurate. With less than i n f i n i t e l y large money markets, however, such models should be expanded to include these markets as well. 87 CHAPTER III NOTES Keynes, Monetary Reform. 2 See Chapter II for an exposition of some of the better known models. 3 This approach might be useful for empirical research. See, for example, H e l l i w e l l , "Structural Model," pp. 99-100. H e l l i w e l l recognizes the bias which this s i m p l i f i c a t i o n might introduce into his estimates. In theory, however, this sort of treatment carries the risk of s t i p u l a t i n g directions of causation for which there may not exist any theoretical foundation. In the present case, for example, the notion evolved that changes in interest rates are always the cause, never the effect of changes in forward rates. The forward rate appears therefore as the sole e q u i l i b r a t i n g agent. See also Einzig, Dynamic Theory, Chapter XIV, who is especially c r i t i c a l of this misconception. 4 Dunn, Canada's Experience, p. 3. 5To be precise, whether short-term cap i t a l transfers lead to changes in the money supply, depends partly on the mode of financing foreign exchange operations and partly on the e l a s t i c i t y of the supply of credit. Under normal c i r -cumstances, the rules set forth above are v a l i d , however. See F r i t z Machlup, International Payments, Debts, and Gold: Collected Essays (New York: Charles Scribner's Sons, 1964), pp. 34-39, 45-46, for a thorough examination of this question. Hereafter referred to as International Payments. Dunn, Canada's Experience, pp. 15-16, provides a good summary of the Canadian arrangement which d i f f e r s in some respects from conventional systems of exchange s t a b i l i z a t i o n . He points out, however, that in the end the Canadian system produces the same results as conventional systems. Einzig, Dynamic Theory, pp. 190-91, examines the impact of capi t a l transfers on interest rates in the context of interest arbitrage. His conclusions are i d e n t i c a l with ours . S o m e authors have acknowledged the existence of this relationship, though they f a i l e d to include i t in their model. Others allude to i t en_ pass ant . See, for example: Richard E. Caves, e_t al_. , Capital Trans fers and Economic Policy: C anada, 1951-1962 (Cambridge, Mass.: Harvard University Press, 1971), pp. 103, 127-29, 138, hereafter referred to as Capital  Trans fers; Grubel, Forward Exchange, p. 18; Holmes and Schott, New York Foreign Exchange Market, pp. 55-56 ; Peter B. Kenen, "Trade, Speculation, and the Forward Exchange Rate," in Trade, Growth, and the Balance of Payments, ed. by R. E. Baldwin et_ a_l_. (Chicago: Rand McNally, 1965), pp. 143-69 ; 88 CHAPTER III NOTES (cont'd) Kindleberger, "Speculation and Forward Exchange," p. 181; Oscar Morgenstern, International Financial Trans actions and  Business Cycles (Princeton, N. J.: Princeton University Press, 1959), pp. 124-25; Marcus Nadler, Sipa Heller, and Samuel S. Shipman, The Money Market and i t s Institutions (New York: Ronald Press Company, 1955), pp. 16-17; Neldner, Devisenter-minmarkt , p. 42; Stein, Nature and E f f i c i e n c y , p. 30. Balbach i l l u s t r a t e s another interesting mechanism which is generally overlooked in the discussion of interest arbitrage. Suppose that holders of U.S.-denominated assets convert their holdings into some foreign currency. Foreign recipients of these dollars may s e l l them to the i r central bank. The cen-t r a l bank, in turn, invests these dollars in U.S. treasury b i l l s , thereby r a i s i n g prices and lowering yields on TB's. The net effect of short-term c a p i t a l outflows on the U.S. interest rate depends therefore on the type of assets in which the funds were o r i g i n a l l y invested. If a l l purchasers of foreign assets had previously held TB's, the net effect would be n i l . I f , on the contrary, these funds had been i n -vested in equities, the U.S. rate of interest would actually f a l l ; a result which is contrary to our predictions. It i s assumed, in the following, that central banks r e f r a i n from investing their acquisitions of reserves i n the country of o r i g i n . See Anatole Balbach, "Will Capital Reflows Induce Domestic Interest Changes?" Federal Reserve Bank of St. Louis (July, 1972), pp. 2-5. g Given, for example, the spot and the interest rate, the arbitrage curve in the forward market intersects the ordinate at the point where the forward rate assumes a value such that the net interest d i f f e r e n t i a l i s zero. At this point, arbitragers would neither demand nor supply forward exchange. If the foreign interest rate i s increased, cet. par•, then the forward rate at which net incentive equals zero must be lower. The argument would be exactly analogous i f the domestic interest rate was lowered. The A^ schedule would s h i f t up i f the foreign interest rate was lowered or i f the domestic one was increased. 9 It must be assumed, furthermore, that the continued support of the spot rate by monetary authorities is in some doubt in the minds of market participants. Otherwise, the existence of a forward market could hardly be j u s t i f i e d . In r e a l i t y , forward markets exist because (1) the spot rate is allowed to vary to some extent at least and (2) the con-fidence in the maintenance of existing p a r i t i e s is not absolute. CHAPTER III NOTES (cont'd) Our examination of the polar case of r i g i d l y pegged exchange rates is j u s t i f i e d in that we wish to determine th maximum impact that arbitrage can exert on interest rates. The opposite case where spot rates are fre e l y fluctuating i examined in the following chapter. It w i l l emerge that transfers of funds by arbitragers have no monetary effect at a l l . A l l other cases of fixed spot rates l i e somewhere between these extremes. "^In the context of our analysis in Chapters III and IV, non-arbitragers include a l l those market participants whose a c t i v i t i e s are not r e f l e c t e d in the arbitrage schedul ^See for example Haberler, "Market for Foreign Exchange," pp. 118-21. 12 Earl Francis Beach, Economic Models: An Expos i t ion (New York: John Wiley § Sons, Inc., 1957), pp. 25-27, on the appropriate use of slopes versus e l a s t i c i t i e s . 13 Paul F. Smith, E conomi cs of Financial Institutions  and Markets (Homewood, 111.: Richard D. Irwin, Inc., 1971), pp. 163-66. 14 Ibid., p. 164, my i n s e r t . Obviously, Smith's contentions apply equally well to foreign exchange markets as they do to money markets. Smith also notes that the stated r e l a t i o n holds only to the point where some monopolistic aspects may outweigh the gain due to greater e f f i c i e n c y . *^OECD, Capital Markets Study, Formation of Savings, Vol. II (Paris, 1967), p. 99. 9 0 CHAPTER IV INTEREST ARBITRAGE UNDER FLEXIBLE EXCHANGE RATES 4 . 1 Introduction A system of f l e x i b l e (or fluctuating) exchange rates is characterized "by the condition that monetary authorities never intervene at a l l by purchases or sales of foreign exchange."''' In other words, exchange rates are allowed to respond fre e l y to market forces of demand and supply. Under such a system, the spot and forward rates must be determined simultaneously. Tsiang and others have demonstrated how this 2 can be accomplished a n a l y t i c a l l y . Taking Tsiang's and Sohmen's analyses as a point of departure, two questions w i l l be discussed in this chapter. The f i r s t relates to the burden that spot and forward rates must bear respectively in the process of adjustment to equi-librium. In analogy to the analysis in the preceding chapter, we hypothesize that the re l a t i v e variations of the two rates 3 depend on the size of the two markets involved. S i m i l a r l y , the analysis w i l l be carried out f i r s t in terms of the slopes of demand and supply curves. The results are l a t e r interpreted in terms of market size. The second question arises in connection with the process of adjustment to equilibrium. A comparison of Tsiang's con-clusions with those obtained by Auten in an a r t i c l e which deals with some related aspects leads to a confusing paradox. According to Tsiang's treatment of the adjustment process, i t appears that the variation of the spot rate tends to amplify the variation of the forward rate. Auten's analysis, on the other hand, suggests that the va r i a t i o n of the forward rate is dampened as the spot rate responds to changes in demand and supply. Our analysis of the interaction between the spot and forward rates w i l l enable us to resolve this problem. Before these questions can be discussed, however, i t is necessary to examine the role of capit a l flows in a pure system of f l e x i b l e exchange rates. Our aim i s to demonstrate that under such a system the analysis of interest arbitrage can be legitimately confined to the foreign exchange markets. 4.2 Capital Movements under a. System  of Flexible Exchange Rates The role of capit a l movements under a system of f l e x i b l e rates is closely connected with the issue of economic inde-pendence and i s s t i l l a matter of controversy in the l i t e r a -ture.