Lee R. Thomas III Allianz Global Investors, Newport Beach, California, USA, and

Transcription

1 The current issue and full text archive of this journal is available at Random walk currency futures profits revisited Kuntara Pukthuanthong College of Business Administration, San Diego State University, San Diego, California, USA Lee R. Thomas III Allianz Global Investors, Newport Beach, California, USA, and Carlos Bazan Computational Science Research Center, San Diego State University, San Diego, California, USA Random walk currency futures 263 Abstract Purpose Recent research indicates that the random walk hypothesis (RWH) approximately describes the behavior of major dollar exchange rates during the post-1973 float. The present analysis seeks to examine the profitability of currency futures trading rules that assume that spot exchange rates can be adequately modeled as a driftless random walk. Design/methodology/approach Two random walk currency futures trading rules are simulated over all available data from the period In both cases, the investor buys currencies selling at a discount and sells those selling at a premium, as the RWH implies. The two rules differ only in the way they allocate the hypothetical investor s resources among long and short foreign currency positions. Findings Results show that an investor who used these trading strategies over the past decade would have enjoyed large cumulative gains, although periods of profit were interrupted by periods of substantial loss. Research limitations/implications The findings encourage the hope that profitable random-walk-based strategies for currency futures trading can be devised. The simulation results have important implications for those willing to hedge, borrowers, and speculators. Originality/value This paper provides evidence that purchasing futures contracts on currencies priced at a discount and selling futures contracts priced at a premium has generally been a profitable trading strategy during the last two decades of floating exchange rates. Keywords Random processes, Futures markets, Currency options Paper type Research paper 1. Introduction and literature review A growing body of research indicates that the random walk hypothesis (RWH) approximately describes the behavior of major dollar exchange rates during the post-1973 float[1]. This suggests that the RWH might be a useful foundation upon which to construct a currency futures trading program. Approximately 20 years ago, Thomas (1986) published an article in the Journal of Futures Markets describing a rule for trading foreign exchange futures contracts. The rule was simple: the only thing the speculator needed to do was to examine short-term interest rates in two countries, then The authors really appreciate the helpful comments and suggestions from Richard Roll, Ralf Zurbrugg (the editor), and the anonymous referee. International Journal of Managerial Finance Vol. 3 No. 3, 2007 pp q Emerald Group Publishing Limited DOI /

2 IJMF 3,3 264 buy (respectively, sell) foreign currency if the foreign interest rate was above (respectively, below) the corresponding US rate. This strategy is equivalent to buying foreign currencies that sell at a discount in the futures market, and selling currencies that sell at a premium. In essence, the investor bets the futures market s implicit forecast of a change in the spot rate is incorrect, as it will be if spot rates evolve as random walks. When the price of a commodity describes a driftless random walk, the current spot price is an unbiased predictor of the future spot price. That is, the current spot rate systematically neither overestimates nor underestimates the future spot rate. If the futures market price of such a commodity equals the expected future spot price, then the spot and future prices should be equal. If they are not, either futures market prices contain risk premia, or future-market participants believe they can forecast the future spot price of a commodity better than the random walk model can. Despite its simplicity, the rule had been historically very profitable. As always, one explanation for these favorable results was implicit data mining. When many researchers look historically at many different trading rules, some are bound to work, or at least work when applied to historical data. But post-sample, many such rules collapse. The purpose of this article is to see if the profits reported in 1983 persisted during the 20 years that have passed subsequent to publication of the trading rule in the Journal of Futures Markets. Mounting evidence indicates that it is difficult for any exchange-rate forecasting model to significantly outperform the RWH[2]. Most importantly to our purpose, Bilson (1981), Bilson and Hsieh (1983), Huang (1984), and others, have shown that the economic theory relating interest-rate differences among countries to subsequent exchange rate changes (uncovered interest-rate parity) seems to have broken down during the recent float. As a consequence, exchange-rate changes are no longer governed by international interest differentials. Hacche and Townsend (1981) and Meese and Rogoff (1983a, b) have demonstrated that other plausible economic theories, such as purchasing power parity and the monetary model, also add little to random walk forecasts of exchange rates, at least at horizons of less than a year. These studies all reported strong rejections of uncovered interest-rate parity. Subsequent studies have confirmed these results[3]. There is also an active theoretical literature, which attempts to determine if the failure of uncovered interest parity is due to risk aversion or market segmentation rather than market inefficiency[4]. In contrast, Roll and Yan (2000) suggest that forward exchange rates are unbiased predictors of subsequent spot rates and there is really no forward premium puzzle. The puzzle arises because the forward rate, the spot rate, and the forward premium follow nearly non-stationary time series processes. These theological issues will not be addressed in this paper. If dollar exchange rates evolve as approximate random walks, the mean absolute percentage deviations between spot and future exchange rates (Table I) also represent the expected rates of return from following a random walk trading strategy of buying futures contracts on currencies that are selling at a discount to the spot price, and selling currency futures that are priced at a premium[5]. We compare these predicted profits to the actual profitability of random walk trading and find that during the last two decades of floating rates, on average only about one-third of the observed difference between spot and futures exchange rates has been justified by subsequent exchange rates changes. In short, the trading rule has continued to produce profits that appear disproportionate to their associated risks.

