An efficient market test of the International Monetary Market : implications for hedging strategies by Thomas William Miller A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in APPLIED ECONOMICS Montana State University © Copyright by Thomas William Miller (1979) Abstract: This research examines a problem that international business firms must face — fluctuating exchange rates. These flexible exchange rates create uncertainty as to what the exchange rate will be at some time in the future. An international firm can protect itself from adverse price fluctuations via a futures market in international currencies. But, can a firm predict the exchange rate in the future? If it could, then it can hedge at the most favorable time. The currency futures market is assumed to be an efficient market This means information about future events is contained in today's price. The objective of this research was to test the efficiency of the currency futures market. This is done by. setting up a general model of market equilibrium that specifies returns in currency futures markets is constant. Then, this model is tested by an autoregressive model that states, post returns can be used to forecast future returns It was concluded that some statistical dependence existed in the returns of currency futures markets. Whether or not this dependence could be used to reap gains in the currency futures market is the unresolved question of this research — although several strategies were tested using estimated coefficients and they did not yield gains.  STATEMENT OF PERMISSION TO COPY In presenting this thesis in partial fulfillment of .the requirements for an advanced degree at Montana State University, I agree that the Library shall make it freely available for inspection. I further agree that permission'for extensive copying of this thesis for scholarly purposes may be granted by my major professor, or, in his absence, by the Director of Libraries. It is understood that any copying or publication of this thesis for financial gain shall not be allowed without my written permission Signatu: Date ' AN EFFICIENT MARKET TEST OF THE INTERNATIONAL MONETARY MARKET: IMPLICATIONS FOR HEDGING STRATEGIES by THOMAS WILLIAM MILLER, JR. A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in APPLIED ECONOMICS Approved: Chairperson, Graduate Committee MONTANA STATE UNIVERSITY Bozeman, Montana July, 1979 iii ACKNOWLEDGEMENTS I am grateful to my major advisor. Dr. Gail L. Cramer for his guidance. Also, I am grateful to Drs. Douglas YoungDay and Oscar Burt, and to Professor.Elweyn Blodget for their advice and assistance. I must also express appreciation to the Chicago Mercantile Exchange for their aid in funding this research through a graduate fellowship. I am appreciative of the efforts of Debra Lechner for her help in gathering the data and to Leslie Fort for her typing. I am especially grateful for the programming skill and patience of Dan Chandler. Special thanks to my parents and extra special thanks to my wife, Gail, whose encouragement and support was undaunted. TABLE OF CONTENTS iv CHAPTER • PAGE V I T A ............................................... ii . ACKNOWLEDGEMENTS .............:................. . ill TABLE OF CONTENTS ...................... ........'... iv LIST OF TABLES .................................. vi LIST OF FIGURES ...........................'___ '.___ . x' ABSTRACT ..... xi S STAEMNOFAPMT . ......... ............... ............ ■' i Objectives . ...................................... 2 Procedure ........................................ 3 ,2 THEORETICAL CONSIDERATIONS ........... 5 An Introductory Perspective ...................... 5 Fixed Exchange Rates ................ Flexible Exchange Rates ....... . .... The Mechanics of an Exchange Rate Change The Uncertainty of Exchange Rates ..... Hedging . ............................... 16 3 ' EFFICIENT MARKETS .................................. 24 Forms of the Efficient Market Hypothesis ........ . 24 General Discussion of Efficient Markets .......... 25 Tests of Efficient Markets .:.................... 28 Expected Returns are Positive .................... 29 Expected Returns are Constant ............. 30 Returns Conform to the Capital Asset Pricing Model................................ 32 Returns Conform to a Risk-Return Relationship ..... 34 Methodology ...................... 35 The Model .......... .................. ........ ■ 35 The Data ....................................... 38 RESULTS ......................... '.................. 40 Comparing the Specification Models ........... 67 Developing a Prediction Model .................... 74 . 5 EXTENSION OF THE RESULTS ................. 79 The Multimarket Model .................... 107 m vo Co O1I VCHAPTER PAGE 6 SUMMARY, CONCLUSIONS, AND HEDGING IMPLICATIONS ..... 113 Summary .......................................... 113 Conclusions ............................. 116 Implications to Hedging Strategies ................ 119 Suggestions for Further Research .............. . 128 APPENDICES I ........ .......................................... 130 II ............................................. ;..... 140 . FOOTNOTES ......... ............. ,.................... ........ 183 REFERENCES .................................. '........... ..... 186 vi i LIST OF TABLES TABLE TITLE PAGE 2.1 Income Statement for NBC Some Possible Outcomes ,...., , ii 2.2 Variations in Gains or Losses Resulting from Differences in Cash and Futures Price Movements .... 15 4.1 V eIl = ^12 ^ ‘ ' • = ei5 = 0 Return Specification — Percentage Change ........... 46 4.2 V 6H ' '6U " • • • " »15 " 0 Return Specification — First Differences ........ . ,• 47 4.3 H6: Bg = Bg = - - - = Big = .0 Return Specification — Percentage Change ......... , . 49 4.4 V Sg = 3g = • . • = S15 = 0 Return Specification — First Difference ............. 50 4.5 aO= 6Z ' 63 ■ " 6IS ' 0 Return Specification — Percentage Change..... . ... 52 4.6 V S2 = B3 = . . . B15 = 0 Return Specification — First Differences .........., 53 4.7 V eI = e2 = • • • = S15 = o Return Specification — Percentage Change......... .. . 54 4.8 V 3I = e2 = ••• - = 3I5 = 0 Return Specification — First Differences....... .... 55 4.9 H : a = S- = . . . = S1C = Oo I 15 Return Specification — Percentage Change ......... . 57 4.10 H : Ot = B1 = . . . = B 1C = OO i 15 Return Specification — First Differences ......... . 58 vii TABLE TITLE PAGE 4.11 Significance of Individual Variables Return Specification — Percentage Change Holding Period — Twenty Days ...................... 60 4.12 Significance of Individual Variables 5% Level Return Specification— First Difference ■ Holding Period — Twenty Days ..................... 61 4.13 Significance of Individual Variables Return Specification — Percentage Change Holding Period — Six Days ......................... 63 4.14 Significance of Individual Variables 5% Level Return Specification — First Differences Holding Period — Six Days ......................... 64 4.15 Significance of Individual Variables Return Specification — Percentage Change Holding Period — One Day ............ .............. 65 4.16 Significance of Individual Variables 5% Level Return Specification — First Differences Holding Period — One Day .......................... 66 4.17 Testing the Intercept .............................. .70 4.18 Choosing the Return Specification ......... 71 4.19 First Order Lag — Percentage Change Returns Twenty Day Hold ................................... 76 4.20 First Order Lag — Percentage Change Returns Six Day Hold ....................................... 77 4.21 First Order Lag — Percentage Change Returns One Day Hold .............................. ■........ 78 5.1 Prediction Model Coefficients ...................... 81 5.2 Prediction Values for British Pounds Twenty Day Hold .................................. . . 82 vlii 5.2a Probability of Prediction Values for British Pounds Twenty Day Hold ............................. ....... 83 • 1 5.3 Prediction Values for German Marks One Day Hold ...................................... 84 5.3a Probability of Prediction Values for German Marks One Day Hold ................... ................... 85 5.4 Prediction Values.for German Marks ■ Six Day Hold .......... ........................ . 86 5.4a Probability of Prediction Values for German Marks Six Day Hold .............. ....................... ' 87 5.5 Prediction Values for Japanese Yen One Day Hold ........;.......................... 88 5.5a ■ Probability of Prediction Values for Japanese Yen One Day H o l d ......................................... 89 5.6 Prediction Values for Japanese Yen Six Day Hold. ... . .................................. . 90 5.6a Probability of Prediction Values for Japanese Yen Six Day Hold ......................................... 91 5.7 Prediction Values for Swiss Francs One Day Hold ......................... 92 5.7a Probability of Prediction Values for Swiss Francs One Day Hold ........ ............................ 93 5.8 British Pounds — Twenty Day Hold ..................... 99 5.9 German Marks — One Day Hold ....................... 100 5.10 German Marks — Six Day Hold ........................ 101 5.11 Japanese Yen — One Day Hold ....................... 102 5.12 Japanese Yen — Six Day Hold ........... 103 TABLE TITLE PAGE 5.13 Swiss Francs — One Day Hold ....................... 104 5.14 Summary of Strategies ................... .......... 105 5.15 Estimated Multimarket Model Holding Period — One Day N = 1609 .......... ................................ 109 5.16 Estimated Multimarket Model Holding Period — Six Days N = .260 ............................................ H O 5.17 Estimated Multimarket Model Holding Period -- Twenty Days N = 76 ..................... ....................... H l 6.1 British Pounds — Standard Deviation of Basis and Spot Prices .............................. 120 6.2 Canadian Dollars — Standard Deviation of Basis and Spot Prices ................. ............ 121 6.3 ' German Marks — Standard Deviation of Basis and Spot Prices ...................... ....... 122 6.4 Japanese Yen — Standard Deviation of Basis and Spot Prices . ............................. 123 6.5 Swiss Francs — Standard Deviation of Basis and Spot Prices .............................. 124 6.6 Average Basis During Delivery Month — 1972-1978 .... 126 ix TABLE TITLE PAGE XLIST OF FIGURES FIGURE TITLE PAGE 2-1 SUPPLY AND DEMAND CURVES FOR D-MARKS IN U.S......... 7 2-2 SHIFT IN THE FOREIGN EXCHANGE- RATE ............'.... ' ' 9 2- 3 THE RELATIONSHIP BETWEEN THE EXCHANGE RATE AND PROFIT BEFORE TAXES ............................. 10 ' 2-4 LONG HEDGE IN CURRENCY FUTURES MARKET .......... '--- 12 3- 1 CAPM RISK-RETURN RELATIONSHIP ........................ 34 5-1 STANDARD NORMAL DISTRIBUTION ......................... 96 xi ABSTRACT This research examines a problem that international business firms must face — fluctuating exchange rates. These flexible exchange rates create uncertainty as to what the exchange rate will be at some time in the future. An international firm can protect itself from adverse price fluctuations via a futures market in international currencies. But, can a firm predict the exchange rate in the future? If it could, then it can hedge at the most favorable time. The currency futures market is assumed to be an efficient market This means information about future events is contained in today's price. The objective of this research was to test the efficiency of the currency futures market. This is done by. setting up a general model of market equilibrium that specifies returns in currency futures markets is constant. Then, this model is tested by an autoregressive model that states, post returns can be used to forecast future returns It was concluded that some statistical dependence existed in the returns of currency futures markets. Whether or not this dependence could be used to reap gains in the currency futures market is the unresolved question of this research — although several strategies were tested using estimated coefficients and they did not yield gains. Chapter I Introduction A foreign exchange market facilitates trade between nations by providing a market where currencies can be exchanged. In this market money is treated like any other commodity — it is bought and sold at a price. This market is known as a cash, or spot, market — that is, this market exchanges dollars for other currencies. The price in this cash, or spot, market is called the exchange rate. If a firm in the United States is doing business internationally, it must realize that the exchange rates between the dollar and other currencies are constantly changing. Because of these constant changes in the exchange rate, a firm does not know what the exchange rate will be at some time in the future. The exchange rate at some future date could be such that the international firm is adversely affected. A firm can reduce the risk of adverse price fluctuations via a market where future exchange rates are bought and sold. This market is known as a futures market. Unlike the cash — or spot -- market, the futures market deals with exchange rates in the future: whereas the cash market deals with exchange rates that prevail at present. The futures market in international currencies.is the International Monetary Market division of the Chicago Mercantile Exchange. As in other futures markets, a firm can "lock-in" a price (an exchange rate) by purchasing the exchange rate today. This process, used by 2importing firms, is known as a long hedge and its basic elements are discussed in the next chapter. The existence of a futures market in foreign exchange provides firms with a vehicle to reduce the risk of unfavorable exchange rate changes. An unfavorable exchange rate change for an importing firm would be an increase in the exchange rate for a particular currency — that is, if it took more dollars to buy the same number of Yen, Francs, or D-marks. The next chapter of this paper discusses this problem in detail. Obviously, a firm would like to hedge or "lock-in" a price for a foreign currency at the "most favorable" time. For example, an importing firm would like to purchase the foreign currency necessary to complete international transactions at the lowest price possible. Then, it would like to "lock-in" this price on the futures market. However, can a firm determine when a price is the lowest possible price? If it can, then this implies that a firm can predict prices in the futures market. Objectives The primary objective of this study is to test the efficiency of five foreign currency futures markets on the International Monetary Market. These five currencies are: British Pound, Canadian Dollar, German Mark, Japanese Yen, and Swiss Franc. 3As in the case of most auction markets, the foreign currency I futures market is generally thought of as an efficient market. In short, this is "generally understood" to mean that "all available" information concerning the price of a currency in the future is "fully reflected" in the price quoted at any time for the currency in the future. The statement that the current price of a futures contract "fully reflects" all market information implies that successive price changes, or successive returns, are independent. If this is the case, then a firm (or speculator) could not predict the price (or return) in the futures based upon previous price changes (or returns). This implies that a firm trying to decide when the "most favorable" time to place a hedge faces a difficult task. Procedure The efficient market hypothesis states that, in short, the pre­ diction of futures prices is impossible. The reason for this, accord­ ing to the efficient market hypothesis, is that all information con­ cerning the price of a foreign currency in the future is already reflected in the current price of that foreign currency. Chapter Three of this study discusses the rudimentary elements of the theory of efficient markets. Tests of efficient markets are concerned with the effects of information upon market prices. The information set used in this 4study is the set of past prices in the various currency futures markets. This information set is tested with an autocorrelation model developed in Chapter Three. Basically, this model tests the I degree of statistical dependence between past returns in the futures market and present returns. The results of this model are presented in Chapter Four while Chapter Five presents the results of trying to predict futures prices. Chapter Six contains the study summary, the conclusions, implications to hedgers, as well as suggestions for further research. THEORETICAL CONSIDERATIONS An Introductory Perspective An open economy is an economy that trades with other economies, As no country’s economy in the world is self-sufficient, every country's economy can be classified as open. Trade between economies takes place in both goods and services — Americans buy German cars and Europeans buy American brokers' services — and in assets — Americans buy Japanese stocks and Arabs buy American real estate. To facilitate this trade between nations, there must be a market in which, for example,, an American can trade dollars for another currency in order to make foreign transactions. The market where these currency transactions take place is the foreign exchange market. Simply stated, the conversion of one money into another is the sale of one currency for another. In this market money is treated like any other commodity — it is bought and sold at a price. This price is called the foreign exchange rate. There are two basic foreign exchange rate systems. One of these is the fixed rate system and the other is a flexible or floating system. Fixed Exchange Rates The world was basically on a fixed rate system from 1946 to 1973, though there were occasional adjustments of exchange rates during that period. In the fixed rate system foreign central banks fix the Chapter 2 6 ' dollar price of their currencies, and stand ready to buy and sell U.S. dollars at that price. This means the central bsnk has to finance the excess demands for, or excess supplies of, U.S, dollars. (That . is, the balance of payments deficits or surpluses.I The central banks do this by running down or adding to their reserves of U.S. dollars. Flexible Exchange Rates During 1973, exchange rates between the dollar and other currencies were allowed to float. This is the current system of exchange rates. In this system the exchange rate is determined by the supply of and demand for foreign exchange. Under the flexible ex­ change rate system the demands for, and supplies of, foreign currency can be made equal through movements in exchange rates. Under "clean" floating there is no central bank intervention and the balance of payments is zero. However, central banks sometimes intervene in a floating rate system — this is engaging in so-called "dirty" float. By no means is floating a universal practice. Only a small minority of the world's currencies float, yet these currencies are the major currencies in the world; they account for seventy percent of world trade. Of 122 member nations of the International Monetary Fund, 18 nations allow their currency to float (seven of these were 2 tied together in a joint float). The other 104 were pegged to the dollar, pound, mark, or some other currency. It must be remembered. 7however, that when a currency is pegged to the dollar, it will float along with the dollar vis-a-vis other currencies. Figure 2-1 is a representation of how the foreign exchange rate is determined. S (D-mark out-payments) D (D-mark in-payments) Quantity Figure 2-1. Supply and Demand Curves for D-marks in U.S. The demand for D-marks in the U.S. (D-mark outpayments) repre­ sents such things as: a) Purchasing German^ goods (Imports) b) Buying German services c) Transfer of capital to Germany for investment The supply of D-marks in the U.S. (D-mark outpayments) repre­ sents such things as: a) Selling domestic goods to Germany (exports) b) Selling domestic service to Germany c) inflow of German capital The prediction of an exchange rate is very difficult. The supply of and the demand for foreign currencies depends on economic condi­ tions and policies relative to the rest of the world. Anything that 8affects prices, incomes, interest rates, consumer tastes and prefer­ ences, or any other economic variable at home or abroad influences the position of the supply and demand curves for foreign exchange and therefore affects the exchange rate. The exchange rate is deeply rooted in the economic conditions of the countries involved. It does bear emphasizing that in all cases it is the changes inside the country relative to changes in the rest of the world that counts. The Mechanics of an Exchange Rate Change Assume an exogenous change, in American tastes toward lederhosen (leather walking shorts). Further, assume that I) there are no arti­ ficial barriers to trade in this market and 2) Germany is the only exporter of lederhosen. (Of course, this is to keep, within the two- country framework.) What will happen to the foreign exchange rate [ between Germany and the United States, ceteris paribus? ; Figure 2-2 shows this change. The Americans must pay the German producers in Deutschemarks (D-marks). Americans must convert their dollars to D-marks, therefore there will be a shift in the demand for D-marks in the United States. The exchange rate, the price of one D-mark in terms of dollars, will increase due to the change in consumer tastes that stimulated imports of lederhosen. Note that this means a dollar will buy fewer D-marks. When this happens, the dollar has "weakened" or depreciated 9relative to the D-mark. Or, the D-mark has "strengthened" or appre­ ciated relative to the dollar. Price of one DM in terms of dollars Quantity Figure 2-2. Shift in the Foreign Exchange Rate The Uncertainty of Exchange Rates The increase in the exchange rate in the preceding example will affect all firms dealing with Germany. If a firm is doing inter­ national business it must realize that it is doing so in a system of exchange rates that are constantly changing. What effect do fluctuating exchange rates have on international business firms? Assume an importing firm — National Blivit Company. This firm imports blivits from Germany for resale in the United States. Also, assume that there is a rather constant demand for blivits in the United States market. Further assume NBC signs a contract with a 10 German blivit producer under the following conditions: 10,000 blivits delivered in 90 days to the United States, payment in D-marks, at a price of 12.50 DM/blivit. At the time of the contract signing the exchange rate was $.50/1 DM. Situation (I) in table 2.1 is a pro forma income statement with the expected profit in 90 days represented at the bottom line. If the exchange rate happens to be $.50/1 DM, the firm will realize this profit. This situation is depicted in (2) in table 2.1. Suppose, however, the exchange rate in 90 days is $.55/1 DM. This situation is shown in (3) in table 2.1. Note that the profit before taxes is lower. In general, the relationship between the exchange rate and profit before taxes can be shown as: Exchange rate Profit Figure 2-3. The Relationship Between the Exchange Rate and Profit Before Taxes. 