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L -.'.* ‘JI’II'IJJI: IJTI'I1-"I'EI'FJ .. .‘ .. y l g,JJ11(IJ “a; ,IJIEJIIJJJJJIJJJ'JJJJJJHJJJJJ-‘JJJJJJ1J1J'JI5JJIM .3. “ ?,'1J.11‘J1.r:1|"JJJ ’ N111 """. J'J . " :JJII'J} ".9 'I' 1.9.” :11 'fJIIIJ‘ .JJ :J'Il.“ . ' ". - 331.: I' 'PJEI um I'Ul<”(”vl'.W1-?h F .dt ffifi'IP 1‘.“ 'nnn - . :1 I I. III “III II I I m»... .. I . . . ' 'I.‘ ’1'}.wa W -.| E‘I " I JIM "’ I' . ’-'J I "I‘ J'JI’I lllllllllllllllll"INHIIHIIIIIIIJIIHJIIIIIUIHIHlillHll 193 10370 4858 This is to certify that the thesis entitled MARKET EFFICIENCY AND SPECULATIVE ACTIVITY IN THE FOREIGN EXCHANGE MARKET presented by THELMA SUSAN POZO has been accepted towards fulfillment of the requirements for Ph.D. degree in Economics Major pmf% (i Date—NMQWQBO 0-7539 j‘w ‘ LIBRAR Y Michigan State University OVERDUE FINES: 25¢ per day per item RETURNING LIBRARY MATERIALS: Place in book return to remove charge from circulation record MARKET EFFICIENCY AND SPECULATIVE ACTIVITY IN THE FOREIGN EXCHANGE MARKET By Thelma Susan Pozo A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1980 ABSTRACT MARKET EFFICIENCY AND SPECULATIVE ACTIVITY IN THE FOREIGN EXCHANGE MARKET By Thelma Susan Pozo This dissertation examines market efficiency in the context of foreign exchange markets as this concept has been employed to date. The necessary conditions for efficiency to hold in all markets are derived. It is found that it is unlikely that all conditions hold simultaneously. Three alternative models of foreign exchange pricing are developed which invoke the efficiency concept. The necessary conditions in these cases are deemed more viable. In the first model developed, risk neutral behavior on the part of speculators ensures efficiency in all markets. It is found that opportunities never arise for covered interest arbitrageurs. Also, in order that efficiency holds through time it must be that expectations of tomorrow's price incorporates expectations of prices expected to prevail through all future periods. A second model of foreign exchange market determination is derived which assumes risk averse behavior on the part of market participants. When both spot and forward exchange risk are present the following relation— ships are found to emerge. The interest rate parity condition cannot hold. Spot speculation will contribute more risk to the investors portfolio than forward specula— tion despite the fact that the latter form of speculation includes two sources of risk and the former only one. It is also found that the inclusion of transactions costs will not alter the relative relationships. Lastly, it is verified that prices in an efficient market will under— estimate expectations of future prices if investors are risk averse or if transactions costs are present. Finally, a model explicitly using expectations of prices expected to prevail for a series of future periods is developed. The coefficients with respect to the‘ different time periods tend to be inversely related to the period for which that expectation is relevant. A policy implication of this formulation is that the official authorities have some influence in regard to "managing the float". However, a contradiction arises. To be effective for an extended period of time without intervening directly during each and every period, it is necessary that policymakers alter expectations of prices expected to prevail at some distant period, say t+j. The policy shock will be magnified as time passes, having its maximum effect during the period t+jvl. At t+j its effect will be reduced to zero. Hence, intervention may tend to make exchange rate fluctuations more volatile ultimately resulting in the Opposite of what exchange rate management is purported to accomplish. ACKNOWLEDGMENTS I would like to express my profound gratitude to Lawrence H. Officer, my major adviser. Professor Officer was always available for consultation, and went to great pains to help me through the difficult periods. Next, I would like to thank Norman P. Obst for being an excellent sounding board, and for performing equally well in the motivation department. Special thanks are also in order for James J. Johannes and Dennis L. Warner for their useful comments. I am also appreciative of Daniel S. Hamermesh, Director of Graduate Programs. During my stay at Michigan State University, Professor Hamermesh was always willing to provide advise and encouragement. Thanks are also in order for my parents, my brothers, Eddie and Lee and my sister Maria. Last, but not most certainly not least, I am grateful to Terie L. Snyder. Terie was most instrumental in helping me meet deadlines. ii CONTENTS INTRODUCTION . . CHAPTER ONE- REVIEW OF THE LITERATURE Introduction A. Definition of an Efficient Market. B. Interpretations of Market Efficiency in the Foreign Exchange Market. Spot Market Efficiency. Forward Market Efficiency Other Forms of Efficiency C. Conclusions. TWO- CRITIQUE OF THE THEORY Introduction . . A. Four Versions of Foreign Exchange Market Efficiency Model 1 Model 2 Model 3 Model H B. Conclusions. THREE- EFFICIENCY ASSUMING RISK Introduction A. A Model of the Simultaneous Determination of Spot and Forward Prices . The model Assuming Efficient Markets NEUTRALITY B. C. Efficient Markets Through Time . . D . . . . Conclusions. FOUR- EFFICIENCY ASSUMING RISK AVERSE BEHAVIOR Introduction . . A. The Capital Asset Pricing Model. . B. Application of the Capital Asset Pricing Model to the Pricing of Forward and Spot Exchange . . C. Capital Asset Pricing with Forward Exchange Risk . . . . . . . . . D. Conclusions. . . . . . . . . iii PAGE \10'3 1” 1M 18 2O 21 23 2H 2H 27 28 30 31 31+ 35 RR 56 59 El 63 76 83 89 CONTENTS (cont'd.) CHAPTER FIVE-—AN ALTERNATIVE MODEL OF EFFICIENT FOREIGN EXCHANGE RATES Introduction . . . . A. Are Forward Prices Biased or Unbiased Estimators of Expectations? Version I - . . . . . . Version II - B. Using a Time Series of Expected Prices to Explain Current Prices C. Rational Expectations - . D. Relationships Between Current Spot Prices and Expected Spot Prices . . . - E. Conclusions . - - . - . . - SIX-—POLICY IMPLICATIONS OF THE EFFICIENT MARKET HYPOTHESIS Introduction - . A. The Management of Exchange Rates Assuming Risk Neutrality. . . . . . . . . B. The Management of Exchange Rates Assuming Risk- Averse Behavior C. Conclusions. . . . . . . . . SEVEN-CONCLUSIONS. . . . . . . . . . FOOTNOTES . . . . . . . . . . . . . . . . . . WORKS CONSULTED . .'. . . . . . . . . . . . . . . . . iv PAGE 92 97 97 99 101 103 107 113 115 116 120 122 12” 131 136 INTRODUCTION The purpose of this thesis is to deal with the subject of foreign exchange market efficiency. In the first chapter various definitions of market efficiency are presented. These definitions may be categorized in the following manner. On the one hand,we may characterize a market as efficient subject to the type of equilibrium return that is expected to prevail. Alternatively, a market may be said to exhibit a particular degree of efficiency according to the subset of information that is ultimately reflected in prices. The remainder of Chapter One completes the review of the literature. This includes categorizing the many interpretations that have been given to efficiency as this concept relates specifically to the foreign exchange market. In Chapter Two an examination of the major inter— pretations of foreign exchange market efficiency is conducted. We attempt to seek answers to the following questions. Are the major interpretations of exchange market efficiency consistent with one another? If they are not, under what conditions can inconsistencies be reconciled with one another? What assumptions remain hidden behind the interpretations of foreign exchange market efficiency? Are these assumptions realistic? In the third chapter of this thesis we deal with the issue of foreign exchange market efficiency in an alternative manner. As Opposed to using a partial equilibrium approach, a general equilibrium approach is used. A simple model of exchange rate determination is presented. In this model spot and forward rates of exchange are simultaneously determined. We find that in this model, the conditions for efficiency in the foreign exchange market are easily found. By making a few modifications of the behavior assumptions regarding the activities of market participants we can easily derive a consistent model of exchange rate determination which insures efficiency in all markets. Three major results are derived in this chapter. First, covered interest arbitrageurs need never participate in the market. Secondly, expectations of future exchange rates incorporate expectations of all future interest rates as well as future spot prices. The third conclusion is that the forward rate is an unbiased predictor of the future spot price. The model of exchange rate determination of Chapter Three is deficientinone respect. This model assumes that investors and speculators are risk neutral. It is suggested that such an assumption may not coincide with actual behavior. The purpose of Chapter Four is to build a model that assumes risk aversion on the part of market participants, and is also consistent with respect to the efficiency concept. The Sharp—Linter capital asset pricing model is used to accomplish this. Three different variants of the "internationalized" capital asset pricing model are presented. In the first instance only one form of risk is incurred. We find that when assets are risky only in regard to the uncertainty of future exchange rates, it must be that the expected return from engaging in a spot transaction is identical to that when engaging in a forward transaction. It is also found that covered interest parity holds exactly. These results are similar, though not identical,to the results obtained in the model assuming risk neutral behavior. In the second model we allow for the existence of forward exchange risk. In this instance we prove that forward transactions are less risky than spot transactions and that interest rate parity does not hold exactly. Another result is that a forward bias exists. The forward rate is negatively biased with respect to the expected future spot price. Margin requirements are imposed in the third model. It is found that the incorporation of transactions costs does not alter the general results. In the fifth chapter of this thesis we use results from the previous four chapters to build an alternative model of exchange rate determination. It is suggested that a sequence of expected future prices is most appropriate for describing current exchange rates. Coefficients describing the relative importance of these variables are derived using the results from Chapter Four. It is suggested that a testable model is easily derived. Forward prices, adjusted for biases that may exist can easily be used as estimators of future expectations. The final contribution of Chapter Five is to point out the similarities, differences, and improvements of our model with respect to the asset approach to exchange rate determination. Our model is similar to that using the asset approach in that expectations play a major role in exchange rate determination. Our model differs in that we need not assume rational expectations. Expectations of this form automatically result. A major improvement is that the coefficients decline in value with time as a result of sound economic theory. This contrasts with the common practice of assuming such a "weighing scheme". In Chapter Six we investigate the policy implica- tions of foreign exchange market efficiency. We attempt to answer the following questions. Can policymakers alter exchange rates so that they do not correspond to their "true" values? Which methods for intervention are most useful? How effective are these tools? Is it necessary for tmflicymakers to intervene continuously or will a one period alteration be selfwsustaining for an extended period of time? The conclusion to this dissertation summarizes the contributionscfi’this study. CHAPTER ONE REVIEW OF THE LITERATURE Introduction