3 Currency Mean absolute differences between spot and future exchange rates (percent) Number of observations Random walk currency futures Euro British pound Australian dollar Swiss franc Japanese yen Canadian dollar Notes: There are 465 samples from the first quarter of 1984 to the fourth quarter of 2003 for Euro, British pound, Australian dollar, Swiss franc, Japanese yen, and Canadian dollar. All prices are quoted as US dollar per one unit of foreign currency. Standard deviations are reported under mean absolute differences between spot and future exchange rates 265 Table I. Mean absolute percent differences between spot and futures market exchange rates (annualized, ) In section 2 we review the trading rule, and its economic foundations. Section 3 contains data description and examines the historical data. Section 4 extends the original analysis, considering more sophisticated ways of implementing the basic strategy. The concluding section discusses the implementations for hedgers and traders. 2. Research methodology 2.1 Predicted and actual profits from random walk trading The trading strategies described in section 1 are all based on the RWH-derived assumption that the current exchange rate is an approximately unbiased estimate of the future spot rate. If this is so, the expected profit from buying/selling a currency using a futures contract equals its associated discount/premium. This section compares the expected and actual rates of return realized by an investor who followed such a RWH-based strategy, using each of the six major currencies that typically sell at a significant discount or premium from the spot. 2.2 Random walk trading Suppose that starting at time t, a spot foreign currency exchange rate evolves independently of its previous values, or of any economic variables that were observable at time t. In common parlance (if somewhat inaccurately), the exchange rate evolves as a random walk. We can easily derive the expression describing the profits from currency futures speculation if exchange rates evolve this way. The profit or loss on a foreign exchange position (R tþ1 ) depends on the domestic interest rate, i t, and the foreign interest rate, i * t, and also on the change in the exchange rate. The return to holding foreign exchange R tþ1, can be written: R tþ1 ¼ FX tþ1 FX t ðt þ i * t Þ 2 ð1 þ i tþ: ð1þ