11 Table 2.1 Income Statement for NBC Some Possible Outcomes At the Exchange Exchange Exchange time the rate is the rate is rate is lower contract same (.50) higher (.55) (.45) in was signed in 90 days in 90 days 90 days ill 121 ill ill Sale of 100,000 10,000 blivits (at $10/blivit) 100,000 100,000 100,000 Cost of goods sold (at 12.50) DM/blivit) 62,500 62,500 68,750 56,250 Gross Profit from Sales 37,500 37,500 31,250 43,750 EXPENSES 20,000 20,000 20,000 20,000 PROFIT BEFORE TAXES - 17,500 17,500 11,250 23,750 12 Figure 2-3 shows that when.the exchange rate is higher (i.e. if it takes more dollars to buy the same amount of D-marks), profits before taxes will be lower. The National Blivit Company can protect itself from this adverse price movement by a type of insurance called a hedge. The firm must buy D-marks in 90 days to pay for the imported blivits. To insure a favorable price, the firm could enter the futures market today and buy the D-marks. This transaction, known as a long hedge, is shown in Figure 2-4. SPOT MARKET FUTURES MARKET NOW BUY* 90 days from now BUY SELL Figure 2-4. Long Hedge in Currency Futures Market In order to do this all a firm has to do is call a commodity broker and inform him of their desire to purchase a D-mark futures contract. (Specifications of IMM futures contracts, can be found in . Appendix I). The firm must issue a performance bond, called margin, which is kept in the firm’s account with a broker. Margin must be put up to show good faith in meeting contract specifications. At this time the firm is long in the futures market. To close their position the firm calls the broker and informs him of their desire to sell a 13 contract. This action concludes what is known as a round-turn. At this time the firm must pay the broker a commission — usually . amounting to $45-$55. Note that the transactions 90 days from now, more or less, cancel each other out. The firm, then, has effectively "locked in" a price for the D-marks now. (* above) The firm would then be protected against an adverse price change, i.e. the dollar price of Deutschemarks increasing. The transactions labeled "90 days from now" do not cancel each other out precisely. This is because the price on the spot market and the price on the futures market might be different, at that point, in time. The difference between, the futures price and the spot price is known as the basis. In order for a hedge to eliminate price risk, the futures price, must move precisely parallel with the cash price — which it rarely does. In the context of this research, the basis is the difference between the futures market exchange rate and the cash, or spot, exchange rate. In actuality, the basis, tends toward zero as the delivery day for the futures contract approaches. This is reasonable because, in the end, the futures delivery becomes a spot delivery. They become nearly perfect substitutes for each other, hence they should have equal value. This principle is very important in a futures market, as it is the essence of the use of the futures for . , I 14 forward pricing, or fixing costs in advance. In the foreign currency futures market the basis primarily reflects I) the interest rate differential among countries and 2) expectations. If stable conditions exist and there are no restrictions for trade and capital flow, forward rates will vary inversely with the interest rate differential between two countries. For example, if interest rates are 10 percent higher in Japan than in the United States, the forward price for Japanese yen should be at a 10 percent discount in terras of United States dollars. This is known as interest-rate parity: seldom is it achieved because world monetary conditions have been relatively unstable, and barriers to free trade and capital flow exist. Consequently, expectations have played a large role in the foreign exchange basis. As table 2.2 indicates, basis movements can earn a firm either a profit or a loss: even if the firm is hedged or unhedged — long in the cash (or spot) market or short in the cash market. (See .Figure 2-4) Note that this profit or loss is either added to or sub­ tracted from the profit a firm "locked-in" by hedging. This is because a firm will enter the futures market to guarantee itself a price, a price that will earn a profit for the firm. Basis movements can either add to or reduce this profit. What if the price of Deutschemarks had, in fact, gone down? The 15 Table 2.2 Variations in Gains or Losses Cash and Futures Resulting from Differences in Price Movements5 Price Movements Results to one who is "long" to one who is "short" in the Cash Market in the Cash Market Cash Price Future' Pr ice Unhedged Falls Falls by same amount as cash Loss Falls Falls by greater amount than cash Loss Falls Falls by smaller amount than cash Loss Falls Rises Loss Rises Rises by same amount as cash Profit Rises Rises by greater amount than cash Profit Rises Rises by smaller amount than cash Profit Rises Falls Profit Hedged Unhedged Hedged Neither profit nor loss Profit Neither profit nor loss Profit Profit Loss Loss but smaller than unhedged loss Profit Profit but smaller than unhedged profit Loss but greater than unhedged loss Profit Profit but greater than unhedged profit Neither profit nor loss Loss Neither profit nor loss Loss Loss Profit Profit but smaller than unhedged profit Loss Loss but smaller than unhedged loss Profit but greater than unhedged profit Loss Loss but greater than unhedged loss 16 effect on the firm's profit is shown in (4) of table 2.1. The firm's profit will be greater than expected. If the firm had placed a hedge, it would have done so needlessly because the price of D-marks had gone down. In fact, the firm would have been better off profit-wise if it had not hedged. But — this is an ex post realization, as the firm did not know whether the price of Deutschemarks would be higher or lower than it is now in 90 days. Could the firm have predicted these price movements? If it could, then it could devise hedging strategies to place hedges at the most favorable time. However, it is the contention of this research that a hedging strategy is a risk reduction tool. Trying to earn additional revenues from the movement of foreign exchange rates should be a secondary consideration. An exporting or importing firm knows it can make money by export­ ing or importing goods. Such a firm should realize it faces uncertainty in the foreign exchange market and use futures markets to reduce this risk. Hedging The predominant role of the futures market has been, and is today, facilitating hedging. In reviewing the development of hedging. Gray and Rutledge (1971) categorized hedging into four classes. They concede that classification is difficult, but each of their 17 categories represent a different set of motives from the hedger's point of view. The first class of hedging is that done to eliminate price fluctuations — a,n archaic view of hedging. Early in the century Hardy and Lyon (.1923) spoke of risk elimination. However, since cash and futures prices do not move strictly parallel, the risk of price fluctuations cannot be eliminated. The second class of hedging under Gray and Rutledge's classification is the reduction of risk connected with price fluctuation. Gruen (I960) pointed out that the effective usefulness of a futures contract depends upon the relationship between the futures and cash price. A third type of hedging is hedging to profit from basis move­ ments. The basis is the difference between the futures price of a commodity and its price on the cash market. Working (1953), using wheat price data, said that basis fluctuations are predictable. In effect, Working was saying that hedging is undertaken and inspired by the profit motive rather than solely as a means of reducing risk. Risk reduction gained through hedging is an advantage but, according to Working, it is usually not the factor that instigates the hedge. By being able to predict basis movements, agents in the market hope to profit from changes in the basis. Brogan (19.771 tried to predict the Great Falls, Montana winter wheat basis on two grain exchanges, with generally negative results. 18 Working also talked about the ideas of selective hedging and anticipatory hedging. These types of hedging are done in response to expected prices. An example of anticipatory hedging would be a cattle feeder who long hedges grain for cattle he has yet to place in the feedlot but anticipates he will have the animals in the feed- lot. Selective hedging is the placement of a hedge when the market participant expects prices to move against him. The remaining classification of hedging is hedging that is done J to maximize expected returns for a given risk (variability of returns) or minimize risk for a given expected return. This view of hedging has been studied from the viewpoint of portfolio theory by Steing (1961). Billings (1978) found that cattle feeders in Montana could have increased returns or reduced risk by judicious use of feeder and fat cattle hedges. The International Monetary Market of the Chicago Mercantile Exchange was conceived as a futures market specializing in monetary vehicles whether they be currencies, precious metals (including gold), or other financial instruments such as United States Treasury Bills. The International Monetary Market began trading in foreign currencies on May 16, 1972. There was already a world-wide forward currency market maintained by the largest banks, but it catered primarily to multinational businesses and a few wealthy individuals. In that market dealings usually start at $1 million. On the other end,. 19 brokers had been critical of the International Commercial Exchange because the contracts were so small.that they had little appeal to larger hedgers. The International Monetary Market was aimed to split the middle.^ Most banks that deal in foreign exchange can, enter a contract to buy or sell any amount of a currency at any date in the future, i.e., for forward delivery to its customers. Overall, the modern foreign exchange market has been beneficial in promoting world trade. ^ Higinbotham and Telser (1977) state that the benefits of an organized futures market are an increasing function of the number of potential participants. Baron (1976) points out that forward exchange markets or their equivalents permit investors to perform their own covering of the uncertain returns from share ownership, and consequently, firms are able to plan their outputs using the forward exchange rate as a planning equivalent. There is indeed a new literature developing which focuses on the effects of an exchange rate on futures cash flows, that is, the degree to which the present value of a firm is affected by a given exchange rate change.^ The major conclusions are that the sector of the economy a firm operates in (exports, imports, or pure domestic) and the sources of its inputs Cimports or domestics) are important in delineating a q firm’s true economic exposure to exchange rate fluctuations. 20 The issue addressed in this research is the predictability of exchange rates and its implications to hedging strategies. If firms . ' I would be hedging selectively, i.e., hedging when exchange rates are expected to move against them. Conceivably, firms might also be hedging to profit from basis changes, implying that risk reduction motives are superceded by profit motives. The issue of whether futures prices are predictable or not,has been a long standing argument between academicians and traders. The traders, broken into two basic groups, argue that futures prices are, in fact, predictable — at least to a certain degree. These two groups agree in philosophy but disagree in practice. One group, the fundamentalists, believe that the fundamental laws of supply and demand underlying the futures market are responsible for the price formation process in the futures market. The fundamenta­ lists then claim they are able to predict futures prices via their ability to forecast shifts in supply and demand. The other group, the technicians, doesn’t discredit the process by which the fundamentalists predict. However, they point out that this information.is in past price data and that future futures prices can be predicted via past futures market price data. The academicians argue that all new information regarding the commodity occurs randomly. If this is the case there is no way to predict a change in the price because of the random nature of new 21 information being released. Also, argue the academicians, this new information is quickly absorbed by the market. Since prices respond to information so quickly, there will be no possibility to observe price movements and take advantage of these price movements through any sort of mechanical trading rules. The academicians argue, in essence, if you expect the price to be higher tomorrow, it will be higher today. (Samuelson, 1965) It is interesting to note that Carl Menger is reported to have made careful studies of the Vienna Stock Market before, in.1871, Kdeveloping the concept of marginal utility and his theory of prices. However, the take off point for most arguments concerning the random nature of price series' usually begins with the pioneering work of the French mathematician, M. L. Bachelier. BacheHer, in his Theorie de la Speculation (1900), elegantly derived the theory of the random character of stock market prices. Basically, Bachelier's argument took this form: (D' Pt = V l + et Equation (I) says that the price in time t is equal, to the price at time t-1 plus a.random element, Efc. Bachelier also argued that RIC ) = 0. The logic behind this goes as follows. If in fact t E(et) > 0, traders would buy now, assuming they could always sell later.at a higher price at a later date. However, this buying 22 ■ Because Bachelier's work was so advanced for its time, it took the rest of the academic world years to follow up on his outstanding work. Economists have applied his theory to price series and have had mixed results. Kendall (1963) and Alexander (1961) both concluded that price changes were random, while Carson (1960) found that prices change following a moving average process. Samuelson (1965) elegantly demonstrated that properly anticipated prices move randomly. However, Smidt (1965) and Stevenson and Bear (19.70) rejected the random walk as an explanation of futures price behavior. Holbrook Working (1958) offered another explanation of the move­ ment of futures prices. His model tried to explain a gradual effect of information on futures prices. The random element, Et, in any period is affected not only by the new information but also by the series of past information, according to Working. Working’s theory can be put into the framework of time series analysis according to the following finite random shock .formula:"*""*" • <« ’ Zt - Dt + fIuC-I + ?2Ut_2 + ' • • • + V t - n ■ where Z is the price change and the U .’s are random variables which t 1 explain the process of price formation through hew development and activity at time t-1 would force the t-1 price up to the level where E(e^) =0. A similar argument could be constructed if E(e^) < 0. new information. 23 To apply time series analysis it is necessary to assume that the "I2t-I + V t - 2 + ' + TT Z + E P t-p t IL 's are independently, identically distributed. With this assumption, the process becomes a linear discrete stochastic process (Nelson, 1973). Equation (2) can be approximated as an autoregressive process of order P as follows: (3) . Zt where the APY 1 s are functions of the Y^'s in (2) above and the Zfc^ ' s are previous observations. Equation (3) is the fundamental form of an' equation that will be developed in chapter three that will be used to test the efficiency of currency futures markets on the International Monetary Market. Expression (3) is a test of the efficient market theory. In the next chapter the rudimentary elements of the efficient market theory is discussed as well as the model used to test the efficiency of currency futures markets on the International Monetary Market. Chapter 3 EFFICIENT MARKETS Forms of the Efficient IIarket Hypothesis I Market prices react to supply and demand conditions. The effi­ cient market hypothesis attempts to explain some factors affecting supply and demand. There are three forms of the efficient market hypothesis: weak form, semistrong form, and strong form. All describe the effects of information, they differ in degree of the information set. The weak form says that markets are efficient enough to make the study of past prices a worthless way of trying to determine future prices. The information set in the weak form test is the series of past prices. The semistrong form says that as soon as information affecting the market becomes public, it is instantaneously and accu­ rately taken into account by the market. The strong form says that these adjustments are not predictable enough to permit superior re­ turns. The strong form test says all information, not just public information is taken into account and the market prices move accord­ ingly. The form of the efficient market hypothesis tested in this study is the weak form test. It is appropriate here to present 12 general discussion of efficient markets. a 25 General Discussion of Efficient Markets An efficient market is a market that processes information "correctly". The prices observed at any time in the market are based on "correct" evaluation of all information available at that time.. In an efficient market, prices "fully reflect" available information. (Fama. 1970) Information includes not only such things as hard data, factual reports, or informed opinion, but all elements affecting a market — such as misinterpretation, misinformation, fads, trends, and faulty expectations. (Jones, Tuttle, Heaton, 1977) The weak form test of the efficient market hypothesis assumes such factors are reflected by past prices. To start the discussion of efficient markets, assume that all events of interest take place at discrete points in time — t-1, t, 13t+1, etc. Then define the following: (J)fc ^— the set of information available at time t-1, which is relevant for determining currency futures prices at t-1. t is a subset of (J)^ 1 . P, — price of currency j in dollars at time t-1; J'^-1 j=l,2,..., n where n is the number of currencies. fm,A ^available at time t-1 includes what might be called the "state of the world" at time t-1: e.g. current and past values of any relevant variables, like wages, prices, and "political climate", the tastes of consumers and investors, etc. Since included the past history of all relevant variables, includes 6 ; equivalently, is a subset of ck •, (for n > I). In.t“n t-n L-JL addition to current and past values of relevant variables, is also assumed to include whatever is knowable about relationships among variables. This includes relationships among current and past values of the same or different variables, and also whatever can be predicted from future "states of world" from the current state. In short, , the information available at t-1, includes not only the state of the world at t-1, but also what is knowable about the process that describes the evolution of the world through time. (Fama, 1976) One of the things that is assumed to be knowable about the process is the implication of the current state of the world for the joint probability distributions of exchange rates at future times. Thus, is assumed to imply the joint density functions f^ .(P^ . , 5 Pn,t+A I' *t-V; A 0, I, 2, . . . . The process of price formation at time t-1 is then assumed to be 27 as follows. On the basis of the information ™ , the market assesses a joint distribution of exchange rates for time t, , ^nt I t^t-I ^ w i^ere ^it = ^it* ^2t ’ **’’ ^nV From this assessment of the distribution of exchange rates at t, the market then determines appro­ priate current exchange rates, ........ P^ for individual currencies. The appropriate current exchange rates are determined by some model of market equilibrium — that is, by a model that determines what equilibrium market exchange rates should be on the basis of 14characteristics of the joint distribution of exchange rates at t. (Fama, 1976) A market equilibrium at time t-1 is achieved when the market sets exchange rates P^ t_^, . . . , nE for individual • currencies at which, the demand for each currency is equal to the supply of the currency. In other words, a market equilibrium implies a market-clearing set of exchange rates for all currencies. If the currency futures market is hypothesized to be efficient (1> +:-! = +t_l ' That is, the information used by the market to determine futures . prices is equal to all the information available. Having correctly assessed the joint distribution of exchange rates for t, the market then uses some model of equilibrium to set exchange rates at t-1. The model says what the current exchange rate of curren­ cies, npr t„j, . . . , Pn t_-p should be in the light of the correctly 28 15assessed joint distribution of exchange rates for t. (Fama9 1976) Tests of market efficiency are concerned with whether or not the market does correctly use available information in setting prices. Most common are tests that try to determine whether prices fully reflect specific subsets of information. For.example, one possible source of information about future currency futures prices is the ' history of past currency futures prices. A nontrivial segment of the empirical literature of efficient markets is concerned with whether current prices fully reflect any information in past prices. ,(Fama9 1976) This has a major empirical implication — it rules out the possibility of trading systems based only on information in past prices that have expected profits or returns in excess of equilibrium expected profits or returns, (Fama9 1970). » Tests of Efficient Markets The primary objective of this study is to test the efficiency of foreign currency futures markets on the International Monetary Market. To test the efficiency of a market, a general model of market equili­ brium must be devised. It is against this model that efficient market . ’ tests are conducted. There are four basic types of these market models and each will be briefly presented. 