The intent of this chapter is to convey two major points. The first point is a definition of the efficient market hypothesis (EMH). The second point regards its role in the foreign exchange market literature. We begin the analysis of efficiency by presenting an intuitive explanation of this concept. Next, more formal definitions of the efficient market hypothesis are reviewed. These definitions fall into two major groups. Markets may be regarded as efficient subject to the nature of expected equilibrium returns. Alternatively, markets may exhibit varying degrees of efficiency subject to the subset of information being reflected in final equilibrium values. In part B of this chapter we review the concept of efficiency as it pertains to the foreign exchange market. Though the major econometric tests concerning the EMH are presented, we do not dwell on the statistical methods employed. Review of the existing theoretical and empirical literature serves to highlight the major interpretations that have been developed in regard to foreign exchange market efficiency. The results of this chapter are used as a basis for a critique of the existing interpretations of foreign exchange market efficiency, the subject of Chapter Two. A. Definition of an Efficient Market Before developing a rigorous definition of the EMH, it might be useful to intuit this concept. In an efficient market, prices fully reflect the available information.1 This results from the premise that market participants use all available information while conducting transactions. This information is used to formulate expectations regarding future prices. Hence, under rational behavior, today's price will be set such that it closely resembles tomorrow's expected price and no "unusual" profit oppor— tunities remain unexploited. If this is not the case, if today's equilibrium price does not reflect tomorrow's expected price, "unusual" profit opportunities remain. All information is not being reflected in prices. By definition the market is not efficient. The above points to two basic statements that can be made concerning efficiency. In an efficient market, prices are such that no "unusual" profit Opportunities remain unexploited. Furthermore, expectations of future prices are reflected in today‘s price. The first statement implies that certain technical conditions must be satisfied if a market is to be efficient. The second statement characterizes the EMH as being derived from asset market theory.2 Both approaches to efficiency prove to be useful in our analysis. Using the technical approach, efficiency implies the elimination of unusual profit Opportunities. It follows that on average, returns in excess of those expected are zero. Let Et(Rt+lI¢t) be the expected return in time period t+l given the information available at time t. If Rt+l is the actual return realized in period t+l then we would require in an efficient market that: m (1) B{zt} — HR“1 - Et(Rt+l|¢t)} 0 where E the expectations operator ¢t the information available at time t N indicates a random variable In addition it is necessary that the Zt‘s are serially uncorrelated. This amounts to requiring that the sequence of excess returns {Zt}’ follows a fair game with respect to the information sequence {Ot} for efficiency in the market. Intuitively, the market will set today's price such that no unusual profit opportunities remain unex- ploited given the information at hand at time t. Hence today's price is set conditional on tomorrow‘s expected price, allowing for a fair return. This does not preclude the possibility that tomorrow's actual price (Pt+l) deviate from the expected price (EtIPt+l|¢tl). This may result from new information that becomes available at time t+l. However, new information must be random, otherwise it is not new. It follows then that deviations of actual prices from expected prices are also random and on average sum to zero. It is therefore argued that deviations of actual returns from expected returns will also be charac— terized by an expectation of zero, hence E{Rt+l-E(Rt+l|¢t)} = 0. We have established that in an efficient market prices are set such that market participants expect at the margin to earn a fair though not an "unusual" return. Before characterizing a market as efficient or not, it is necessary to specify the exact nature of this fair return. Once we have done so, we can by examining sequences of prices establish whether in fact the market behaves as posited in equation (1). Richard Levich points out the joint hypothesis problem that arises from this.“ When one looks for evidence to accept or reject the EMH, one is in fact testing a joint hypothesis. On the one hand, a Specific market equilibrium condition (expected fair return) is posited. Secondly, conditions for market efficiency in a technical sense are being tested. Following are examples of differing market equili- brium conditions that may be posited.5 If expected returns are positive, then: 10 |¢ ) = Bt(pt+ll¢t) - P > 0 t+1 t Pt (2) Et(R Given that the market determines Pt in the manner just described, it must be true for market efficiency that the sequence of excess returns, {Z }, follows a fair game I with respect to the information set, {¢t}' That is: (3) E(% ) = E{R - E (R |¢ )} = 0 t t+l t t+l t Substituting equation (2) into (3) we find that: P - P E (P |¢ ) - P E(%t) : E, t+é t _ t t+1P t t} z 0 t t or m P - E (P |¢ ) (u) E(Zt) = E{ t+l Pt t+1 t } = o t Since we do not observe Et(%t+ll¢t) we must hypothesize its nature. An alternative hypothesis concerning equilibrium expected returns is that they are constant. E P - P (5) E (R |¢ ) = t( t+1|¢t> t = o t t+l t Pt where a is a nonnegative constant. The EMH claims that the sequence of excess returns, defined as: 11 has an expected value Of zero and is serially uncorrelated. Once again, by the substitution of (S) into (6) we claim that for efficiency to hold: P - P E (P |¢ ) - P E{%t} : E, t+1P t _ t t+1P t t} = 0 t t or P - E (P |¢ ) E{%t} = E{ t+1 ; t*1 t } = o t Hence tests of the EMH should be viewed with skepticism. If tests do not reject the hypothesis it is possible that in fact the market is not efficient since the equilibrium condition has been misspecified. If efficiency is rejected, it can on the other hand be argued that the market is truly efficient but again a misspecified market equilibrium condition is the cause of rejection. Keeping in mind the issue of jointness, we will review the different manners by which we may expect sequences of prices to behave in a market characterized by efficiency. One common formulation is that of a random walk. Using Giddy and Duffey's6 formulation, a formal statement of the random walk may proceed as follows: (7) Et(Pt+lth) = E(Pt+ll¢t) = Pt+l where G is the series of present and past information. t o I o o o o Th1s 1nformat1on lS spec1f1cally,present and past prices for this market, i.e. Pt’ Pt—l’ Pt_2,... l2 ¢t is all publically available information at time t. Et is the expectations operator. Lett1ng Rt+l = lnPt+l - lnPt, equat1on (7) 1mp11es that: (8) Et(Rt+1|Gt) = Et(Pt+l|¢t) = 0 The random walk, further requires that the Rt's are serially independent and identically distributed. Hence 3(Pt) = 0 and cov (Pt,Pt_j) = 0 for all 3 ¢ 0. The argument in favor of using a random walk is explained by Roll.7 When a market is competitive in the classic sense, every trader has perfect information and serial dependence in price changes is immediately discovered. Serial dependence in price changes implies that costless mechanical trading rules earn positive profits. But economic profits cannot persist in a competitive market. If such profits should arise, they will soon be erased, along with their cause (the serial dependence), by competition. The random walk is more restrictive than the submartingale, as defined below. Mandelbrot8 notes that mechanical trading rules will not be profitable even if prices follow a submartingale. Hence, as Rollg points out, we should not expect a stronger condition to hold when a weaker suffices. The submartingale claims the following: _ > (9) ECPt+l|Gt) - ECPt+ll¢t) 2 Pt and so 13 > t+l|¢t) Z = E(P (10) E(P 0 ttllet) The submartingale differs from the random walk in that it is no longer necessary to assume that the Rt'8 f01lOW any particular distribution and that they are identically distributed. We need only posit that a finite variance for Rt exists. A special case of the submartingale is the martingale, which merely states that (9) holds with an equality. That is: = E(P : P ‘ (11) P(P t+1|¢t) t t+lth) Often efficient markets are characterized by the specific subset of information being reflected in prices. In a sense, differing "magnitudes" of efficiency are being defined.10 In a weakly efficient market, the relevant subset of information is the set of past prices. A semi-strongly efficient market uses all publically available information to determine today‘s equilibrium price. By contrast, a market characterized by numbers of individuals with priviledged information, and hence marked by monopolistic elements, is a strongly efficient market. It should be noted that the statement that prices reflect all information has been proven inconsistent by Grossman and Stiglitz.ll Information is not costless. If prices should contain all available information, the individuals who gathered the information (and thus paid a 1M price) would not be compensated for their activities. Consequently, there is no incentive for traders to gather information. Hence how can the market price be informa— tionally efficient? They conclude that costless informa- tion is a necessary condition for efficiency to hold. B. Interpretations of Market Efficiency in the Foreign Exchange Market A voluminous set of literature on testing the EMH with respect to the foreign exchange market exists.12 Though general test results will be presented, the purpose of this chapter and those to follow is got to accept or reject the concept of efficiency on the basis of empirical work. The purpose of this study is to investigate the various interpretations that have been given to the concept of efficiency in the foreign exchange market. Presentation of interpretations will proceed as follows. First we shall discern the major conclusions derived with respect to behavior of spot prices. Next we shall concern ourselves with expected relationships between spot and forward prices. The last conclusions to be drawn are those that describe the behavior of prices in ways that do not fall into the above two categories. Spot Market Efficiency The most common interpretation of spot market efficiency is to claim that spot prices follow a random walk. The reasoning used to arrive at this conclusion is 15 as follows.13 If speculators are rational they will attempt to set today's price so that it is identical to tomorrow's expected price, presumably so that no unexploited profit opportunities remain. Hence, successive spot prices should change only as a result of new information concerning the expected values of future prices. New information is unpredictable and is independent of past information. It follows then that price changes are random and serially uncorrelated. In essence, it is being claimed that equilibrium returns are zero, and that given that such is the case, in an efficient market we would observe that successive prices follow a random walk. To test for serial dependence in price changes, filter tests are often conducted. Filter testslu discern whether mechanical trading rules can be used to earn profits in the market. One attempts to find a rule which claims that buying when spot prices rise by X percent from a trough and selling when theyfall by X percent from a peak will earn positive profits. Poole, by conducting filter in addition to other tests of serial correlation finds evidence to reject the random walk model for spot exchange. He suggests that this might result partly from transactions costs. Also, regarding whether filter rules can be of help, it should be noted that rules need be discerned