^ Part of the argument arises because the participants in the discussion do not have the same model in mind. Some of the confusion can be eliminated, i t would appear, i f the d i s t i n c t i o n between a system of f l e x i b l e exchange rates on one hand and the f l e x i b i l i t y of exchange rates on the other hand is made. Accordingly, we have defined a system of f l e x i b l e rates by the condition that monetary authorities do not intervene; i.e., o f f i c i a l foreign reserves remain unchanged. This does 92 not necessarily imply, however, that exchange rates do actually vary. 6 We can therefore distinguish two cases. In the f i r s t case the exchange rate responds to changes in demand and supply of foreign exchange. Suppose the world rate of interest i s lowered. There w i l l be a tendency for arbitragers to transfer spot funds to the domestic economy. If there are no speculators and i f banks are f u l l y loaned up, arbitragers can s e l l t heir foreign balances to commodity importers only. The foreign exchange rate must consequently f a l l to the point where the amounts of foreign exchange supplied by capital importers equal the demand by commodity traders. The volume of bank deposits remains unchanged; there i s simply a transfer of domestic funds from the accounts of commercial traders to those of capital importers. Accord-ingly, there is no expansion of the money supply and hence 7 no tendency for interest rates to change. In the second case i t is assumed that commercial banks and foreign exchange dealers or speculators undertake to peg g the exchange rate. The a c q u i s i t i o n of foreign exchange by commercial banks reduces th e i r excess reserves. Likewise, speculators and foreign exchange dealers must borrow money either from commercial banks or on the money market. In either case, an inflow of cap i t a l w i l l tend to tighten domes-t i c credit conditions. Unless banks dispose of i n f i n i t e l y e l a s t i c supply of c r e d i t , interest rates w i l l tend to r i s e . Thus we are l e f t with the somewhat paradoxical conclusion that an inflow of foreign exchange leads to--temporari1y--93 9 increased interest rates. As foreign investors redirect t h e i r funds to the money market, however, the demand for s e c u r i t i e s is restored to i t s previous level and interest rates assume their i n i t i a l value. We conclude, therefore, that so long as monetary authorities do not intervene on the spot exchange market (and i f foreign exchange is excluded as a reserve asset) the a c t i v i t y of interest arbitragers cannot affect the going rate of i n t e r e s t . If t his i s true, then the analysis of interest arbitrage can be r e s t r i c t e d to the foreign exchange markets. Interest rates appear as exogenous variables or data. Spot and forward rates share the f u l l burden of adjusting the system to equi-librium and appear therefore as the only endogenous variables in a model of interest arbitrage under f l e x i b l e exchange rates. 4. 3 The Process of Adj ustment 4.31 Autonomous Change in Interest Rates It w i l l be r e c a l l e d that the main purpose in this chapter is to determine the burden that spot and forward rates must bear respectively in the process of adjusting to equi-librium. This requires a detailed study of the adjustment process. It is convenient to divide the analysis into two parts. This section explores the consequences of an autono-mous change in interest rates. Section 4.32 deals with direct intervention in the forward market. In the process i t w i l l become clear why f l e x i b l e spot rates can have a dampening 94 effect on forward rates in some cases and an amplifying effect in others. A further d i s t i n c t i o n is he l p f u l . With respect to the forward market, primary and secondary effects of changes are distinguished. Primary effects are the changes in the forward rate which prevail i f the forward market is analysed in i s o l a -t ion. Secondary effects are the further changes which are induced through simultaneous changes in the spot r a t e . 1 1 Let us examine the primary effects of changes in foreign 12 interest rates on the forward market. where A£ .. arbitragers' excess demand for forward exchange M£+S£ .. demand for forward exchange by importers plus excess demand by speculators X £ .. supply of forward exchange by exporters Fig. 13. Determination of equilibrium in the forward market 95 The i n i t i a l equilibrium rate is FX^. The assumption is made that the spot rate (not shown) equals the forward rate and that foreign and domestic interest rates are equal at the 13 outset. Net arbitragers' demand is therefore zero at FX^. Suppose now that the foreign interest rate is increased. Arbitragers are now induced to transfer funds from the domes-t i c economy to the foreign center. This gives r i s e to an increase in the supply of forward exchange. Diagrammatically, this is shown by a downward s h i f t of the A^ . schedule to A^. The magnitude of the s h i f t is in direct proportion to the increase of the foreign interest rate. This means that at the point Z, where A^ intersects the ordinate, the net i n -centive for arbitrage is again z e r o . ^ A new equilibrium w i l l be established when the amount of forward exchange supplied by arbitragers equals the excess demand of importers and speculators over exporters' supply. The condition for equilibrium can also be expressed by the following equations: I. -Af = (M f +S f)-X f (4-1) or A f+S f +(M f-X f) = 0 1 5 (4-1') The condition is f u l f i l l e d at FX 2, where the amount of for-ward exchange traded equals A. Inspection of Fig. 13 w i l l y i e l d the following observations: 1. Given the M+S and X schedules, the forward rate i s subject to wider fluctuations the more e l a s t i c the A^ schedule. 1^ When the arbitrage schedule i s per f e c t l y e l a s t i c , 96 the forward rate f a l l s to Z, at which point equilibrium i s consistent with parity. 2. Given the slope of the A g schedule, the forward rate varies more strongly as both the X and the M+S schedules become steeper. In the extreme case where both schedules are perfectly i n e l a s t i c , the forward rate w i l l change in direct proportion to the s h i f t in the arbitrage schedule. As in the case of a pe r f e c t l y e l a s t i c arbitrage schedule, the new equilibrium rate would be Z, which would again be consistent with parity. The difference between the two cases l i e s in the amount of forward exchange traded. If the arbitrage schedule was per f e c t l y e l a s t i c , the amount traded would be B; this quan-t i t y is determined by the slopes of the X and M+S schedules. If these schedules were perf e c t l y i n e l a s t i c - - t h e second case--the amount traded would be n i l . It is worthwhile to note that these are the only instances where IRPT holds systematically. As soon as spot rates are also affected, the equilibrium forward rate w i l l be different from Z even in the extreme cases and parity i s not attained, except by coincidence. It appears, therefore, that IRPT requires the following, very r e s t r i c t i v e , as sumptions: 1. that supply and demand curves are perfectly e l a s t i c on the spot market (so that no secondary effects occur on the forward market) 97 2. that the arbitrage schedule is p e r f e c t l y e l a s t i c or that demand and supply of forward exchange by non-arbitragers 17 are perfectly i n e l a s t i c . In other words, IRPT holds systematically only when the for-ward rate i s the sole adjusting agent and when the schedules 18 in the forward market exhibit certain properties. We can now include the spot market in our analysis. The secondary effects of variations of the spot rate on the forward rate can then be examined. Given that arbitragers operate simultaneously in the spot and forward markets, f u l l equilibrium must be j o i n t l y determined. The solution can be approximated with the aid of Fig. 14 (page 98). The dia-gram i s drawn under the assumption that, i n i t i a l l y , domestic and foreign interest rates are equal. It is assumed, further, that i n i t i a l l y demand and supply by non-arbitragers are equal to each other in both markets at rates which are consistent with parity. Suppose now that the foreign interest rate is increased. Arbitragers are induced to buy spot exchange and to s e l l an equivalent amount of forward exchange--plus interest earned--on the forward market. The primary effects consist in the forward rate changing to FX,, . But now the secondary effects must be taken into consideration. The additional demand for spot exchange (shown as an upward s h i f t of the schedule to A^) increases the spot rate to SX2. As a result of this change, the AL schedule s h i f t s back up in direct proportion to the increase in the spot rate. A new arbitrage schedule A1' 98 Q spot exchange/t Q forward exchange/t where A M +S s s arbitragers' excess demand for spot exchange supply of spot exchange by exporters whose transactions c a l l for immediate settlement demand for spot exchange by importers whose transactions c a l l for immediate settlement, plus excess demand by speculators Fig. 14. Equilibrium after autonomous change in interest rates and the new forward rate FX^ are obtained. This increase in the forward rate leads in turn to a downward s h i f t of the A^ schedule to A^. The system o s c i l l a t e s in this manner u n t i l the new rates of equilibrium are found. Joint equilibrium is established when the demand for spot exchange and the supply of forward exchange by arbitragers equal the excess supply of spot exchange and the excess demand for forward exchange by non-arbitragers. 99 Omitting interest accrual, these amounts must a l l be equal. The conditions for simultaneous equilibrium can therefore be written as -Af = ( M £ - X £ ) + S £ (4-2) II. A £ = ( M s - X s ) + S s (4-3) -Af = A s (4-4) A comparison of the solutions to systems I and II w i l l enable us to i s o l a t e the secondary eff e c t s . 1. The most notable difference l i e s in the value of the equilibrium forward rate. Although i t is d i f f i c u l t to obtain graphically, i t is clear that the new equilibrium rate w i l l be in the neighbourhood of FX^. When the spot rate i s f l e x i b l e , the difference between the i n i t i a l and the f i n a l equilibrium value for the forward rate ( i . e . iFX^-FX^j ) is always smaller than i t would be i f only the primary effects are considered ( i . e . |F X - FX 2| )• In other words, the varia-tion of the forward rate i s dampened by the variation of the spot rate. But this is true only when the i n i t i a l equilibrium is disturbed by an autonomous change in an exogenous variable. This was achieved in our example by assuming that the foreign interest rate was increased by, say, the monetary authorities 21 through open market interventions. 