4 IJMF 3,3 266 It is convenient to rewrite this as: R tþ1 ¼ FX tþ1 2 FX t ð1 þ i * t FX Þþði* t 2 i tþ; ð2þ t where FX tþ1 is the spot rate at next time period and FX t is the spot rate at this time period. The spot rate is quoted as domestic currency per unit of foreign currency. The first term on the right-hand side is the capital gain or loss associated with exchange rate changes, and it depends on the percentage change in the exchange rate and on the quantity of foreign exchange rate owned. The latter, in turn, depends on the foreign interest rate, i *. The second sum on the right-hand side, (i* 2 i), is often called the carry on the trade. It represents the interest income or expense associated with:. borrowing the domestic currency; and. lending the foreign currency out. When the foreign rate exceeds the domestic rate, buying foreign currency is called a positive carry trade. When the foreign currency interest rate is less than the domestic rate, buying foreign currency is a negative carry. If exchange rates evolve as different random walks, the expected change in the exchange rate is zero. Then the expected return equals the carry. To put it succinctly, EðR tþ1 Þ¼i* 2 i. There is an economic theory, the uncovered interest parity hypothesis, that asserts that one cannot treat the spot exchange rate as being a simple, driftless random walk. Instead, according to uncovered parity, on average, the capital gain or loss associated with foreign exchange rate changes will just offset the carry. Or, to put it more succinctly, EðR tþ1 Þ¼0, because ð1 þ i * t ÞðFX tþ1 2 FX t Þ=FX t equals 2ði* 2 i Þ. If this is correct, then a speculator who buys the higher interest currency and funds herself in the lower interest currency will not earn profits (except by chance) even though the carry on her position is positive. We can test this by estimating b in the following regression equation: R tþ1 ¼ a þ bði * t 2 i tþþu t ; where a and b are parameters and u t is a normally distributed random error term. The left-hand side represents the actual rate of return from the purchase of a currency futures contract. The right-hand side variable represents the expected profit if the RWH is correct. Thus equation (3) compares the RWH-predicted profits from holding a currency contract during the quarter preceding its expiration to the profit actually realized by following that strategy. According to the RWH, b ¼ 1 or exchange rate changes are unrelated to spreads; according to the uncovered interest parity condition, b ¼ 0 and then there are no profits to be made using the random walk trading rule, except by chance. 3. Data description We collected quarterly future and spot prices from Commodity Research Bureau (CRB) InfoTech CD. Our samples are from the first quarter of 1984 to the fourth quarter of 2003 and for the British pound, Euro, Swiss franc, Canadian dollar, Australian dollar, and Japanese yen. All prices are quoted per US dollar. The French franc was dropped, as it vanished on January 1, 2000 when the Euro came into ð3þ

5 existence. The Euro is treated as the successor currency for the Deutschmark. Despite ample empirical evidence advancing the random walk model as the best tool available for forecasting dollar exchange rates, for most currencies, IMM futures prices seldom equal the contemporaneous spot exchange rates. Table I shows the mean absolute percentage deviations between spot and the futures exchange rates, at a quarterly horizon, over the periods. The differences have been substantial for all currencies. The regression statistics, in each case estimated using all available data from , are summarized in Table II. As predicted, the estimated slope coefficients are generally close to zero. Unfortunately, the low coefficients of determination and the high standard errors of the estimated slope terms demonstrate that the ordinary least squares (OLS) estimates are imprecise. For four of the six currencies the regression estimates are not powerful enough to decisively reject either hypothesis i.e. random walk or unbiasness in favor of the other. However, it is fair to say that the weight of the evidence presented in Table II favors the RWH. Uncovered interest parity can be rejected in favor of the RWH for one of the currencies, the Japanese yen. In contrast, RWH can be rejected in favor of uncovered interest parity for the Canadian dollar. The British pound, Euro, Swiss franc, and Australian dollar are not rejected for both hypotheses. Zellner s Seemingly Unrelated Regression (SUR) procedure will yield more precise estimates of the parameter values if the u t from equation (1) are contemporaneously correlated across different currencies. It is likely that the error terms for various currencies are related. First, changes in any dollar exchange rate can be imagined to result from real and monetary disturbances in either the US or the foreign country. Random walk currency futures 267 Currency Estimated value of the intercept a Estimated value of Number of the slope b Adjusted R 2 observations Euro British pound Australian dollar * 1.72 * Swiss franc * Japanese yen ** Canadian dollar Notes: OLS regressions of returns of holding foreign exchange on the difference between foreign interest rate (i * ) and US interest rate (i ). Return of holding foreign exchange is given by equation (2). There are 465 samples from the first quarter of 1984 to the fourth quarter of 2003 for Euro, British pound, Australian dollar, Swiss franc, Japanese yen, and Canadian dollar. All prices are quoted as US dollar per one unit of foreign currency. The regression model is given in equation (3). Hansen s heteroscadasticity consistent estimator of the covariance matrix is shown under each coefficient. Asymptotic standard errors are presented beneath coefficients. * statistical significance at the 10 percent level; ** statistical significance at the 5 percent level; *** Statistical significance at the 1 percent level Table II. OLS regression statistics testing for random walk hypothesis