ex—ante.15 Using tests we discern them ex-post. It is possible that rules often change and hence specula- tors cannot easily earn profits if they cannot find the 16 correct filter before the fact. Burt, Kaen and Booth,16 find that serial dependence exists for the Canadian dollar but accept the random walk hypothesis for the British pound and the German mark. Perhaps of greater concern for the acceptance or rejection of the EMH is the question concerning whether spot prices follow a submartingale. A variety of tests have been conducted with the submartingale in mind. Recall that each version is presupposing a specific equilibrium expected return. Most tests are close in spirit to those proposed by Giddy and Duffey.l7 In one formulation, they suggest that the following relationship holds: m (12) Et(St+1|¢t) - st : i - i ' St $,t X,t where i$ t and iX t are the one period interest ’ ’ rates in the two countries under consideration. Presumably, differential nominal interest rates measure differential expected inflation rates given that nominal rates are composed of the real interest rate and expected inflation rates. The exchange rate is expected to depreciate or appreciate directly in relation to differen- tial price level movements. Giddy and Duffey also proposed testing for a martingale, under the assumption that equilibrium expected returns could be zero. Assuming a martingale, the 17 following relationship would be observed: E (P |¢ ) s (13) t t+l t t = 0 St It was found that both models fit the data well, and therefore one could not be considered more accurate than the other. To conclude, some writers feel that for spot market efficiency to exist, it must be that prices follow a submartingale. Though often a random walk is specified, the random walk is really just a special case of the submartingale and the general consensus is that a stronger condition is unnecessary when a weaker one will do as well. It suffices that spot prices follow some version of the submartingale. Hence we might expect that the best predictor of tomorrow's spot price is today's price, or some predetermined function of that price. In other words we expect that: (1n) Pt(§t+l|¢t) = st or m (15) Et(st+1|¢t) = Stf(X) where f(X) specifies a particular equilibrium return relationship, for example, accounting for differential interest rates for the two countries under consideration, as in Giddy and Duffey's formulation of equation (12) above. 18 Forward Market Efficiency In regard to forward market efficiency, many writers have claimed that forward prices are unbiased estimator of future spot rates. Hence the following relationship would be expected to prevail: : + + (16) St a th-l,t “t where St is the spot price at time t, F is the forward price set a t-l for delivery t-1,t at t, ut's are serially uncorrelated, and a and b are not significantly different from zero and unity respectively.18 It has been argued that if a bias does exist, all profit opportunities have not been exploited and hence the conditions for market efficiency do not exist. Levichlg documents fourteen tests of this form. For the most part, it appears that the forward rate is a biased predictor of the future spot price.20 Note however, that biasedness should not be taken a priori as evidence of inefficiency. If risk is systematic, then a risk premium might be consistent with a forward bias and not 21 One must take necessarily indicative of inefficiency. care to not allow an incorrectly specified equilibrium condition disprove efficiency if such a conclusion is unwarranted. 19 In general, proponents of unbiasedness claim that forward prices correctly reflect expectations of future prices. Moreover, these expectations are on average realized. Hence: Ft-l,t = Et-l (St|¢t_l) and so on average, Et-l(§tlet-l) = St Geweke and Feige22 approach the efficiency of forward markets in a novel way. They propose that as with spot prices, forward prices should follow a martingale. However, they also suggest that one seeks for multimarket efficiency. This is distinguished from single market efficiency in that the information being reflected in prices encompasses more variables. Specifically, the test for multimarket efficiency involves examining the following: i ' ' . . (l7) E(qt|q%_l, q%_2, ...) for j = 1, l, ... ,N i i i _ St ' Ft-1,t where qt - . , sl t-l i is the $/i exchange rate, j is the $/j exchange rate. In other words, Geweke and Feige reason that the dollar/ pound exchange rate will depend not only on information concerning the dollar and the pound, but also in informa— tion regarding the dollar in relation to the yen, lira, etc. A test of single market efficiency would on the other hand involve testing the following form: .. , .. LIII’PMIIIIII! . . , .y __ .— 20 (18) E(q%lq:_l, q:_2, ...) In this case, the conditional variables or the information set is a subset of those used in equation (17). It includes $/i but ignores $/j exchange rates. Geweke and Feige in regard to their econometric results conclude that the efficiency hypothesis must be rejected. They hypothesize that rejection is a result of the existence of transactions costs and risk averse behavior. Other Forms of Efficiency Often it is claimed that in an efficient foreign exchange market, the interest rate parity condition ought to hold.23 Pursuing covered interest arbitrage Opportunities is essentially riskless, thus equilibrium expected returns in excess of transactions costs should not exist in an efficient market.2u In every period the following should approximately hold: r -r (19) £Z§ - a b = a = o S l—r b where ra is the n period domestic nominal interest rate prevailing at time t, rb is the n period foreign nominal interest rate prevailing at time t, S is spot price at time t, F is the forward price set at t for delivery at t+n. 21 Deviations from interest parity (d), should be zero in every period, or in the presence of transactions costs not exceed these costs. In general, results from such tests are ambiguous. Frenkel and Levich25 find with respect to Euro-currencies, interest rate parity is accurate. For other markets, they find the results to be less promising. Presumably, this is because political risk, absent in Euro—currency trans— actions, is an important factor with respect to less integrated markets. C. Conclusions In this chapter we covered two major points. We reviewed the theoretical literature regarding the efficient market hypothesis. Both formal and informal analyses of market efficiency were presented. It was noted that when one seeks for evidence of efficiency, or inefficiency it is necessary that the joint hypothesis problem not be ignored. What may seem as evidence to accept or reject the hypothesis of efficiency must be dealt with cautiously. This is because it may be that equilibrium returns are expressed incorrectly. A true test of the theory has not been conducted. In addition we reviewed the ways by which the EMH has been interpreted in the foreign exchange market literature. Efficiency has been dealt with on various levels and with respect to differing sets of prices. 22 Efficiency has been interpreted as implying that sequences of spot prices behave in particular manners. Efficiency has also been taken as establishing particular relation- ships between current spot and past forward rates. Also, other particular equilibrium relationships are claimed to hold in an efficient foreign exchange market such as the interest rate parity relationship. The review of the existing literature, as presented in this chapter does not pretend to be exhaustive. It is felt that the major themes of foreign exchange market efficiency have been presented. With this in hand we are ready to pursue a critique of these developments, the subject of Chapter Two. CHAPTER TWO CRITIQUE OF THE THEORY Introduction The purpose of this chapter is to investigate further foreign exchange market efficiency. Most often a segmented approach is used when studying this issue. It might be claimed, for instance, that spot prices need behave in a particular manner while forward prices follow some other rule. Often these hypotheses are inconsistent with one another. It is not possible that both be true at the same time. In the sections to follow, we shall be examining various interpretations of market efficiency. In our analysis we shall attempt to uncover the necessary conditions that need hold so that the interpretations be consistent with one another. If these conditions cannot be expected to prevail, then it will be argued that some— thing is lacking in the theory of efficiency of foreign exchange markets as it has evolved to date. If individual market participants can discern inconsistencies there exists ways to "beat the system" to obtain "unusual" profits. Following are four versions of the theory of 23 214 foreign exchange market efficiency. This is not an exhaustive list, but conclusions regarding the inadequacies of the current theory are nonetheless derived. A. Four Versions of Foreign Exchange Market Efficiency Model 1 It was concluded in the previous chapter that it is often stated that spot prices of forward exchange under the EMH will follow a martingale sequence.l Expected returns for engaging in further spot transactions yield no additional returns. mt |¢)— (1) t t+1 St t z 0 t This implies that today's equilibrium spot price is identical to tomorrow's expected spot'price, or (2) s = Et(S t t+1|¢t) The above hypothesis assumes that transactions costs are zero and that individuals do not expect to be compensated for risks they incur when engaging in spot speculative transactions. Following the same type of reasoning and maintaining the assumptions of no transactions costs and zero compensation for risk we can expect the forward price of foreign exchange to be set to reflect tomorrow's expected spot price. foreign exchange market efficiency. This is not an exhaustive list, but conclusions regarding the inadequacies of the current theory are nonetheless derived. A. Fpur Versions of Foreign Exchange Market Efficiency Model 1 It was concluded in the previous chapter that it is often stated that spot prices of forward exchange under the EMH will follow a martingale sequence.l Expected returns for engaging in further spot transactions yield no additional returns. E(S |¢)- (1) t t+l St t = 0 t This implies that today's equilibrium spot price is identical to tomorrow's expected spot'price, or (2) s = Et(S t t+1|¢t) The above hypothesis assumes that transactions costs are zero and that individuals do not expect to be compensated for risks they incur when engaging in spot speculative transactions. Following the same type of reasoning and maintaining the assumptions of no transactions costs and zero compensation for risk we can expect the forward price of foreign exchange to be set to reflect tomorrow's expected spot price. 