2. The steeper the demand and supply curves of non-arbitragers in the spot market, the greater the impact of arbitragers on the spot market. But the greater the varia-tion of the spot rate, the more pronounced the backward s h i f t 100 of the arbitrage schedule in the forward market. Consequently the secondary effects are also stronger, i . e . the distance | F X 2 - F X i s larger as the variation of the spot rate increases. This means that the dampening effect of the spot rate on the forward rate is more pronounced. In other words, the larger the share of the spot rate in the equi l i b r a t i n g process, the smaller the variation of the forward rate. To i l l u s t r a t e , consider two extreme cases. If the Xg and M g +S s curves are perfectly e l a s t i c , arbitragers can buy or s e l l unlimited amounts of spot exchange without affecting the spot rate. Hence the forward market must bear the f u l l burden of adjustment. The changes in the forward rate would then be i d e n t i c a l with what we have calle d the primary e f f e c t s . If the X g and Ms+Ss schedules are perfectly i n e l a s t i c , any attempt by arbitragers to buy or s e l l spot exchange w i l l immediately turn the spot rate to their disfavor. No funds can be transferred and no funds can be offered on the for-ward market. Consequently, the forward market is not affected 2 2 at a l l . The dampening effect i s t o t a l . 4.32 Autonomous Change in the Forward Rate The adjustment process is again examined in this section. In contrast to the preceding analysis, i t is assumed here that the i n i t i a l equilibrium is disturbed by an autonomous change which affects the forward rate d i r e c t l y . The case of direct intervention in the forward market by monetary authorities i s 2 3 used as an example. Although this analysis i s inspired by 101 Tsiang, the model employed here is i d e n t i c a l to that used in 24 Section 4.31. Fig. 15 (page 102) shows the system in i n i t i a l equilibrium at SX^ and FX^. For s i m p l i c i t y i t is again assumed that non-arbitragers' demand and supply are equal at these rates in both markets and that domestic equal foreign interest rates. Let us now assume that monetary authorities intervene in the forward market by offering forward exchange. The X £ schedule moves to the right to become X£. The primary effects in the forward market consist in the f a l l of the forward rate from F l 1 to FX 2 > Now consider the effects on the spot market. The A g schedule f a l l s as arbitragers are induced to supply spot exchange. Given the new level of the forward rate, FX^ j A s h i f t s to A'. Hence the spot rate f a l l s to SX„; the difference SX^-SX2 is greater the steeper the X g and Ms+Ss schedules in the spot market. But i f the spot rate f a l l s , the A £ schedule must also s h i f t down. (A lower spot rate implies decreased net incentives for interest arbitrage. This gives r i s e to decreased demand for forward exchange by arbitragers.) The relevant arbitrage schedule is then A £. Thus a new forward rate FX^ is established which causes an upward s h i f t in the A^ schedule, since arbitragers' supply of spot exchange decreases. This i t e r a t i v e process continues u n t i l the quantities of forward exchange demanded and of spot exchange supplied by arbitragers are equal to the quantities supplied and SX SX Q spot exchange/t Q forward exchange/t Fig. 15. Equilibrium after autonomous change in the forward rate o 103 demanded by non-arbitragers (equations 4-2, 4-3, 4-4 apply in this case as well). At that point there remains no inherent tendency for change and equilibrium prevails in both markets. It can be shown that so long as the system converges to a stable equilibrium and i f the forward rate i s lowered by direct intervention, the new equilibrium forward rate w i l l always l i e below the rate which would pre v a i l i f the forward 2 5 rate was the sole adjusting agent ( i . e . FX < FX ). It can be concluded, therefore, that the variation of the spot rate has an amplifying effect on the forward rate when the system is disturbed by an intervention in the for-ward market. It i s demonstrated in the appendix that this amplifying effect i s larger the steeper the demand and supply schedules for non-arbitragers in the spot market. The same relationship holds for the slopes of demand and supply sched-ules for non-arbitragers in the forward market. F i n a l l y , the amplifying effect is smaller as the arbitrage schedule becomes more e l a s t i c . 4.4 Reinterpretation The connection between the notions of slopes of market schedules and market size was discussed in the preceding chapter. It was found, in b r i e f , that large markets (mea-sured in terms of volume of transactions) should be represented by f l a t schedules, small markets by steeply sloped curves. The results of the foregoing analysis can thus be summarized and reinterpreted in terms of market size. 104 Following the development in the main part of this chapter, consider f i r s t the case of adjustment to equilibrium following a change in interest rates. Here the variation of the spot rate w i l l tend to dampen the variation of the forward rate. The smaller the spot market, the stronger the impact of interest arbitrage on the spot rate. Therefore a small spot market exerts a strong dampening effect on the forward rate. Thus, the smaller the spot market r e l a t i v e to the for-ward market, the greater the variation of the spot rate and the smaller the impact of interest arbitrage on the forward 2 6 rate. The reverse is also true. The interactions of the two markets are a b i t more complicated when the i n i t i a l disturbance occurs in the for-ward market. Given the level of arbitragers' demand, the intervention w i l l be more successful i f the forward market is large. This is so because arbitrage tends to exert an of f s e t t i n g effect with respect to the i n i t i a l change of the forward rate. This o f f s e t t i n g effect is smaller the larger the mark et. The secondary e f f e c t s , however, tend to push the forward rate in the direction of the intervention. They are more pronounced the smaller both the spot and the forward markets. The net effects of the two counteracting forces are therefore d i f f i c u l t to ascertain in a general way. The results are unambiguous for two extreme cases, however. The f i r s t case involves a very large forward market and a very small spot market. Suppose monetary 105 authorities wish to lower the forward rate by a certain amount. Unless the demand by arbitragers is highly e l a s t i c , the dampening effect of interest arbitrage is small. The secondary, amplifying effects are pronounced, however, such that the f i n a l equilibrium forward rate is l i k e l y to be con-siderably lower compared to i t s level under p a r t i a l equi1ibrium. If the forward market i s very small compared to the spot market, a desired change of the same amount of the forward rate w i l l be largely offset by the dampening impact of arbitrage. Moreover, the secondary effects tend to be smaller. The equilibrium forward rate i s therefore l i k e l y to be only marginally lower than the one established in the f i r s t round. The actual change in the forward rate w i l l f a l l considerably short of the desired one. 106 CHAPTER IV NOTES Sohmen, Flexible Exchange Rates, p. 4. 2 It should be noted that Tsiang's is not a pure system of f l e x i b l e exchange rates. He l i s t s monetary authorities among the many participants in both markets. Their supply and demand schedules are not perf e c t l y e l a s t i c at any point, however, as would be the case under a system of pegged exchange rates. Tsiang's system might therefore be called one of managed f l e x i b l e rates. See Tsiang, "Theory of Forward Exchange." A summary of Tsiang's exposition is given by Glahe, Empirical Study, pp. 12-22. The pure case of f l e x i b l e exchange rates has been analysed by Neldner, Devisenterminmarkt, pp. 62-64. 3 It i s possible to confine the analysis to the spot and forward markets under a system of f l e x i b l e exchange rates, as w i l l be demonstrated in Section 4.2. 4 Auten, "Forward Exchange Rates," p. 14. A^ c r i t i c a l appraisal of the l i t e r a t u r e on this topic can be found in Robert D. McTeer, J r . , "Economic Independence and Insulation Through Flexible Exchange Rates," in Money, the Market, and the State: Economic Essays in Honor of  James Muir Walter, ed. by Nicholas A. Beadles and L. Aubrey Drewrey, J r . (Athens: University of Georgia Press, 1968), pp. 102-33. Our conclusions do not appear to be in agreement with his view; his arguments are not as e x p l i c i t as one might wish, however. ^Sohmen, Flexible Exchange Rat es, pp. 4-5. 7 I b i d . , p. 37, 39. Only our conclusions are i d e n t i c a l with h i s ; the line of reasoning i s not given by Sohmen. He notes, further, that there may be a tendency for interest rates to change following a capital transfer. But i t is occasioned by changes in the transactions - demand for cash which come about after the realignment of commodity - exchange and i t s influence on incomes. We exclude such r e a l , or secondary, effects from our analys i s . g Machlup, International Payments, pp. 40-47, examines this case i_n extenso. 9 I b i d . , p. 45. Delbert A. Snider, Introduction to International  Economics (5th ed.; Homewood, 111.: Richard D. Irwin, Inc., 1971), p. 338, note, concludes that an inflow of private, short-term cap i t a l which is induced by a ri s e in the exchange rate leads to a reduced money supply. 107 CHAPTER IV NOTES (cont'd) •^Except for secondary effects which we have excluded here; see note 7. 11 This is a purely a n a l y t i c a l d i s t i n c t i o n . It is possible because the effects of a disturbance of the pre-v a i l i n g equilibrium are traced through the dif f e r e n t markets sequentially. In r e a l i t y , such changes occur simultaneously, of course, and only the net effects can be observed. The primary effects correspond to the effects of changes i n , e.g., interest rates on forward rates in models such as Grubel's. The equilibrium of the forward market can only be p a r t i a l , in his model, since under a system of f l e x i b l e exchange rates, spot and forward rates are determined j o i n t l y . 12 The following analysis i s adapted from Auten, "Forward Exchange Rates." 