6 IJMF 3,3 268 Since all exchange rates are expressed here in terms of the dollar, disturbances originating in the US will influence them all. Equation (1) was re-estimated using SUR over the 69 observations common to all of the six currencies, which yielded 414 pooled data points. The SUR results in Table III show that all of the beta coefficients are greater than unity and all are significantly greater than zero at standard levels of statistical significance. Uncovered interest parity can be rejected in favor of RWH. These results confirm the early studies of Hansen and Hodrick (1980), Bilson (1981), and Fama (1984) and the many subsequent studies of the uncovered interest rate parity condition. A joint test of the hypothesis that all of the regression coefficients are zero is firmly rejected with this data. The fact that the regression coefficient is greater than unity means that the expected return from this strategy exceeds the interest rate differential. In other words, the trader not only benefits from the higher foreign interest rate (dividend yield) but also from the appreciation of the exchange rate against the dollar (the capital gain). When the foreign interest rate is below the US rate, the trader will borrow the foreign currency and invest in dollars. Since the regression coefficient is greater than unity, this strategy will also benefit from a dividend yield and the appreciation of the dollar against the foreign currency. Unfortunately, even using SUR the estimates of slope remained imprecise. To secure a further gain in efficiency, equation (1) was re-estimated using the SUR procedure while constraining the estimated slope terms to be equal for all six currencies. The result was: R tþ1 ¼ 0:012 þ 0:61ði * ½0:0075Š ½0:27Š t 2 i tþþu t ; ð4þ Currency Estimated value of the intercept a Estimated value of Number of the slope b Adjusted R 2 observations Table III. Seemingly Unrelated Regression statistics testing for random walk hypothesis Euro *** British pound ** Australian dollar * 1.05 * Swiss franc ** Japanese yen * Canadian dollar ** Notes: Regressions of returns of holding foreign exchange on the difference between foreign interest rate (i * ) and US interest rate (i ). Return of holding foreign exchange is given in equation (2). There are 465 samples from the first quarter of 1984 to the fourth quarter of 2003 and for British pound, Euro, Swiss franc, Canadian dollar, Australian dollar, and Japanese yen. All prices are quoted as US dollar per one unit of foreign currency. The regression model is given by equation (3). Hansen s heteroskedasticity consistent estimator of the covariance matrix is shown under each coefficient. Asymptotic standard errors are presented beneath coefficients. * statistical significance at the 10 percent level; ** statistical significance at the 5 percent level; *** Statistical significance at the 1 percent level

7 where the standard error of the coefficient is reported in the brackets under each coefficient. As was the case 20 years ago, the regression result is inconsistent with the uncovered interest rate parity (b ¼ 0), but consistent with the random walk hypothesis (b ¼ 1 and a ¼ 0). The failure of the uncovered interest parity to hold is one of the great puzzles in international finance[6]. According to this regression, in an average quarter, exchange rates moved by approximately 39 percent of the amount indicated by the futures-market discount or premium; therefore, on average 61 percent of the quarterly discount or premium has not been justified by subsequent exchange-rate changes. This makes us optimistic that the random walk trading rule may have continued to work after it was published in To confirm this, we have examined the profitability of the trading rule. Random walk currency futures Simulated trading results For many economists and for all investors, the profitability of trading rules is likely to be more persuasive than that of regression statistics. This section simulates two random walk currency future trading rules over all available data from the period. The trading rule is simple: compare the spot exchange rate to the futures price of the same currency. If the foreign currency futures contract trades at a premium (respectively, a discount) to the spot exchange rate, then the trading rule sells (respectively, buys) it. This is equivalent to buying foreign currencies where the short-term interest rate is greater than the corresponding US short-term rate. All trades occur as close as possible to the Friday, 13 weeks prior to contract expiration, at the daily settlement price. Each position is held for the entire 13 weeks, with no intermediate trading permissible. Thus, for each currency, four positions are taken each year, except in the unusual case where the spot and futures prices are identical. In this event, no trade is made for that currency in the quarter in question. Two versions of the trading rule are compared, differing in how portfolios of positions are sized. Both approaches to portfolios construction are naïve. In the first strategy, equal-weighted, the investor s resources are divided equally among the six currencies. The resulting portfolio may be risky if the trading rule calls for simultaneously buying/simultaneously selling all or almost all of the foreign currencies. Then, a general dollar rise/dollar decline could produce large losses. We assume the investor has $1M in capital available, and posts $0.15M in margin for each contract bought or sold. This gross over-margining is to preclude any margin calls. In the second trading rule, square-dollar, the investor divides her resources equally between buying and selling foreign currencies. This is an attempt to insulate the portfolio s return from the general movement of dollar exchange rates. In each period, the total dollar value of currency the investor purchases for future delivery equals the dollar value of foreign currency she sells for future delivery. Accordingly, $0.5M is deployed to sell currency futures that are offered at a premium to spot, and $0.5M is allocated to buying foreign currency futures. If both buy and sell candidates are not available, then no position is taken. As before, margin of $0.15M is allocated to each contract. We assume the investor can buy or sell fractional currency futures contracts, and the brokerage fee is assumed to be $10 per contract bought or sold, or $20 per round trip. This strategy helps to show whether the profits associated with random walk currency trading are exclusively associated with the US dollar exchange