29 Expected Returns are Positive The first model of market equilibrium says that at any time t-1 the market sets the price of any security j in such a way that the market's expected return on the security from time t-1 to time t is positive. Therefore, let (2) Rj,t pj,t pj Pi, t-1 t-1 where: t - the rate of return in currency futures market j during some * time period t-1 to t. P . - the futures market price of currency j at time t.i, t P 1 1 - the futures market price of currency j at time t-1. 3 > t-l This model of market equilibrium says that the market always sets Pj so that the mean of the resulting distribution of Rjt is strictly positive. (Fama, 1976) That is, (3) Em j t-1 > 0 Expression (3) states that the market sets the price of security j at time t-1 such that the true expected return on security j is positive. In other words, the market has correctly assessed all available information at time t-1 to set the price of security j such that its return from time t-1 to time t is positive. Note that if the expected return on D-marks in terms of dollars is positive, then 30 the expeqted return on dollars in terms of D-marks is negative. .Filter tests are the only tests of market efficiency which test the general equilibrium market model that assumes returns are positive (Fama, 1976) A filter test is a type of trading rule that says once a security has started to rise it will continue to rise; once a security has started to fall it will continue to fall. The basic trading rule of a filter test is: if the price of a security moves up at least x percent, buy and hold the security until its price moves down at least x percent from a subsequent high, at which time simultaneously sell and go short. The short position is maintained until the price rises at least x percent above a subsequent low, at which time the short position is covered and a long position is established. Moves less than x percent in either direction are ignored. Such a system is' called an x percent filter. (Stevenson and Bear, 1970) Stevenson and Bear (1970) found that, in fact, filter rules could be successfully employed in corn and soybeans futures markets. Alex­ ander (1961), and Fama and Blume (1966) conclude the filter rules cannot, for the most part, beat the simple strategy of buy and hold. Expected Returns are Constant Another type of test is based on a model in which the expected return is assumed to be constant through time. Specifically, this 31 model says that at any time t-1 the market sets the price of any security j in such a way that the market's expected return on the security from time t-1 to time t is constant. This model of market equilibrium says that the market always sets if\ ^^ so that the mean of the resulting distribution of R. is constant. (Fama, 1976) That is, <4) V V I C 1)- ' ^ t-1 pj,t-i E(Rj) This return, Rj, is constant through time, according to the model. Note that the model says that the return is constant, which implies that the return, Rj, could be positive, negative, or zero. Equation (4) implies that for all E(Rj 11 t_^) , the regres­ sion function of Rjt on t_i is the constant E(Rj). This means, that if the return, Rj is regressed on any of the elements from the set of information available at t-1, all the coefficients except for the intercept should be. indistinguishable from zero. If some of the variables have nonzero coefficients, (4) must be rejected. That is, • the joint hypothesis that the market is efficient and that it sets prices so that equilibrium expected returns are constant through time is rejected. (Fama, 1976) Stevenson and Bear (1970) tested against this type of model using serial correlation coefficients. They concluded.that, in corn 32 and soybean futures markets, the information in past returns could be used to predict future returns. Larson (1960) also concluded that there is a discernable pattern to commodity futures prices. Returns Conform to the Capital Asset Pricing Model A third type of general equilibrium market model that is tested is the capital asset pricing model. Sharpe (1964) originally proposed a model of the equilibrium pricing of portfolio assets. Lintner (1965), Mossin (1966) and Fama (1971) later extended the model. Sharpe showed that conditions exist under which the equilibrium risk-return relation for any capital asset i can be represented as: 4 So(Ru) (5) E(R1) - Rf + E< V - Rf 0 cty where: - the random rate of return on asset i. E(R^) - the mathematical expectation of R^ . R^ - the pure time return on capital or the so-called riskless rate of interest. R - the random rate of return on a representative dollar of total wealth or, equivalently, the return on a portfolio containing all existing assets in the proportions, x^, in which they are actually outstanding. E(Rw) - the mathematical expectation of Rw . 33; a(Rw) - the standard deviation of the return on total wealth — a measure of the risk involved in holding a representative dollar of total wealth. the marginal contributions! asset i to the risk of the 3 Xj return on total wealth O(Rw). Expression (5) says that, in equilibrium, the expected rate of return on any asset i will be equal to the riskless rate of interest plus a risk premium proportional to the contribution of the asset to the risk of the return on total wealth. In general, the test conducted with the capital asset pricing model follows this type of procedure. First let R. be some return Jt on asset j at time t and further let R be the returns to the market IUt at time t. The market model can then be estimated from time series data on Rjfc and R^_, using ordinary least squares. (Fama, 1976) The form of the test is: (6) R.t - + B. Rmt + E.t the error term, e. , is assumed to be normally distributed with a ■ J t mean of zero and a constant variance. With an efficient market, the deviation of e from zero results solely from new information that J t becomes available from t: there is no way to use information avail­ able at t-1 as the basis of a correct nonzero assessment of the expected value of e. . (Fama, 1976) J t The first study that uses the market model as the basis of a test of market efficiency is the work on stock splits by Fama, Fisher, 34 Jensen, and Roll (1969). Dusak (1973) uses the market model to test market efficiency In several commodity futures markets. These studies concluded that such a relationship existed in the market. Returns Conform to a Risk-Return Relationship The fourth type of general equilibrium market model is the theory that returns conform to some risk-return relationship. Note that the capital asset pricing model also posits a risk-return relationship. However, the risk-return relationship described in the capital asset pricing model has some restrictive assumptions. The capital asset pricing model (CAPM) presents this type of a risk-return relationship: Return Risk Figure 3-1 CAPM RISK-RETURN RELATIONSHIP The relationship presented in Figure 3-1 says that in order to achieve a higher return an investor must assume more risk. Figure 3-1 also says that the return to an asset is equal to some risk free rate (point A) plus some risk premium. The capital asset pricing model posits that the risk-return relationship is linear. Other general equilibrium market models 35 could hypothesize a relationship between risk and rate of return that is direct but could be quadratic, exponential, or some other nonlinear I specification. Also, the assumptions of the capital asset pricing 17model such as I) unrestricted borrowing and lending at a risk free rate, 2) costless information and 3) costless transactions could be modified or eliminated in another general equilibrium market model. Methodology The Model The general model of market equilibrium tested against here is the model that posits that expected returns are constant. The subset of (|> ^ (the information available to all market participants at time t-1) that is used in this study is the potential information about expected returns that appears in time series of past returns. If the market is efficient and equilibrium expected returns are constant through time, the past returns on security j are a source of infor­ mation about E(IL). (Fama, 1976) But, if the market is efficient the past returns on security j.cannot be used to estimate the devia­ tion of R from E(R ' ).jt jt If an efficient market exists, there is no way to use any infor­ mation available at time t-1 as the basis for a correct assessment of an expected value of R which is different from the assumed constant equilibrium expected return. (Fama, 1976) 36 Therefore, for any sequence of past returns, R. R. the conditional expected value is <7>. E(EjtlKj,t-V • • • • ) - E(Kj) Statement (7) can be tested in this manner. If the correct assessment of the expected value of R is E(R1), then for any R n.J JjtU (8) To test (8) an alternative hypothesis is Introduced, one that proposes that the regression function is linear in Rj t .^n* This hypothesis .has the general form: (9) m Jt|Rjit_n> - »n + If the market is efficient, and if equilibrium expected returns are constant through time, this implies that the coefficients oh the return on any security j are zero for all values of the lag n. The general form of the model to be estimated is do) EJ>t - Cj > S1Rj i w + S2Rj>t„2 +---+BnRj i w + Sjjit. where: otj - the intercept term. 8 ^- estimated regression coefficients i - l.,,,,n. R - the return to currency j during the time period t-1 to t. Rj - its return from time period t-2 to t-1. R4j. - its return from time period t-n-1 to t-n.J,t-n t - a random error term that is assumed to be normally distri- ’ buted with a mean of zero and a constant variance. Expression (10) will be lagged an arbitrary n time periods. If the last time period does not contribute significantly to the 37 estimation of R , the variable will be dropped. This process J > t" thwill continue until a model of some n order autoregressive process remains. The.null hypothesis to be tested is (11) Hq: P1 => P2 -Pn « 0 Also, the constant term, a , will be tested to determine if J 9 L it is significantly different from zero. Two. different return specifications will be tested; this is because a priori neither is a clear choice. The first defines the return to currency futures market j as (12) P1,t P1,t-1 Pj,t-1 where: R - the return to currency futures market j during time period J’ t-1 to t. P . - the futures market price of currency j at time t. P . , - its price at time t-1. Jjt- I The second defines the rate of return to currency futures market j during the time period t-1 to t as (13) where: pj t-1 jjt - the rate of return to currency futures market j during the time period t-1 to t. 38 P . - the futures market price of currency j at time t. J > ^ ' P. 1 - its price at time t-1.J > t-i Note that this model measures absolute changes in the price rather than percentage changes. If one of the return specifications can be deemed superior to the other, another model will be specified and run. This model will contain lagged returns from other currency futures markets as well as lagged values of the currency in question on the right hand side. This is a multimarket model, that tests whether returns, in currency market i are dependent upon returns in currency market j. The Data The five most actively traded currency contracts on the Inter­ national Monetary Market comprise the data set for this study. These five currencies include: British pounds, Canadian dollars, Japanese yen, Swiss francs, and West German marks. The data set covers seven years of trading on the IMM: . 1972 to 1978. Each year is broken down into four contracts — March, June, September, and December. Therefore the data set for each currency starts with the September 1972 contract and ends with the December 1978 contract. For any particular contract, there are about 62 observations. Consequently, this results in a data sample of 1,636 trading days. 39 The contract nearest maturity is used in the data set. For example, when the June contract closes, the September contract com- . prises the data set until it closes, then the December contract comprises the data set until, in turn, it closes, etc. In this manner a continuous time horizon is achieved; a futures price for each currency on every trading day from June 19, 1972 to December 18, 1978 is thus included in the data. These futures price observations then generated returns. The difference between the futures price at time t and at time t-1 com­ prises a first-difference return while the percentage change of these two futures prices comprise the percentage change return. These two return specifications are computed over three different time periods, i.e., for a one day return, six day return, and a twenty day return. Note that these returns are computed over trading days. Chapter 4 RESULTS The general equilibrium market model that was tested was one that stated the returns over time are constant. The type of model con­ structed to test against the general equilibrium market model is an autocorrelation model of this form: a) ^ > t - « + B1Rjjt_1 + e2K.jt.2 + 6nRj,t-n + =t where R. -.the return to currency market j at time t. R^ t ^- its return at time t-i, i = I, , n . a - the intercept term. Sgi•••> - estimated autocorrelation coef ficients. - a random error term which is assumed to have the classic properties, i.e. E^ ~ n (0,a^) . Two different specifications of the independent variables were tested. After the results from each are presented, these specifica­ tions will be assessed one against the other to determine if one is superior. One, the percentage return model, defines the independent variables as a percentage return, that is (2) xj,t P - P ■i,t *j,t-r p j, t-i The other, the difference model, defines the independent variables as a change in the price, that is (3) R =P' - P j,t j,t j,t-l 41 Tfye two specifications were lagged an arbitrary fifteen time periods. The time periods used were one day, six day, and twenty day. Each specification and time period combination was estimated for each currency. The maintained general equilibrium market hypothesis tested against is (4) EOSt I R . Expression (4) implies that this null hypothesis be tested for equation (I): (5) P1 = ... = » 3 = 0o x n If (5) is true then (I) becomes (6) R = a + e- , that is, the return to currency j at time t is some constant (a) plus a random error term (e^). If (5) cannot be rejected, then (4) cannot be rejected. In cases where (5) is rejected, (4) is rejected, In cases where (5) is not rejected another hypothesis will be tested. This hypothesis states that the returns to currency futures market j over time are constant and, further, these constant returns are zero. Thus, this hypothesis will then be tested: (7) . a - P1 - ... = P15 - 0 Note expression (7) is conditional upon (5) and is tested by whether the mean return is equal to zero, that is if the mean*of the . dependent variables is different from zero. 42 Expression (5) has broad implications. If it cannot be rejected then it appears that Information contained in last time period’s return is not much help in forecasting this time period’s return. If it is rejected, then information in the last time period’s return can be used to forecast this time period's return. This, then, would lead to the rejection of the joint hypothesis that the market is efficient and that it sets prices so that equilibrium returns through time are constant. Although (5) is the null hypothesis that tests the theory of whether returns are constant, it would be beneficial to test to see where any statistical dependence might be concentrated. If (5) is rejected, it is because high order lags are different from zero or is it because low order lags are different from zero? So, for each of the currencies, time periods, and independent variable specifica­ tions, the following tests were made. First, this hypothesis was tested: (8) Ho: Bu = B12 - B13 - B14 - B15 - 0 then. (9) Bg = gg = . then, = S15 - 0 (.10) Hq: B2 B3 = ... - B15 = 0 and finally, CU) Hq: B1 = B2 = ... = B15 = 0 43 The most extensive hypothesis is expression (10), wherein the maintained hypothesis is that none of the explanatory variables has an influence on the mean of the dependent variable. This hypothesis is tested against the alternative that the maintained hypothesis is hot true; that is, that at least one of the regression coefficients is different from zero. If (11) is true then the variation of the depen­ dent variable from observation to observation is not affected by changes in any one of the explanatory variables, but is a constant term plus a random element. The appropriate test for evaluating (11) is SSR/(k-l_) SSE/(n—k) k—l,n—k where SSR — the sum of squares due to regression. k — the number of estimated regression coefficients. S S E--the residual sum of squares. n — the number of sample observations. Y ^n k — distributed as an F with k-1 degrees of freedom in * the number and n-k degrees of freedom in the deno­ minator . The calculated F is compared to a critical F value. The critical value changes with the number of degrees of freedom. If the calculated F exceeds the critical value, then the null hypothesis, (11), is rejected. 44 Testing the expressions (8), (9), and (10), is somewhat different. In general, these tests concern the influence of additional expla­ natory variables on the mean of the dependent variable. In particular, note expression (8). This expression tests two different theories. The first states . (12) Rj,t = “ + 3lRj.,t-l + ••• + 3SeEstt-10 + Et where the terms in (11) have the same definition as they do in (I). The second theory states (13) Rj,t ^ a + 3lRj, t-1 + ••• + ei5Rj,t-15 + et /. where the terms in (13) have the same meanings as the terms in (12). Note the difference in the two theories is that (13) includes five more independent variables. To test expression (8) is to test between the two theories. That is, do the five added regressors as a group affect the mean of the dependent variable? To formulate the appropriate test, a new notation is introduced, using the subscript k to denote the values pertaining to the original set of explanatory variables, and the subscript Q to denote the values of the extended set of explanatory variables. If the null hypothesis is true then Iigha Illfu - SSRr) /'(Q- K), ^ p llRu / (n - Q) Q-K,n-Q 45 where SSRg — the sum of squares due to regression of the original set of explanatory variables. Q — 1 the number of explanatory variables in the extended set. K — the number of explanatory variables in the original set. S S E — the residual sum of squares in the extended set of ^ explanatory variables. n — the number of sample observations. rV F g n _ — distributed as an F with K-I degrees of freedom in ^ ^ the numerator and n-K degrees of freedom in the denominator. (Kmenta, 1971) Expression (14) can be used to test (8), (9), and (10). Again, a calculated F statistic is compared to a critical value of an F statistic to make the decision of whether or not to reject the main­ tained hypothesis. Tables 4.1 and 4.2 present the test results from expression (8). This is the hypothesis that the independent variables eleven to fifteen are not significantly different from zero. Of the thirty cases tested, the maintained hypothesis that the eleventh, twelfth, thirteenth, fourteenth, and fifteenth lag of the dependent variable have no effect on the mean of the dependent vari­ able was only rejected once. This was the first difference return specification for the Japanese yen during a six day holding period. SSR — the sum of squares due to regression of the extended set vT of explanatory variables; 46 TABLE 4.1 Return Specification — Percentage Change Currency I Time Period (days) 6 20 British Pounds NO NO NO Canadian Dollars NO NO NO German Marks NO NO NO Japanese Yen NO NO NO Swiss Francs NO NO NO YES — the null is rejected NO — the null is not rejected 47 TABLE 4.2 Return Specification — First Differences Currency I Time Period (days) 6 20 British Pounds NO NO NO Canadian Dollars NO NO NO German Marks NO NO NO Japanese Yen NO YES NO Swiss Francs NO NO NO YES — the null is rejected NO — the null is not rejected 48 At the 95 percent confidence level, the null is expected to be ■rejected when it is in fact true 5 percent of the time. Therefore, (8). might be true for a first difference return specification for a holding period of six days for the Japanese yen. However, sampling may. have called for a rejection .of this, hypothesis in this one case. Therefore, it is concluded that CS) cannot be rejected. Tables 4.3 and 4.4 show the test results of expression (9). This hypothesis states that the eighth through the fifteenth lag of ■ the dependent.variable do not, as a group, affect the mean of the dependent variable. Of the thirty cases where (9) was tested, two were rejected. These two were: German marks, percentage return specification, one day hold and Japanese yen, first difference return specification, six day hold. The same argument used previously could be used in this case. Five percent of the time, the null will be rejected when it is, in fact, true. . Therefore, two cases out of thirty is.not extraordinary. However, it must be remembered that these tests are not independent. That is, if CS) is rejected, C9) will also be rejected. But, only Japanese yen six day hold with a first-difference return specification has rejected CS). This could be nothing more than chance, especially in the light that the time periods were arbitrarily chosen. For any given sample. 