13 Equilibrium does not require that arbitragers' demand be zero in a l l cases. The assumptions were made for convenience only. 14 The A £ schedule would s h i f t in the same direction i f domestic interest rates were lowered. It would s h i f t up i f the domestic rate was increased or the foreign rate decreased. ^See also Glahe, Empirical Study, pp. 16-17. ^Note that no matter what the slope of the arbitrage schedule, i t would always s h i f t down such that i t s intersec-tion with the ordinate i s situated at Z. If the schedule is steep, however, the additional supply of forward exchange is small so that FX^ would be cloase to FX^. 17 Under pegged exchange rates, the further assumption that money markets are i n f i n i t e l y large is required. 1 8 See also Glahe, Empirical Study, p. 6. 19 Fig. 14 i s a modified version of Sohmen's treatment; Flexible Exchange Rat es, p. 99. The component S is not a function of the current spot rate, but i s a resiaual of past commitments. If i t was shown separately i t would be perfectly i n e l a s t i c . Sohmen also admits the p o s s i b i l i t y that monetary authorities might intervene in both markets. 2 0 These assumptions are made for si m p l i c i t y only and do not r e s t r i c t the analysis. Such a situation would occur in r e a l i t y only by coincidence, of course. 108 CHAPTER IV NOTES (cont'd) 2 1 The case of autonomous changes in the forward market is examined in Section 4.32. 2 2 The above statements hold only within certain l i m i t s . It must be assumed that stable equilibrium can be attained. The conditions for equilibrium and the limits within which our statements hold are the same as those defined in the appendix. 2 3 Compare Tsiang, "Theory of Forward Exchange," pp. 95-102 and Glahe, Empirical Study, pp. 16-22. It is worth emphasizing that the case of o f f i c i a l intervention is used here only as one of many possible ex-amples for a general case. A change in taste on the part of foreign consumers of domestic exports would have i d e n t i c a l e f f e c t s . 2 4 Differences with Tsiang's treatment are mainly no-tat i o n a l and are not relevant to the analysis at hand. Tsiang's, and in p a r t i c u l a r Glahe's, graphical exposition and choice of notation appear at f i r s t sight to imply that trade and speculation are after a l l a function of the current spot rate, although both groups are assumed to base their decisions on the forward rate in his model. What appears on the spot market would then simply be a residual of past com-mitment. But because supply and demand schedules on the spot market comprise some other components which are a function of the current spot rate, those schedules retain some degree of e l a s t i c i t y . Regretfully, Glahe has over-looked this proviso by excluding a l l non-lagged variables from his market schedules, thereby v i t i a t i n g the v a l i d i t y of his exposition; Empirical Study, p. 16, n. 14. For a further discussion of these problems see Sohmen, Flexible Exchange Rates, pp. 97-99. 2 5 For a mathematical proof of this proposition see the appendix. If the intervention results in an increase of the forward rate, i t would always l i e above the rate which would preva i l i f primary effects alone were considered. 2 6 See also Glahe, Empirical Study, p. 7. 109 CHAPTER V MULTIPLE INTEREST DIFFERENTIALS AND THE THEORY OF INTEREST RATE ARBITRAGE 5.1 Interest Rate D i f f e r e n t i a l s in the Real World A l l the models reviewed in Chapter II indicate that forward rates are influenced to some extent at least by interest d i f f e r e n t i a l s between the markets concerned. These models are a l l based on the assumption that i t i s possible to speak of a single interest d i f f e r e n t i a l ; that i s , the assumption is made that there is only one type of security in any one market. Arbitragers are thought to compare the rate of interest on these securities with the p r e v a i l i n g rate of interest in another market. In r e a l i t y , of course, there is a large variety of s e c u r i t i e s in most developed money markets. Consequently, there exist not one but many interest d i f f e r e n t i a l s between two markets at any p a r t i c u l a r moment. S p e c i f i c a l l y , i f there are m types of s e c u r i t i e s in one market and n types in another, m*n interest d i f f e r e n t i a l s can be computed. This contrast between real world conditions and s i m p l i f i e d models gives r i s e to two major problems. The f i r s t one concerns the v a l i d i t y of the conclusions derived from the various models of arbitrage. To what extent must these conclusions be altered in the context of multiple interest d i f f e r e n t i a l s ? 110 The second problem i s faced mainly by empirical researchers and makers of monetary policy. Of the many possible combinations they must select the pair of securi-ti e s which is most relevant to arbitragers. Regretfully, neither of these problems have received much attention in the l i t e r a t u r e . 1 Yet the v a l i d i t y of empirical research and the effectiveness of o f f i c i a l i n ter-vention in money or foreign exchange markets depend on a sat i s f a c t o r y solution to these problems. In the following, the emphasis w i l l l i e on the analysis of the implications of multiple interest d i f f e r e n t i a l s for the theory of arb itrage. The related aspect of the selection of relevant i n t e r -est d i f f e r e n t i a l s cannot be treated as extensively mainly because the detailed information required for such a discussion is not available. 5.2 Single Interest Rate Models This section reviews b r i e f l y conclusions derived from models incorporating single interest d i f f e r e n t i a l s in order to contrast these findings with the implications of the existence of multiple interest d i f f e r e n t i a l s . There are two basic propositions to be derived from IRPT and related models. 1. Funds w i l l flow from country A (the domestic country) to country B i f S < Q (5-1) I l l where S = (1+RD) Q = |f(l+RF) Conversely, inward arbitrage occurs i f S > Q (5-2) Since there i s only one RD and only one RF, S < Q and S>Q cannot be true simultaneously. Consequently funds can flow in one direction only at any given moment. 2. If S = Q, the system i s in equilibrium. There is no net incentive for arbitrage in either d i r e c t i o n . Accord-ingly, no funds are transferred from one country to another. Modern models of arbitrage y i e l d s l i g h t l y d ifferent r e s u l t s . 1. Funds w i l l flow in the same directions as under IRPT . 2. Given arbitrage schedules which are less than per f e c t l y e l a s t i c , equilibrium can be attained before net incentives are completely eliminated. That i s , arbitragers may r e f r a i n from transferring c a p i t a l to another market even though parity is not established. 5.3 Multiple Interest Rat e Mode Is 5.31 Risk as Discriminating Factor The models are now expanded to allow for the presence of multiple rates of inte r e s t . Suppose there are two money markets, market X and market Y. There are three types of equivalent s e c u r i t i e s in both markets, as follows: 112 Market X Market Y Securities A A Both series are ranked in terms of risk of default and in ascending order. Differences in risk of default between two types of s e c u r i t i e s in the same market are assumed to be reflected in the interest rate on those assets. Thus B x commands a premium over A , and so does C over B ; the same relationship holds for market Y. Furthermore, let rep-resent the rate of interest on assets of type A , b is the J r x x y i e l d of B and so on. x Although i t is t h e o r e t i c a l l y possible to compute m-n interest d i f f e r e n t i a l s , a large number of the possible combi-nations can be eliminated. IRPT and related models were seen to be s p e c i f i c applications of the general economic p r i n c i p l e that two assets with comparable properties cannot s e l l for 2 different prices in the same market. Consequently, only interest d i f f e r e n t i a l s between "equivalent" securities are relevant in this context. In r e a l i t y , the integration of national money markets is of course far from perfect. Accordingly, securities which appear to be equivalent at f i r s t glance may d i f f e r in terms of risk of default and in a number of other dimensions. The aim in this section is to explore the implications of d i f f e r -e n t i a l r i s k of default. The relevance of other considerations w i l l be discussed in Section 5.4. 1.13 IRPT and other models discussed previously could be applied to real world conditions in a r e l a t i v e l y straight-forward manner i f one of the following conditions held: 1. Arbitragers confine their a c t i v i t i e s to a single pair of s e c u r i t i e s , 2. The absolute y i e l d d i f f e r e n t i a l s between se c u r i t i e s of d i f f e r e n t r i s k - l e v e l s are i d e n t i c a l in both markets. Many researchers seem to hold the b e l i e f that the f i r s t condition applies in r e a l i t y . This i s , at any rate, the most plausible explanation for the popularity of treasury b i l l 3 rates as standards for comparison. However, at least in the case of U.S. - Canadian markets these securities are largely neglected.^ At any rate, the hypothesis that arbitrage is only carried out on the basis of a single interest d i f f e r e n t i a l when a whole realm of debt instruments are available in both markets is hardly tenable. If investors d i f f e r in their acquisition p o l i c i e s for domestic s e c u r i t i e s , there is l i t t l e reason to assume that a l l arbitragers suddenly converge on one type of issue when i t comes to foreign investments.