26 (3) F = Et(S t,t+l t+ll¢t) To complete the analysis we might add that the interest parity condition must hold.2 Again, this assumes that there are no transactions costs. In addition individuals are not being compensated for "political risk", the risk that exchange controls may be imposed to render incomplete or partial inconvertibility of holding of foreign currency. Under interest rate parity we could expect (H) to hold. F l+r _ a m “Is—“17;; Now examine the three propositions together. Under what conditions can all three be expected to prevail? By substituting (2) and (3) into the left—hand side of (N) we find the following: Et(§t+1|¢t) _ (5) g - Bt( t+l'¢t) l+r a l+r For all three propositions to be true at once, it is required that nominal interest rates be identical in the two countries under consideration. Though we might expect that real interest rates are identical across countries, to assume that nominal interest rates are identical requires stronger assumptions. One possible scenario is that expected inflation rates are zero. Then nominal interest rates only reflect real interest rates. This is 27 inadequate however, since we do not observe nonpositive inflation rates in today's world. Alternatively, we might assume that the expected inflation rate in country A is identical to the expected inflation rate in country B. This would make ra = rb and hence Ft,t+l = St' To conclude, propositions (2), (3), and (H) can be expected to prevail simultaneously only under rather restrictive assumptions. These assumptions are that transactions costs are zero, market participants are risk neutral, and expected differential inflation rates are zero. Assuming away transactions costs is a simplifying assumption and not crucial to the analysis. Risk neutrality on the part of speculators may not be entirely adequate but can be ignored for the moment. However, assuming expected differential inflation rates amounting to zero is a crucial assumption. It implies subscribing to the law of one interest rate, and hence accepting the views of global monetarism.3 The crucial assumption here is that capital is perfectly substitutable across countries and hence in reality there is only one capital market. Model 2 Now let us examine the hypothesis that spot prices follow a submartingale.” Assume the expected returns are constant and reflect compensation for risk taking. Once again we shall assume no transactions costs so as to concentrate on the more important aspects of this scenario. 28 With constant expected returns Et(§t+1l¢t) ’ St _ (6) S - a t 01" E (P |¢ ) _ t t+l t (7) st - 1+a Similarly, we might expect that forward prices are settled in the same manner. That is, to compensate individuals for risk taking, today's forward price is set such that the following holds: Et(gt+ll¢t) (8) Ft,t+l = 1+8 Again, assuming that interest rate parity holds exactly: Ft,t+1 _ l+ra (9) ——§——— - I??— t b Substituting (7) and (8) into (9) yields the following: l+r (10) ——— = Hence, it is necessary that for the three propositions to hold exactly that the ratio of nominal interest rates exactly reflect the ratio of expected returns. If expected returns are a result of risk premiums, then premiums in some way exactly correspond to differential inflation rates. Model 3 Another common proposition made by proponents of the EMH is that forward prices are unbiased estimators of 29 future spot rates.5 The forward price set at time t—l, for delivery at t is an unbiased estimator of S the spot .t’ price prevailing at t. Hence, on average the following relationship holds: : -|- + (11) St a th-l,t e where a is not significantly different from zero, b is not significantly different from unity, and the 6's are serially uncorrelated. This relationship presumably is true because the forward rate set at t-l for delivery at t was based on the price expected to prevail at t given the information at hand during time period t—l. (12) F = Et_l(S t-l,t t|¢t—1) where Et-l is the expectations operator relevant for time period t-l, is the information at hand at time period cb t‘l t-l. In addition, expectations on average are unbiased estimators of actual future spot price. Suppose now that we theorize on the formation of St’ According to the EMH, it is formed based on the premise that it contains all of the information at hand in time period t and should reflect t+l's expected price. Hence, (13) st = Et(Pt+l|¢t) 30 For equality of F and St’ it must be that equations t-1,t (12) and (13) are identical or, (1n) Et_l(§t|¢t_l) = Et(St+1|¢t) An EMH proponent would argue that the expectation formed at time t—l and that one formed at t will differ only as a result of the arrival of new information. Since new information is random, the expectation at time t (and thus t's actual price) will differ from t-l's expectation by only a random variable. Hence unbiasedness of equation (11) ought to be the norm. Upon examining (1n) however, we are left with the impression that if market participants are to act in the manner prescribed above, their expectations incorporate the notion that expectations of future prices will never vary. In other words, it is assumed that prices follow a random walk with zero drift parameter (no trend). It has been established by Mandelbrot and Roll6 that a random walk of such specifications is unnecessary and thus not descriptive of foreign exchange markets. Model H An alternative version of the EMH claims that the percentage difference of today's price and tomorrow's expected price will exactly reflect differential nominal interest rates in the two countries under consideration. Hence: 31 2(3’ 1 ) s t t+l ¢t ' t _ (15) - r - r St a b Thus by rearranging we find: m (16) s = Bt(st+1l¢t) t l + ra - r b Now letting forward prices reflect expectations of the future spot price we have: ’b (17) F = E (S t,t+l t t+1|¢t) Having the interest rate parity condition satisfied as well as (16) and (17) would require that: F 1+r t,t+1 : 1 + r _ r = a St a b l+rb With exogenously determined interest rates, this could occur only by accident. Specifically, it is necessary that ra = rb. Differential inflation rates must be zero. B. Conclusions In this chapter, it has been shown that unless rather stringent assumptions are made, efficiency of foreign exchange markets has to date been inadequately described. Studies of this topic have for the most part examined specific variables in the foreign exchange market and reasoned that efficiency would require specific patterns to emerge. However, by studying for example spot prices in isolation of other variables and conditions that might be expected to hold in foreign exchange markets, it is shown 32 that inconsistancies arise. A segmented approach to the issue of the efficiency of foreign exchange markets is evidently not a proper approach. One must consider the fact that equilibrium prices of spot and forward exchange cannot be isolated from one another. If we disregard this fact, we may reason that efficiency holds regardless of the existence of "unusual" profits to be made via arbitrage. This obviously should not be the case, and ultimately contradicts the EMH. It was found however,that under special conditions the interpretations could be consistent with one another. If we ignore transactions costs and risk premiums one might expect that forward and spot prices exactly reflect tomorrow's expected price, and that the interest rate parity condition holds. We need require in addition that inflation rates be zero. If we relax the assumption that risk premiums are zero, i.e. that investors are risk neutral, we might claim that the interest rate parity condition can be fulfilled simultaneously with the require- ments of spot and forward market efficiency. This requires however that differential nominal interest rates exactly reflect risk premiums. For forward prices to be unbiased estimators of future spot rates it is necessary that spot rates follow a random walk with zero drift parameter (no trend). In order that interest rate parity holds in addition to the 33 requirement that changes in spot prices reflect differences in nominal interest rates,a special condition must be imposed. This is that differential inflation rates are zero. It is concluded that the above conditions are rather strong and often unrealistic. A case can be made for the need to seek for other approaches to the issue of efficiency in the foreign exchange market. In the chapter to follow this is accomplished by devising a general equilibrium approach to exchange rate determination which incorporates the efficient market hypothesis. CHAPTER THREE EFFICIENCY ASSUMING RISK NEUTRALITY Introduction In this chapter we eliminate the assumption that spot and forward exchange rates are set independently of each other. A model of the simultaneous determination of spot and forward foreign exchange prices is presented. Behavioral assumptions are then incorporated into the model, such that market participants behave in accordance to the efficient market hypothesis. It is found that in such a world, covered interest arbitrageurs do not have the opportunity to engage in transactions. Spot speculators arbitrage all profit opportunities away, leaving no room for covered interest arbitrageurs to participate actively in the determination of exchange rates. This is interesting in that it points to the possibility that covered interest arbitrageurs are not as prevalent in the market as is often thought. Also, it raises questions with respect to the idea that these individuals have the "last word" in regard to the determination of forward premiums and thus exchange rates. In the third section of this chapter, conditions for 3H 35 the efficient market hypothesis to hold through time are derived. It is found that market participants must formulate expectations on all future prices for the model to be consistent. It is suggested that the modeling of expectations conditional on an infinite time horizon is unrealistic and thus poses a severe problem for the proponents of the efficient market hypothesis. The reader is referred to Chapter Five for a solution to this problem. A. A Model of the Simultaneous Determination of Spot and Forward Prices In the following section, a freely fluctuating exchange rate regime is assumed. We examine the foreign exchange market as if three classes of transactions exist. These are spot transactions, forward transactions and covered interest arbitrage transactions. In the spot market we encounter traders who buy and sell spot foreign exchange so as to effectuate trade contracts due at that time. We also find spot speculators attempting to make profits from discrepancies between today's price and tomorrow's expected price. The forward market consists of hedgers and forward speculators. Hedgers are traders who having entered trade contracts due at a future date, wish to insure themselves against uncertain future exchange rates. They agree to purchase or sell foreign exchange at some future date at the price set today. Speculators in the forward market have 36 alternative motives for engaging in such transactions. When entering a forward contract they anticipate being able for example, to buy the foreign currency on the spot market at the time of maturity at a cheaper rate than that stipulated in the forward contract. Thus, the speculator expects to meet the terms of the contract while earning a positive return. A third type of transaction is conducted by covered interest arbitrageurs (CIAs). These market participants take advantage of riskless profit opportunities that may exist as a result of the relationships between spot prices, forward prices and interest rates between two countries. The model determining equilibrium exchange rates consists of four equations.1 The notion to be used is as follows: A Country A. Its unit of currency is the dollar ($) B Country B. Its unit of currency is the pound (L). St Spot exchange rate at time t. The exchange rate is defined as the number of dollars that will exchange per pound ($/L)- Ft t+1 Forward exchange rate set at time t, for ’ delivery at time t+l. E8: Excess supply of spot exchange at time1;,ES§ < 0 implies an excess demand for spot exchange at time t. ED: Excess demand for forward exchange at time t. ED; < 0 implies an excess supply of forward exchange at time t. r The nominal interest rate in country A. 37 rb The nominal interest rate in country B. l+ra rt :: (IIFE) ; a def1n1tion. t The analysis begins by positing the existence of an excess demand curve for forward exchange and an excess supply curve for spot exchange. _ f (l) Ft,t+l - f(EDt) where f(0) (2) st = g(ES:) where g(0) Ft,t+1 a -EDforESf t* 4* t t O St A). S ESt B s s .1 -ESt or EDt ‘ 0 FIGURE ONE = a> O, f'<(l : B>O,g'<0 f t f EDt S t 38 Equations (1) and (2) may be rewritten as follows, so as to explicitly include the intercept term. (3) Ft (H) St For simplicity, f + a h(EDt) S B + k(ESt) where h' < 0 where k' > 0 functions (3) and (H) may be approximated by the following linear equations: (5) Ft = a - aED: 0, a > O (6) st = 8 + cES: B, c > 0 t,t+l ED: f f f -BDtox~Est - 0 -BDt St 8 Est s s - s -EStcm)EDt . 