49 TABLE 4.3 Ho= Gg" *9= ' ' ' * Gl5 " O Return Specification — Percentage Change Time Period (days) Currency I 6 20 British Pounds NO NO NO Canadian Dollars NO NO NO German Marks YES NO NO Japanese Yen NO NO NO Swiss Francs NO NO NO YES — the null is rejected NO — the null is not rejected 50 TABLE 4.4 Return Specification — First Difference Time Period (days) Currency I 6 20 British Pounds NO NO NO Canadian Dollars NO NO NO German Marks NO NO NO Japanese Yen NO YES NO Swiss Francs NO NO NO YES — the null is rejected NO — the null is not rejected 51 there exists a model that will fit it well. It could be that for the Japanese yen, the six-day first difference return model happens to fit the data. Alternatively, this could be an indication qf statistical dependence in the Yen market. Tables 4.5 and 4.6 show the test results of the null hypothesis in expression (10). Of the thirty cases, eight rejected the main­ tained hypothesis that the second through the fifteen lag of the dependent variable, as a group, have no effect on the mean of the dependent variable. Xt is interesting to note that in no case has a twenty day holding period shown any statistical dependence. In this particular case, two six period holds, the Japanese yen and the Swiss franc (both with the first-difference retprn specifica­ tion) rejected the null. Six one period holds rejected the null: three in the first-difference return specification (Canadian dollars, German marks, Swiss francs), and three in the percentage return speci­ fication (Canadian dollars, German marks, Japanese yen). Again, the Japanese yen six day hold with a first-difference returp specifica­ tion rejected the null. The last hypothesis tested was expression (11). Tbis maintained hypothesis theorizes that none of the lags of the dependent variable, from the first to the fifteenth, have an effect on the.mean of the dependent variable. Tables 4.7 and 4.8 show the results. (Appendix Il contains the numerical test results for (8), (9), (10), and (11)). 52 TABLE 4.5 Return Specification — Percentage Change Time Period (days) Currency I 6 20 British Pounds NO NO NO Canadian Dollars YES NO NO German Marks YES NO NO Japanese Yen YES NO NO Swiss Francs NO NO NO YES — the null is rejected NO — the null is not rejected 53 TABLE 4.6 V B2 " 63 B15 - O Return Specification — First Differences Currency I British Pounds NO Canadian Dollars YES German Marks YES Japanese Yen NO Swiss Francs YES Time Period (days) 6 20 NO NO NO NO NO NO YES NO YES NO YES — the null is rejec t e d NO — the null is not rejected 54 TABLE 4.7 Hq: B1 = B2 = • • • = S15 = O Return Specification — Percentage Change Time Period (days) Currency I 6 20 British Pounds NO NO NO Canadian Dollars YES NO NO German Marks YES NO NO Japanese Yen YES YES NO Swiss Francs YES NO NO YES — the null is rejected NO — the null is not rejected 55 H Return Currency British Pounds Canadian Dollars German Marks Japanese Yen Swiss Francs YES — the null TABLE 4.8 0 Specification — First Differences Time Period (days) I 6 20 NO NO NO YES NO NO YES NO NO YES YES NO YES YES NO is rejected NO — the null is not rejected 1 56 Of the thirty cases, 11 reject the null. None of these rejections were in the twenty-day holding period. Three were in the six-day holding period and eight of the ten cases in the one-day hold were rejected. They were the same for both return specifications, namely, the first-difference return and the percentage return specification. For the cases where (11) was not rejected another hypothesis was tested. This is expression (7) above. This hypothesis is tested conditionally that (11) is not rejected and it states that the returns over time are zero. The test was conducted in this manner. After (11) was not rejected, (I) became (6), with (6) stating the return to currency j is a constant plus a random term. Expression (7) tests this constant term to judge if it is significantly differ-, ent from zero. Since.the intercept is the estimate of the meand of the dependent variable when the independent variables have no effect upon the mean of the dependent variable, (7) tests the mean of the dependent variable to judge if it is significantly different from zero. The results are presented in tables 4.9 and 4.10. Note that in all cases where CU) was not rejected, the mean of the dependent variable was judged not significantly different from zero. Thus, in the majority of the cases, the short run returns in the currency futures markets are zero. However, there are some cases where these short run returns are not zero, i.e. there is some statistical 57 TABLE 4.9 Return Specification — Percentage Change Time Period (days) Currency I 6 20 British Pounds NO NO NO Canadian Dollars * NO NO German Marks * NO NO Japanese Yen * * NO Swiss Francs * NO NO * - V 6I = 62 = • - S15 - 0 was rejected YES — the null is rejected NO — the null is not rejected 58 H : O = B1 = . . . = B ic = Oo I 15 Return Specification — First Differences TABLE 4.10 Time Period (days) Currency I 6 20 British Pounds NO NO NO Canadian Dollars * NO NO German Marks * NO NO Japanese Yen * * NO Swiss Francs * * NO * — H : e = e ;O l Z = . . . = B15 = 0 was rejected YES — the null is rejected NO — the null is not rejected 59 dependence in some of the cases. It is apparent that if any statistical dependence exists, it is not in the twenty-day holding period. None of the forty cases rejected a null that a group of variables was not different from zero. Therefore, it appears that any statistical dependence is concentrated in the one d.ay hold and the six-day hold. Thus, it is of interest to determine the order, of the autocorrelation. Tables 4.11 and 4.12 summarize the autocorrelation functions for both return specifications for the twenty day holding period. Note two things. First, only seven of the 150 estimated coeffi­ cients were significantly different from zero (using a t-test). Further, when a fifteenth order autoregressive function is estimated, the probability that all fifteen estimated coefficients will be judged not significantly different from zero is approximately .54; if, in fact, there is no significant relationship. That is, some of the coefficients can appear to be significantly different from zero, when there is no relationship. This is assuming a 5 percent confidence level. The probability of observing no significant coefficients when there is no true dependence is given by (15) I - (I - c)n where c is the confidence level and n represents the number of variables. In this case c is .05 and n is 15, giving a probability 60 TABLE 4.11 Significance of Individual Variables Return Specification — Percentage Change Holding Period — Twenty Days Currency Lag British Pounds Canadian Dollars German Marks Japanese Yen Swiss Francs I + 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 - 0 0 5 0 0 0 0 0 6 0 0 - 0 0 7 0 0 0 0 0 8 0 0 0 0 0 9 0 0 0 0 0 10 0 0 0 0 0 11 0 0 0 0 0 12 0 0 0 0 0 13 0 - 0 0 0 14 0 0 0 0 0 15 0 0 0 0 0 Key: 0 not significant + significant positive - significant negative 61 TABLE 4.12 Significance of Individual Variables 5% Level Return Specification — First Difference Holding Period — Twenty Days Currency Lag British Pounds Canadian Dollars German Marks Japanese Yen Swiss Francs I + 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 4 0 0 - 0 0 5 0 0 0 0 0 6 0 0 - 0 0 7 0 0 0 0 0 8 0 0 0 0 0 9 0 0 0 0 0 10 0 0 0 0 0 11 0 0 0 0 0 12 0 0 0 0 0 13 0 0 0 0 0 14 0 0 0 0 0 15 0 0 0 0 0 Key: 0 not significant + significant positive - significant negative 62 of .54. In fact, of the ten equations estimated for a twenty day holding period, five of them had every estimated coefficient not different from zero, complying with expectations. Tables 4.13 and 4.14 show the results of the six day holding period. Two of the estimated equations have no significant coeffi­ cients. Of the remaining eight equations, four have lags significant that are third order or higher. Of these four, two have only the third lag significant and two have the third and fourth lag signifi­ cant. In the context of this study, this says the return from the eighteenth trading day ago to the twelfth trading day ago will help the trader forecast the return from today to six trading days from now. This seems a bit implausible since the information from this time period should have been adapted into the price by the market. It makes more sense, in terms of information, to talk about returns that occurred only one period ago. The remaining four, however, show a significant coefficient on the first variable. This might lead to the conclusion that this model might be used to reap gains from this seemingly inefficient market. Tables 4.15 and 4.16 have five estimated equations that have the first order lag significant. (The numerical results of 4.11 - 4.16 appear in Appendix II.) 63 TABLE 4.13 Significance of Individual Variables Return Specification — Percentage Change Holding Period — Six Days Currency Lag British Pounds Canadian Dollars German Marks Japanese Yen Swiss Francs I 0 0 4* + 0 2 0 0 + 0 0 3 + 0 0 0 - 4 0 0 0 0 + 5 0 0 0 0 0 6 0 0 0 0 0 7 0 0 0 0 0 8 0 0 0 0 0 9 0 0 0 0 0 10 0 0 0 0 0 11 0 0 0 0 0 12 0 0 0 0 0 13 0 0 0 0 0 14 0 0 0 0 0 15 0 0 0 0 0 Key: 0 not significant + significant positive significant negative 64 TABLE 4.14 Significance of Individual Variables 5% Level Return Specification — First Differences Holding Period — Six Days Currency British Canadian German Japanese Swiss Lag____________Pounds______Dollars____Marks________Yen______Francs I 0 0 + + 0 2 0 0 0 0 0 3 + 0 0 0 - 4 0 0 0 0 + 5 0 0 0 0 0 6 0 0 0 0 0 7 0 0 0 0 0 8 0 0 0 0 0 9 0 0 0 0 0 10 0 0 0 0 0 11 0 0 0 0 0 12 0 0 0 - 0 13 0 0 0 0 0 14 0 0 0 0 0 15 0 0 0 0 0 Key: 0 not significant + significant positive significant negative 65 TABLE 4.15 Significance of Individual Variables Return Specification — Percentage Change Holding Period — One Day Currency Lag British Pounds Canadian Dollars German Marks Japanese Yen Swiss Francs I 0 0 0 - + 2 0 - 0 - 0 3 0 + 0 - 0 4 0 0 0 0 0 5 0 0 0 + 0 6 0 0 0 + 0 7 0 + 0 0 + 8 0 0 0 0 0 9 0 0 + + 0 10 + 0 + + + 11 0 0 0 0 0 12 0 0 0 0 0 13 0 0 0 0 0 14 0 0 0 0 0 15 0 0 0 0 0 Key: 0 not significant + significant positive significant negative 66 TABLE 4.16 Significance of Individual Variables 5% Level Return Specification — First Differences Holding Period — One Day Currency Lag British Pounds Canaidan Dollars German Marks Japanese Yen Swiss Francs I 0 0 + - + 2 0 - + 0 0 3 0 + 0 - + 4 0 0 0 0 - 5 0 + 0 0 - 6 0 0 0 0 0 7 0 + 0 0 + 8 0 0 0 0 0 9 0 0 + + + 10 + 0 + + 0 11 0 0 0 0 - 12 0 0 0 0 0 13 0 0 0 0 0 14 0 0 0 0 0 15 0 0 0 0 0 Key: 0 not significant + signifleant positive - significant negative 67 This could be explained in part by the discussion concerning (15). However, since the first order lag would be the lag that . theoretically would contain the most information, these results might point to some market inefficiency. In the one day hold models, every estimated equation had at least one coefficient sig­ nificantly different from zero. Consequently, it appears that there is some dependence in the one-day hold models. Comparing the Specification Models Throughout these results, two models of return specifications have been presented. However, if only one of the return specifica­ tions is correct, then faulty estimates may have been made. There­ fore, it is best to determine if one of the models is superior to the other. In order to do this two different null hypotheses must be tested. The first is (16) » - pJ.t j,t-l Ej,t where P . — the price in currency market j at time t. J »t- — its price at time t-1. e. — a random error term assumed to have the classic ordinary least squares properties. 68 Note.that (16) says that returns in currency futures market j are just due to a random error term, which according to tests of (5) and (7) appears plausible. If (16) is true, then it implies (17) Pj,t ' Pj,t-1 " Pj,t-lej,t The right-hand term in (17) can be defined as a new error term, U , that is assumed to be normally distributed with mean zero and J » t 2 2variance a P. . Thus (17) becomesJ jt-i (1S) Fj,t - pJlt- I = Y t The variance of the error term in (18) will be proportional to the square of the last period's price. That is, if the percentage return specification is correct and a first difference return specifi­ cation is estimated, a heteroskedastic error term is introduced. To test (16) the squared residuals from the first-order first-difference regression were regressed on the squared price one period ago. Therefore this equation was estimated (19) 5 ^ = ; + B + Vt 2 The estimated coefficient of the term is the key to deter­ mining if the error term in (18) is heteroskedastic. Note, from (18), (20) =(Ujit)2 - " X t-! arid from (19) (21) E(U. )2 = a + g (P2 ) ],t J»t-1 69 -v 2 Thus, the coefficient 3 in (19) is an estimate of a in (20). The maintained hypothesis (16) states that this coefficient should be positive in sign. If 3 is significantly greater than zero, then the error term in (18) is heteroskedastic and the first difference model with homoskedastic error must be rejected. Intuitively, a positive, value for 3 lends credence to the null hypothesis of a percentage change return specification. Two hypotheses from (19) are tested (22) Hq: a = 0 ' and (23) Hq: 3 = 0 Expression (23) was tested against the alternative that the estimated coefficient in (21) is greater than zero. The results of (22) are in table 4.17 and the results of (23) are in table 4.18. The numerical results are in Appendix II. The other null hypothesis that is tested is (24) V Fj,t - * =3,« where P . — the price in currency market j at time t. ]»t P. , — its price at time t-1. J * t X E. — a random error term assumed to have the classical ^’ properties. 70 Testing the Intercept TABLE 4.17 Dj,t- I + Kt-V ^ jlt Y t ' “ + saW 2 + K H : a = 0o Holding Period I 6 20 H : a = 0 o Holding Period I 6 20 BP Yes No No No No Yes CD Yes No No Yes No No DM Yes Yes No Yes Yes Yes JY Yes Yes Yes Yes Yes Yes SF Yes Yes Yes Yes Yes Yes Yes — The null is rejected N o -- The null is not rejected 71 Choosing the Return Specification TABLE 4.18 (A) " L t ' “ + + "j.t A2* , ,-2 * uj,t ' a + 6 0a H : 8 = 0o H : 8 > 0a Holding Period I 6 20 Holding Period I 6 20 BP No No Yes Yes No No CD Yes No No Yes No No DM Yes Yes Yes No No No JY Yes Yes Yes No No No SF Yes Yes Yes No No No Yes — The null is rejected in favor of the alternative hypothesis No — The null is not rejected 72 The argument of mu is exactly reversed of the argument presented by Also, (24) states that returns in currency market j are due to a random element. From tests of (5) and (7), this appears plausible. If (24) is true, then it implies E , (25) Fi-t - j»t P j,t-lj,t"l The right hand term in (25) can be defined as a new error term, A U . , that is assumed to be normally distributed with mean zero and J > t 2 variance a . Expression (25) then becomes P 2 j,t-l Pj,t-Pj,t_l Pj,t-1 3tt The variance of the error term varies inversely with the square of the last period's price. If the first-difference specification is the correct specification, but the percentage change return specifica­ tion is estimated, a heteroskedastic error term is introduced. To test (24) the squared residuals from the percentage change regression were regressed on the inverse of the squared price one period ago. Therefore this equation was estimated "*2 " " , ,-2 * (27) Djjt- O t e C P jjt^) + V t 73 The estimated coefficient of the term (P. n ^is the key to 3 j t_i determining if the error term in (26) is heteroskedastic. Note, from (.26) (.28) E (U ) = O s ot- M P d,t-i and from (27) 9 -2 (.29) ECU ) = a + 8 (P ) ' Jst J»t-i Thus, the coefficient 8 in (27) is an estimate of in (28). The maintained hypothesis (24) states that this coefficient should be positive in sign. Xf (3 is significantly greater than zero, then the error term in (26) is heteroskedastic and the percentage change model with homoskedastic error must be rejected. Intuitively, a positive value for 6 lends credence to the null hypothesis of a first differ­ ence return specification. Two hypotheses from (27) are tested (30) Hq: Ot = 0 and (31) : 8 - 0 Expression (31) was tested against the alternative hypothesis that, the estimated coefficient in (29) is greater than zero. The results of (30) are in table 4.17 and the results of (31) are in table 4,18. The numerical results are in Appendix II. Table 4.18 shows some interesting results. It appears that the 74 error term resulting from a first-difference specification in hetero- skedastic. This is shown by the number of yeses in (A). These yeses say that the first difference returns specification results in a heteroskedastic error term. Further, in all but two cases of (B) in table 4.18, the null hypothesis was not rejected. This suggests that the percentage change return specification is not heteroskedastic. Therefore, the specification of a first difference return will be rejected in favor of a percentage change return specification. Note, though, these two different return specifications generated basically the same results (see tables 4.1 to 4.16). Developing a Prediction Model Tables 4.11 through 4.16 suggest that most of the statistical dependence on past returns occurred in the first lag i.e., the return from one period ago. A more powerful test should be made to determine the magnitude of this dependence. Therefore, this equation was estimated for all currencies for all time periods: (32) Rj,t ~ aj + ejRj,t-l + Ej.,t where R j Rj, t-1 the percentage change return in currency market j during the time period t-1 to t. — its return from time period t-2 to t-1. /oy — the intercept in currency market j. 3 . — autoregression coefficient to be estimated. 3 e. — a random error term assumed to have the classical ’ properties. The results from (.32) are presented in tables 4.19, 4.20, and 4.21. Only one of the coefficients in the twenty day hold was significant, the first order coefficient on British pounds. The six day hold had significance in two of the first order lags and the one day hold had three. These are the most interesting theoretically because all the information contained in all past prices is considered to be contained in the most recent time period. From Tables 4.19, 4.20, and 4.21 as well as tables 4.11 - 4.16, it appears there is some statistical dependence in past returns in the currency futures market. Whether or not this dependence can be used to produce future returns resulting in a positive gain is ' examined in Chapter Five. 76 TABLE 4.19 First Order Lag — Percentage Change Returns Twenty Day Hold British Pound Canadian Dollar a .0011 -.0014 (.1224) (.1285) t 2.556* -.3813 R2 .0982 .0024 N 62 62 Currency German Mark Japanese Yen Swiss Franc .0016 .0048 .0062 (.0036) (.0036) (.0046) .4444 1.333 1.3478 .0266 .0451 .0463 (.1290) (.1302) (.1297) .2062 .3464 .3570 .0007 .0020 .0021 62 62 62 * ** *** Significant at the Significant at the Significant at the .05 level .02 level .01 level 77 TABLE 4.20 First Order Lag — Percentage Change Returns Six Day Hold British Pound Canadian Dollar a .0005 -.00048 (s;) (.0007) (.00036) t .7143 -1.3333 6 .0164 .05877 (.0637) (.06402) t .2575 .9181 R2 .00027 .0056 N 246 246 Currency German Mark Japanese Yen Swiss Franc .00108 .00118 .00221 (.00097) (.00081) (.00123) 1.1134 1.4568 1.7967 .1837 .2905 .05763 (.06298) (.06127) (.06393) 2.917** 4.740*** .9015 .0337 .08433 .0033 246 246 246 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level 78 TABLE 4.21 First Order Lag — Percentage Change Returns One Day Hold British Pound Canadian Dollar Currency German Mark Japanese Yen Swiss Franc a .00008 -.00008 .00020 .00033 .00031 (.00012) (.00005) (.00014) (.00019) (.00016) t .6631 -1.4881 1.4286 1.7368 1.9375 8 .02899 -.01892 .05287 -.2451 .1327 (S-) (.02513) (.02506) (.02510) (.02431) (.01249) t 1.154 -.7549 2.106* -10.08*** 5.332*** R2 . 