** The second case where models incorporating a single interest d i f f e r e n t i a l could be applied to real world condi-tions with ease refers to the comparative risk structure in the two markets. Assume, for example, that the risk premium between different types of s e c u r i t i e s in both markets is a constant amount of equal magnitude. That i s , let 114 and b -a = c (5-3) Cx" bx = d ^ ~ 4 ^ b -a = c (5-3') y y c -b = d (5-4') y y Then the following must also be true: a - a = b - b = c - c r r x y x y x y (5-5) Since the forward margin is the same for a l l pairs, i t follows that the covered interest d i f f e r e n t i a l i s i d e n t i c a l for a l l pairs of equivalent s e c u r i t i e s . Accordingly, simpli-f i e d models can be applied to such a situation without further revisions. Any pair could be chosen for empirical work. Moreover, funds would flow only in one direction at any given moment. F i n a l l y , whenever there is parity between one type of s e c u r i t i e s , the same must be true for a l l other pairs of equivalent s e c u r i t i e s . But is such an assumption about the comparative r i s k structure reasonable? Grubel contrasted the hypothesis dis-cussed above with the following one: 6 The risk premium between different types of sec u r i t i e s is a constant and equal percentage of the basis rate in both markets. Hence the y i e l d d i f f e r e n t i a l between, say, B and A depends on the X X absolute magnitude of the rate of interest on A . Under such circumstances there is no unique interest d i f f e r e n t i a l between the two markets, unless a = a . Rather, there are as many x y margins as there are pairs of equivalent s e c u r i t i e s . Similar conclusions would be reached i f the risk structure in the two international centers were unrelated. 115 Evidence collected by Grubel indicates that in the case of New York and London one of the l a t t e r hypotheses is 7 supported. He compared the arbitrage margins for the follow-ing pairs of sec u r i t i e s over a period of six years: New York-London treasury b i l l s , New York finance papers - London prime commercial papers, New York-London bankers' prime acceptances. There appeared to be a good deal of correlation between the f i r s t two series, though the lines intersected frequently. Differences between these two series did in general not exceed %% . The t h i r d series frequently digressed by as much as 2-3%, however. What is even more s i g n i f i c a n t , the t h i r d series often favored one market when the f i r s t two series favored the other market. Grubel's data quite c l e a r l y point to the fact that in the instance of New York and London i t is not possible to speak of a unique interest d i f f e r e n t i a l . Grubel relates such discrepancies to the comparative ris k structure in the two markets. There are, however, several other factors which help to explain such deviations. These factors are discussed in the next section. 5.32 Other Discriminating Factors The argument to be advanced here is similar in nature to one put forth by Fair and Malkiel in their recent study of y i e l d d i f f e r e n t i a l s between debt instruments of the same g maturity. They note that debt instruments of the same risk level and maturity are not in a l l cases perfect substitutes for one another. They attribute variations in y i e l d to 116 certain c h a r a c t e r i s t i c s which distinguish debt instruments of one kind from si m i l a r s e c u r i t i e s . The three discriminating factors found to be most relevant are: 1. The extent to which the bonds are needed to meet various legal requirements, 2. The "window dressing" quality of the bonds, 3. The quality of the after-market. Depending on their p a r t i c u l a r requirements, investors would prefer one type of bond over another. Si m i l a r l y , i t is postulated here, there are a number of attributes which d i f f e r e n t i a t e short-term investments of the same risk class in the minds of investors. Such factors may be subsumed under the general heading of market imperfections. The f i r s t group of these attributes includes r e s t r i c t i o n s imposed on the investor which l i m i t his f i e l d of choice. These constraints may be of a legal nature where certain i n -vestors such as pension funds are prohibited from acquiring certain types of s e c u r i t i e s for their p o r t f o l i o s ; more common is the exclusion of foreign investors from certain 9 kinds of domestic investments. A l t e r n a t i v e l y , corporate policy may dictate that investments be r e s t r i c t e d to a number of instruments with which operators are f a m i l i a r . Management may also pursue goals other than profit-maximization. 1*^ P o l i t i c a l and psychological factors may also help to d i f f e r e n t i a t e s e c u r i t i e s which are e s s e n t i a l l y s i m i l a r , 1 1 not to mention loss of investors' confidence as a result 12 of spectacular business f a i l u r e s . Thus external and 117 self-imposed r e s t r i c t i o n s may prevent e s s e n t i a l l y similar s e c u r i t i e s from being perfect substitutes for one another. Lack of accurate information may int e r f e r e with the evaluation of equivalent s e c u r i t i e s . Grubel relates a case where an investment o f f i c e r of a large i n s t i t u t i o n was w i l l i n g to compare the y i e l d on money-market paper issues by U.S. finance companies (in which he regularly invested) only with the y i e l d on U.K. treasury b i l l s . The covered-yield margin of U.K. commercial paper over U.S. commercial paper did not interest him because of the lack of i n f o r -mation on the soundness of U.K. debtors.*"* Many investors w i l l also consider the quality of the after-market. A we 11-organized secondary market guarantees that an investor can liquidate his holdings prematurely without too much of a loss. A l l things being equal, he w i l l therefore prefer a security which can be sold readily p r i o r to maturity. F i n a l l y , consideration should be given to d i f f e r e n t i a l tax treatment. An arbitrager must obviously compare rates of return net of withholding taxes to account for d i f f e r e n t i a l tax 14 rates. Such rates may not only vary from country to country; they also depend in some cases on the type of security within the same market. Thus, in the United States, interest and dis-counts are not treated in the same manner for taxation pur-poses.1*' Furthermore, different treatment under the U.S. Inter-est Equalization Tax Act e f f e c t i v e l y precludes a number of businesses from acquiring certain types of foreign s e c u r i t i e s . In Canada a special procedure in the administration of the 15% withholding tax on treasury b i l l s has discouraged 118 many foreign investors. Instead, their attention has shifted to finance papers and similar s e c u r i t i e s where a simpler procedure i s fo1lowed--though the same rate a p p l i e s . 1 6 The foregoing considerations bring out the fact that, in r e a l i t y , arbitragers must evaluate several factors which are not embodied in models designed for empirical research. Two conclusions may be drawn: 1. Many sec u r i t i e s which on the surface appear to be perfect substitutes are in fact considered to be different by arbitragers. 2. The degree of s u b s t i t u t a b i l i t y varies from one pair of s e c u r i t i e s to another; this depends on the severity of the r e s t r i c t i o n s which apply to a p a r t i c u l a r kind of security 1 7 but not to others. These conclusions tend to give additional support to our contention that there are multiple arbitrage margins at any one time between any two markets. 5.4 Implications for the Theory of  Interest Rate Arbitrage The conclusions derived from models which are based on the assumption that only one interest rate prevails in any market hold no longer when multiple margins are introduced. They d i f f e r in at least two respects. F i r s t , equilibrium can only prevail with respect to a pa r t i c u l a r pair of s e c u r i t i e s . In the context of IRPT this means that when parity is attained for one interest d i f f e r -e n t i a l , net incentives s t i l l exist for a l l other margins. 119 Modern models do not require parity as a necessary condition for equilibrium, but the conclusion remains the same. That i s , one margin may be reduced to a point where arbitragers consider i t no longer advantageous to transfer funds for the acquisition of the type of security concerned. At the same time, other margins may s t i l l exceed that cut-off rate. The following i l l u s t r a t i o n may c l a r i f y this point. There are two markets and two types of similar, but not equivalent, s e c u r i t i e s . Market X Market Y Securities A A B X B>^  x y Suppose the rates of return on these assets are as follows: a = 3% a = 4% b x = 5% b 7 = 7% x y Assume further that the forward rate (FX) equals the spot rate (SX) at the outset. The net incentives for s e c u r i t i e s A equals 1% and favors market Y; for securities B the net incentive is 2%. Funds are now transferred to market Y. As a r e s u l t , the forward rate depreciates to a 1% discount (currency Y in terms of currency X). There is now equilibrium for s e c u r i t i e s A, i . e . there is no further incentive for arbitrage between and A . But there is s t i l l a positive margin of 1% in the case of securities B. Therefore, funds w i l l continue to flow from X to Y although equilibrium was attained with respect to s e c u r i t i e s A. This marks the f i r s t instance where a model incorporating multiple interest 120 d i f f e r e n t i a l may y i e l d d ifferent results compared to one 1 8 based on a single arbitrage margin. But the process is not yet completed. As a result of continuing arbitrage between securities B the forward rate w i l l depreciate further, thereby decreasing the net incentive for s e c u r i t i e s B. At the same time there emerges a net i n -centive for arbitrage between sec u r i t i e s A. But now market X is favored. Consequently funds w i l l flow in both directions simultaneously, a result which is impossible under si m p l i f i e d 19 assumptions. The analysis could be carried farther, but the con-clusion is obvious by now: under the conditions which have been set out s t a t i c equilibrium cannot be attained. Indeed, there i s no value which would s a t i s f y the condition for equilibrium as expressed in the equation (1 + RD) = ||(1.