8 IJMF 3,3 270 rate change. Moreover, because dollar exchange rates have generally been more volatile over the past decade than cross exchange rates have, the second rule reduces the risk of currency futures trading from the perspective of a dollar-based investor. The trading simulations that follow are based on the following assumptions:. Each simulated position is opened 13 weeks before the final Friday on which the contract trades. Whenever possible, simulated foreign currency positions are opened and closed at Friday settlement prices. If these data are unavailable, settlement prices from the nearest available trading day are used. Each futures position is closed on the last Friday on which the contract trades. No mid-quarter trading is permitted.. In each quarter, futures contracts on a total of $1,000,000 of foreign currency are bought or sold, valued at the futures prices on the date the option is opened. This generally requires assuming that fractional contracts can be purchased. The rate of return computations assume that an investor posts $150,000, or 15 percent, of the initial dollar value of her position as margin. This represents approximately three times the required initial margin and approximately five times the required maintenance margin for the typical portfolio simulated. This relatively conservative strategy is chosen to avoid margin calls.. Normally, large account holders will satisfy percent of their margin requirements by posting Treasury bills. The interest income they will earn is not generally accounted for in what follows; that is, the reported rates of return are based on trading profits only. Therefore, to find the approximate total return to capital invested in trading strategy, about percent of the three month Treasury bill rate should be added to the reported trading profit.. Transaction costs are assumed to be $20 per round trip, per contract. This probably exaggerates their drag on trading profits. Currently, discount brokers typically charge retail customers as little as $8 for a round-trip trade. In the first simulation, imaginary equal-dollar-valued positions are taken in each of the six currencies. The random walk strategy adopts a contrarian stance, selling strong and purchasing weak currencies. Foreign currency futures selling at discounts are always purchased, and those selling at a premium are always sold. The resulting portfolio positions are shown for each quarter, , in Table IV. Often, Euros, Swiss francs, and Japanese yen are sold, while Australian dollars, Canadian dollars, and British pounds are purchased. That is, the futures markets often predicted that the Euro, Swiss franc, and Japanese yen will appreciate and the pound, Canadian dollar, and Australian dollar will depreciate, against the US dollar. A total of 465 trades are simulated; 274 (or 59 percent) are profitable. Each currency produces a cumulative profit over the 20-year period. Table V shows the summary of the profit data. Gross trading profits total $622,060. After commission of $101,046, net profit is estimated to have been $521,014 or $6,513 per quarter. Based on an investment of $150,000, this represents an average annualized rate of return of percent, exclusive of the interest earned on Treasury bills posted as margin. Table VI shows the quarterly profits and losses earned in each currency and the portfolio s profit or loss. It is immediately obvious that the random walk strategy earned large profit after 1995, but that it sustained substantial losses during the mid to

9 British pound Euro Canadian dollar Australian dollar Japanese yen Swiss franc Contract Equal Square Equal Square Equal Square Equal Square Equal Square Equal Square 03/16/ /15/ /14/ /14/ /15/ /14/ /13/ /13/ /14/ /13/ /12/ /12/ /13/ /12/ /11/ /11/ /11/ /10/ /16/ /16/ /10/ /16/ /15/ /15/ /16/ /15/ /14/ /14/ /15/ (continued) Random walk currency futures 271 Table IV. Positions taken