0008 .0004 .0028 .05998 .01754 N 1595 1595 1595 1595 1595 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level Chapter Five EXTENSION OF THE RESULTS In the preceding chapter, some statistical dependence was found in the returns of currency futures markets. The purpose of this chapter is to determine if this dependence can be used to predict returns that will yield a positive profit. Also, a multimarket model will be estimated. This model will include hot only lagged values of the dependent variable but also lagged returns from other curren­ cies as independent variables. The currencies to be tested are: British pounds (20 day hold), German marks (six day and one day hold), Japanese yen (six day and one day hold), and Swiss francs (one day hold). The prediction model used is a first order autoregressive equation. Therefore this model is estimated for all currencies and all holding periods: a) Rj,t ~ aj + ej*j,t-l + ej,t where I. — the percentage change return in currency figures 9 market j during the time period t-1 to t. R — its return from t-2 to t-1. oy — the intercept term for currency j . S. — the estimated autocorrelation coefficient. J e. — a random.error term assumed to have the classic ^’ properties. Equation (I) was estimated for all currencies and all holding periods. The results are presented in tables 4.19, 4.20, and 4.21, 80 The currencies named above have a coefficient that is significantly, different from zero. Table 5.1 presents these currencies. Since none of the intercept terms were significant, the prediction model becomes (2) Rj,t+1 " 3jRj,t Ri t+l — the predicted return in currency futures market j J’ during the time period t to t+l. R. — the return in currency futures market j during the time ^’ period t-1 to t. EL — the estimated first order autoregression coefficient for J currency j. Expression (2) predicts the return during the time period t to t+l is equal to the return in time period t-1 to t multiplied by an estimated coefficient. For a participant in the futures market the prediction model can be used to predict a return during the time period t to t+l that will yield a gross gain of some amount of money. In order to do this the return of the previous time period must be of a certain magnitude in order to predict this gross gain. Tables 5.2 to 5.7 show what return must be observed last time period in order to predict a gross gain of a certain amount of dollars. Tables 5.2a to 5.7a show what the probability is of. observing this return. The method in which these probabilities are calculated is as follows. The returns to the currency futures 81 Prediction Model Coefficients Table 5.1 British Pounds 20 day German Marks one day German Marks six day Japanese Yen one day Japanese Yen one day Swiss Francs one day a .0011 .00020 .00108 .00033 .00118 .00031 S'1 a (.0027) (.00014) (.00097) (.00019) (.00081) (.00016) t .4074 1.4286 1.1134 1.7368 1.4568 1.9375 K .3128 .05287 .1837 -.2451 .2905 .1327 % (.1224) (.02510) (.06298) (.02431) (.06127) (.01249) t 2.556* 2.106* 2.917** -10.08*** 4.740*** 5.332*** R2 .0982 .0028 .0337 .05998 .08433 .01754 N 62 1595 246 1595 246 1595 * _ Significant at the .05 level ** — Significant at the .02 level *** — Significant at the .01 level 82 Table 5.2 Prediction Values for British Pounds — Twenty Day Hold If the Price of BP is: The value of to give an $X return would have to be: $50 $100 $150 $200 $250 $500 1.80 .0036 .0071 .0107 .0142 .0178 .0355 1.81 .0035 .0071 .0106 .0141 .0177 .0353 1.82 .0035 .0070 .0105 .0141 .0176 .0351 1.83 .0035 .0070 .0105 .0140 .0176 .0349 1.84 .0035 .0069 .0104 .0139 .0176 .0347 1.85 .0035 .0069 .0104 .0138 .0173 .0346 1.86 .0034 .0069 .0103 .0138 .0172 .0344 1.87 .0034 .0068 .0103 .0137 .0171 .0342 1.88 .0034 .0068 .0102 .0136 .0170 .0340 1.89 .0034 .0068 .0101 .0135 .0169 .0338 1.90 .0034 .0067 .0101 .0135 .0168 .0337 1.91 .0033 .0067 .0100 .0134 ,0167 .0335 1.92 .0033 .0067 .0100 .0133 .0167 .0333 1.93 .0033 .0066 .0099 .0133 .0166 .0331 1.94 .0033 . 0066 .0099 .0132 .0165 .0330 1.95 .0033 .0066 .0098 .0131 .0164 .0328 1.96 .0033 .0065 .0098 .0130 .0163 .0326 1.97 .0032 .0065 .0097 .0130 .0162 .0325 1.98 .0032 .0065 .0097 .0129 .0161 .0323 1.99 .0032 .0064 .0096 .0129 .0161 .0321 2.00 .0032 .0064 .0096 .0128 .0160 .0320 2.01 .0032 .0064 .0095 .0127 .0159 .0318 2.02 .0032 .0063 .0095 .0127 .0158 .0317 2.03 .0031 .0063 .0094 .0126 .0157 .0315 2.04 .0031 .0063 .0094 .0125 .0157 .0313 2.05 .0031 .0062 .0094 .0125 .0156 .0312 2.06 .0031 .0062 .0093 .0124 .0155 .0310 2.07 .0031 .0062 .0093 .0124 .0154 .0309 2.08 .0031 .0061 .0092 .0123 .0154 .0307 2.09 .0031 .0061 .0092 .0122 .0153 .0306 2.10 .0030 .0061 .0091 .0122 .0152 .0304 Mean BP price during the sample period: $2.02 Range of BP prices during the sample period: $1.54 - 82.51 83 Table 5.2a P r o b a b i l i t y of P r e d i c t i o n Values for B r i t i s h Pounds — Twen t y Day Hold If the Price of BP is: The Probability oi: Observing the Value of to $X return is give an $50 $100 $150 $200 $250 $500 1.80 .4356 .3746 .3149 .2612 .2113 .0549 1.81 .4374 .3746 .3165 .2627 .2126 .0559 1.82 .4374 .3763 .3181 .2627 .2139 .0569 1.83 .4374 .3763 .3181 .2641 .2139 .0580 1.84 .4374 .3780 .3197 .2656 .2166 .0590 1.85 .4374 .3780 .3197 .2671 .2179 .0596 1.86 .4391 .3780 .3213 .2671 .2193 . 0606 1.87 .4391 .3797 .3213 .2686 .2206 .0617 1.88 .4391 .3797 .3230 .2701 .2219 .0628 1.89 .4391 .3797 .3246 .2716 .2233 .0639 1.90 .4391 .3814 .3246 .2716 .2246 .0645 1.91 .4409 .3814 .3262 .2731 .2259 .0656 1.92 .4409 .3814 .3262 .2746 .2259 . 0668 1.93 .4409 .3831 .3278 .2746 .2273 .0680 1.94 .4409 .3831 .3278 .2761 .2287 , 0686 1.95 .4409 .3831 .3294 .2776 .2300 .0698 1.96 .4409 .3848 .3294 .2791 .2314 .0710 1.97 .4427 .3848 .3311 .2791 .2328 .0716 1.98 .4427 .3848 .3311 .2806 .2342 .0728 1.99 .4427 .3866 .3327 .2806 .2342 .0741 2.00 .4427 .3866 .3327 .2821 .2355 .0747 2.01 .4427 .3866 .3344 .2836 .2369 .0760 2.02 .4427 .3883 .3344 .2836 .2383 .0767 2.03 .4445 .3883 .3360 .2852 .2397 .0780 2.04 .4445 .3883 .3360 .2867 .2397 .0793 2.05 .4445 .3900 .3360 .2867 .2411 .0800 2.06 .4445 .3900 .3376 .2882 .2425 .0813 2.07 .4445 .3900 .3376 .2882 .2439 .0820 2.08 .4445 .3917 .3393 .2898 .2439 .0834 2.09 .4445 .3917 .3393 .2913 .2454 .0840 2.10 .4463 .3917 .3409 .2913 .2468 .0854 84 Table 5,3 Prediction Values for German Marks ■— One Day Hold If the Price of DM is: The Value of Rt- to Give an $X Return would have to be: $50 $100 $150 $200 $250 $500 .30 .0252 .0504 .0757 .1009 .1261 .2522 .31 .0244 .0488 .0732 .0976 .1220 .2441 .32 .0236 .0473 .0709 .0946 .1182 .2364 .33 .0229 .0459 .0688 .0917 .1146 .2293 .34 .0223 .0445 .0668 .0890 .1113 .2225 .35 .0216 .0432 .0648 .0865 .1081 .2162 .36 .0210 .0420 .0630 .0841 .1051 .2102 .37 .0204 .0409 .0613 .0818 .1022 .2045 .38 .0199 .0398 .0597 .0796 .0995 .1991 .39 .0194 .0388 .0582 .0776 .0970 .1940 .40 .0189 .0378 .0567 .0757 .0946 .1891 .41 .0185 .0369 .0554 .0738 .0923 .1845 .42 .0180 .0360 .0540 .0721 .0901 .1801 .43 .0176 .0352 .0528 .0704 .0880 .1759 .44 .0172 .0344 .0516 .0688 .0860 .1719 .45 .0168 .0336 .0504 .0673 .0841 .1681 .46 .0164 .0329 .0493 .0658 .0822 .1645 .47 .0161 .0322 .0483 .0644 .0805 .1610 .48 .0158 .0315 .0473 .0630 .0788 .1576 .49 .0154 .0309 .0463 .0618 .0772 .1544 .50 .0151 .0303 .0454 .0605 .0757 .1513 .51 .0148 .0297 .0445 .0593 .0742 .1483 .52 .0145 .0291 .0436 .0582 .0727 .1455 .53 .0143 .0285 .0428 .0571 .0714 .1427 .54 .0140 .0280 .0420 .0560 .0701 .1401 .55 .0138 .0275 .0413 .0550 .0688 .1376 .56 .0135 .0270 .0405 .0540 .0676 .1351 .57 .0133 .0265 .0398 .0531 .0664 .1327 .58 .0130 .0261 .0391 .0522 .0652 .1304 .59 .0128 .0256 .0385 .0513 .0641 .1282 .60 .0126 .0252 .0378 .0504 .0630 .1261 Mean DM price during the sample period: $.41 Range of DM prices during the sample period: $.31 - $.58 85 Table 5.3a Probability of Prediction Values for German Marks — One Day Hold If the Price of DM is : The Probability of Observing $X Return is: the Value of Rt to give an $50 $100 $150 $200 $250 $500 .30 .000004 * * A A A .31 .000007 * * A A A .32 .000013 * * A A A .33 .000023 * * A A A .34 .000036 * * A A A .35 .000060 * * A A A .36 .000093 * * A A A .37 .000141 * * * A A .38 .0002 * A A A A .39 .0003 * A A A A .40 .0004 * A A A A .41 .0005 * A A A A .42 .0007 * * A A A .43 .0009 * A A A A .44 .0011 * A A A A .45 .0014 * A A A A .46 .0018 * A A A A .47 .0021 * A A A A .48 .0025 * A A A A ,49 .0031 * A A A A .50 .0036 * A A A A .51 .0042 * A A A A .52 .0049 * A A A A .53 .0055 * A A A A .54 .0064 * A A A A .55 .0070 * A A A A .56 .0081 * A A A A .57 .0090 * A A A A .58 .0103 * A A A A .59 .0114 * A A A A .60 .0125 .000004 A A A A *VaIue is more than 4.5 standard deviations from the mean. 86 Table 5,4 Prediction Values foi: German Marks — Si:< Day Hold If the Price of DM is: The Value of to give an $X Return would have to be: $50 $100 $150 $200 $250 $500 .30 .0073 .0145 .0218 .0290 .0363 .0726 .31 .0070 .0140 .0211 .0281 .0351 .0702 .32 .0068 .0136 .0204 .0272 .0340 .0680 .33 .0066 .0132 .0198 .0264 .0330 .0660 .34 .0064 .0128 .0192 .0256 .0320 .0640 .35 .0062 .0124 .0187 .0249 .0311 .0622 .36 .0060 .0121 .0181 .0242 .0302 .0605 .37 .0059 .0118 .0177 .0235 .0294 .0589 .38 .0057 .0115 .0172 .0229 .0287 .0573 .39 .0056 .0112 .0167 .0223 .0279 .0558 .40 .0054 .0109 .0163 .0218 .0272 .0544 .41 .0053 .0106 .0159 .0212 .0266 .0531 .42 .0052 .0104 .0156 .0207 .0259 .0518- .43 .0051 .0101 .0152 .0203 .0253 .0506 .44 .0049 .0099 .0148 .0198 .0247 .0495 .45 .0048 .0097 .0145 .0194 .0242 .0484 .46 .0047 .0095 .0142 .0189 .0237 .0473 .47 .0046 .0093 .0139 .0185 .0232 .0463 .48 .0045 .0091 .0136 .0181 .0227 .0454 .49 .0044 .0089 .0133 .0178 .0222 .0444 .50 .0044 .0087 .0131 .0174 .0218 .0435 .51 .0043 .0085 .0128 .0171 .0213 .0427 .52 .0042 .0084 .0126 .0167 .0209 .0419 .53 .0041 .0082 .0123 .0164 .0205 .0411 .54 .0040 ,0081 .0121 .0161 .0202 .0403 .55 .0040 .0079 .0119 .0158 .0198 .0396 .56 .0039 .0078 .0117 .0156 .0194 .0389 .57 .0038 .0076 .0115 .0153 .0191 .0382 .58 .0038 .0075 .0113 .0150 .0188 .0375 .59 .0037 .0074 .0111 .0148 .0185 .0369 .60 .0036 .0073 .0109 .0145 .0181 .0363 Mean DM price during the sample period: $.41 Range of DM prices during the sample period: $.31 - $.58 87 Table 5.4a Probability of Prediction Values for German Marks ■— Six Day Hold If the The Probability of Observing the Value of R to give an Price of $X Return is: L DM is: $50 $100 $150 $200 $250 $500 .30 .3188 .1748 .0798 .0307 .0096 * .31 .3258 .1832 .0867 .0349 .0118 * .32 .3304 .1901 .0941 .0396 .0141 .00001 .33 .3351 .1972 .1007 .0443 .0166 .00001 .34 .3398 .2045 .1077 .0493 .0195 .00002 .35 .3446 .2119 .1138 .0541 .0224 .00003 .36 .3493 .2175 .1215 .0592 .0257 .00005 .37 .3517 .2232 .1267 .0647 .0289 .00007 .38 .3565 .2291 .1336 .0698 .0320 .0001 .39 .3589 .2350 .1406 .0751 .0359 .0002 .40 .3638 .2410 .1465 .0798 .0396 .0002 .41 .3662 .2470 .1525 .0857 .0431 .0003 .42 .3686 .2511 .1571 .0909 .0474 .0004 .43 .3711 .2573 .1634 .0952 .0513 .0005 .44 .3760 .2615 .1698 .1007 .0555 .0007 .45 .3784 .2657 .1748 .1054 .0592 .0009 .46 .3809 .2700 .1798 .1114 .0631 .0011 .47 .3833 .2743 .1849 .1163 .0672 .0014 .48 .3858 .2786 .1901 .1215 .0715 .0017 .49 .3883 .2829 .1954 .1254 .0760 .0021 .50 .3883 .2873 .1990 .1308 .0798 .0025 .51 .3907 .2917 .2045 .1350 .0847 .0029 .52 .3932 .2939 .2081 .1406 .0888 .0034 .53 .3957 .2984 .2137 .1450 .0930 .0040 .54 .3982 .3006 .2175 .1495 .0962 .0047 .55 .3982 .3051 .2213 .1540 .1007 .0053 .56 .4007 .3074 .2252 .1571 .1054 .0060 .57 .4032 .3120 .2291 .1618 .1089 .0069 .58 .4032 .3142 .2330 .1666 .1126 .0078 .59 .4057 .3165 .2370 .1698 .1163 .0086 . 60 .4082 .3188 .2410 .1748 .1215 .0096 *Value is more than 4.5 Standard Deviations from the Mean 88 Table 5.5 Prediction Values for Japanese Yen — One Day Hold If the Price of JY is: The Value of to give an $X Return would have to be: $50 $100 $150 $200 $250 $500 .0030 -.00544 -.01088 -.01632 -.02176 -.02720 -.05440 .0031 -.00526 -.01053 -.01579 -.02106 -.02632 -.05264 .0032 -.00510 -.01020 -.01530 -.02040 -.02550 -.05100 .0033 -.00495 -.00989 -.01484 -.01978 -.02473 -.04945 .0034 -.00480 -.00960 -.01440 -.01920 -.02400 -.04800 .0035 -.00466 -.00933 -.01399 -.01865 -.02331 -.04663 .0036 -.00453 -.00907 -.01360 -.01813 -.02267 -.04533 .0037 -.00441 -.00882 -.01323 -.01764 -.02205 -.04411 .0038 -.00429 -.00859 -.01288 -.01718 -.02147 -.04295 .0039 -.00418 -.00837 -.01255 -.01674 -.02093 -.04185 .0040 -.00408 -.00816 -.01224 -.01632 -.02040 -.04080 .0041 -.00398 -.00796 -.01194 -.01592 -.01990 -.03980 .0042 -.00389 -.00777 -.01166 -.01554 -.01943 -.03886 .0043 -.00380 -.00759 -.01139 -.01518 -.01898 -.03795 .0044 -.00371 -.00742 -.01113 -.01484 -.01855 -.03709 .0045 -.00363 -.00725 -.01088 -.01451 -.01813 -.03627 .0046 -.00355 -.00710 -.01064 -.01419 -.01774 -.03548 .0047 -.00347 -.00694 -.01042 -.01389 -.01736 -.03472 .0048 -.00340 -.00680 -.01020 -.01360 -.01700 -.03400 .0049 -.00333 -.00666 -.00999 -.01332 -.01665 -.03331 .0050 -.00326 -.00653 -.00979 -.01306 -.01632 -.03264 .0051 -.00320 -.00640 -.00960 -.01280 -.01600 -.03200 .0052 -.00314 -.00628 -.00942 -.01255 -.01569 -.03138 .0053 -.00308 -.00616 -.00924 -.01232 -.01540 -.03079 .0054 -.00302 -.00604 -.00907 -.01209 -.01511 -.03022 .0055 -.00297 -.00593 -.00890 -.01187 -.01484 -.02967 .0056 -.00291 -.00583 -.00874 -.01166 -.01457 -.02914 .0057 -.00286 -.00573 -.00859 -.01145 -.01432 -.02863 .0058 -.00281 -.00563 -.00844 -.01126 -.01407 -.02814 .0059 -.00277 -.00553 -.00830 -.01106 -.01383 -.02766 .0060 -.00272 -.00544 -.00816 -.01088 -.01360 -.02720 Mean JY price during the sample period: $.0037 Range of JY prices during the sample period: $.0033 - $.0056 89 Table 5.5a Probability of Prediction Values for Japanese Yen — One Day Hold If the Price of JY is: The Probability of an Observing $X Return the Value is: of Rt to Give $50 $100 $150 $200 $250 $500 .0030 .2445 .0832 .0190 .0028 .0003 * .0031 .2518 .0903 .0223 .0037 .0004 * .0032 .2583 .0973 .0258 .0047 .0006 * .0033 .2645 .1042 .0296 .0059 .0008 * .0034 .2708 .1111 .0335 .0073 .0011 * .0035 .2767 .1177 .0376 .0088 .0015 * .0036 .2823 .1244 .0418 .0106 .0020 * .0037 .2874 .1310 .0462 .0124 .0025 * .0038 .2927 .1373 .0507 .0144 .0032 * .0039 .2975 .1436 .0552 .0166 .0035 * .0040 .3019 .1497 .0598 .0190 .0047 * .0041 .3064 .1557 .0644 .0215 .0057 * .0042 .3104 .1615 .0691 .0241 .0067 * .0043 .3144 .1672 .0737 .0268 .0079 * .0044 .3185 .1727 .0785 .0296 .0092 * .0045 .3222 .1783 .0832 .0325 .0106 * .0046 .3258 .1833 .0880 .0356 .0120 * .0047 .3295 .1887 .0926 .0387 .0136 .00001 .0048 .3327 .1936 .0973 .0418 .0153 .00001 .0049 .3360 .1985 .1020 .0451 .0171 .00001 .0050 .3392 .2031 .1066 .0484 .0190 .00001 .0051 .3420 .2078 .1111 .0518 .0209 .00002 .0052 .3448 .2122 .1155 .0552 .0230 .00003 .0053 .3476 .2167 .1200 .0606 .0251 .00005 .0054 .3505 .2212 .1244 .0621 .0273 .00006 .0055 .3528 .2254 .1288 .0656 .0296 .00008 .0056 .3557 .2292 .1332 .0691 .0319 .0001 .0057 .3580 .2331 .1373 .0727 .0343 .0001 .0058 .3604 .2370 .1415 .0761 .0368 .0002 .0059 .3623 .2409 .1456 .0798 .0393 .0002 .0060 .3647 .2445 .1497 .0832 .0418 .0003 *Value is more than 4.5 Standard Deviations from the Mean. 90 Table 5.6 Prediction Values for Japanese Yen — Six Day Hold If the Price of JY is: The Value of Rtto Give an $X Return Would Have to be: $50 $100 $150 $200 $250 $500 .0030 .00459 .00918 .01377 .01836 .02295 .04590 .0031 .00444 .00888 .01333 .01777 .02221 .04442 .0032 .00430 .00861 .01291 .01721 .02151 .04303 .0033 .00417 .00835 .01252 .01669 .02025 .04173 .0034 .00405 .00810 .01215 .01620 .02002 .04050 .0035 .00393 .00787 .01180 .01574 .01967 .03934 .0036 .00382 .00765 .01147 .01530 .01912 .03825 .0037 .00372 .00744 .01116 .01489 .01861 .03721 .0038 .00362 .00725 .01087 .01449 .01812 .03624 .0039 .00353 .00706 .01059 .01412 .01765 .03531 .0040 .00344 .00688 .01033 .01377 .01721 .03443 .0041 .00336 .00672 .01008 ,01343 .01679 .03358 .0042 .00328 .00656 .00984 .01311 .01639 .03278 .0043 .00320 .00640 .00961 .01281 .01601 .03202 .0044 .00313 .00626 .00939 .01252 .01565 .03129 .0045 .00306 .00612 .00918 .01224 .01530 .03060 .0046 .00299 .00599 .00898 .01197 .01497 .02993 .0047 .00293 .00586 .00879 .01173 .01465 .02930 .0048 .00287 .00574 .00861 .01147 .01434 .02860 .0049 .00281 .00562 .00843 .01124 .01405 .02810 .0050 .00275 .00551 .00826 .01102 .01377 .02754 .0051 .00270 .00540 .00810 .01080 .01350 .02700 .0052 .00265 .00530 .00794 .01059 .01324 .02648 .0053 .00260 .00520 .00779 .01039 .01299 .02598 .0054 .00255 .00510 .00765 .01020 .01275 .02550 .0055 .00250 .00501 .00751 .01001 .01252 .02504 .0056 .00246 .00492 .00738 .00984 .01229 .02459 .0057 .00242 .00483 .00725 .00966 .01208 .02416 .0058 .00237 .00475 .00712 .00950 .01187 .02374 .0059 .00233 .00467 .00700 .00934 .01167 .02334 .0060 .00229 .00459 .00688 .00918 ,01147 .02295 Mean JY price during the sample period: $.0037 Range of JY prices during the sample period: $.0033 - $.0056 91 Table 5.6a Probability of Prediction Values for Japanese Yen — Six Day Hold If the Price of JY is: The Probability of Observing $X Return the Value is; of Rt to give an $50 $100 $150 $200 $250 $500 .0030 .36461 .24437 .14953 .08309 .04175 .00027 .0031 .36887 .25153 .15738 .09010 .04697 .00040 .0032 .37286 .25807 .16513 .09716 .05238 .00059 .0033 .37658 .26444 .17254 .10407 .06336 .00082 .0034 .38002 .27065 .17976 .11091 .06555 .00113 .0035 .38347 .27642 .18676 .11761 .06898 .00150 .0036 .38664 .28200 .19352 .12428 .07466 .00196 .0037 .39953 .28737 .20000 .13073 .08024 .00251 .0038 .39243 .29227 .20618 .13725 .08589 .00314 .0039 .39504 .29721 .21225 .14347 .09158 .00387 .0040 .39765 .30193 .21798 .14953 .09716 .00472 .0041 .39998 .30615 .22357 .15557 .10272 .00566 .0042 .40231 .31040 .22902 .16141 .10822 .00672 .0043 .40465 .31467 .23431 .16701 .11364 .00787 .0044 .40670 .31843 .23943 .17254 .11895 .00915 .0045 .40875 .32221 .24437 .17798 .12428 .01051 .0046 .41080 .32573 .24913 .18334 .12946 .01200 .0047 .41256 .32927 .25370 .18839 .13462 .01356 .0048 ,41432 .33255 .25807 .19352 .13975 .01525 .0049 .41609 .33584 .26247 .19831 .14467 .01704 .0050 .41785 .33888 .26667 .20297 .14953 .01890 .0051 .41933 .34192 .27065 .20769 .15432 .02087 .0052 .42080 .34469 .27466 .21225 .15902 .02291 .0053 .42227 .34747 .27844 .21665 .16363 .02504 .0054 .42375 .35026 .28200 .22088 .16814 .02724 .0055 ,42523 .35278 .28557 .22515 .17254 .02949 .0056 .42641 .35530 .28891 .22902 .17700 .03184 .0057 .42759 .35783 .29227 .23315 .18114 .03423 .0058 .42907 .36009 .29565 .23686 .18535 .03670 .0059 .43026 .36235 .29878 .24060 .18941 .03919 .0060 .43144 .36461 .30193 .24437 .19353 .04175 92 Table 5.7 Prediction Values for Swiss Francs -— One Day Hold If the Price of SF is: The Value of to Give an $X Return Would Have to be: $50 $100 $150 $200 $250 $500 .35 .0086 .0172 .0258 .0344 .0431 .0861 .36 .0084 .0167 .0251 .0335 .0419 .0837 .37 .0081 .0163 .0244 .0326 .0407 .0815 .38 .0079 .0159 .0238 .0317 .0397 .0793 .39 .0077 .0155 .0232 .0309 .0386 .0773 .40 .0075 .0151 .0226 .0301 .0377 .0754 .41 .0074 .0147 .0221 .0294 .0368 .0735 .42 .0072 .0144 .0215 .0287 .0359 .0718 .43 .0070 .0140 .0210 .0280 .0351 .0701 .44 .0069 .0137 .0206 .0274 .0343 .0685 .45 .0067 .0134 .0201 .0268 .0335 .0670 .46 .0066 .0131 .0197 .0262 .0328 .0655 .47 .0064 .0128 .0192 .0257 .0321 .0641 .48 .0063 .0126 .0188 .0251 .0314 .0628 .49 .0062 .0123 .0185 .0246 .0308 .0615 .50 .0060 .0121 .0181 .0241 .0301 .0603 .51 .0059 .0118 .0177 .0236 .0296 .0591 .52 .0058 .0116 .0174 .0232 .0290 .0580 .53 .0057 .0114 .0171 .0227 .0284 .0569 .54 .0056 .0112 .0167 .0223 .0279 .0558 .55 .0055 .0110 .0164 .0219 .0274 .0548 .56 .0054 .0108 .0161 .0215 .0269 .0538 .57 .0053 s 0106 .0159 .0212 .0264 .0539 .58 .0052 .0104 .0156 .0208 .0260 .0520 .59 .0051 .0102 .0153 .0204 .0255 .0511 . 60 .0050 .0100 .0151 .0201 .0251 .0502 .61 .0049 .0099 .0148 .0198 .0247 .0494 .62 .0049 .0097 .0146 .0194 .0243 .0486 .63 .0048 .0096 .0144 .0191 .0239 .0478 . 64 .0047 .0094 .0141 .0188 .0235 .0471 .65 .0046 .0093 .0139 .0185 .0232 .0464 93 Table 5.7a Probability of Prediction Values for Swiss Francs — One Day Hold If the Price of SF is: The Probability of Observing the Value of $X Return is: Rt to Give: an $50 $100 $150 $200 $250 $500 .35 .0927 .0040 .00004 * * * .36 .0979 .0051 ..00006 * * * .37 .1061 .0060 .00009 * * * .38 .1119 .0072 .0001 * * * .39 .1179 .0085 .0002 * * * .40 .1241 .0100 .0003 * * * .41 .1272 .0118 .0003 * * * .42 .1338 .0133 .0005 .000005 * * .43 .1405 .0155 .0006 .00001 * * .44 .1440 .0174 .0008 .00001 * * .45 .1511 .0195 .0010 .00002 * * .46 .1547 .0218 .0012 .00003 * * .47 .1622 .0244 .0016 .00004 * * .48 .1660 .0262 .0019 .00006 * * .49 .1699 .0291 .0022 .00008 * * .50 .1778 .0312 .0027 .0001 * * .51 .1818 .0346 .0032 .0001 * * .52 .1859 .0370 .0037 .0002 .000004 * .53 .1900 .0396 .0042 .0002 .00001 * .54 .1943 .0423 .0051 .0003 .00001 * .55 .1985 .0451 .0058 .0004 .00002 * . 56 .2028 .0481 .0066 .0005 .00002 * .57 .2072 .0513 .0072 .0005 .00002 * .58 .2116 .0546 .0081 .0007 .00003 * .59 .2161 .0581 .0092 .0008 .00004 * .60 .2207 .0618 .0100 .0010 .00006 * .61 .2253 .0637 .0113 .0011 .00007 * .62 .2253 .0676 .0123 .0014 .00009 * .63 .2299 .0697 .0133 .0016 .0001 * .64 .2346 .0739 .0150 .0019 .0001 * .65 .2394 .0761 .0162 .0022 .0002 * * Value is more than 4.5 Standard Deviations from the Mean. 94 markets &re assumed to be normally distributed in the long run. The unconditional mean of this distribution is given by (3) E(R1) = Ct1 + B1E(R1) which is equal to (4) E(R1) I — B, where E(R1) — the expected return in currency futures market i. i Ct1 — the intercept term of currency i. P1 — the regression coefficient of currency i. Therefore, if a is zero the mean of the long run distribution is zero. Note from tables 4.19, 4.20, and 4.21 the intercept term, a is never significantly different from zero. Thus, the mean of the long run distribution of returns in the currency futures markets examined in this study is not different from zero. The variance of this distribution is given by 2 (5) V(R1) = B1V(R1) + V(E1) which is equal to (6) V(R1) i - e? where V(R1) — the variance of the return in currency futures market i. 95 V(e^) — the square of the standard error of the estimate for currency i. 2 fL — the square of the regression coefficient for currency i. After the mean and variance of the long run distribution is,cal­ culated the probability of observing a particular return is calculated in this manner. First, (7) Rj. - E(R1) V(Ri)1/2 Z where A Ri — the return in currency futures market i necessary to predict an $x return. E(Ri) — the expected long run return in currency futures market i: : which, based upon the results of tables 4.19, 4.20, and 4.21, is not different from zero. 1/2 V(Ri) — the standard deviation of the return to currency futures market i. Z — a measure of how many standard deviations from the mean A R . lies, i Note that the distribution of long run returns has now been trans­ formed into a standard normal distribution having a mean of zero A and a unit variance. After a Z is computed for an Ri the probability of observing a value that exceeds this. Z value is shown by Figure 5-1. 