+ R F ) 2 0 (5-6) for both pairs of sec u r i t i e s simultaneously. It would appear that at some point between the values of -1% and -2% for the forward margin a dynamic equilibrium i s established such that funds moving from X to Y are just s u f f i c i e n t in proportion to offset the impact of funds flowing from Y to X on the forward rate. The results for modern models are only s l i g h t l y d i f f e r -ent. The forward rate may assume a value such that both margins are in a zone where net incentives s t i l l exist; but they are not s u f f i c i e n t l y large to induce arbitragers to 121 transfer funds. A l t e r n a t i v e l y , s t a t i c equilibrium is attained when arbitrage funds are exhausted. In neither case is parity consistent with equilibrium. This b r i e f analysis shows that models which assume the existence of a single interest d i f f e r e n t i a l are not easily adaptable to real world conditions. It also points to p i t -f a l l s which a researcher may encounter. For example, the responsiveness of arbitragers to changes in the interest rate of a p a r t i c u l a r security may be seriously underestimated i f o f f s e t t i n g a c t i v i t i e s are not considered. Nor can the direction of the net flow of funds be predicted on the basis of net incentives alone, as is the case under IRPT and similar mode 1s. 5.5 Prob1 ems of Empirical Validation To overcome these theoretical d i f f i c u l t i e s , Grubel suggests that for purposes of empirical validation or recom-mendations for o f f i c i a l intervention " i t may be convenient to assume that the differences are of a second order of 2 1 smallness and can be neglected." Accordingly, he assumes that the relevant rate for arbitragers is an average of a l l existing d i f f e r e n t i a l s . Einzig agrees that a theoretical 2 2 parity should be a weighted average of the several margins. He recognizes, however, that i t would be impossible to assign the proper weight to each margin. It is pertinent, moreover, to point out that even i f such weighting could be properly done, the results of tests carried out on that basis would be 122 q u a l i t a t i v e l y different from the predictions of the models discussed previously. S p e c i f i c a l l y , c a p i t a l would continue to be transferred even i f theoretical parity was established. Yet arbitrage models can y i e l d useful predictions. Though i t i s true that there are several margins at any given moment, the a c t i v i t y of arbitragers i s l i k e l y to be concen-trated on a small number of issues; possibly a single pair of sec u r i t i e s dominates a l l other combinations. The case of U.S.-Canadian money markets where finance papers and commer-2 3 c i a l papers predominate has already been mentioned. Simi-l a r l y , Einzig points out that "for a long time most arbitragers lost interest in Interest P a r i t i e s other than those between Euro-currency r a t e s . " 2 4 If i t is true that one or two pairs of securities dominate a l l others in their impact on the forward rate, then the problem can be reduced to determining the relevant pair. How can i t be identi f i e d ? Einzig notes that: "Evidently, the rates which mainly matter are those which 2 5 are used systematically in the course of interest arbitrage." But the determination of the relevant securities must be carried out in a methodologically sound fashion. Grubel appears to violate this p r i n c i p l e when he computes the devia-tion from zero of these different margins "because i t is important to know which of the three series best predicts the 2 6 level of the forward rate." Such a procedure is not sound because i t presupposes the v a l i d i t y of arbitrage models. If such a model were to be tested on the basis of a series of 123 data selected in the above manner, the conclusions would inevitably confirm i t s v a l i d i t y . A superior approach would be to garner outside information about the a c t i v i t i e s of investors in the money and foreign exchange markets. Reliable outside information is essen t i a l , moreover, because fashions in arbitrage may change from one period to another. One possible reason for changes in arbitragers' preference is the emergence of a s u f f i c i e n t l y large interest 2 7 d i f f e r e n t i a l on other s e c u r i t i e s . Legal and structural developments in international 2 8 centers may also provoke a change of fashions. Fashions may not only change over time, they are also l i a b l e to vary from one pair of money markets to another. Secondary markets for commercial papers, for example, are hardly developed in Continental countries so that arbitragers are not l i k e l y to consider such issues. '5.6 Summary and Conclusion The aim in this chapter was to examine the v a l i d i t y of conclusions derived from models incorporating single interest d i f f e r e n t i a l s in the context of a more r e a l i s t i c s ituation where various arbitrage margins can be computed at any given moment. It was found that such models are s t r i c t l y applicable only to situations where 1. Arbitragers confine t h e i r a c t i v i t i e s to a single pair of s e c u r i t i e s , disregarding a l l other combinations , or 2. A l l margins assume i d e n t i c a l values. 124 Neither of these assumptions was seen to be reasonable. A more r e a l i s t i c proposition is that there is a variety of issues used by arbitragers at any given moment. Consequently there exist several arbitrage margins between two markets. It i s probable, however, that certain types of securities are p a r t i c u l a r l y well suited to the needs of arbitragers. Such pairs would therefore tend to exert a predominant influence on the formation of the forward margin. The suggestion was made, furthermore, that arbitragers' preference may change over time and from market to market. The existence of several interest d i f f e r e n t i a l s was found to a l t e r the conclusions of t r a d i t i o n a l models of interest arbitrage both q u a l i t a t i v e l y and quantitatively. Funds may continue to flow although theoretical parity has been attained with respect to the sec u r i t i e s selected. Moreover, funds may be transferred in both directions simul-taneously. Because of this combination of opposite forces the direction of the net flow of funds i s unpredictable unless more is known about the readiness with which investors respond to changes in p a r t i c u l a r covered interest d i f f e r e n -t i a l s . By the same token, the net amount of funds transferred is l i k e l y to be smaller than i t would be i f only one type of security was used. It was also suggested that arbitragers consider several factors in t h e i r decisions which are not included in the various models. Such factors include taxation, legal and corporate r e s t r i c t i o n s and the quality of information. As a 125 res u l t , s e curities of a similar type tend to be less perfect substitutes than would appear at f i r s t glance. These factors may also shape investors' preference for certain kinds of se c u r i t i e s . A l l these considerations point to the fact that the process of arbitrage is i n f i n i t e l y more complex and ephemeral than the various models would have i t . They must be applied to real world conditions with due consideration given to the large number of circumstantial factors which tend to obscure the fundamental relationships. 126 CHAPTER V NOTES The only known publications to deal with the questions raised above are: Grubel, Forward Exchange, pp. 50-52, 114-16; Einzig, Dynamic Theory, pp. 144-61. Grubel adresses him-s e l f to the choice of relevant interest d i f f e r e n t i a l s for empirical tests; Einzig emphasizes the limited a p p l i c a b i l i t y of s i m p l i f i e d models. 2 See also S t o l l , "Causes of Deviation," p. I l l ; Grubel, Forward Exchange, p. 51. 3 Another reason for the use of treasury b i l l rates could be that these s e c u r i t i e s are t y p i c a l l y vehicles for o f f i c i a l interventions in money markets. This reasoning was suggested by Stein, "International Short-Term Capital Movements," p. 51. It may be desirable, therefore, to select treasury b i l l rates as a basis when the impact on the foreign exchange market of o f f i c i a l intervention in the money market is under study. This peculiar c h a r a c t e r i s t i c of TB's cannot in i t s e l f j u s t i f y t h e i r selection for purposes of validating a p a r t i c u l a r model of interest arbitrage, however. 4 Royal Commission of Banking and Finance, Report, p. 321. The same conclusion was reached by several money market dealers in Vancouver interviewed by the author. According to one dealer, interest arbitrage between these two markets is mainly confined to commercial papers and finance company papers. ^It i s argued below that different types of securities possess attributes other than rate of return and risk of default which distinguishes them in the minds of investors. It is unlikely, therefore, that one type of sec u r i t i e s s a t i s -f i e s the needs of a l l arbitragers. ^Grubel, Forward Exchange, pp. 51-53. 7 I b i d . , pp. 114-16. \ Grubel does not consider the p o s s i b i l i t y that the two series may not be systematically related. o Ray C. Fair and Burton G. Malkiel, "The Determination of Yield D i f f e r e n t i a l s Between Debt Instruments of the Same Maturity," Journal of Money, Credit, and Banking, III (Nov., 1971), 733-49. 9 For example, German and Swiss bankers are not allowed to pay interest on term deposits of foreigners. Banks engaging in arbitrage for purposes of prestige or as loss-leaders are a case in point. See Einzig, Dynamic  Theory, pp. 167-68. 127 CHAPTER V NOTES (cont'd) "In the f i r s t half of 1962, for example, despite at t r a c t i v e net y i e l d s , the flow of U.S. funds was discouraged by the psychological effects of the withholding tax combined with an unsettled p o l i t i -cal s i t u a t i o n and fears about external f i n a n c i a l p o l i c i e s . " Royal Commission on Banking and Finance, Report, p. 321. 12 The collapse of A t l a n t i c Credit Corporation in June, 1965 discredited Canadian finance papers in the eye of the investing public. As a r e s u l t , there was l i t t l e - - i f any--arbitrage in such se c u r i t i e s for a prolonged period. See also H e l l i w e l l , "Structural Model," p. 100. 