0 *7 Est FIGURE TWO 39 Equations (5) and (6) depict the behavior of spot and forward prices when covered interest arbitrageurs do not participate in the market. Equilibrium will be attained where excess demand and excess supply are equal to zero. Thus forward and spot prices will be set at O and B when only traders, hedgers and speculators deal in foreign exchange. However, a and 8 need not necessarily be such that CIAs are without profit opportunities. Once we include CIAs as market participants the additional condition, interest rate parity must be met. The following is a derivation of the interest rate parity condition. Assume transactions costs are zero, there is no currency inconvertability risk, interest rates are fixed, there are no liquidity time preferences, and unlimited arbitrage funds exist. Then an American (resident of country A) with one dollar has two nearly risk free options to undertake if he or she chooses to place that dollar in a financial asset. The first Option is to place that dollar in a U.S. interest bearing asset. At maturity the investor will have 1+ra, principle plus return. Alterna- tively, the investor may exchange the dollar for pounds, place it in a British interest earning asset of equivalent maturity, and simultaneously exchange it for dollars in the forward market. These alternatives are depicted in Figure Three on the next page. U0 mummfi mmeHm Ann + C u. H+#.# m m “9+: H1 The two options yield identical returns when A; a = St (1+rb) F t,t+l If interest parity does not hold, if for example the return from keeping funds in the U.S. is less than the return that can be earned from placing covered funds abroad, then funds will flow out of the United States assets into British denominated assets. The resulting activity in the exchange market, exchanging dollars for spot pounds, will cause St to rise. In the forward market we would expect Ft to fall as the volume of contracts to exchange pounds for dollars rises. Thus, prices will adjust such that the excess returns from transferring funds abroad is eliminated. The interest rate parity relationship may be rewritten as follows: Ft,t+1 : (1+ra) St l+rb t l+ra Lett1ng iTF; : rt we have, F t,t+l _ (7) ——§——— — rt U2 So far the model consists of equations (5), (6), and (7). Equation (5) depicts the behavior of forward market participants. The resulting forward price is set indepen- dently of St (except for the IRP condition) and is obtained via the activities of spot speculators and traders as described by equation (6). The ratio of forward to spot prices necessary to eliminate all profit opportunities is indicated by (7). A fourth equation links the setting of prices in the spot and forward markets to the activities of CIAs and thus to each other. CIAs will demand as much forward exchange as they demand spot while pursuing arbitrage transactions. H'H) u t") (D (8) ED The complete system of four equations is depicted in Figure Four.5 In the upper quadrants excess demand and excess supply for forward and spot exchange are shown. In the lower quadrants the equilibrium curve (EC) indicates at each quantity of flow of funds from country A to B, what the ratio of forward to spot price must be. For a flow of funds of quantity X, for example, so that there be an excess supply of spot funds of this amount and an excess demand for forward funds of the equivalent amount, prices (8 ) and t . x (Ft,t+l)x must prevail. The ratio (Ft)X/(St)x is then plotted against flow of funds of quantity x in the lower (F f EDt f 4- f 4% EDt or Est EDf t -E85 0} E85 I s t t Pst t SAF it (F) t. t x """" EC (St)x 4- flow of funds from B to A FIGURE FOUR A flow Of funds from A to B uu quadrant. The supply of arbitrage funds (SAF) curve maps the interest rate parity relationship, equation (7). The SAF curve is horizontal throughout as a resultcfi’assumptions made while deriving the interest rate parity condition. These are, no currency inconvertability risk, fixed interest rates, unlimited funds and no liquidity preferences. Where EC and SAF intersect, interest rate parity holds and prices in the spot and forward market are such that enough funds are available to effectuate the transfer of covered funds from country A to B or vice versa. In Figure Two there is a positive flow of funds from A to B at E*. In order for CIAs to purchase quantity E* of pounds in the spot market, the spot price must be St*. To obtain that same quantity of forward dollars (or to be able to sell quantity E* of forward pounds), the price of forward exchange needs be Ft*. B. The Model Assuming Efficient Markets The efficient markets hypothesis states that prices are set such that no unusual profit opportunities exist. If this were not the case market participants would take advantage of the discrepancy. Thus, the activities of speculators and other investors in the foreign exchange market will ensure that prices will conform to those necessary in an efficient market. In order to introduce the efficient markets hypothesis in our model, it is useful to elaborate on the HS behavior of the various participants in the spot and forward markets. Assume for the moment that traders are the only participants. Traders, who enter contracts at time t due that same period, buy or sell spot exchange, while those that have contracts due at time t+1 hedge and thus trade in the forward market. In this case, demand and supply of foreign exchange and thus equilibrium prices, simply reflect import demand functions for the residents of the two countries. Now allow spot and forward speculators to enter the market. These participants will make inferences concerning tomorrow's spot price based on all the relevant economic information available today. If today's prices, set by spot traders and hedgers, differs from the expectation of tomorrow's price, profits will be expected from buying or selling foreign exchange. Thus trading on the part of speculators will not cease until today's price is equal to tomorrow's expected price if speculators are risk neutral. To reiterate, an initial equilibrium is set by market participants with non—speculative motives. Speculators observe this price and compare it to the expected future price formulated from all the publically available information. This expectation may be formally expressed as follows: (9) Et(st+ll¢t) where Et is the expectations operator H6 ¢ is the information set at time t. o includes all publically available information which may have an effect on foreign exchange rates such as interest rates, projected trade volumes, etc. (For rmflational simplicitity, the tilda (W), has been dropped from St. Hereafter it should be understood that Et(3t+ll¢t) is actually a random variable.) If St° or Ft°, the initial equilibrium prices, differ from the above expectation expected profits from engaging in foreign exchange transactions remain and speculators effectively do so until the expectation is attained. This implies that where excess demand for forward exchange is zero and excess supply of spot exchange is zero (the intercept terms), prices will be equal to the above expectation. The new system of equations is then: _ f (10) Ft,t+l — Et(St+1|¢t) - ant (11) s = E (s |¢ ) + cEDf ' t t t+l t t . f s (8) PDt - Est P . (7) iéjgi : rt t where (9) has been substituted in for o and B, the inter- cept terms, to establish prices where excess demand = excess supply = 0. This results because speculators will then be satisfied that all profit opportunities have been removed. For example, with regard to equation (5) a — aEDf excess demand for forward funds is zero t,t+l t’ at a. Hence a must correspond to the expectation of the F spot price at time t+l. H7 Solving for the system of four equations we find: c+l (12) st“ = Et(St+1l¢t) 515;; :': - ___C_:_:_l (13) Pt - rtEt(St+lI¢t) a+crt where * denotes final equilibrium values. Equilibrium is attained at a price that is biased with respect to expected future values. Without knowing the specific values of c, a and rt we cannot define the bias. We do observe, however that St* and Ft* will tend to overestimate the future expected spot price, given values for the slopes of the excess demand and excess supply curves, as rb rises relative to ra. Alternatively, equilibrium St and Ft tend to underestimate the expected future spot price as rb falls relative to ra. One might attempt to explain this bias in a mean-variance framework. However, we have been employing the assumption of risk neutrality. It follows then that the above equilibrium values cannot be explained as ones which compensate_ investors for incurring greater risk. An alternative explanation might be that prices are biased because CIAs are selling and buying additional amounts of forward and spot exchange to traders which are now intimaposition as a result of arbitrage activities to export and import more goods and services. However, when prices no longer equal their expected values, profitable opportunities remain for speculators to act upon and then H8 prices will once again be bid up to their expected values. But this leaves us with opportunities for CIAs once again, and so prices will once again change. Thus, equations (12) and (13) leave us with a disequilibrium situation. An equilibrium might be attained if one group exhausts its supply of funds. Since we have assumed that the supply of funds is unlimited, such a situation is not possible. No unique equilibrium exists and hence there are inherent inconsistencies in this model. The model is inconsistent in an alternative sense as well. We cannot expect forward and spot speculators to continue acting as posited in the previous system of equations. Suppose we allow (12) and (13) to remain at their initial equilibrium value. Then we can show that behavior will change over time by examing equilibrium values in time period t+l. Equations (1H) through (17) depict for the period t+l the relevant system Of equations. They are identical to t's system of equations with the exception of the substitution of an expectations term relevant to time period t+2. (1H) Ft+l t+2 = Et+1(St+2l¢t+l) ' aEDi+1 . <15) st+l = Et+1(st+2|¢t+l) + OEDi+1 (16) ED£+1 = ES:+1 (17) figliiiz = rt.1 t+1 H9 Solving for the spot price we find, SP+1 = Et+l(st+2l¢t+1) c+1 a+cr t In order for speculators to continue behaving as posited two conditions must hold. First, it must be true that expectations are met on average. If there is a consistent bias, market participants will eventually discern such and will change their behavioral patterns. The second condition which must be fulfilled is that the 'forward price be a "reasonable" predictor of the future spot price. Being a "reasonable" predictor implies that on average unbiasedness between the forward price and next period's spot price exists. If there is a consistent bias we encounter a discrepancy. In this model transactions costs have been ignored and thus cannot be used to explain a bias. Also since risk neutrality has been assumed, one cannot explain a bias as a result of risk compensation. The first condition implies that: EtCSt+ll¢t) = Si+l or _ c+l (18) Et(st+l|¢t) ‘ Et+1(st+2|¢t+l) a+crt The second condition implies that: 0+1 _ .21;— (19) rtEt(St+1|¢t) ates; ' Et+l(st+2l¢t+l) a+crt For both (18) and (19) to hold it must be that: 50 _ c+l (20) Et(St+1l¢t) ' PtEt(St+1l¢t) a+crt Equation (20) will be true only under very special condition. For example, if rt = l and a = l, (20) will be true. Requiring rt = 1 implies that ra = rb, that nominal interest rates are identical across countries. Even though it may be argued that the law of one real interest rate holds, to assume that the law of one nominal interest rate holds implies subscribing to the notion that capital is perfectly substitutable across national boundaries. This is a rather strong assumption and implies assuming global monetarism.7 If we do not agree with this view of the world and feel that expected inflation rates across countries are not identical, then markets behaving in the above manner cannot be efficient. We ought not be unduly discouraged however. The above model used behavioral assumptions as presented by most writers of efficient markets. In our system of equations the substitution of such behavior results in a serious theoretical flaw. Recall equation (11): _ S (11) st - Et(st+1l¢t' + cPst The intercept is based on the premise that spot speculators engage in pure spot speculation. These market participants buy or sell spot currency until today's price is equal to tomorrow's expected price. This is inconsistent with rational behavior. To see this, consider the following. 