96 Figure 5-1. Standard Normal Distribution The probability of observing a Z less than some -Z or of observing a Z greater than some Z is shown by the shaded area in Figure 5-1. Note that this prediction model can be used by the participant to open either a long or short position. If the previous return was negative and of a certain magnitude, a negative return is predicted for the next time period. The participant would then sell short. If the previous return was positive and of a certain magni­ tude, a positive return is predicted for the next time period. The participant would then buy, i.e. open a long position. The above strategy holds for a positive value of the estimated coefficient: a negative value of the estimated coefficient would use a strategy opposite of the positive coefficient strategy. The decision rule used to invest is: If the price of the currency is between any two values on the tables and if the previous time period's return exceeds the return necessary to predict a gross 97 gain of a certain amount, a position will be opened. For example, if the price of British pounds is between $2.00 and $2.01 the investor will open a long position if the previous period’s return was greater than .0032 and will open a short position if the pre­ vious period’s return was less than -.0032. It is possible to construct many different strategies from the table based upon how much of a gross gain the investor wants and the value of ttie return necessary to open a position. In the following example, two different strategies will be used. The first will open a position if the previous period’s return falls within the values corresponding to the gross gain column of $50 or if the previous period's return exceeds the largest value. The position is assumed to be closed out at the start of the next time period. The second will open a position based upon the values in the $100 column and close it in the same manner at the $50 gain strategy. Note that these strategies are similar to filter rules in that a minimum price movement must occur before a position is opened. However, these strategies assume that the position is closed the next trading day regardless of how the price moves; whereas a filter rule specifies the price must move a specified amount before the position is closed. Both strategies will be compared to a buy and hold and a sell and hold strategy for control purposes. The data used is March 1979 futures prices in the 98 various currencies from December 19, 1978 to March 19, 1979: a time period that lies outside of the sample time period used for the esti­ mation of regression coefficients. The results are presented in tables 5.8 to 5.13. Table :5.14 shows the comppsite results of the various strategies. During the time period the prediction models were tested, it appears that prices in these, markets were falling. This is shown by the success of the sell-and-hold strategy and |)y the failure of the buy-and-hold strategy. The sell-and-hold strategy netted the greatest return, the buy-and-hold strategy netted the smallest return, and the two prediction strategies were in between. The commission charge ($50 a round turn), or transactions cost, was sufficient to make the $50 gain strategy a negative net strategy. It must be noted that these transactions costs could be reduced if the contract position was held longer than the assumed one time period. However, as previously stated, there are many different strategies that could be constructed: with the two tested being conservative strategies with respect to exposure. The opportunity cost of capital figure did not affect the out­ comes of the $50 gain and $100 gain strategies as much as the transactions costs did. The buy-and-hold and sell-and-hold strategies were affected more by the opportunity costs of capital than by Table 5.8 British Pounds — Twenty Day Hold Strategy Number of Trades R 0 1 S L0 n 1 Trades Resulting in Gains Trades Resulting in Losses Dollar • Amount Gained Dollar Amount Lost Com­ mis­ sion Opp. Cost of Capital Net re­ sult of Strategy Buy and Hold I 0 I 0 0 300.00 —• 0— 50 35.63 214.37 Sell and Hold I I 0 0 I —0— 300.00 50 35.63 (385.63) $50 Gain I I 0 0 I -0- 62.50 50 7.92 (120.52 $100 Gain I I 0 0 I -O'- 62.50 50 7.92 (120.52) Table 5.9 German Marks — One Day Hold Strategy Number of Trades 6 O E S L On § Trades Resulting in Gains Trades Resulting in Losses Dollar * Amount Gained Dollar Amount Lost Com­ mis­ sion Opp. Cost of Capital Net re­ sult of Strategy Buy and Hold I 0 I 0 I O'- 2362.50 50 47.50 (2460.00) Sell and Hold I I 0 I 0 2362.50 —O— 50 47.50 2265.00 $50 Gain 2 I I 2 0 437.50 —0— 100 .53 336.97 $100 Gain 0 0 0 0 0 —0— —0— 0 —0— -O'- 100 Table 5.10 German Marks — Six Day Hold Strategy Number of Trades 0 1 S L0n 1 Trades Resulting in Gains Trades Resulting in Losses Buy and Hold I 0 I 0 I Sell and Hold I I 0 I 0 $50 Gain 8 5 3 2 6 $100 Gain 7 5 2 2 5 Dollar Amount Gained Dollar Amount Lost Com­ mis­ sion Opp. Cost of Capital Net re­ sult of Strategy —0— 2362.50 50 47.50 (2460.00) 2362.50 -0- 50 47.50 2265.00 2912.50 4300,00 400 25.23 (1812.73) 2912.50 2987.50 350 22.17 (447.17) TO T Table 5.11 Japanese Yen — One Day Hold Strategy Number of Trades O E S L 0 n 1 Trades Resulting in Gains Trades Resulting in Losses Dollar - Amount Gained Dollar Amount Lost Com­ mis­ sion Opp. Cost of Capital Net re­ sult of Strategy Buy and Hold I 0 I 0 I —0— 6062.50 50 47.50 (6160.00) Sell and Hold I I 0 I 0 6062.50 —O— 50 47.50 5965.00 $50 Gain 27 8 19 11 15 3200.00 3687.50 1350 14.25 (1851.75) $100 Gain 11 3 8 6 5 2487.50 1475.00 550 5.81 456.69 102 Table 5.12 Japanese Yen — Six Day Hold Strategy Number of Trades 0 1 S L0 n 1 Trades Resulting in Gains Trades Resulting in Losses Dollar ' Amount Gained Dollar Amount Lost Com­ mis­ sion Opp. Cost of Capital Net re­ sult of Strategy Buy and Hold I 0 I 0 I ~0— 6062,50 50 47.50 (6160.00) Sell and Hold I I 0 I 0 6062.50 -0- 50 47.50 5965.00 $50 Gain 10 8 2 6 3 5537.50 2450.00 500 31.67 (2555.83) $100 Gain 8 6 2 4 3 3250.00 2450.00 500 25.33 274.67 103 Table 5.13 Swiss Francs — One Day Hold Strategy Number of Trades I O E S L On § Trades Resulting in Gains Trades Resulting in Losses Dollar • Amount Gained Dollar Amount Lost Com­ mis­ sion Opp. Cost of Capital Net re­ sult of Strategy Buy and Hold I 0 I 0 I ■*-()*— 4387.50 50 71.25 (4508.75) Sell and Hold I i 0 I 0 4387.50 —0— 50 71.25 4266.25 $50 Gain 20 12 8 9 9 3087.50 3537.00 1000 15.83 (1465.53) $100 Gain 8 5 3 6 2 1312.50 325.00 400 6.33 581.17 104 Table 5.14 Summary of Strategies Strategy Number of Trades § 0 1 S L 0n 1 Trades Resulting in Gains Trades Resulting in Losses Dollar • Amount Gained Dollar Amount Lost Com­ mis­ sion Opp. Cost of Capital Net re­ sult of Strategy Buy and Hold 6 0 6 I 5 300.00 21237.50 300 296.88 (21534.38) Sell and Hold 6 6 0 5 I 21237.50 300.00 300 296.88 20340.62 $50 Gain 68 35 33 30 34 16562.00 14037.30 3400 95.43 (1970.63) $100 Gain 35 20 15 18 16 9962.50 7300.00 1750 67.56 844.94 105 106 transactions costs. This can be explained by the nature of the two groups of strategies; with one group trading very often and the other group being exposed by a long period of time. The opportunity cost of capital was computed in this manner: The required margins used were the margin requirements set by the Chicago Mercantile Exchange. This figure varies for different currencies, (contract sizes in parentheses): British pounds (25,000 BP) and Canadian dollars (100,000 CD), $1500; German marks (125,000 DM) and Japanese yen (12,500,000 JY), $2000; and Swiss francs (125,000 SE), $3000. The required margin was multiplied by an assumed interest rate of 9-1/2 percent — giving an annual interest figure. This yearly interest was then allocated to the amount of time the margin was committed. For example, a buy-and-hold strategy during the prediction period would use the margin money for three months. Therefore, the interest foregone by investing in the futures market is given by the margin times the interest rate times the amount of time the margin was committed. The margin requirements set by the Chicago Mercantile Exchange are the minimum amount that can be charged by a brokerage house. An individual speculator would have to pay a much higher margin -— $5000 versus the $1500 a large firm would have to pay in the British pound market, for examplei A large firm trading many contracts has 107 another option available to it. The purpose of margin is to act as a performance bond. It is permissable to deposit Treasury Bills instead of cash. The interest paid on these Treasury Bills accrues to the original holder of the T-Bill — the firm in this case. In this instance, the opportunity cost of capital figure is negligible because the firm is posting no cash for margin but instead an interest-bearing note. The opportunity cost of capital figure becomes meaningful if the firm can post no Treasury Bills and has to pay cash for margin. Also, from an individual speculator’s point pf view the opportunity cost of his margin is higher than shown in tjhe tables; simply because a broker generally requires more margin from individual speculators than from firms. The Multimarket Model One other extension of the model which was not fully developed in this paper was the idea of multimarket models. In the previous models, the theory stated that all worldly events were accounted for by the market and the price of the currencies reflected these events. That is, inflation in West Germany would already be accounted for in the price of Japanese yen. The multimarket model tries to estimate the percentage change in the price of currency j from lagged values of the percentage change in the price of currency j and from lagged percentage changes in the price of other currencies. 103 Since events in the world offset all currencies these events are no longer assigned to be picked up by the error term but by other regressors. This model was then estimated for each currency: (3) R aj + + nj ,:t-i This model was run for all three holding periods, with the results in tables 5.15, 5.16, and 5.17 with the ratio of the independent variable to its standard error in parentheses. Note that the estimates, unlike the previously estimated coefficients in the single market test, are efficient. The efficiency of the estimators in the single market equations is "questionable" because each equation was estimated separately and independently. This is because the mutual correlation of the disturbance terms is disregarded. It is reasonable to expect that the disturbance terms could be correlated because of the following: There are events taking place in the world of economy that could affect the prices, and thereby the returns, of the currencies in question and since these events could affect all the currencies simultaneously the disturbance terms across equating could be correlated. This problem . could be eliminated by estimating these equations simultaneously. Also, the "questionableness" of the efficiency of these estimators is reduced if the sample size is "large". A special case of Table 5.16 Estimated Multimarket Model Holding Period — Six Days N = 260 Dependent British Canadian German Japanese Swiss ^ (Mean)_______ Intercept Pounds_______Dollars_____Marks___________Yen______Francs_____R British Pounds (.000300) -.0001306 (-.0123) -.04728 (-.7158) -.2657 (1.937) Canadian Dollars (-.0004919) -.0004278 (-.0403) -.0006874 (-.02253) .06519 (1.029) German Marks (.001140) .0005766 (.0542) -.02405 (-.2935) -.2790 (-1.640) Japanese Yen (.001469) .0009908 (.0932) .1390 (2.002)* -.1118 (-.7760) Swiss Franks (.002136) .001734 (.1625) .1095 (1.067) -.2506 (-1.177) .1006 (1.197) .1045 (1.499) .01667 (.2602) .05572 -.001456 (-.3747) .01998 (.6204) -.02891 (-.0768) .01086 .1117 (1.071) .2144 (2.478)** -.01060 (-.1334) .06832 -.08034 (-.9090) .2756 (3.761)** .03372 (.5008) .1033 .1442 (1.104) (.2862) -.1544 (2.643)***(-.1552) .05993 *25,230 * Significant at the .05 level ** Significant at the .02 level Joint Test F- „ _ = 9.6044 *** Significant at the .01 level Critical = 1.52 109 Table 5.15 Estimated Multimarket Model Holding Period — One Day N = 1609 Dependent (Mean) Intercept British Pounds Canadian Dollars German Marks Japanese Yen Swiss 2 Francs R British Pounds (.0005606) .00005336 (.0250) .03644 (1.343 -.05565 (-.9930) .01709 (.5410) -.01215 (-.7688) -.01011 .002498 (-.3799) Canadian Dollars (-.0000821) -.00008765 (-.0412) .002899 (.2398) -.01606 (-.6434) .01192 (.8474) .004310 (.6122) .001304 .002124 (.1100) German Marks (.0002181) .0002032 (.0953) .07001 (2.230)* .09658 (1.490) -.01255 (-.3435) .02706 (1.480) .04275 .01066 (1.389) Japanese Yen (.0003871) .0003093 (.1448) .05556 (1.307) -.03118 (-.3551) .1187 (2.399)** -.2754 (-11.12)*** .07097 .07838 (1.702) Swiss Francs (.0003603) .0003149 (.1476) .08288 (2.315)* .08753 (1.184) .1488 (3.573)*** .05721 (2,744)*** ,007431 .03854 (.2117) Joint Test F25 ^579 = 65,94 F25,1579 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level Critical = 1.52 H O Table 5.17 Estimated Multimarket Model Holding Period - Twenty Days N = 76 Dependent (Mean) Intercept British Pounds Canadian Dollars German Marks Japanese Yen Swiss Francs R2 British Pounds (.001446) .001322 (.0308) .3510 (2.732)*** -.1877 (-.8368) -.08403 (-.5454) .0007676 (.006810) -.002049 (-.1513) .1222 Canadian Dollars (-.001613) -.001859 (-.0536) -.1303 (-1.911) -.08593 (-.7220) -.03375 (.4128) .01055 (.1763) -.002816 (-.3918) .05334 German Marks (.003972) .004602 (.1320) .3511 (1.768) -.07375 (-.2127) .03152 (.1324) .0005474 (.003142) -.1293 (-.6179) .04923 Japanese Yen (.004545) .004865 (.1397) .4661 (2.534(** -.003377 (-.01051) -.3614 (-1.638) -.05514 (-.3416) .1491 (.7685) .1045 Swiss Francs (.007563) .007786 (.2229) .4209 (1.806) .2103 (.5167) -.4375 (-1.565) -.08063 (-.3942) .2611 (1.062) .06644 * Significant at the .05 level ** Significant at the .02 level Joint Test F = I■9936 *** Significant at the .01 level ’ ------ Foc . , Critical = 1.79 25,46 111 112 simultaneous equation estimation arises when all of the independent variables are the same in each question. This is the case with (3). In this special case the estimates from a simultaneous estimation are the same as the ordinary least squares estimates from estimating each equation separately and independently. The results in the multimarket tests were similar to those in the single .market tests. The same pattern of time period significance was observed with six variables being significant in the one day hold, four in the six day hold, and two in the twenty day hold. The 2 R was "low" in these tests, with one being over .125. It is interesting to contrast two of the currencies in the multimarket tests. The Canadian dollar did not have a single significant variable when it was an independent variable trying to explain the variance of some other currency. Further, no other currency, in any of the time periods, could explain any variance in the returns to Canadian Dollars. The Japanese yen was significant a total of five times as an independent variable — half of the total numbet of significant variables. In three other cases, other currencies were able to.explain some of the variance in Japanese yen. This has the possible implication of using, the Japanese yen to predict other currency movements. SUMMARY, CONCLUSIONS, AND HEDGING IMPLICATIONS . Chapter 6 Summary A firm in the United States that does business internationally must realize the exchange rates between the dollar and other currencies are constantly changing. This has at least one clear implication — the results realized from forward contracts to purchase goods or services from foreign countries may differ from their intended purpose. This is illustrated in Chapter 2. The reason the realized result of the forward contract differed from. the intended result of the forward contract was a change in the exchange rate. The process by which the exchange rate is determined and how it changes is.then described in Chapter 2. The firm can reduce the risk of adverse price fluctuations via a market where future foreign exchange rates are bought and sold — a futures market. The futures market in international currencies is the International Monetary Market of the Chicago Mercantile Exchange.. As in other futures markets, a firm can "lock-in" a price (an exchange rate) by purchasing the exchange rate today. This process is known as a long hedge. The extent to which the hedge can reduce risk depends upon the variation in the basis versus variation in the spot price of the currency. 114 Suppose a firm places a hedge and the price (exchange rate) did not move such that the firm was protected by the hedge, but in fact would have been better of financially if it had not placed the hedge. Could the firm have predicted this price movement? This, then, is the objective of the study. Can the firm predict short run exchange rate movements so as to place hedges at the most favorable time? Being able to place hedges at the most favorable time is pre­ dicted upon the ability to forecast short run exchange rate movements. As in the case of most auction markets, the futures market in foreign currencies is thought to be an efficient market. In short, this is "generally" understood to mean that "all available" information concerning a price in the future is "fully reflected" in the price quoted at any time for the price in the future. The statement that the current price of a futures contract "fully reflects" all market information implies successive price changes, or successive returns, are independent. Many different opinions on why a firm should Hedge as well as the predictability of futures market-prices were examined. Tests of efficient markets concern themselves with testing how information affects the market. This test of information is broken down into three types: weak-form tests, semi-strong form tests, and strong form tests. The test used in this study is the weak.form test — where the information set is the series of past prices. 115'. The weak form test says that markets are efficient enough to make the research of past prices a worthless way of trying to determine future prices. Further, tests of market efficiency are concerned with a general equilibrium market model. It is against this model that alternative models of the market are tested. The general equilibrium market model can take many specific forms, however, there are four general specifications of the general equilibrium market models. All of these concern themselves with the returns to a market. The first form says that the returns in the market are some constant rate, the second states that returns are zero, the third says returns conform to the Sharpe-Litner model, and the fourth says the returns conform to some other risk-return rela­ tionship. The general equilibrium market model tested against in this study is the one that says returns in the market are constant. There are many different alternative models that could be con- . structed to test the maintained model that returns are constant. The model used is an autoregressive model that is tested for three different time periods - one day, six day, and twenty day. That is the returns over one day are calculated and regressed on previous returns over one day. Similar runs were made for six day and twenty day holding periods. 116 The results of the various runs are presented in Chapter 4. Also, two different model specifications were tested and the results of each model are presented. Both models showed the same basic results, but the percentage change return model was deemed superior to the first-difference model. Conclusions The results of the models tested showed two things. First, there was more statistical dependence in the shorter time period "than in the longer time periods. The return from last time period explained more of the variance in this time period if the time period was short. Twenty day time periods, with one exception, did not show much statistical relationship. There was more in the six day periods and there was the most in the one day time periods. Secondly, the . higher the order of the lag the less it explained the additional variance of the dependent variable. The lags from many time periods ago were not significant, in general. However, lags that were first or second order appeared significant. These lags did explain some of 2 the variance in the dependent variable. However, the R in the 2various runs was never higher than .33 and in most cases the R ranged from .02 to .20. Even though some of these lags had signifi­ cant t-values they did not explain much of the variance of the dependent variable. 117 The fact that some statistical dependence exists contradicts the theory that there should be no statistical dependence. The value of this dependence to a market participant was the subject of Chapter 5. In this chapter some prediction models were used to try to exploit the statistical dependence. The results of this were negative. The statistical dependence could not be used to contruct prediction models that would yield significant returns. Also, a multimarket model was specified and tested. The results of this model were consistent with the single market model results in that there was more statistical dependence in the one day hold that in the six day hold and more in the six day hold than in the twenty day hold. It must be kept in mind that many different strategies could be constructed from the prediction models and even more could be con­ structed from the data used in this study. After trying many different prediction models, al least one would be found that would yield significant profits. In some sense, the general model of market equilibrium that states returns through time are zero in the currency futures market must be rejected. This model implies that no strategy could consis­ tently yield positive returns. However, the $100 gain prediction model did just that. But, in another sense, the general model of 118 market equilibrium that states returns through time are zero in the currency futures market must not be rejected. In the strict sense, the $100 gain model did not yield positive returns across every currency tested in this study. Also, the $50- gain strategy did not yield consistent gains across currencies and looking at the sum total of the $50 gain strategy it netted a negative amount. In order to not reject the maintained hypothesis that returns are zero in the currency futures market, models of many variations must be tested. In order to reject this model, a strategy must be found that will consistently yield positive returns. In this study, neither of these conditions were met. Certainly an infinite number of strategies were not tested, and, of the ones tested, there was not one that consistently yielded positive returns. The concrete conclusion is that some statistical dependence was found in the series of past returns in the various currency futures markets tested. This statistical dependence was found using autocorrelation coefficients, that is, laggec values of some variable. In this case the lagged variable was the percentage change in the price of a currency futures market from time t-1 to time t. The majority of the dependence was found in the first or second lag and in the one day holding period. Whether or not this statistical dependence can be exploited into 119 positive returns is the unresolved question arising from this research. Implications to Hedging Strategies Gruen (1960) pointed out that if a futues market is to be useful there must be a relationship between the futures price and cash price. In order to determine if a hedge can be useful, the degree of basis change must be compared to the degree of cash price change. Usually, this is done in this method: (A) Measure the variability of the price of the cash commodity over a period of time; (B) Measure the variability of the basis over the same period of time; and (C) If the variability of the basis is less than the variability of the cash price, an effective hedge can be placed in that commodity. Tables 6.1 - 6.5 present these calculations for twenty-six contracts in five currencies. These calculations were made during the last three months of the contract. The contracts marked by an asterisk (*) have basis variability greater than spot price variability. The cases numbered thirteen, or ten percent of the total cases. ,In these instances, a hedge would not have reduced.the risk of price variability because the basis variability was greater. Note, however. 