13 Grubel, Forward Exchange, p. 51, n. 14 This is a consideration most researchers seem to overlook. 1 5 E i n z i g , Dynamic Theory, p. 155. 1 6Royal Commission on Banking and Finance, Report, p. 321. 1 7 In the instance of U.S.-Canadian money markets, commercial papers are obviously better substitutes for each other than treasury b i l l s . 18 Following conventional analysis, i t is s t i l l assumed that interest rates are not affected by the a c t i v i t y of arbitragers. If this assumption were to be dropped, the analysis would become rather complicated. Our conclusions would hold only i f the supply and demand curves for the various short-term s e c u r i t i e s exhibited exactly i d e n t i c a l e l a s t i c i t i e s . It must also be assumed that other "factors" such as marketability and taxation are i d e n t i c a l for a l l pairs of s e c u r i t i e s under consideration. If this was not the case, equilibrium could conceivably p r e v a i l under arbitrage mar-gins of various magnitudes. This would c a l l , however, for an exceptional constellation of a l l these factors which i s quite unlikely to be realized. It i s suggested, therefore, that the above analysis would s t i l l be v a l i d . But in order to ascertain i t s v a l i d i t y for a s p e c i f i c case, more would have to be known about the various factors that influence the decisions of arbitragers. 19 According to Einzig, situations where arbitrage is carried out i n both directions simultaneously have become the rule rather than the exception. Dynamic Theory, p. 161. 128 CHAPTER V NOTES (cont'd) 2 0 Compare Chapter II, pp. 16-17. 2 1 Grubel, Forward Exchange, p. 52. 2 2 Einzig, Dynamic Theory, p. 147. 2 3 Compare note 4. H e l l i w e l l , "Structural Model," also finds that "during the sample period the forward exchange premium or discount has moved so as to maintain interest parity almost exactly with respect to finance paper d i f f e r e n t i a l , and p a r t i a l l y with respect to the Euro-dollar d i f f e r e n t i a l . " The treasury b i l l d i f f e r e n t i a l was not found to have any material impact. H e l l i w e l l did not examine the commercial paper d i f f e r e n t i a l . 2 4 Einzig , Dynamic Theory, p. 161. 2 5 I b i d . , p. 146. 2 6 Grubel, Forward Exchange, p. 116. 2 7 H e l l i w e l l , "Structural Model," p. 100. See page 117 for an example; also Einzig, Dynamic  Theory, p. 161. 129 CHAPTER VI CONCLUSION In this paper i t was undertaken to assess the v a l i d i t y of the contemporary theory of interest arbitrage. Our dis-cussion of this question was divided into three parts corres-ponding to different aspects of the problem. The f i r s t part, Chapter II, was devoted to an exposition of the h i s t o r i c a l development of the theory of forward exchange in which interest arbitrage assumes a pivotal position. It was our intention to demonstrate that although recent models yielded different results than Keynes* Interest Rate Parity Theory, t h e i r v a l i d i t y could not constitute a refutation of this theory as such. Rather, they should be seen as s p e c i f i c applications of the general economic p r i n c i p l e which forms the basis of Keynes1 theory. It says that, in a perfect mar-ket, two assets of the same kind must command the same price. So long as Keynes' theory is seen as a formulation of this p r i n c i p l e in i t s application to the foreign exchange market, i t i s c e r t a i n l y v a l i d . But i t is important to r e a l i z e that i t s predictions hold systematically only under the circum-stances for which the theory was formulated, namely the ideal case where a l l markets are perfect and where funds are un-limited. In more recent contributions to the theory of for-ward exchange these assumptions were dropped in favor of more 130 r e a l i s t i c ones. In this sense, these models only specify the consequences of various market imperfections for Keynes1 theory. They are, therefore, extensions of that theory, not competing ones. Thus, while we believe that Keynes' theory i s v a l i d insofar as i t assigns to arbitrage an e q u i l i b r a t i n g force within the monetary system, the mechanism which is operative appears to be somewhat dif f e r e n t from the one envisioned by Keynes. To examine alternative systems of exchange rate determination formed the second part of our enquiry. Keynes' contention that, under a system of f l e x i b l e spot rates, the forward rate acts as the sole adjusting agent after a change in bank rate i s correct only i f the unreasonable assumption is made that arbitragers are the only participants in the forward market. In a more r e a l i s t i c setting where merchants and speculators are also active in the forward market, adjustment to equilibrium comes about through changes in both the spot and the forward rates. In this connection i t was found in Chapter IV that the variation of the forward rate is amplified when there is direct intervention in the forward market. The variation is dampened, however, when intervention occurs in one of the other markets . Under a system of fixed exchange rates the process of adjustment i s l i k e l y to be quite d i f f e r e n t . The transfer of funds w i l l t y p i c a l l y affect the interest rate in both coun-t r i e s . The burden of adjustment is then borne not only by the 131 forward rate, but also by the interest rates in both countries. It was found that the size of various markets is a major factor in the assignment of the r e l a t i v e v a r i a t i o n . Small markets are much more exposed to the influence of interest arbitrage than large ones. It follows that monetary authori-t i e s in large markets enjoy a greater degree of freedom in pursuing p o l i c i e s which are commensurate with internal goals. Small countries, on the contrary, must closely follow the p o l i c i e s of larger ones. This dependence is mitigated to some extent by the variation of the forward rate. The greater the contribution of the forward rate in the process of adjust-ment to equilibrium, the smaller the burden borne by interest rates. The t h i r d part of this paper was concerned with one of the most r i g i d assumptions of IRPT that has hardly ever been challenged. The theory assumes that securities which are used by arbitragers are perfect substitutes for each other. In r e a l i t y , however, there exists a variety of short-term s e c u r i t i e s in each money market. They are distinguishable in terms of ris k of default, marketability, taxation and so on. The existence of such a large variety of money market i n s t r u -ments led us to postulate in Chapter V that there is in effect a considerable degree of market segmentation between the many possible pairs of short-term s e c u r i t i e s . The theory was advanced, moreover, that arbitragers exhibit preferences for one or few p a r t i c u l a r sets of instruments, while other short-term s e c u r i t i e s are largely neglected by arbitragers. 132 These hypotheses require empirical substantiation. Their implication for the theory of interest arbitrage is clear, however. The choice of a pair of securities both for purposes of empirically validating a p a r t i c u l a r model of interest arbitrage and for policy recommendations must be guided by a knowledge of the preferences of arbitragers. The t y p i c a l procedure of selecting treasury b i l l s i s not adequate in this l i g h t , p recisely because, as a preliminary survey indicates, arbitragers tend to avoid treasury b i l l s in many markets. Empirical validation is complicated, moreover, by the fact that the preferences of arbitragers tend to be un-stable over time; they also tend to vary from market to mar-ket which i s a r e f l e c t i o n of variations in i n s t i t u t i o n a l arrangements. S t r i c t l y speaking, therefore, the theory of interest arbitrage in i t s many expressions which are known today is only applicable to a p a r t i c u l a r pair of short-term securi-ti e s which is systematically used by arbitragers. It i s not amenable, however, to generalizations about the level of interest rates in one market as compared to another and i t s manifestations on the forward rate. If i t i s true, .further-more, that arbitragers' preferences change over time, current models of interest arbitrage are useful for short-term predictions only. None of the available models has so far been very successful in explaining the behavior of the forward rate. But i f a model of the foreign exchange markets which was 133 based on interest arbitrage were ever to be successful, i t would constitute a confirmation, and not a refutation, of Keynes' theory, as so many have claimed. There are three important factors which have not been considered in this study, just as they have been largely neglected in the l i t e r a t u r e of interest arbitrage. The f i r s t one is related to the method of analysing processes in the foreign exchange markets. Most of our dis-cussions were held in terms of s t a t i c equilibrium analysis. It has been convincingly argued by Einzig, however, that a theory of forward exchange must of necessity be dynamic. 