51 An individual in the U.S. with a dollar to speculate has an alternative Option. He or she may exchange that dollar for pounds, place it in a British interest bearing asset and then convert the principle plus return into dollars in the spot market on the maturity date. This yields an expected return on a one dollar investment equivalent to: E (s |¢ ) (21) 1 - t t+l t (1+r ) st b The expected return from one dollar of "pure" spot speculation is: Et(st+1l¢t) St (22) 1 - Equations (21) and (22) will yield identical expected returns only if rb - 0. Alternatively, if we begin with pounds, the two options are:' S t (23) l - Bt(st+ll¢ t7 (1+ra) for an investment in the U.S. securities market, and St Ettst+ll¢ty (2H) 1 - for a pure speculative transaction. Again, (23) and (2H) are equivalent only if the nominal interest rate ra is zero. The expected return from pure spot speculation can be greater only if the nominal interest rate is negative. Thus, as long as nominal interest rates are positive, speculators will never choose to pursue pure spot 52 speculation. Instead, speculators will choose between holding domestic interest earning assets and foreign interest earning assets in a manner analogous to that of covered interest arbitrageurs. Thus they ensure that "uncovered interest rate parity" is attained. The uncovered interest parity condition may be derived in a manner analagous to that of covered interest parity. We shall again assume that transactions costs are zero, there is no currency inconvertability risk, interest rates are fixed, and funds are unlimited. Observe Figure Five. This is identical to figure One with the exception that in placeof the forward price we have substituted expectations corresponding to the expected spot price on the day of maturity. Individuals choose between placing funds in domestic denominated assets and foreign denominated assets thereby setting prices such that the return from the two activities coincide. l+r —— Et(st+ll¢t) U) l+r FIGURE FIVE 53 This implies: _ 1 (25) (1+ra) - g; (1+rb) Et(st+ll¢t) OP Et(St+l|¢t) _ 1+ra _ (26) S - 1:??— - Pt t b Spot speculators will indulge in uncovered interest arbitrage so that: Et(st+ll¢t) t r (27) S t or uncovered interest parity is attained. Thus if spot speculators behaveas posited above, the uncovered interest parity condition is substituted in for the intercept term of equation (11). At this point there is no excess supply or demand for spot exchange. The excess supply curve for spot exchange is now: Et(st+1l¢t) t r + cESi t (28) S Does the same argument apply to forward speculators? Do they have the Option to commit their funds where they can earn a higher return? It may be that this is the case; however, assume that forward commitments are opportunity cost-free until the contract is due. By this we mean that contracts to buy or sell forward exchange require only a promise. No funds need be committed until the contract becomes due. 5H The system of equations under the efficient market hypothesis and under the assumption that spot speculators pursue uncovered interest arbitrage is as follows: _ f (10) Pt - Et(st+1'¢t) - aEDt E (S |¢ ) (28) st = t t+1 t + cESS r t t f _ s (8) EDt - ESt F .t (7) —— = r St t The solution is: s * : Et(St+ll¢t) t rt 3': : Ft Et(st+ll¢t) EDf‘ = BSS" = o t t Examine Figure Six: Spot speculators set the spot price such that uncovered interest parity is attained. Since forward speculators insure that the forward price is equal to next period's expectation, no covered interest arbitrage opportunities remain.9 EC intersects SAF at a zero flow of arbitrage funds. Spot and forward speculators are also satisfied since no more profit opportunities exist in these markets. A word ought to be said here concerning the actual sequence of events. If CIAs enter the market before F ‘F f {Tf % -EDt or ESt St ESS EDf t ‘t -ESS orIEDS F +8 t t _t ESt St SAF EC flow of funds from flow of funds from B to A A to B FIGURE SIX 56 speculators do, the above arguments will run in reverse. IRP will be attained and hence forward and spot speculators will not have a motive to participate. The answer to this indeterminancy lies in which set of participants has the lower threshold to begin pursuing their respective activities. This might be explained by transactions costs. Since we have assumed away these costs this cannot be used to verify that speculators begin transacting first. However, speculators pursue only one transaction, while CIAs simultaneously engage in two. Hence, it follows that speculators' transactions costs are lower than those for CIAs. It follows then, that the incorporation of transactions costs would ensure that speculators initiate trading. C. Efficient Markets Through Time Now we investigate the conditions under which the market is efficient through time. We will retain the assumption that spot speculators behave as uncovered interest arbitrageurs, and we will allow exogenously determined interest rates to differ according to the time period. The system of equations applying to period t+1 is obtained by substituting in place of period's t+l expectation, that expectation concerning period t+2 formulated 1n period t+l; Et+l(st+2l¢t+l)' _ f (29) Ft+1,t+2 ‘ Et+1(st+2'¢t+1' ' aEDt+1 57 (30) s = Et+lcst+2|¢t+l) + as8 t+1 r t+1 f _ s (31) EDt+l - ESt+l F t+1 (32) = r l+ra : 0 O + where rt+l .. (ITFE) in period t l. The solutions to this system of equations are: * - Ft+l’t+2 ' Et+i(st+2|¢t+1) s k : E(St+2|¢t+l) t+l rt+l f * _ S * - EDt+l — ESt+l - 0 Two conditions must be found to ensure market efficiency. First, we must investigate whether the forward rate is an unbiased predictor of the future spot rate. Also, expectations must be met so that market participants continue to act as posited, that spot speculators ensure that uncovered interest arbitrage is attained, that forward speculators trade until the forward rate is equal to tomorrow's expected spot rate, and that CIAs ensure that covered interest arbitrage is attained. The first condition is met if: or t+1 . > E(Rl) - E(Rz) - 0 depending on whether 81m 2 < 82 - 0. Then E(Rl) Z E(R?) or m Suppose 81m 1 82m' Et(st+ll¢t) Et(St+l|¢t) F i (l + rb) S t,t+l t (l + r ) a Rearranging, we see that this implies that: ) Ft,t+l St > (1 + r ) (l + r a .— b This is a contradiction since it would imply that pursing a risky investment alternative leads one to expect to earn at most as much as the riskless return. Thus, we must conclude that 82m > 81m and that E(R2) > E(Rl). This is interesting in that it implies that though investment alternative 1 incurs two forms of risk, while alternative 2 incurs only one, alternative 2 is in fact riskier. Presumably, by introducing forward exchange risk, part of 86 the risk associated with exchange rate uncertainty is cancelled. asset 2 moves with the market asset more than does asset l, i.e. B2m-> Blm’ Intuitively, actual R2 is subject to greater variations than is R1. In regard to a forward transaction, default (if the contract would at most imply that the investor earns only the risk-free rate of return. His or her funds have not been converted into foreign exchange and hence actual losses will only consist of the difference between expected returns and the riskless return. In regard to the spot speculator, the variability of his or her return is much greater and more closely follows the market portfolio. If actual St+l differs by much from the expected spot price, his or her returns will vary by much. If on the other hand actually St varies by little from the expected spot price, his or her returns will vary by little. These differences will be closely associated with the variability of the market portfolio. A second relationship may be derived from the above model. The return from any of the three risky alternatives must be greater than the riskless return. For example: Pt(st+1|¢t) E(R ) = (l + r ) - l > r l a Ft,t+l a so I E (S ¢ ) (l + ra) tF t+l t > 1 + ra t,t+l or Et(st+il¢t) > 1 Ft,t+1 87 This, of course implies that F < Et(S t,t+l t+1l¢t)' Observe the difference in risky alternatives 2 and H. ' E:‘lT(St‘*'ll¢’C) Ft,t+1 ) - (l-tr ) St b St E(R?) - E(R”) (l + r b [B - B l A. 2m Hm Since we have concluded that F,C < Et(St+l|¢t)’ it must be the case that the difference is positive; i.e. 82m > BHm and therefore risky investment alternative 2 is in fact riskier than H. It might be argued that one can make the analysis more appropriate by introducing margin requirements for forward exchange contracts. This is done below by subtracting the margin requirement q, from expected returns l and H. The new equations describing returns are as follows: E (s |¢ ) (38) E(Rl) = (1 + ra) tr t+1 t - (1 + q) t,t+l E (8 ¢ ) (39) E(R ) = (1 + r ) t t+l t - 1 2 b St ' Ft t+l (H0) E(R ) = (l + r ) ——i—~— H b St We know that the expected return from a risky investment alternative is in equilibrium greater than the expected return from a riskless alternative. It follows then that E(Rl) > E(R3) = ra, or: E (s |¢ ) t t+1 t (l + ra) F - (l + q) > ra. t,t+1 88 Hence: E (s |¢ ) (1 + r ) t t+1 t > 1 + r + q a a t,t+1 or (”1) Et(St+l|¢t) > 1 + I‘a + q (1 + r ) F e t,t+l a The right hand side of (H1) is greater than one, provided that transactions costs are positive. It follows then that the left hand side is greater than one. As before, t+l|¢t) > Ft,t+1 Et(S A second relationship is easily derived. This is that the expected return from pursuing investment alternative 2 is greater than that from pursuing 1. This can be proven by contradiction. Suppose this were not the case; i.e. E(Rl) : E(R2). From (38) and (39) we have: (1+rH)E |¢) (1+rH)E |¢) St+l t _ (1 ,q) 3 s St+l _ 1 F t t,t+l OI" l S ¢ ) (l-tr )E (S ) (1.,ra) F t+1 t + q i D St t+1 urn t or Ft t+l ’ (1+ra )1: (Stfllcbt) + th _>_ (1+rb)Et(St+1|¢t T or F (.2) (1+ ) + t’ “l (1 + ) Ft’t” ra q > r ——————— E tst+1|¢')— b st Examine the left hand side. From p. 87 we know that 89 F t,t+l . Ei5t+1 ra or Ft,t+1 t,t+l (l + r ) b (l + q) > ra equivalently Ft,t+1 St (H3) = (l + r ) b >l+ra+q It therefore follows that F F t t+1 t t+1 (HH) (1 + r ) ——i——— 1 + r + q ’ 47 , Since the right hand side of (HH) is even smaller than the right hand side of (H3) Equation (H2) is a contradiction. It follows then that E(R?) > E(Rl) . D. Conclusions In this chapter the Sharpe-Linter capital asset pricing model is used as an alternative to the model of Chapter Three. The CAPM allows us to account for risk aversion on the part of investors. In addition, we are able to derive final equilibrium values such that no unusual profit Opportunities remain unexploited. Hence the market complies with the conditions necessary for efficiency. 