10 IJMF 3,3 272 Table IV. British pound Euro Canadian dollar Australian dollar Japanese yen Swiss franc Contract Equal Square Equal Square Equal Square Equal Square Equal Square Equal Square 06/14/ /13/ /13/ /11/ /12/ /11/ /11/ /12/ /11/ /10/ /10/ /11/ /10/ /16/ /16/ /10/ /16/ /15/ /15/ /15/ /14/ /13/ /13/ /14/ /13/ /12/ /12/ /13/ /12/ (continued)

11 British pound Euro Canadian dollar Australian dollar Japanese yen Swiss franc Contract Equal Square Equal Square Equal Square Equal Square Equal Square Equal Square 09/11/ /11/ /12/ /11/ /10/ /10/ /10/ /16/ /15/ /15/ /16/ /15/ /14/ /14/ /15/ /14/ /13/ /13/ /14/ /13/ /12/ /12/ Total contracts Notes: Number of future contracts traded based on equally weighted currency and square-dollar positions in each of the six currencies. The random walk strategy adopts a contrarian stance, selling strong and purchasing weak currencies. Foreign currency futures selling at discounts were always purchased, and those selling at a premium were always sold. Equal presents equally-weighted currency strategy, which is implemented by investing equally in all six currencies, and Square presents square-dollar strategy, which is implemented by investing resources equally divided between long and short dollar positions. The resulting portfolio positions are shown for each quarter, Random walk currency futures 273 Table IV.

12 IJMF 3,3 Currency Number of trades Number of profitable trades Number of futures contracts traded Cumulative profits 274 Table V. Equally weighted portfolio profit summary British pound , Euro , Canadian dollar , Australian dollar , Japanese yen , Swiss franc , All , , Notes: Number of trades, number of profitable trades, number of futures contracts traded, and cumulative profits in each of the six currencies from based on equal-dollar-valued. The random walk strategy adopts a contrarian stance, selling strong and purchasing weak currencies. Foreign currency futures selling at discounts were always purchased, and those selling at a premium were always sold. Equally weighted currency strategy is implemented by investing equally in all six currencies late 1980s. This pattern corresponds to the major US dollar movements in recent years; the dollar generally declined in the late 1980s and recovered from 1990 to 2000 and then has declined from 2000 until recently. This suggests that the trading rule may be sensitive to general dollar trends. The second random walk trading rule is designed to show whether the profitability of the first resulted only from capturing the dollar bull market of It accomplished this by constructing square dollar positions[7] the investor s resources are equally divided between long and short dollar positions. The square dollar portfolio is both long and short of foreign currencies, so that the net dollar value of the portfolio is zero. In essence, the investor places a series of bets on how cross exchange rates for example, the Euro/Swiss franc or British pound/japanese yen rates will evolve, rather than betting on the course of dollar exchange rates. This should substantially insulate the portfolio s return from general dollar exchange rate trends. Incidentally, it also should reduce the risk of random walk trading from a dollar-based investor s perspective. The quarterly positions taken by the square dollar portfolio are also documented in Table IV. If a currency was bought and sold according to the first trading rule, then it was also bought and sold during the corresponding period by the square dollar rule. The scale of the overall portfolio $1,000,000 worth of currency bought or sold was also the same. Thus, the second trading rule differed from the first only in the division of resources among currency positions. Table VII summarizes the square dollar trading results; the quarterly performance is presented in Table VIII. The square dollar portfolio simulated 465 trades involving 872 futures contracts. The portfolio produced a gross, 20-year profit of $316,066 before estimated commissions of $87,203, or a net trading profit of $228,863. The average quarterly profit of $2,861 corresponds to an annualized rate of return of 7.85 percent (above Treasury bill interest rate). This is roughly less than half the rate of return registered by the equally-weighted portfolio (18.53 percent)[8]. This suggests that the random walk strategy benefits from taking outright dollar positions, but that dollar exchange rate trends do not explain all of the observed profits. Betting against