1 In our analysis i t was indeed necessary to make the quite unreasonable assumption that supply and demand curves for non-arbitragers could be treated as given. In r e a l i t y , of course, such a system would be subject to continual external shocks. The interval between these shocks would probably be too short for the establishment of a new equilibrium. It is doubtful, therefore, that predictions about the behavior of the forward rate could be made on the basis of s t a t i c models such as the one employed in Chapters III and IV. It is more appropriate to view our analysis as an experiment conducted in a controlled environment. Undoubtedly, the theory of interest arbitrage requires further elaboration along the lines proposed by Einzig. The second omission concerns the somewhat a r t i f i c i a l d i s t i n c t i o n which is made between speculation, trade, and arbitrage. It was seen in Chapters II, III and IV that the 134 amount of funds demanded and supplied by the various groups of non-arbitragers in a l l markets is c r u c i a l for the determi-nation of the v a r i a b i l i t y of the different prices. But the theory of interest arbitrage f a i l s to explain the re l a t i v e amounts of speculative, arbitrage and trade a c t i v i t i e s . A comprehensive theory, however, should enable us to predict when speculation dominates or when arbitrage is p a r t i c u l a r l y strong. Research in this area appears to be a necessary condition for the advancement of the theory of forward exchange. The last observation relates to expectations about future prices which arbitragers may hold. Aside from a l l other factors which are operative and interfere with the process of arbitrage, parity may not be established in the sense that i t is not observable. Kenen has suggested that arbitragers consider marginal interest rates--which d i f f e r from lender to lender--to be relevant as opposed to s t a t i s -2 t i c a l l y recorded data. In the same vein, arbitragers may base their decisions not on prevailing rates but on the rates which they expect to obtain i f they actually carry out the i r transactions. If this were true i t would be i l l o g i c a l to expect parity to be established systematically even i f a l l other conditions were met, since i t would imply that some arbitragers are w i l l i n g to transfer funds to another country although no measurable gain could be made. Following this line of reasoning, one would expect deviations from interest parity to be much larger 135 between small countries than between large ones. This is so because changes in demand in small markets can be expected to lead to greater price variations than in large ones. Future research may provide an answer to this problem. 136 CHAPTER VI NOTES "'Einzig, Dynamic Theory, pp. 127 f f . , 278-83. 2 Kenen, "Trade, Speculation and the Forward Exchange Rates . " 137 APPENDIX In this appendix i t is attempted to ascertain the effects of direct intervention on the forward exchange rate. We are p a r t i c u l a r l y interested in the s e n s i t i v i t y of forward rates under various assumptions about the slopes of the constituent curves. The following diagram shows the system to be analysed. 1 A l l schedules have been expressed in terms of excess supply of the p a r t i c u l a r commodity. This was done in order to attain closer conformity with the convention of measuring the dependent variable on the ordinate. S F 0 + 0 + Fig. 16. Determination of spot and forward exchange rates after autonomous change in forward rate 138 To avoid confusion a simpler notation was adopted for this appendix: S ... spot exchange rate F ... forward exchange rate cx . . . slope of excess supply schedule of non-arbitragers (forward exchange) (i . . . slope of arbitragers' supply curve of forward exchange ^ ... slope of non-arbitragers excess supply curve of spot exchange2 Convention: Q>0 i f commodity in excess supply We would l i k e to determine the equilibrium spot and forward rates before intervention takes place: Given that Q a = <F-V (1) Q b = C F - F B ) (2) Q = (F-S ) c c (3) we require F = F' and S = S' such that % xb c (4) O (F- F.) = _ / 9 ( F . _ F b ) (5) and we obtain the equilibrium forward rate F . = F a + Fb CX+ f3 (6) Simi1arly /3 (F' -F b) = ^ ( S - S C ) (7) and we obtain the equilibrium spot rate S ' (F'-F b)+ S C (8) or S ' = | ( F ' - F B ) + S C . (8') Let us now consider the effects of an intervention in the forward market. Assume that the forward rate is lowered. The excess supply curve of forward exchange by non-arbitragers 139 s h i f t s down. The ensuing steps can best be i l l u s t r a t e d with a flow diagram. We adopt the following notation: A 2 Bl B2 > B3 C the i n i t i a l (a) curve (a) curve after intervention the i n i t i a l (b) curve (arbitragers) successive (b) curves in i t e r a t i o n (c) curve (non-arbitragers in spot market) This curve remains unchanged throughout. A1 B1 C r 1 1 A 2 B 1 C Fl S l - , F2 So. A 2B 2C S3 A 2 B 3 C S4 _^  A 2B 4C Fig. 17. Flow diagram i t can be seen that as the i t e r a t i v e process continues, only the excess supply curve of arbitragers continues to s h i f t . We can now write the general equation for this excess supply curve: (F-F*) ^ n' where F = F . + k(S -S .) for n > 2' n n-1 v n n-1'. (9) (10) and F n i s the intersection of the nth arbitrage curve with the ordinate. F^ i s a given constant. It corresponds to 140 in the diagram. We also have, from above, A 2 Q a = d>(F-F^) (11) where £> slope of non-arbitragers' excess supply of forward exchange after intervention Oremains constant for the rest of the analysis; and C from above We can now write F' and S' generally for n. F - ^ F<* + F " - l S = 4(F _ F * . )+S n 0 n n-1' c and repeating F* = F* ,+k(S -S n n-1 n n-1 n > 2 n » 2 (3) (12) (13) (10) where constant, given (constant) F - °<F~ V*F1 1 cx + (3 S l BfC F l - F P + Sc-We wish to solve for Fe>o , the equilibrium forward rate, Foo = F 2 + A ( F 2 " F 1 : ) 2 /*a (14) (15) (16) where A = bty+fty +/3) d ( l + a) /c?k c> where a For a stable solution we require 0 < a < l . In other words, s t a t i c equilibrium can be obtained i f a is smaller than 1, but larger than zero. The v a r i a b i l i t y of the forward rate can now be conven-i e n t l y analysed with the help of the following diagrams.** 141 • F, 1 _'F2 }A(F _ V F ; since 0< a < 1 for convergence A -max 2o A . m m Fig. 18. Variation of the forward rate Fig. 20. Amplifying effect of 2d y then cb has minimum value i f j_ k 2b r then & has no minimum value m m Fig. 22. Amplifying effect of c> 143 APPENDIX NOTES ^Adapted from Glahe, Empirical Study, p. 21. 2 Subscript a refers to non-arbitragers in the forward market, b to arbitragers, c to non-arbitragers in the spot market. 3 k i s the proportion by which the arbitrage curve s h i f t s in response to a change in the spot rate. 4 The author would l i k e to thank Martin C r i l l y , Joseph E. Spuller and Dana G. Schroeder for their assistance in the solution of this problem. 144 BIBLIOGRAPHY References Cited BOOKS Anion, Tagmir. "The Forward Exchange Market: A Capital Market Approach." Unpublished Manuscript, University of Chicago, 1969. 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Die Kursbi1 dung auf dem Devisenterminmarkt  und die Devisenterminpolitik der Zentralbanken. Berlin: Walter de Gruyter $ Co., 1970. OECD. Capital Markets Study. Formation of Savings, Vol. II. Paris, 1967. Smith, Paul F. E conomi cs o f Financial Institutions and Markets. Homewood,. 111. : Richard D. Irwin, Inc., 1971 Snider, Delbert A. Introduction to International Economics. 5th ed. Homewood, 111.: Richard D. Irwin, Inc.; Georgetown, Ont.: Irwin Dorsey Limited, 1971. Sohmen, Egon. Flexible Exchange Rat es. Rev. ed. Chicago and London: University of Chicago Press, 1969. Stein, Jerome L. The Nature and E f f i c i e n c y of the Foreign  Exchange Mark et. Princeton Studies in International Finance, No. 40. Princeton, N.J.: International Finance Section, 1962. Young, John P. The International Economy. 4th ed. New York Ronald Press Company, 1963. JOURNALS Altman, Oscar L. "Foreign Markets for Dollar, Sterling and Other Currencies." International Monetary Fund Staff  Papers, VIII (1960-1961), 313-52. . "Recent Developments in Foreign Markets for Dollars and Other Currencies." International Monetary Fund  Staff Papers, X (1963), 48-96. Auten, John H. "Forward Exchange Rates and Interest-Rate D i f f e r e n t i a l s . " Journ al of Finance, XVIII (March, 1963), 11-19. Balbach, Anatol. "Will Capital Reflows Induce Domestic Interest Changes?" Federal Reserve Bank of St. Louis, (July, 1972), 2-5. Black, Stanley S. "Theory and Policy Analysis of Short-Term Movements in the Balance of Payment." Yale Economic  Essays, VIII (Spring, 1968), 5-78. Fair, Ray C , and Malkiel, Burton G. "The Determination of Yield D i f f e r e n t i a l s Between Debt Instruments of the Same Maturity." Journal of Money, Credit, and Banking, III (Nov., 1971), 734-37. Fre'vert, Peter. "A Theoretical Model of the Forward Exchange Part I." International Economic Review, VIII (June, 1967), 153-67. . 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" Southern Economic Journal, XXII (July, 1955), 1-21. O f f i c e r , Lawrence H., and W i l l e t t , Thomas 0. "The Covered-Arbitrage Schedule: A C r i t i c a l Survey of Recent Develop-ments." Journal of Money, Credit, and Banking, Vol. II, No. 2 (1970), 244-55. Reading, B. "The Forward Pound 1951-59." Economic Journal, LXX (June, 1960), 304-19. Spraos, John. "The Theory of Forward Exchange and Recent Practices." Manchester Schoo1 of Economics and Social  Studies, XXI (May, 1953), 83-117. Stein, Jerome. "International Short-Term Capital Movements." Ame r i can E conomi c Review, LV (March, 1965), 40-66. S t o l l , Hans. "An Empirical Study of the Forward Exchange Market under Fixed and Flexible Exchange Rate Systems." Canadian Journal of Economics, II (Feb., 1968), 55-78. . "Causes of Deviation from Interest-Rate Parity." Journal of Money, Credit, and Banking, VI (Feb., 1972), 113-17. Tsiang, Sho Chieh. 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"Capital Movements among Major OECD Countries: Some Preliminary Results." Journal of  Finance, XXVI (May, 1971), 251-68. Canterbery, Ray E. "A Dynamic Theory of Foreign Exchange." National Banking Review, IV (June, 1967), 347-415. . "A Theory of Foreign Exchange Speculation Under Al-ternative Systems." Journal of P o l i t i c a l Economy, LXXVII (May/June, 1971), 407-36. 151 Caves, Richard E. "Flexible Exchange Rates." American Eco-nomic Review, Papers and Proceedings, LIII (May, 1963), 120-29. Henderschott, Patrick C. "Structure of International Interest Rates: The U.S. Treasury B i l l and the Euro-dollar Deposit Rate." Journal of Finance, XXII (Sept, 1967), 455-65. Johnson, Harry G. "The Taxonomic Approach to Economic Policy, Economic Journal, LXI (Dec, 1951), 812-15. . "Monetary Theory and Policy." American Economic Review, LII (June, 1962), 335-84. "Conditions of International Monetary Equilibrium: Equilibrium under Fixed Exchanges." 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