120 Table 6.1 British Pounds — Standard Deviation of Basis and Spot Prices Contract Basis Spot N Sept 1972 .0105 .0349 64 Dec 1972 .0060 .0367 62 Mar 1973 .0128 .0542 61 June 1973 .0089 .0412 63 Sept 1973 .0073 .0550 63 Dec 1973 .0091 .0434 64 Mar 1974 .0161 .0487 61 June 1974 .0184 .0338 63 Sept 1974 .0374 .0523 63 Dec 1974 .0066 .0094 63 Mar 1975 .0185 .0320 62 June 1975 .0115 .0414 63 Sept 1975 .0083 .0548 63 Dec 1975 .0065 .0200 64 Mar 1976 .0256 .0390 61 June 1976 .0081 .0555 63 Sept 1976 .0112 .0123 63 Dec 1976 .0145 .0371 63 Mar 1977 .0132 .0138 62 *June 1977 .0072 .0038 63 Sept 1977 .0060 .0113 68 Dec 1977 .0034 .0364 63 Mar 1978 .0044 .0224 57 June 1978 .0042 ,0308 69 Sept 1978 .0039 .0379 63 Dec 1978 .0074 .0349 62 * Basis standard deviation exceeds spot standard deviation 121 Table 6,2 Canadian Dollars — Standard Deviation of Basis and Spot Prices Contract Basis Spot N Sept 1972 .0007 .0024 64 Dec 1972 .0005 .0067 62 Mar 1973 .0013 .0042 61 *June 1973 .0106 .0103 63 Sept 1973 .0138 .0151 63 Dec 1973 .0023 .0047 64 Mar 1974 .0018 .0107 61 June 1974 .0017 .0055 63 Sept 1974 .0018 .0064 63 Dec 1974 .0022 .0035 63 Mar 1975 .0018 .0051 62 June 1975 .0023 .0109 63 Sept 1975 .0012 .0040 63 Dec 1975 .0024 .0064 64 Mar 1976 .0030 .0110 61 *June 1976 .0038 .0036 63 Sept 1976 .0025 .0075 63 Dec 1976 .0052 .0210 63 Mar 1977 .0028 .0146 62 June 1977 .0024 .0036 63 Sept 1977 .0018 .0068 68 Dec 1977 .0008 .0013 63 Mar 1978 .0009 .0082 57 June 1978 .0013 .0088 69 Sept 1978 .0040 .0108 63 Dec 1978 .0009 .0049 62 * Basis standard deviation exceeds spot standard deviation 122 Table 6,3 German Marks ■— Standard Deviation of Basis and Spot Prices Contract Basis Spot N Sept 1972 .0012 .0018 64 *Dec 1972 .0010 .0009 62 Mar 1973 .0030 .0190 61 June 1973 .0028 •0118 63 Sept 1973 .0038 .0132 63 Dec 1973 .0022 .0163 64 Mar 1974 .0016 .0096 61 June 1974 .0020 .0082 63 Sept 1974 ,0016 .0074 63 Dec 1974 .0012 .0104 63 Mar 1975 .0014 .0084 62 June 1975 .0010 .0045 63 Sept 1975 .0027 .0149 63 Dec 1975 .0013 .0049 64 Mar 1976 .0009 .0039 61 June 1976 .0126 .0131 63 Sept 1976 .0004 .0043 63 Dec 1976 .0007 .0047 63 Mar 1977 .0004 .0039 62 June 1977 .0004 .0029 63 Sept 1977 .0367 .0379 68 Dec 1977 .0014 .0119 63 Mar 1978 .0025 .0101 57 June 1978 .0038 .0098 69 Sept 1978 .0028 .0099 63 Dec 1978 .0046 .0162 62 * Basis standard deviation exceeds spot standard deviation Japanese Yen — Standard Deviation of Basis and Spot 123 Table 6,4 Contract Basis Spot *Sept 1972 ■ .000047 .000012 *Dec 1972 .000041 .000005 Mar 1973 .000041 .000239 *June 1973 .000037 .000017 *Sept 1973 .000078 .000013 Dec 1973 .000032 .000091 Mar 1974 .000069 .000095 *June 1974 .000074 .000031 Sept 1974 .000010 .000086 Dec 1974 .000011 .000016 Mar 1975 .000017 .000071 June 1975 .000007 .000019 Sept 1975 .000005 .000017 *Dec 1975 .000021 .000020 Mar 1976 .000009 .000023 *June 1976 .000017 .000011 Sept 1976 .000005 .000046 Dec 1976 .000004 .000044 Mar 1977 .000006 .000056 June 1977 .000013 .000034 Sept 1977 .000014 .000035 Dec 1977 .000014 .000138 Mar 1978 .000018 ,000038 June 1978 .000025 .000093 Sept 1978 .000051 .000194 Dec 1978 .000387 .000407 Prices N 64 62 61 63 63 64 61 63 63 63 62 63 63 64 61 63 63 63 62 63 68 63 57 69 63 62 * Basis standard deviation exceeds spot standard deviation 124 Table 6.5 Swiss Francs — Standard Deviation of Basis and Spot Prices Contract Basis Spot N Sept 1972 .00076 .00113 64 *Dec 1972 .00091 .00070 62 Mar 1973 .06451 .06936 61 June 1973 .00154 .00748 63 Sept 1973 .00367 .01194 63 Dec 1973 .00158 .00871 64 Mar 1974 .00169 .01047 61 June 1974 .00156 .00734 63 Sept 1974 .00135 .00352 63 Dec 1974 .00267 .01549 63 Mar 1975 .00281 .00875 62 June 1975 .00093 .00556 63 Sept 1975 .00083 .01111 63 *Dec 1975 .05306 .05276 64 Mar 1976 .00238 .00319 61 June 1976 .00123 .00582 63 Sept 1976 .00180 .00214 63 Dec 1976 .00139 .00241 63 Mar 1977 .00124 .00627 62 June 1977 .00265 .00393 63 Sept 1977 .00099 .00581 68 Dec 1977 .00194 .01792 63 Mar 1978 .00404 .01824 57 June 1978 .00318 .01820 69 Sept 1978 .00618 .03271 63 Dec 1978 .00812 .00349 62 * Basis standard deviation exceeds spot standard deviation 125 that most of these instances occur during the early years of the IMM when volume was very low. This low volume resulted in a very stable futures price — hence, a small variance. Another implication to hedging in the currency futures market also has to do with the basis. Table 6.6 shows the average basis near the end of a currency contract. Assume that a firm, such as National Blivit Company, has agreed to import some goods sometime in the delivery month but before the expiration date of the contract. If the firm has decided to hedge the currency, the price the firm has "locked in" is the futures price on the day the hedge was placed minus the expected basis. If, at any time the firm can contract a price in the cash, market that is "better" than the "lockin-in" price, the firm could close its future position and take a cash position. For example, assume that National Blivit Company is going to import some goods from West Germany sometime during the first week of June. At the time the agreement was made, assume that the June futures price is .5400. The expected basis is .0005; therefore the firm has "locked-in" a price of .5395. Note that in this example the "lock-in" price is a price the firm can buy D-marks. If the firm sees the cash and futures price for D-marks fall below this "locked-in" price, it could close its futures position. If the cash price rises Again.to Table 6,6 Average Basis During Delivery Month — 1972-1978 126 Month Currency March June September December British Pound -.0052 -.0023 .0034 -.0011 Canadian Dollar -.00003 -.0040 .0001 -.0002 German Mark .00002 .0005 -.0001 -.00001 Japanese Yen .00004 -.000001 .0000 .0000 Swiss Franc .0007 .0004 .00004 .0003 127 some level the firm sets, a futues position can again be opened. However, when the firm has closed its futures position, it again becomes a speculator in the currency market. Also, the firm might watch the forward contract market if the. firm is more conservative. If the price in the forward market reaches a "favorable" level, then the firm can close its futues position and buy its exchange rate on. the forward market. However, this leads to less flexibility for the firm than by dealing in the futues market and watching basis movements, which is a more risk-taking position. The important thing is that once a hdge .is placed it should not be completely ignored. That is, conditions change in the market that would make it undesirable to be hedged. For example, if prices are falling, the firm that has a long position in the futures market might be better off unhedged. But, this again introduces some sort of specula tion or risk taking. An importing firm can successfully reduce the risk of adverse price movements via the futures market. When a firm sees a price for a foreign currency in the futures market that allows the firm a profit on the international transaction, the firm can hedge that price. When the firm starts lifting and placing hedges to take advantage of price movements, it must realize that it is taking more risk and speculating on future price movements. Suggestions for Further Research The object of this study was to determine if futures prices in currency futures markets are predictable. The conclusion was that, in general, they were not. The implications to hedgers are: (I) trying to place hedges based upon the ability to predict futures prices will not work, and (2) successful hedges can still be placed even though futues prices are not predictable. This research used the information set of past prices to try to predict future prices in the futures market. Perhaps other infor­ mation sets could be tested. For example, in this study such economic variables as relative inflation rates, trade balances, and rates of money supply changes were assumed to be reflected in the futures price. Perhaps these economic variables could be tested as indepen­ dent variables that determine futures prices. Further, this study looked at relatively short run futures price changes. A longer time horizon could be tested, for example, the length of the contract for one year or longer. A model using lagged returns of several currencies to predict returns of a specific currency could be developed. Clearly, many different types of models could be tested using different data sets, independent variables, assumptions and functional forms. Certainly, one point above could be expanded. • This is the fact that successful hedges can still be placed despite the fact that. futures market prices are, in general, unpredictable. Basis movements in currency futures markets could, be modeled. From basis movements heding strategies can be constructed and simulated. Further, the concept of using a combination of futures markets and forward markets in a hedging strategy could be examined. 129 APPENDIX I Table I-I Summary of Contract Specifications British Canadian Specification Pounds Dollars Ticker Symbol BP CD Contract Size 25,000 BP 100,000 CD Approx. U.S. Dollars $51,SOOa $86,000a Minimum Fluctuation .0005 .0001 ($12.50) ($10) Daily Limit (Normal) .0500 .0075 ($1,250) ($750) Commissions * * Margins * * All currency contracts' delivery months are: a As of May 30, 1979 Currency German Japanese Swiss Marks Yen Francs DM JY SF 125,000 DM 12,500,000 JY 125,000 SF $65,000a $57,000a $73,000a .0001 .000001 .0001 ($12.50) ($12.50) ($12.50) .0060 .000060 .0060 ($750) ($750) ($750) * * * * * * March, June, September, December * Consult your broker Source: International Monetary Market Yearbook 1977-1978 131 132 Table 1-2 Volume of British Pound Futures Market Number of Contracts Traded Month 1978 1977 1976 1975 1974 1973 1972 Janauary 15,576 2,239 599 1,218 3,900 5,441 — February 11,360 2,155 434 313 978 2,728 — March 17,360 2,561 2,143 508 698 1,087 — April 22,667 1,658 2,679 789 391 981 —- May 22,983 1,283 2,187 1,094 476 2,143 1,271 June 21,956 1,851 2,686 2,456 775 6,385 1,655 July — 3,598 1,279 1,778 442 1,774 2,710 August — 6,059 984 1,690 970 490 1,104 September — 7,997 4,986 1,864 1,692 579 1,876 October ---- — 17,712 6,063 1,274 362 1,026 1,948 November — 17,808 5,513 660 1,158 368 911 December —— 13,870 3,912 1,371 2,191 8,410 3,312 TOTAL 111,902 78,701 33,465 15,015 14,033 31,412 14,787 Source: International Monetary Market Yearbook 1977-1978 133 Table 1-3 Volume of Canadian Dollar Futures Market Number of Contracts Traded Month 1978 1977 1976 1975 1974 1973 1972 Janauary 7,791 11,769 297 1,568 831 8,529 — February 7,083 18,946 248 16 314 2,565 — March 8,984 34,170 381 10 238 1,155 — April 17,470 10,584 492 113 202 832 — May 14,511 3,248 1,499 106 123 2,621 7,074 June — 13,424 1,001 93 539 5,398 9,672 July — 8,528 607 73 75 4,707 3,334 August — 11,447 837 42 49 1,322 4,450 September — 6,449 1,032 77 188 688 1,306 October — 19,466 1,243 92 250 11 919 November — 9,167 3,333 186 172 26 3,462 December — 13,941 7,098 301 718 1,327 8,587 TOTAL 55,839 161,139 17,068 2,677 3,699 29,161 38,804 Source: International Monetary Market Yearbook 1977-1978. 134 Table 1-4 Volume of German Mark Futures Market Number of Contracts Traded Month 1978 1977 1976 1975 1974 1973 1972 Janauary 12,864 9,646 2,921 7,470 12,839 4,680 — February 14,410 7,167 3,522 3,628 676 2,488 — March 22,901 6,764 4,481 5,306 1,010 3,195 — April 28,721 9,008 2,317 4,172 1,547 1,494 — May 32,464 8,691 3,070 3,143 2,892 2,258 2,906 June 24,707 6,551 2,451 2,425 2,667 1,055 3,785 July — 18,441 1,495 4,292 2,398 989 1,237 August — 11,416 3,373 3,527 3,808 2,149 2,299 September — 8,097 4,269 4,335 2,300 5,234 1,004 October — 14,922 8,304 5,265 3,323 1,384 1,114 November — 16,434 3,705 5,382 2,804 24,388 1,094 December ---- — 17,231 4,979 5,848 13,183 27,810 5,879 TOTAL 136,067 134,368 44,887 54,793 49,447 77,264 19,318 Source: International Monetary Market Yearbook 1977-1978. 135 Table 1-5 Volume of Japanese Yen Futures Market Number of Contracts Traded Month 1978 1977 1976 1975 1974 1973 1972 Janauary 11,634f 216 14 629 1,951 17,196 — February 18,751 443 198 153 172 8,410 — March 35,098 3,107 38 174 173 20,487 —- April 22,242 4,149 165 82 238 9,463 — May 25,479 1,142 22 27 84 5,377 486 June 38,290 5,006 75 40 842 21,767 1,273 July — 4,757 109 12 668 6,499 147 August — 3,378 158 70 491 12,035 3,900 September — 5,470 157 258 900 3,993 1,813 October — 28,387 138 228 80 15,050 4,063 November — 13,921 157 28 353 3,422 9,571 December — 12,285 218 89 1,287 1,954 22,736 TOTAL 151,494 82,261 1,449 1,790 7,239 125,653 42,989 Source: International Monetary Market Yearbook 1977-1978 136 Table 1-6 Volume of Swiss Francs Number of Contracts Traded Month 1978 1977 1976 1975 1974 1973 1972 Janauary 18,992 3,301 2,972 8,474 3,309 4,152 — February 23,486 4,864 2,223 3,482 2,181 949 — March 23,985 8,733 3,467 5,686 3,018 3,729 — April 22,242 8,057 2,377 5,462 3,661 762 — May 27,776 4,160 4,099 7,443 5,978 2,920 984 June 24,603 4,457 5,607 3,656 3,686 1,686 2,641 July — ---- 5,845 3,519 9,086 4,133 767 2,010 August — 8,939 2,411 6,254 1,881 1,080 1,275 September — 9,629 2,582 6,617 1,537 315 4,409 October — 14,992 3,514 6,758 2,373 1,444 1,572 November — 13,215 2,104 3,179 3,664 3,316 1,458 December — 20,776 2,371 3,836 7,084 893 3,372 TOTAL 141,084 106,968 37,246 69,933 42,505 22,013 17,721 Source: International Monetary Market Yearbook 1977-1978. Table 1-7 Volume of All Futures Markets — CME and AMM Month Number of Contracts Traded 1978 1977 1976 1975 1974 1973 1972 January 583,327 461,105 521,782 446,630 454,319 561,918 — February 699,337 391,485 409,231 348,098 384,202 539,925 — March — 545,667 461,650 379,634 448,720 642,770 — April — 652,306 536,745 445,883 442,672 447,543 — May — 617,824 481,508 523,336 500,440 435,446 357,953 June — 585,198 526,983 573,178 379,724 407,001 413,907 July — 482,667 470,518 550,351 464,971 457,249 336,416 August — 502,380 400,270 477,258 397,494 564,167 391,084 September — 426,765 413,438 525,828 330,591 337,165 384,118 October — 453,310 505,354 563,267 458,355 463,294 371,008 November — 470,600 429,055 439,046 360,269 405,720 360,464 December — 472,257 400,761 486,252 355,855 361,094 457,176 TOTAL I1,282,664 6,061,565 5,557,295 5,758,761 4,977,612 5,623,292 3,072,116 Source: Chicago Mercantile Exchange Yearbook 1977-1978. 137 Volume oi Table 1-8 : All Futures Markets -— IMM Month Number oJc Contracts Traded 1978 1977 1976 1975 1974 1973 1972 January 298,311 85,302 53,804 89,161 63,820 42,914 — February 246,453 89,529 46,446 39,377 7,871 18,656 — March 327,468 164,016 44,492 46,329 9,679 29,981 April 324,223 124,103 41,478 43,951 12,919 16,308 —- May 316,571 94,566 44,190 50,477 15,399 18,684 12,799 June 351,437 99,225 48,207 40,259 18,948 43,817 19,124 July — 108,831 45,831 56,561 14,475 15,515 9,553 August — 117,458 43,751 58,150 17,149 18,848 13,421 September — 128,849 60,465 62,025 22,018 13,968 10,417 October — 289,565 57,015 56,798 42,257 30,654 9,750 November — 272,698 82,212 51,656 28,869 57,378 21,020 December — 242,540 76,029 48,451 62,864 129,651 48,844 TOTAL I,909,463 1,816,682 664,370 643,195 316,268 436,374 144,928 Source: International Monetary Market Yearbook 1977-19/S. 138 Table 1-9 IMM Volum As A Percent of CME and AMM Volume 139 Year Percent 1978 N.A. 1977 .2997 1976 .1160 1975 .1117 1974 .0635 1973 .0776 1972 .0472 APPENDIX II 141 Currency British Pounds Canadian Dollars German Marks Japanese Yen Swiss Francs Table II-I Return Specification — Percentage Change V eIl = 612 = ••• = 615 = 0 Holding Period I-Day F Calc. F Grit. .67 2.21 .53 2.21 .48 2.21 .30 2.21 .52 2.21 6-Day F Calc. F. Grit. .34 2.21 1.05 2.21 .22 2.21 1.40 2.21 .29 2.21 20-Day F Calc. F Grit. .39 2.45 1.22 2.45 .41 2.45 .86 2.45 .21 2.45 142 Table II-2 Return Specification — First Difference Ho: 3Il = f3I2 = = B15 = O Holding Period Currency 1- F Calc. -Day F Crit. 6- F Calc. -Day F. Crit. 20- F Calc. -Day F Crit British Pounds 1.02 2.21 .45 2.21 .43 2.45 Canadian Dollars .49 2.21 1.16 2.21 1.22 2.45 German Marks .20 2.21 .16 2.21 .31 2.45 Japanese Yen .40 2.21 2.88* 2.21 .98 2.45 Swiss Francs 1.58 2.21 .60 2.21 .11 2.45 * Reject mu 143 Table II-3 Return Specification — Percentage Change V B8 = Sg = ... = S15 = 0 Holding Period Currency I- F Calc. -Day F Grit. 6- F Calc. -Day F. Crit. 20- F Calc. -Day F Crit. British Pounds 1.33 1.94 .32 1.94 .87 2.18 Canadian Dollars .52 1.94 .79 1.94 1.23 2.18 German Marks 2.35* 1.94 .23 1.94 .42 2.18 Japanese Yen 1.57 1.94 1.32 1.94 .55 2.18 Swiss Francs 1.05 1.94 .31 1.94 .22 2.18 * Reject mu 144 Table II-4 Return Specification — First Difference H0 : Bg = 69 = ... = S15 = 0 Holding Period Currency 1- F Calc. ■Day F Grit. 6- F Calc. -Day F. Grit. 20- F Calc. -Day F Crit British Pounds 1.66 1.94 .39 1.94 .89 2.18 Canadian Dollars .47 1.94 .93 1.94 1.20 2.18 German Marks 1.85 1.94 .41 1.94 .35 2.18 Japanese Yen 1.87 1.94 2.45* 1.94 .63 2.18 Swiss Francs 1.89 1.94 .97 1.94 .19 2.18 * Reject mu 145 Table II-5 Return Specification — Percentage Change Hq : 62 = 33 = ... = B15 = 0 Holding Period Currency 1- F Calc. -Day F Grit. 6- F Calc. -Day F. Crit. 20- F Calc. -Day F Crit British Pounds .88 1.67 1.16 1.67 .64 1.92 Canadian Dollars 2.15* 1.67 .91 1.67 .97 1.92 German Marks 2.31* 1.67 .58 1.67 1.62 1.92 Japanese Yen 2.06* 1.67 .81 1.67 .58 1.92 Swiss Francs 1.62 1.67 1.25 1.67 .57 1.92 * Reject mu 146 Table II-6 Return Specification — First Difference Ho : B2 . . qqq . IhPd = 0 Holding Period Currency I- F Calc. -Day F Crit. 6- F Calc. -Day F. Crit. 20- F Calc. -Day F Crit British Pounds 1.08 1.67 1.09 1.67 .77 1.92 Canadian Dollars 2.06* 1.67 .96 1.67 .99 1.92 German Marks 1.93* 1.67 .51 1.67 1.26 1.92 Japanese Yen 1.65 1.67 1.98* 1.67 .84 1.92 Swiss Francs 3.03* 1.67 3.10* 1.67 .57 1.92 Reject mu* 147 Table II-7 Return Specification — Percentage Change mu v £>1 = $2 = • ■ • = ^15 = 0 Holding Period I-Day 6-Day 20-Day Currency F Calc. F Crit. F Calc. F. Crit. F Calc. F Crit British Pounds .91 1.67 1.98 1.67 1.34 1.92 Canadian Dollars 2.05* 1.67 .90 1.67 .91 1.92 German Marks 2.48* 1.67 1.10 1.67 1.51 1.92 Japanese Yen 8.76* 1.67 2.24* 1.67 .55 1.92 Swiss Francs 3.51* 1.67 1.22 1.67 .54 1.92 * Reject mu 148 Table II- 8 Return Specification — First Difference V S1 = e2 = ... = e15 = 0 Holding Period Currency I- F Calc. -Day F Crit. 6- F Calc. -Day F. Grit. 20- F Calc. -Day F Crit British Pounds 1.06 1.67 1.03 1.67 1.09 1.92 Canadian Dollars 1.95* 1.67 .97 1.67 .93 1.92 German Marks 2.33* 1.67 .87 1.67 1.19 1.92 Japanese Yen 6.78* 1.67 2.36* 1.67 .81 1.92 Swiss Francs 5.81* 1.67 2.90* 1.67 .54 1.92 * Reject mu 149 H : a = B, = . . . = Sic = Oo I 15 Return Specification — Percentage Change Table II-9 Currency I 6 20 British Pounds Mean .00009116 .0005687 .001402 (S.D.) (.004818) (.01220) (.02204) t .0189 .0466 .0636 Canadian Dollars Mean -.0005077 -.001366 (S.D.) (.005594 .01193 t -.0908 -.1145 German Marks Mean .001311 .001613 (S.D.) (.01546) (.02813) t .0848 .0573 Japanese Yen Mean .005071 (S.D.) (.02830) t .1792 Swiss Franc Mean .002341 .006481 (S.D.) (.01927) (.03584) t .1215 .1808 150 H : O = P 1 = . . . = B ic = Oo l 15 Return Specification — First Differences Table II-IO Currency I 6 20 British Pounds Mean .0001485 .0009508 .001831 (S.D.) (.009878) (.02508) (.04505) t .0150 .0379 .0406 Canadian Dollars Mean -.0004931 -.001305 (S.D.) (.005423) (.01159) t -.0909 -.1126 German Marks Mean .0004829 .0006532 (S.D.) (.006674) (.01228) t .0724 .0532 Japanese Yen Mean .00001935 (S.D.) (.0001212) t .1597 Swiss Franc Mean .002647 (S.D.) (.01727) t .0533 151 British Pounds Table II-ll.a ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — TWENTY DAYS LAG I 2 3 4 5 6 .3721 -.05622 -.1256 -.07248 .09373 se .1457 .1586 .1621 .1639 .1661 t 2.555** -.3569 -.7750 .4423 .5644 LAG 6 7 8 9 10 6 -.1690 .06039 .1935 -.2731 .2602 se .1636 .1658 .1681 .1681 .1666 t -1.033 .3692 1.151 -1.625 1.561 LAG 11 12 13 14 15 6 .08280 -.06249 .01935 .04419 -.1834 8B .1694 .1699 .1704 .1649 .1413 t .4886 -.3678 .1136 .2679 -1.298 * Significant at the .05 level ** Significant at the .02 level * *** Significant at the .01 level N = 62 R2 = .2453 Mean of the Dependent Variable: Intercept: .0005518 .001402 152 Table II-ll.b ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — TWENTY DAYS Canadian Dollars LAG I 2 3 4 5 B .09749 .01640 .1267 .1548 .05800 < CO. .1468 .1451 .1396 .1460 .1522 t -.6643 -.1130 .9080 1.060 .3810 LAG 6 7 8 9 10 B -.08601 -.08560 .01751 -.1760 .2312 S .1531 .1554 .1571 .1691 .1634 t -.5619 -.5510 .1114 -1.087 1.411 LAG 11 12 13 14 15 B .02373 -.05701 -.3194 -.2250 .01447 5B .1647 .1683 .1629 .1770 .1788 t -.1440 -.3388 -1.967* -1.272 .08096 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 62 R2 = .2297 Mean of the Dependent Variable: -.001932 Intercept: -.001932 153 Table II-ll.c ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — TWENTY DAYS German Marks * LAG I 2 3 4 5 6 .003773 .03057 .1312 -.4006 -.04442 se .1483 .1549 .1420 .1414 .1597 t .02558 .1973 .9244 -2.833*** -.2781 LAG 6 7 8 9 10 6 -.4139 .2717 -.05252 1.1513 -.1528 S. 6 .1582 .1654 .1733 .1701 .1516 t -2.616*** 1.642 -.3030 .8898 -1.008 LAG 11 12 13 14 15 6 .03858 -.1340 .09976 .07452 .02672 S 6 .1533 .1472 .1444 .1405 .1334 t .2516 -.9108 .6907 .5304 -.2004 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N - 62 R2 = .3302 Mean of the Dependent Variable: .001613 Intercept: .002855 154 Table II-ll.d ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — TWENTY DAYS JAPANESE YEN LAG i 2 3 4 5 6 -.04332 .05234 -.1139 -.1381 -.1503 S .1501 (.1570 .1602 .1681 .1729 t -.2886 .3334 -.7109 -.8533 -.8694 LAG 6 7 8 9 10 6 -.03263 .08979 .08578 .04328 .07307 5B .1774 .1752 .1480 .1643 .1557 t -.1840 .5126 .5795 .2634 .4693 LAG 11 12 13 14 15 B .06696 .1580 .09425 .2651 .09718 s; .1547 .1527 .1537 .1624 .1662 t .4327 1.035 .6131 1.632 .5847 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 62 R2 = .1516 Mean of the Dependent Variable: .005071 Intercept: .003550 155 Table II-ll.e ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — TWENTY DAYS SWISS FRANCS LAG I 2 3 4 5 6 .07246 .1527 -.1085 -.1710 .03627 < C Q cz> .1463 .1615 .1646 .1663 .1919 t .4954 .9451 -.6590 -1.028 .1890 LAG 6 7 8 9 10 8 -.09183 .1394 .02448 - .1112 -.05993 sS .1896 .1934 .2101 .2221 .1982 t -.4843 .7211 .1165 - .5009 -.3024 LAG 11 12 13 14 15 8 -.1703 .03033 .01300 .01198 0.1021 lc .1998 .1933 .1904 .1853 .1820 8 t -.8526 .1596 .06829 .06468 -.5608 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 62 R2 = .1490 M e a n of the D e p endent Variable: .006481 Intercept: .009250 156 British Pounds Table 11-12.a ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — TWENTY DAYS LAG I 2 3 4 5 B .3687 -.04164 -.1218 .03905 .1413 s; .1453 .1575 .1609 .1617 .1634 t 2.538** -.2644 -.7573 .2414 .8649 LAG 6 7 8 9 10 B -.2467 .04146 .2111 -.2872 .2417 8B .1620 .1666 .1680 .1659 .1620 t -1.523 .2489 1.256 -1.731 1.492 LAG 11 12 13 14 15 B .09093 -.04441 -.02162 .08186 -.1780 B^ .1641 .1647 .1651 .1575 .1309 t .5540 -.2697 -.1310 .5198 -1.360 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 62 R2 = .2626 Mean of the Dependent Variable: .001831 Intercept: .0003256 157 Canadian Dollars Table 11-12.b ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — TWENTY DAYS LAG I 2 3 4 5 3 -.09899 -.008083 .1448 .1560 .01842 s£ .1466 .1451 .1401 .1463 .1517 t -.6751 -05569 1.033 1.066 .1214 LAG 6 7 8 9 10 3 -.09752 -.08818 .04213 -.1641 .2078 8B .1521 .1545 .1555 .1599 .1609 t -.6410 -.5708 .2709 -1.027 1.291 LAG 11 12 13 14 15 3 -.04822 -.07354 -.2981 -.2167 .01376 S3 .1611 .1639 .1586 .1711 .1728 t -.2993 -.4487 -1.880 -1.266 .07963 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 62 R2 = .2334 Mean of the Dependent Variable: -.001305 Intercept: -.001902 158 German Marks Table 11-12.c ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — TWENTY DAYS LAG I 2 3 4 5 e -.07183 .06911 .1468 -.3713 -.08708 s B .1478 .1556 .1488 .1486 .1679 t -.4859 .4442 .9863 -2.499** -.5185 LAG 6 7 8 9 10 B -.3756 .2707 -.03980 .1859 -.1588 s B .1666 .1721 .1838 .1852 .1672 t -2.254** 1.573 -.2165 1.004 -.9496 LAG 11 12 13 14 15 B .04640 -.1351 .09056 .07889 -.008324 SB .1706 .1631 .1593 .1558 .1511 t .2719 -.8283 .5685 .5062 -.05509 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 62 R2 = .2796 Mean of the Dependent Variable: .0006532 Intercept: .001002 159 Table 11-12.d ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — TWENTY DAYS Japanese Yen LAG I 2 3 4 5 e -.1416 .01842 -.1668 -.01252 -.3239 si .1537 .1663 .1671 .1689 .1837 t -.9216 .1107 -.9984 -.07410 -1.764 LAG 6 7 8 9 10 B .1089 .1530 -.06219 .1999 .008746 3B .1959 .1936 .1791 .2073 .1992 t .5561 .7903 -.3473 .9646 .04391 LAG 11 12 13 14 15 B .08606 .2300 .2160 .1491 .2516 8B .1986 .2023 .2014 .2091 .2065 t .4334 1.136 1.073 .7128 1.218 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 62 R2 = .2100 Mean of the Dependent Variable: .00001935 Intercept: .00001386 160 Table 11-12.e ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — TWENTY DAYS Swiss Francs LAG I 2 3 4 5 6 .02503 .1812 -.2752 -.1077 .09169 s£ .1465 .1669 .1752 .1805 .2138 t .1609 1.086 -1.571 -.5968 .4289 LAG 6 7 8 9 10 B .09757 .1768 -.01541 -.1518 -.1015 8B .2120 .2158 .2543 .2760 .2526 t .4603 .8195 -.06060 -.5498 -.4018 LAG 11 12 13 14 15 B -.1379 -.002717 .01680 -.06273 -.08518 < C Q CQ .2578 .2508 .2483 .2458 .2464 t -.5351 -.01084 .06766 -.2552 -.3457 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 62 R2 = .1490 Mean of the Dependent Variable: .002647 Intercept: .003669 161 Table 11-13.a ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — SIX DAYS BRITISH POUNDS LAG I 2 3 4 5 6 -.002496 .06909 .1577 .04902 .03286 .06577 .06574 .06594 .06751 .07057 B t -.03795 1.051 2.392** .7261 .4656 LAG 6 7 8 9 10 B .05321 -.1348 -.04636 .04215 -.01574 .07125 .07179 .07224 .07198 .07195 B t .7467 -1.877 -.6418 .5856 -.2187 LAG 11 12 13 14 15 B .01162 -.06406 -.05547 .04234 .001352 qA .07208 .07199 .07116 .06989 .06898 B t .1612 -.8899 -.7795 .6058 .01960 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N I= 246 R2 = .06627 Mean of the Dependent Variable: .0005687 Intercept: .0005094 162 CANADIAN DOLLARS Table 11-13.b ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — SIX DAYS LAG I 2 3 4 5 6 .06361 .07048 -.07007 .001540 -.09817 Sg .06587 .06600 .06595 .06612 .06628 t .9658 1.068 ■1.062 .02329 -1.481 LAG 6 7 8 9 10 B .07666 -.01778 ■ .002633 -.0003867 .06929 lc .06811 .06830 .06838 .06838 .06844 6 t 1.126 -.2603 -.03850 -.005655 1.012 LAG 11 12 13 14 15 B .09932 -.01757 .09078 -.01578 .05652 lc .06838 .06885 .06884 .06880 .06884 B t * ** *** N = R2 1.453 -.2552 Significant at the .05 Significant at the .02 Significant at the .01 246 = .05554 1.319 level level level -.2294 .8210 Mean of the Dependent Variable: -.0005077 Intercept: -.0003493 163 Table 11-13.c ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — SIX DAYS GERMAN MARKS LAG I 2 3 4 5 6 .1641 .1497 -.02501 -.03812 -.07418 S6 .06595 .06689 .06754 .06794 .07204 t 2.488** 2.239* -.3703 -.5612 -1.030 LAG 6 7 8 9 10 6 .03647 -.005604 -.03272 .02497 -.05017 5S .07348 .07413 .07400 .07399 .07409 t .4964 -.07560 -.4421 .3375 -.6772 LAG 11 12 13 14 15 6 .03033 .01795 -.07545 -.004640 .02362 sS .07406 .07438 .07471 .07363 .07377 t .4095 .2413 -1.010 -.06301 .3202 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 246 R2 = .06692 Megn of the Dependent Variable: .001311 Intercept: .001143 164 JAPANESE YEN Table 11-13.d ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — SIX DAYS LAG I 2 3 4 5 S .3017 .001476 .009426 -.02043 .03062 Se .06558 .06830 .06824 .06861 .07005 t 4.600*** .02162 .1381 -.2978 .4372 LAG 6 7 8 9 10 B -.004178 -.03371 -.09814 .1064 -.005758 5B .07028 .07047 .07014 .07034 .07077 t -.05944 -.4784 -1.399 1.513 -.08136 LAG 11 12 13 14 15 B .004908 -.1344 .01584 .1049 -.09624 < Q Q LO .07075 .07070 .07163 .07194 .06895 t .06938 -1.901 .2211 1.459 -1.396 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 246 R2 = .1275 Mean of the Dependent Variable: .001638 Intercept: .001371 165 Table II-13.e ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — SIX DAYS SWISS FRANCS LAG I 2 3 4 5 6 .08612 .1248 -.1317 .6143 -.05302 s i .06598 .06623 .06673 .06843 .07133 t 1.305 1.884 -1.973* 2.401** -.7433 LAG 6 7 8 9 10 B - .09081 .02062 -.01914 .06228 -.02850 < aa^ zn .07160 .07186 .07214 .07339 .07429 t -1.268 .2870 -.2652 .8486 -.3838 LAG 11 12 13 14 15 B - .03278 -.07919 .01103 .008026 -.009522 sS .07428 .07499 .07578 .07557 .07506 t -.4413 -1.056 .1455 .1062 -.1269 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 246 R2 - .07356 Mean of the Dependent Variable: .002341 Intercept: .002281 166 Table 11-14.a BRITISH POUNDS ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — SIX DAYS LAG I 2 3 4 5 B .01133 .07004 .1388 .07587 .01959 se .06566 .06555 .06578 .06699 .07017 t .1725 1.069 2.110* 1.133 .2791 LAG 6 7 8 9 10 6 .05129 - .1351 -.05468 .04127 -.007251 sS .07082 .07126 .07168 .07143 .07139 t .7242 -1.896 -.7629 .5778 -.1016 LAG 11 12 13 14 15 B .03337 - .06503 -.06939 .03778 -.003848 S6 .07139 .07121 .07072 .06923 .06768 t .4673 -.9133 -.9813 .5457 -.05686 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 246 R2 = .06322 Mean of the Dependent Variable: .0009508 Intercept: .0008618 167 CANADIAN DOLLARS Table 11-14.b ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — SIX DAYS LAG i 2 3 4 5 6 .06903 .07361 -.07013 -.01162 -.08411 S .06589 .06604 .06588 .06604 .06615 t 1.048 1.115 -1.065 -.1759 1.272 LAG 6 7 8 9 10 6 .08771 - .02115 - .004724 -.002651 .08458 5B .06741 .06766 .06773 .06772 .06766 t 1.301 - .3126 - .06974 -.03914 1.250 LAG 11 12 13 14 15 6 .09362 - .01865 .1097 .01707 .04798 < CO. in .06772 .06811 .06807 .06816 .06815 t 1.383 - .2738 1.6111 - .2504 .7040 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 246 R2 = .05976 Mean of the Dependent Variable: -.0004931 Intercept: -.0003262 168 Table 11-14.c ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — SIX DAYS GERMAN MARKS LAG I 2 3 4 5 6 .1508 .08794 -.02448 -.03616 -.09668 S" .06598 .06681 .06702 .06756 .07467 e t 2.286*** 1.316 -.3653 -.5352 -1.295 LAG 6 7 8 9 10 i .03717 -.01448 -.004388 .06940 - .1066 S" .07715 .07823 .07814 .07807 .07832 6 t .4817 -.1851 -.05616 .8889 -1.361 LAG 11 12 13 14 15 8 .02874 .007846 -.06416 -.005969 - .001314 lc .07850 .07916 .07954 .07897 .07958 6 t .3661 .09911 -.8066 -.07558 - .01651 * Significant at the .05 level *A Significant at the .02 level *** Significant at the .01 level N =; 246 R2 = .05391 Mean of the Dependent Variable: .0004829 Intercept: .0004781 169 Table 11-14.d ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — SIX DAYS JAPANESE YEN LAG I 2 3 4 5 B .1581 .05310 -.01595 .04032 .06879 Sg .06625 .06720 .06697 .06666 .06897 t 2.387** .7901 -.2382 .6049 .9975 LAG 6 7 8 9 10 g - .01230 -.9435 -.06073 .1094 .09207 Sg .06934 .06966 .06985 .06995 .07095 t -.1779 -1.354 -.8695 1.564 1.298 LAG 11 12 13 14 15 g -.04981 - .2473 .09904 .05685 .01138 Sg .07128 .07082 .07309 .07348 .07231 t -.6987 -3.492*** 1.355 .7736 .1574 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 246 R2 = .1343 Mean of the Dependent Variable: .000008943 Intercept: .000006916 170 Table 11-14.e ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — SIX DAYS Swiss Francs LAG I 2 3 4 5 B .06010 .1121 -.2134 .3107 -.03641 8B .06617 .06629 .06675 .07074 .07778 t .9083 1.691 -3.196*** 4.392*** -.4681 LAG 6 7 8 9 10 B -.1359 .01865 -.06901 .1133 -.1073 sS .07778 .07837 .07906 .08109 .08368 t -1.747 .2379 -.8730 1.397 -1.282 LAG 11 12 13 14 15 6 -.04942 -.1260 -.006069 .01024 -.03642 s£ .08403 .08578 .08736 .08755 .08694 t -.5881 -1.469 -.06947 .1169 -.4189 * Significant at: the .05 level ** Significant at: the .02 level *** Significant at: the .01 level N = 246 R2 = .1589 Mean of the Dependent Variable: .0008512 Intercept: .001003 171 Table 11-15.a ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — ONE DAY British Pounds LAG i 2 3 4 5 0 .02496 .0009458 -.01800 -.02442 .01107 s B .02524 .02524 .02523 .02524 .02524 t .9891 .03747 -.7135 -.9676 .4385 LAG 6 7 8 9 10 6 -.009425 -.0009739 .01460 .02023 .06169 s B .02521 .02520 .02519 .02518 .02518 t 0.3739 -.03864 .5795 .8032 2.450** LAG 11 12 13 14 15 0 .02616 -.001719 .01379 .01747 .03066 s B .02522 .02522 .02521 .02521 .02519 t 1.037 0.06818 .5470 .6931 1.217 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .008605 Mean of the Dependent Variable: .00009116 Intercept: 00008406 172 Canadian Dollars Table 11-15.b ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — ONE DAY LAG I 2 3 4 5 e -.02156 -.06255 .05504 .03904 .04801 s§ .02517 .02518 .02522 .02525 .02526 t -.8566 “2.485** 2.183* 1.546 1.900 LAG 6 7 8 9 10 e -.03 300 .07549 .02364 .02121 -.0008888 < Q Q .02529 .02529 .02533 .02526 -.02527 t 0.5139 2.985*** .9332 .8395 -.03518 LAG 11 12 13 14 15 6 .006251 -.006467 .02829 -.008724 -.02501 Sg .02526 .02522 .02521 .02516 .02515 t .2475 -.2565 1.122 -.3468 -.9947 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .01906 Mean of the Dependent Variable: -.00008163 Intercept: -.00006950 173 Table 11-15.c German Marks ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — ONE DAY LAG I 2 3 4 5 6 .03815 .04746 -.001732 .004408 -.01535 S- .02524 .02525 .02527 v02527 .02527 6 t 1.511 1.879 -.06853 .1744 -.6074 LAG 6 7 8 9 10 6 .04109 .04334 .04093 .05125 .07226 H .02522 .02521 .02521 .02521 .02522 t 1.629 1.719 1.624 2.033* * 2.865*** LAG 11 12 13 14 15 8 .0009880 -.007457 .03207 0.2044 -.003362 < C Q CQ .02529 .02530 .02530 .02528 .02527 t .03906 -.2948 1.268 .8083 -.1330 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .02276 Mean of the Dependent Variable: .0002138 Intercept: .0001414 174 Japanese Yen Table 11-15.d ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — ONE DAY LAG I 2 3 4 5 8 -.2682 -.06388 -.05507 -.01167 .05160 S6 .02518 .02607 .02611 .02614 .02612 t -10.65*** -2.450** 2.109* -.4466 1.975* LAG 6 7 8 9 10 6 .07082 .03690 .04635 .07354 .06088 < C Q CD .02612 .02611 .02610 .02610 .02608 t 2.711 1.413 1.776 2.817 2.335 LAG 11 12 13 14 15 B .02202 .007766 -.01883 -.002724 -.007656 5S .02609 .02609 .02608 .02604 .02519 t .8440 .2977 -.7221 -.1046 -.3040 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .07686 Mean of the Dependent Variable: .0002649 Intercept: .0002768 175 Table 11-15.e ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — PERCENTAGE CHANGE HOLDING PERIOD — ONE DAY Swiss Francs LAG I 2 3 4 5 6 .1250 .02569 .04375 -.02717 -.03521 SB .02522 .02541 .02544 .02547 .02546 t 4.956*** 1.011 1.720 -1.067 -1.383 LAG 6 7 8 9 10 B .004400 .06428 .01034 .03071 .05052 sS .02545 .02544 .02549 .02544 .02545 t .1729 2.526** .4054 1.207 1.985* LAG 11 12 13 14 15 B -.02733 -.0007392 .02052 .01386 .01384 ^B .02549 .02548 .02547 .02548 .02528 t -1.072 -.02901 .8058 .5439 .5473 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 •R2 = .03228 Mean of the Dependent Variable: .0003569 Intercept: .0002504 176 British Pounds Table 11-16.a ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — ONE DAY LAG I 2 3 4 5 6 .02014 .001016 -.008683 -.02345 .01798 S6 .02523 .02523 .02521 .02521 .02521 t .7982 .04025 -.3445 0.9302 .7132 LAG 6 7 8 9 10 6 -.01396 -. 0006660 .01194 .01716 .06705 Sg .02517 .02516 .02514 .02513 .02512 t 0.5546 -.02647 .4749 .6828 2.669*** LAG 11 12 13 14 15 g .01688 -.01275 .03664 .01972 .03108 Sg .02516 .02515 .02514 .02515 .02512 t .6709 -.5071 1.457 .7843 1.237 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .009995 Mean of the Dependent Variable: .0001485 Intercept: .0001411 177 Table 11-16.b ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — ONE DAY Canadian Dollars LAG I 2 3 4 5 6 -.02118 -.05235 .04964 .03975 .05095 s£ .02517 .02517 .02520 .02523 .02523 t -.8416 -2.080* 1.970* 1.576 2.019* LAG 6 7 8 9 10 8 -.01223 .08046 .02410 .01670 -.001292 H .02527 .02527 .02532 .02523 .02524 t -.4842 3.185*** .9518 .6619 -.05120 LAG 11 12 13 14 15 I .007479 -.004993 .02698 -.008129 -.02464 S8 .02522 .02518 .02516 .02513 .02512 t .2966 -.1983 1.072 -.3235 -.9809 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .01822 Mean of the Dependent Variable : -.00007774 Intercept: -.00006517 178 German Marks Table 11-16.c ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — ONE DAY LAG I 2 3 4 5 8 .05727 .05945 -.004515 -.01579 -.02086 se .02528 .02532 .02536 .02536 .02537 t 2.265*** 2.348** -.1780 -.6226 -.8222 LAG 6 7 8 9 10 8 .04213 .02487 .04642 .05402 .05322 S8 .02535 .02534 .02532 .02534 .02536 t 1.662 .9815 1.833 2.132* 2.099* LAG 11 12 13 14 15 B -.0003811 -.01292 .01424 .01637 -.007537 8B .02541 .02541 .02542 .02538 .02534 t -.01500 -.5083 .5600 .6450 -.2975 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .02162 Mean of the Dependent Variable: .00008295 Intercept: .00006002 179 Japanese Yen Table 11-16.d ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — ONE DAY LAG I 2 3 4 5 B -.2307 -.03925 -.05939 -.04071 -.006017 s£ .02523 .02588 .02509 .02594 .02596 t -9.142*** -1.517 -2.293*** -1.569 -.2318 LAG 6 7 8 9 10 B .009415 -.01047 .009420 .05431 .08866 8B .02587 .02583 .02584 .02593 .02597 t .3639 -.4054 .3646 2.094* 3.414*** LAG 11 12 13 14 15 B .01807 .006619 .005444 .03565 .01896 8B .02610 .02615 .02610 .02609 .02546 t .6924 .2531 .2086 1.366 .7448 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .06077 Mean of the Dependent Variable: .0000006897 Intercept: .0000007643 180 Swiss Francs Table 11-16.e ESTIMATED AUTOCORRELATION FUNCTION RETURN SPECIFICATION — FIRST DIFFERENCE HOLDING PERIOD — ONE DAY LAG i 2 3 4 5 6 .1577 .04224 .06823 -.06116 -.05556 S6 .02527 .02557 .02564 .02570 .02569 t 6.242*** 1.652 2.661*** -2.379** -2.162* LAG 6 7 8 9 10 3 -.007206 .05708 T.007828 .05487 .04309 sS .02571 .02569 .02573 .02569 .02573 t -.2803 2.222* -.3043 2.136* 1.675 LAG 11 12 13 14 15 3 -.06447 -.02734 -.005390 .03124 -.008366 s; .02575 .02575 .02572 .02573 .02542 t -2.504** 1.061 -.2096 1.214 -.3291 * Significant at the .05 level ** Significant at the .02 level *** Significant at the .01 level N = 1595 R2 = .05235 Mean of the Dependent Variable: .0001374 Intercept: .0001024 Table 11-17 Testing the Intercept H : a = O H : a = O Ho ld C u rre n c y I 6 20 I O 20 BP a : .00007864 .0004633 -.0 0 0 3 4 9 6 .000005595 .00004792 .0006456 iV ( .0 0 0 0 2 5 3 9 4 ) ( .0 0 0 30 8 86 6 ) ( .0 0 1168110 ) ( .0 0 0 00 6 29 6 ) ( .0 0 0073988 ) (.0 0 0 27 9 7 ) t : 3 .0 9 6 8 * 1 .5 0 00 - .2 9 9 3 .8886 .6477 2 .3 0 82 * CD a : .00001698 .00003005 - .0 0 0 00 1 47 5 - .0 0 0 0 1 3 5 8 -.0 0 0007631 .0001662 Sa = (.0 0 0 0 0 2 9 3 7 ) (.0 0 0 0 7 6 4 4 6 ) (.0 0 0 32 3 14 4 ) ( .0 0 0 0 0 2 9 ) (.0 0 0 06 8 35 ) (.0 00293179 ) t : 5 .7 8 1 4 * . 3931 - .0 0 4 6 -4 .6 8 3 3 * - .1 1 1 6 .5669 DM a : -.0 0 0 0 1 0 6 6 -.0 0 0 1 1 9 3 -.0 0 0 2 1 2 8 .0000653 .0004105 .001996 Si = (.0 0 0 0 0 1 6 6 4 ) (.0 0 0 03 9 98 5 ) (.0 0 0 15 8 94 4 ) (.0 0 0 00 8 14 7 ) (.0 0 0 18 3 79 ) (.0 00961349 ) t : - 6 .4 0 6 3 * -2 .9 8 3 6 * -1 .3 3 8 8 8 .0 1 5 2 * 2 .2 3 35 * 2 .0 7 62 * JY -.0 0 0 00 0 00 1 8 03 -.0 0 0000007284 -.00000003557 .0001896 .0005586 .003668 s i= (.0 0 0 0 0 0 0 0 1 ) (.0 0 0 0 0 0 0 0 1 ) (.0 0 0 00 0 08 ) ( .0 0 0 01 5 07 3 ) (.0 0 0 02 3 10 9 ) (.000128034 ) t : - 2 .5 4 6 3 * -5 .2 2 6 4 * -4 .3 3 1 5 * 1 2 .5 78 8 * 2 4 .1 72 4 * 28 .6 486 * SF a : - .0 0 0 0 1 6 0 7 -.0 0 0 2 2 7 7 -.0 0 0 6 7 7 4 .00007709 .0006745 .003126 Si = ( .0 0 0 00 1 60 4 ) ( .0 0 0 03 8 76 4 ) (.0 0 0 19 3 66 1 ) (.1 1 1 11 6 15 6 ) (.0 0 0191201 ) (.0 0 0 91 5 1 ) t : -1 0 .0 2 0 3 * -5 .8 7 4 0 * -3 .4 9 7 9 * 1 2 .5 227 * 3 .5 2 77 * 3 .4160 * R e je c t H0 181 Table 11-18 Choosing the Return Specification H o ld C u rre n c y 5L I , t * ° + 6 ( I j . t - l > Hq : B = 0 H3 : B > 0 6 + J 5 iT.- I H : B - O H g: 8 > 0 C 20 BP B: .000004215 .00003502 .005341 .00007265 .0004082 -.0 0 0 83 7 2 s ;= (.0 0 0 0 0 5 4 9 3 ) ( .0 0 0 0 6 7 8 4 ) ( .0 0 0 2 7 1 7 ) ( .0 0 0 0 2 5 0 2 ) (.0 0 0 29 0 2 ) ( .0 0 1 0 5 6 ) t : .7 6 7 3 * .5 1 6 2 * 1 .966 2 .904 1 .4 0 7 * - .7 9 2 5 * CD t: .00001239 -.0 0 0 0 0 9 3 8 .0001411 .00001717 .00003639 -.0 0 0 02 4 81 Sg = ( .0 0 0 00 3 07 7 ) ( .0 0 0 0 7 9 9 ) ( .0 0 0 3 3 9 8 ) (.0 0 0 00 2 71 9 ) (.0 0 0 06 3 91 ) (.0 0 0 27 2 2 ) t : 4 .3 5 4 - .0 1 1 7 4 * .4 1 5 1 * 6 .3 1 4 .5 6 9 4 * - .0 1 9 1 * DM B: .00009671 .0009426 .002052 - .0 0 0 0 0 5 4 2 6 -.0 0 002891 -.0 0 0 2 1 6 6 Sg = ( .0 0 0 00 9 45 2 ) ( .0 0 0 2 2 5 8 ) ( .0 0 0 8 6 8 0 ) ( .0 0 0 00 1 26 4 ) (.0 0 0 02 9 92 ) (.0 0 0 16 3 4 ) t : 1 0 .2 5 4 .1 7 5 2 .364 -4 .2 9 4 * - .9 9 6 3 * -1 .3 2 6 * JY B: .0001944 .0007117 .003536 -.0 0 0000001719 -.000000005245 -.00000003761 S ' : ( .0 0 0 0 4 8 0 7 ) ( .0 0 0 0 9 3 8 6 ) ( .0 0 0 5 5 1 9 ) (.0 0 00000009243 ) (.000000001409 ) (.0 00000007621 ) t : 4 .0 5 4 7 .583 6 .407 -1 .8 6 0 * -3 .7 2 2 * -4 .9 3 5 * SF 6: .000155 .001854 .005498 -.0 0 0 00 4 86 4 -.0 0 0 04 3 56 - .0 0 0 2 8 8 9 SB: ( .0 0 0 00 8 82 9 ) ( .0 0 0 2 0 9 9 ) ( .0 0 1 0 0 7 ) ( .0 0 0 00 0 77 7 ) (.0 0 0 02 5 5 ) ( .0 0 0 13 5 3 ) t : 1 7 .55 8 .8 3 3 5 .461 -6 .2 6 1 * -1 .7 0 9 * -2 .1 3 6 * Reject 182 FOOTNOTES 184 "*"Fama, Eugene, "Efficient Capital Markets: A review of Theory and Empirical Work," Journal of Finance, May 1970, pp, 469-482, 2 Exchange Rate Practices of IMF Members (as of June 30, 1975) Exchange Rate Regime Number of Countries Percentate of World Trade Floating 18a 69.6 Pegged . 104 30.4 Total 122 100.0 aIncludes seven European nations participating in a joint float. Source: Ingram, James C,, International Economic Problems,,John Wiley and Sons, New York, New York, 1978, p, 90. 3 Ingram, James C., International Economic Problems, John Wiley and Sons, New York, New York, 1978, p. 90. ^Federal Republic of Germany (Bundes Republik Deutschland) i.e. West Germany, hereafter Germany or West Germany. ^Teweles, Richard J.; Harlow, Charles J.; and Stone, Herbert L., The Commodity Futures Game, Who Wins? Who Loses?, McGraw-Hill Book Company, New York, New York, 1974, p. 39. ^"Commodity Traders Play with Currency." Business , Week, April 12, 1972, -j "Understanding Futures in Foreign Exchange," Public Relations. Department, International Monetary Market of the Chicago Mercantile Exchange, Inc., Chicago, IL, 1974. 8Dufey, G,, "Corporate Finance and Exchange Rate Variations," Financial Management, Vol. I, No. 2, (Summer, 1972), Q i Shapiro, A. C., "Exchange Rate Changes, Inflation, and the Value of the Multinational Corporation," Journal of Finance, Vol. 30, No. 2, (May, 1975). 10Granger, C. W 1 J., and Morgenstern, 0., "Spectral Analysis of New York Stock Market Prices," KyKlos, Vol. 16, (1963). 185 11 Brogan, A., "Simulation of Moving Average Hedging Strategies for Winter Wheat," Unpublished M 1S1 Thesis, Montana State Univeristy, Bozeman, MT, June, 19.77, p , 12, 12 This discussion presented can be found in greater detail in Fama, E1F1, Foundations of Finance, Basic Books, Inc1,. New York, New York, 1976, pp, 133-137. 13 Fama, E1F 1, Foundations of Finance, Basic Books, Inc1, New York, New York, 1976, 14Fama, op, cit,, 1976. ^Fama, op, cit,, 1976. "*"^A more detailed presentation is made in Fama, E1F1, Foundations of Finance, Basic Books, Inc1, New York, New York, 1976, pp, 137- 166 and pp. 320-382. 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