90 Three versions of the CAPM are derived. In the first instance we assume that there exists only one form of risk. Spot and forward speculative transactions are risky only with respect to uncertainty regarding tomorrow's exchange rate. We derive the following results. The forward price is a biased estimator of the expected future spot price, i.e. speculators earn a risk premium. Secondly, the expected return from engaging in spot speculation is identical to the expected return of forward speculation. Thirdly, interest rate parity holds exactly. This results from the assumption that there is no forward exchange risk; investments in the covered foreign asset are riskless. The second model assumes that forward exchange risk exists, i.e., forward contracts may not be honored as a result of political risk, for example. Hence, in this instance pursuing covered arbitrage opportunities will earn a positive expected return. Now we have three risky investment alternatives. As in the previous model, the forward rate is negatively biased with respect to the expected future spot price. Another result is that returns expected from spot transactions exceed those expected from forward transactions. Hence despite the fact that spot transactions incur one form of risk, while forward transactions incur two, the spot asset is in fact riskier. In the third model, we incorporate margin require- ments on forward contracts. The results concerning 91 relative returns remain unaffected. It was also noted in this chapter that varying degrees of risk aversion will not change the basic results. If high degrees of risk aversion are present in an aggregate sense, risk premiums will be relatively larger than if lower degrees of risk aversion are present.11 Basic relationships between differing assets remain unaffected, however. We use the results of this analysis in the following chapter where an alternative equilibrium model of exchange rate determination is developed. CHAPTER FIVE AN ALTERNATIVE MODEL OF EFFICIENT FOREIGN EXCHANGE RATES Introduction Past empirical work on the efficient market hypothesis in the context of the foreign exchange market has dealt mainly with the relationship between forward prices sit at past dates due at time t, and the spot price set at time t.1 The following equation is frequently estimated. : + + St Y 8Ft-1,t 5’ where St is the spot price prevailing at time periodtu Ft—l t is the forward price set at time t—l for 3 delivery at t. s is an error term. If Y is not significantly different from zero and B is not significantly different from unity, then thefOrward rate is an unbiased estimator of the future spot price. This implies that the foreign exchange market is efficient since no unusual profit opportunities remain to be exploited. If test results show that a bias does exist, 92 93 then it is claimed that this is a result of a risk premium and the market is still efficient exhuxa investors are being compensated for risk and thus unusual profit opportunities do not exist. Tests ofbiasedness and unbiasedness are plentiful in the literature covering efficient foreign exchange markets. We shall not prove that the above approach to testing is invalid. It is in keeping with testing for weak efficiency. In a weakly efficient market, past prices cannot be used to earn unusual profits. The above approach is testing just for that. Tests which keep more in line with the more elaborate specifications of an efficient market however, can be devised and can better explain the formulation of spot prices and consequently be better tests of the theory. This is the approach that will be used in this chapter. I Three major improvements can be made in devising test of the EMH. First, one should use an observable variable which measures expectations conditional on the information set at time t. Specifically, as opposed to regressing F on St’ one ought to use as the regressor t-l,t F When one measures expectations via the use of t,t+1° forward prices set at a time period prior to t the expecta- tion is based on a past information set. It should not be Optimal in explaining St which in theory is explainable by t's information set. The expectation formulated at t-l given the information available at that time, concerning 93 price at'B{Et_l(Stl¢t_l)}, on two accounts differs from Et(st+ll¢t)' The information sets differ, as does the period during which the expectation is formed. The second improvement which can be made in devising tests ofthe EMH is to discern the specific manner by which expectations are involved in the formulation of forward prices. These prices may measure expectations in a biased manner. If one can theoretically discern these biases they can be accounted for in the estimation process.- The third improvement involves testing for the significance of the time series of expectations. If today's spot price is based on expectations of (t+l)'s price, why should it not be that St is based on expectations of (t+2)'s price? In the study of efficient markets one ought to use expectations of prices expected to prevail at different points of time in the future. One should examine the problem as indicated below. The efficient market hypothesis claims that today's price incorporates all available information and thus should reflect expectations of prices to prevail in future periods. Hence, we can expect some form of the following relationship to hold: (1) St = .E BiEt(St+il¢t) 1-1 m where 2 Bi = 1 The betas give differential i=1 weights to each expectation, and may at some future point in time converge to zero. 95 St is the spot price at time t. E is the expected spot price at t+i given the (s i¢ ) + C O O t t l t information set at time t. If forward prices exactly reflect expectations of future prices then the observable variable, Ft t+i can be used in 3 place of the expectations term to test the hypothesis. Equation (1) would then become: (2) s = Z 3 P i t,t+i is the forward price set at time t for delivery at t+i. where Ft,t+i It may be however, that though forward prices indicate expectations they do not exactly reflect expected future prices. This may be the result of risk premiums earned by holders of forward contracts, or transactions costs. Thus, the correct form for the determination of spot prices would be derived in the following general manner. Suppose it is assumed that forward prices are determined as in (3). (3) P = Bt(st+il¢t) t,t+i x. l The term Xi’ adjust the setting of forward prices such that holders of forward contracts earn a risk premium, and/or are compensated for transactions costs. By rearranging, the following is true: (H) E (S ) = x F t t+il¢t i t,t+i 96 We have hypothesized that spot prices are determined as in equation (2). (2) s = Z (5) St = .Z BiEXiFt,t+i]' 1-1 Letting Bixi = Yi’ equation (5) can be written as follows: (6) St = .2 YiFt t+i . i=1 ’ Under different theories of the formulation of forward prices (the specific forward bias to be encountered with respect to expectations), and of the weighting scheme in the formulation of today's price, specific patterns for the coefficients xi and Bi can be expected to prevail. In this chapter we examine the biases which may exist in using current forward prices as indicators of future prices. In part B the use of many forward prices as regressors is explored. Theories concerning the weights that would be expected to prevail are develOped. Part B is followed by a digression on rational expectations which proves interesting. On the one hand, it is shown that the specification of the model in Chapter Three complies with expectations which are formed under the Rational Expecta- tions Hypothesis. On the other hand, it is shown that it 97 is desirable that one uses a time series of expectations to explain current spot prices. In Part D, the model is further improved by accounting for the fact that just as forward prices may be biased measures of expectations, expectations of future prices may be a biased measure of spot prices. A. Are Forward Prices Biased or Unbiased Estimators of ExpectatiOns? In this section we investigate under varying conditions the question of biasedness of forward prices with respect to expected future prices. Different conclusions are derived using different sets of assumptions. We begin using the results from Chapter Four regarding the setting of prices in a world conforming to the capital asset pricing model. Version I The following is assumed. I. There are margin requirements on purchases of forward exchange. 2. Final equilibrium values are attained using the CAPM. This implies the following assumptions: a. Risk averse behavior. b. Homogeneous expectations on the part of all investors. c. Perfect capital markets. d. An exogenously determined interest rate 98 pertaining to the riskless asset. Using the reSults from Chapter Four we can specify the expected return on purchases of forward contracts to be as follows: E (s l¢ ) (7) E(Rl) = (1 + ra) t, t+1 t (1 + q) t,t+1 where q is the margin requirement per unit of forward exchange contracted. To simplify notation let E(Rl) 2 cl. We know from Chapter Four that the particular value for al will be determined according to the specific conditions that exist in the market simultaneously with the expected returns from alternative investments. We can unequivacally state that with the existance of some risk, a1 is greater than ra, the riskless return. Solving for the forward rate, we find that: (l + ra) Ft,t+1 = (1 + a1 + q) Et(St+1|¢t) 01" (8) Ft’t+l = x1 Pt <8t+ll¢t) (1 + ra) Where x1 = *(1 + a, t q? Since we know that 01 must be greater than ra, and that margin requirements are positive, it must be that x1 is less than 82 > 83 and so forth. The reasoning for such a weighing scheme may be as follows. In a risk-averse world the variance of an expectation for some period further away from today is likely to be higher. Since the time lapse is longer, say between t and t+2 then between t and t+l, it is more likely that actual St+2 be more different from Et(St+2l¢t) than actual St+l be different from Et(8t+ll¢t)' Since there is a greater time difference, more new information is likely to emerge, thus changing the actual price from its expected by a greater amount. Hence individuals are likely to give less weight to expectations concerning prices further from today than those closer to today. There may be other reasons for this type of behavior to be exhibited. We shall return to this again. C. Rational Expectations In this section we shall investigate whether in the model of Chapter Three, expectations are being formulated 10H according to the rational expectations hypothesis (REH). If expectations are in fact rational, then individuals form them according to the underlying model. Individuals use the relevant theory to make predictions concerning future variables.” In Chapter Three it was concluded that equilibrium could be described by the following system of equations. _ f (16) Ft,t+l _ Et(St+ll¢t) - aEDt E (s |¢ ) + (17) s = t t 1 t + as8 t r t t f _ s (18) EDt — ES t F (19) _t_§;t_+_]_'_ : pt t where ED: is excess demand for forward exchange at time t. a is the slope Of the excess demand curve for forward exchange. E8: is excess supply for spot exchange at t. is the slope of the excess supply curve of spot exchange. - 1 + ra . . . . rt :: I-i—YM' , a definition where t pertains b to the period for which these parameters exist. ra the interest rate in country A. rb the interest rate in country B. If investors behave according to the rational expectations hypothesis, it must be the case that in forming expectations concerning next period's spot price, 105 Et(St+ll¢t)’ they use the underlying model. Thus individuals will attempt to solve the following system of equations. f Et(st+2l¢t) - a EDt+l Pt(st+2l¢t) (20) Et(Ft+l|¢t) (21) E (s [q ) = - c ESS t t+1 t Et(rt+1]¢t7 t+1 f _ s (22) EDt+l - ESt+1 (23) :t::t+l::t: : Et(rt+l'¢t) t t+1 t - 1 + r where Et(rt+ll¢t) 2 E1: Tit—F1: ¢t t+1 or the expected ratio of one plus the interest rates expected to prevail at time period t+1 for the two countries under consideration, given the available information at time t. The solution to this system of equations is: E (s |¢ ) _ t t+2 t (2H) Pt(st+1|¢t) - Et(r t+lT¢t) If (2H) is substituted into equations (16) and (17), the resulting solution to that system of equation is: E(S |¢) Et(st+ll¢t) Ptlrt+21¢t (25) St* = r = t t+1 t) t r t One should note the implications of (25). If in fact individuals are behaving according to the rational expectations hypothesis, then the solution is still not evident. Individuals must in this case attempt to solve for (t+3)'s expected price, which will also involve making an inference concerning (t+2)'s expected interest rates and 106 (t+3)'s expected price. The true solution to the actual price at time t is some function of the following form, (25) st* = gEEt(St+ml¢t); rt; Et(rt+i|¢t)l where i = l, ... ,m. The one and only expected future spot price which is relevant, is that one which pertains to the price expected to prevail at time infinity. This indeed is an absurd situation. It might be recalled that (26) is identical to that solution which was determined ought to persist if the market were to remain efficient through time from Chapter Three. In the real world it does not seem plausible that individuals formulate expectations concerning the price at time infinity. Hence, what would the determining expecta- tion be? Is it likely to correspond to the same time period for all individuals? We might venture to say that there must exist individual economic reasons for expecting individuals to differ insofar as the appropriate future determining expectation is concerned. We might for instance propose a preferred habitat theory. Individuals may based on, for instance, their liquidity preferences, the nature and timing of future obligations and their individual degree of risk aversion, choose to invest sums for particular periods of time. If individuals differ insofar as liquidity 107 preferences, obligations and degree of risk aversion are concerned, we can hypothesize that they will be most concerned with the nature of returns for different periods of time. Hence, individual investors will concern them- selves with the expected future spot price for different future periods. For the market as a whole we could expect that St* be some function of the following form: (27) S * gEEt(St+iI¢t); rt; Et(rt+i|¢t)l where i = 1,2, ... ,n. We have reasoned that within the framework of a rational expectations model, today's spot price will involve expectations concerning some time series of future expected prices. In the prior section the same conclusion resulted. However, we still need a theoretical basis for explaining exactly how to specify this type of behavior. The succeeding section which attempts to explain the specific relationship between future expectations and current prices deals with this question. D. Relationships Between Current Spot Prices and Expected Spot Prices Suppose we use the reasoning of Chapter Four and claim that the current spot price and the expected future spot price are related in the following manner: Et(St+lI¢t) St (28) E(R') = (1 + r ) b,l 108 where rb l is the return from an investment in the ’ asset of country B if that investment is made for one period of time Individuals engage in spot transactions until the expected return from a one period transaction, E(R') is as indicated. If we are dealing with a risk neutral world E(R') will be equivalent to the one period domestic riskless interest rate or ra l' Rearranging, (28) becomes 3 Et(st+ll¢t) (29) St = 1 + r a,l 1 + Pb,l because E(R') = ra 1' Note that (29) is identical to the 3 resulting spot price expected to prevailijithe model of Chapter Three. In a risk neutral world, we would expect uncovered interest parity to be the result of spot speculation. Suppose now that we assume a risk—averse world. We know then that E(R') will have to be greater than Pa,l because a spot transaction as described above is a risky venture. In a risk-averse world investors are compensated for incurring risk. Let v1 stand for the compensation in excess of the domestic interest rate for individuals engaging in such transactions for one period. That is, i _ E(R ) - ra,l + Y1 Hence: 31‘ 109 for asset equilibrium. Then the spot price at time t can be described in the following manner: (l + r1) 1) - 9 _. (30) st - (j + r + Y ) Et(St+ll¢t) a,l l (l + 1) Letting ’ = Z , equation (30) becomes: (II+ ra,l + Y1) l (31) st = ZlEt(St+ll¢t) Using the same reasoning, in a risk averse world individuals wishing to pursue spot transaction in regard to a two period time horizon will wish to set the spot price as follows, (32) st = 22Et(St+2I¢t) (l + r ) where 22 (l + rb’2 + I . a,2 Y2 ra 2 is the two period risk-free interest rate for ’ Country A r is the two period risk-free interest rate for b,2 Country B 72 is the two period compensation term for engaging in a risky venture We would expect the following relative values to be true: due to the time preference for money. Also 110 Y2 > Y1 because as the time horizon lengthens, the variance of returns or risk rises. If we assume that the term structure of interest rates in Country A follows that of Country B, then with y2 > Yl’ it might be plausible to say that 22 < 21' For instance suppose that we assume the following values: r = .10, r a,1 = .09 and Y1 = .05. Then a =.9H7. b,l 1 Now assume that in regard to the two period time horizon, interest rates are 100% greater as is compensation for risk. then ra,2 = .20, rb,2 = .18 and 72 = .10. 22 then becomes .907. Hence, 22 is less than 21' Now we are left with the task of formulating one current spot price as a function of two Operations being conducted with regard to two time horizons. Suppose we weigh the determination of today's price by the two operations equally. That is, multiply Z1 and 22 each by one half. Then we may posit that: (33) st = 81Et(8t+1l¢t) + 323t(st+2|¢t) where 81 = 2101/2 82 = Z2°1/2 As a result of equal weighing, and given that Zl > 22 it follows that Bl>'82. 111 In generalized form then, we might expect the following relationship to hold: n (3H) st = .Z BiEt(St+il¢t) 1—1 1 + r . _ b,1 l where Bi - [l + r . + Y-] [H] a,1 1 It should be noted that we are not requiring that the betas sum to one. Actual spot prices need not be an unbiased measure of expectations. Note also that we might further generalize, and conceivably let the normalization coefficient [%}, take some other form. Most probably as the time horizon lengthens, individuals may speculate with smaller sums and hence we might multiply the Z coefficient by a number [l]i where [i]i decreases as 1 increases and .§ [i] = 1. This would strengthen the hypothesis that Th: betas decline as the time horizon lengthens. In order to test the above model, it is necessary to find an appropriate measure of expectations. This can be done by using forward prices adjusted for inherent biases. Suppose we use as our model, that one presented as Version I in this chapter. It was determined that (l + ra l) Ft,t+1 : XiEt(St+1l¢t) Where X1 = (1 + a ’+ q) Ft,t+1 X1 We need also develop a theory for the determination of 1 Thus, Et(St+1I¢t) F Once its bias (x2), is discerned we can adjust t,t+2° the forward rate to account for tomorrow's expected rate. 112 Thus, in general we can test the following relationship, I) . _ l A word ought to be said here concerning the final equilibrium relationship implied by the above equation. By substituting for F in (35) we actually are implying t,t+l that the current spot price is formulated in the following manner: 11 st = 1Z1 BiEt(St+il¢t) The exchange rate is determined in an asset market manner. Frankel and Mussa5 elaborate on this basic approach to market efficiency. They define the asset market theory as one where prices are strongly influenced by expectations of future prices. In general, the asset market approach can be characterized as follows: (36) st = zt + b{Et(St+ll¢t) - st} where St is the logarithm of the spot price at time t. Zt is the ordinary factors of supply and demand that affect the exchange rate at time t. _ 1 “’ b k Et(st+jl¢t) ‘ TFEkEOEITB] Et(zt+j+kl¢t) if rational expectations are assumed. In many respects the determination of exchange rates as proposed by this thesis is similar in spirit. As in the asset approach today's exchange rate is dependent on a time series of expected future prices. However, in 113 the Frankel and Mussa formulation the time series resulted from the rational expectations assumption. In Chapter Three of this thesis it was found that such an assumption was unnecessary. Rational expectation was a natural outgrowth for consistancy of the model. Expectations will be rational if the model is be consistent, the assumption need not be added independently. Secondly, it appears that expectations closer in time carry more weight than these further away in the Frankel and Mussa formulation. This is accomplished by assuming the k coefficient follows the form [IPBJ . The coefficient declines as k increases. This is not developed theoreti- cally in the Frankel and Mussa formulation. In this thesis the coefficients have a theoretical basis for declining in value. The basis is a result of risks and the time preference for money. A third point is that as in the Frankel and Mussa formulation, a Zt+j factor exists in the model of this chapter. It is explicit in the formulation of the coefficient, Bi. E. Conclusions In this chapter, we explored ways to reinterprete the efficient market hypothesis. It was suggested that current empirical work which uses past period's expecta- tions via the use of forward prices set yesterday to explain current prices cxnflxi not adequately explain today's price. 11H A better way for explaining today's price should be found in current forward prices. In deriving this result it was also uncovered that in order to claim that last period's forward price is an estimator Of the current spot price, one needs assume that expectations are formed in a rational manner. Hence, rational expectations is a necessary condition for efficiency of markets according to the specification of such in much of the literature on foreign exchange market efficiency. Prices of forward exchange, set today for delivery tomorrow might not accurately reflect expectations of tomorrow's price. This may be a result of margin require- ments, and risk aversion on the part of market participants. By using the capital asset pricing model developed in Chapter Four we were able to take into account these varying biases. It was suggested then that today's spot price might best be explained by using a time series of expected spot prices. Such might be the case if differing groups of investors prefer to speculate with differing time horizons. A model was then developed where today's spot price is a function of varying expected future prices with biases taken into account. This model was then compared to the asset market model of exchange rate determination. It was concluded that the model developed in this chapter is superior since its economic foundations were deemed superior. CHAPTER SIX POLICY IMPLICATIONS OF THE EFFICIENT MARKET HYPOTHESIS Introduction This chapter is devoted to an examination of the policy implications of efficiency in foreign exchange markets. We examine the models of Chapters Three and Five in the context of the efficient market hypothesis. We discern the alternatives available to policymakers in regard to managing exchange rates. We find that successful intervention is possible in an efficient foreign exchange market. However, questions arise as to the desirability of intervention. Also, it is argued that though interven- tion is theoretically possible, practically speaking the implementation of these policies is questioned. In the section to follow we seek for answers to the following questions. How can policymakers alter the exchange rate so that it corresponds to something other than its "true" value? Secondly, we shall attempt to discern whether successful intervention in one period will affect exchange rates in following periods. Do policymakers need to intervene on a continuous basis, or 115 116 will an exogeneous shock to the system affect the exchange rate for a series of periods. First we shall study the effects of intervention in regard to the model of Chapter Three. Next, we investigate these same issues assuming that exchange rates are deter- mined by a time-series of expected future prices. The models of Chapter Four will not be examined since they are defined for a single time period, and hence we cannot discern long range effects of intervention. A. The Management of Exchange Rates Assuming Risk Neutrality In the model of Chapter Three the equilibrium spot exchange rate at t is: s * : Et(St+ll¢t) t rt where Et(St+1l¢t) = f[Et(St+n|¢ ), Et(rt+il¢t)] for t i = l, ... ,n if a finite time horizon is assumed. Hence it follows that today's exchange rate is some function of the following form: ), r l ), E t 5': : s gIEt(St+n|¢t t