13 Contract month British pound Euro Canadian dollar Australian dollar Japanese yen Swiss franc Portfolio profit/loss Cumulative profit/loss 03/16/ ,374 2, ,014 21, , ,916 06/15/84 8,497 8,881 23, ,535 11,515 31,597 20,681 09/14/84 14,725 18,268 1, ,589 17,904 63,155 83,836 12/14/84 9,278 5, ,396 5,230 23, ,879 03/15/85 215,577 16,116 27, ,674 21,033 23, ,791 06/14/85 33, ,124 3, , ,847 25, ,116 09/13/85 8,412 28, , , , ,069 12/13/85 14, ,071 21, , , ,459 47,610 03/14/86 5, , , , ,450 25,840 06/13/86 8,084 22,428 2, ,444 25,398 27, ,737 09/12/86 23, , , , , ,873 12/12/86 23,929 23,417 2, ,180 2,824 5, ,075 03/13/87 19, ,689 7, , , , ,300 06/12/87 9,716 22,636 22,052 13,881 28,777 24,939 5, ,109 09/11/ ,784 3,251 5,212 1,640 1,437 13, ,240 12/11/87 20, ,359 2,318 21, , , , ,691 03/11/88 1,770 4,837 6,783 6, ,835 26, ,793 06/10/88 22,618 7,244 5,786 18,952 22,003 9,884 37,245 28,547 09/16/88 212,502 15, ,859 17,053 32,461 23,913 12/16/88 15,646 29,190 3,647 13, ,112 29,145 2,005 25,918 03/10/89 27,907 10,437 1,127 23,182 8,710 12,698 21,882 47,801 06/16/89 216,149 11,521 1, ,945 19,784 13,767 18,042 65,843 09/15/89 5, ,977 9,583 2,577 21,503 18,405 84,248 12/15/89 5, ,222 5,101 4, , ,577 62,671 03/16/90 4, ,658 22,333 10,066 4,443 14,761 77,432 06/15/90 12, ,609 7,940 1,702 9,195 33, ,061 09/14/90 21, ,825 2,966 13, , , ,386 97,675 12/14/90 5,983 29,252 2, ,178 4,864 2,482 23,293 94,383 03/15/91 27, ,939 2,087 2,376 25, , ,677 60,706 06/14/91 215, ,297 3, , , ,871 13,835 09/13/91 12,401 12,019 2,012 10,756 9,162 7,863 54,214 68,049 12/13/91 9,941 11, ,162 6,826 9,864 34, ,523 (continued) Random walk currency futures 275 Table VI. Quarterly profits: equally-weighted portfolio strategy

14 IJMF 3,3 276 Table VI. Contract month British pound Euro Canadian dollar Australian dollar Japanese yen Swiss franc Portfolio profit/loss Cumulative profit/loss 03/13/92 27,509 25,849 26,555 21,551 25, , ,931 65,592 06/12/92 16,704 12,032 2,212 2,133 9,409 11,386 53, ,468 09/11/92 8,956 17,547 22,632 26,199 3,382 20,716 41, ,237 12/11/92 229, ,307 26,742 27, , ,172 95,065 03/12/93 211,960 26,271 5,796 6,016 28, , ,680 69,385 06/11/93 11,544 4,713 23,546 26,704 18,862 8,289 33, ,542 09/10/93 4,453 6,039 23,305 25, ,345 10, ,690 12/10/93 25,123 28,455 21,232 6,464 24,477 26, ,476 93,213 03/11/94 1,457 2,420 23,618 10,821 25,837 4,080 9, ,537 06/10/94 1,516 2,624 21,454 5,314 21,640 2,206 8, ,104 09/16/94 8,229 13,599 3,750 2,474 26, ,213 4, ,097 12/16/94 21,921 3,035 24,932 7,083 3,226 6,827 13, ,413 03/10/ ,937 21,441 25, , , ,854 68,559 06/16/95 4,474 21,318 3,651 22, ,783 21,899 28,618 59,941 09/15/95 25,713 8,744 2,426 7,688 32,625 7,400 53, ,111 12/15/ ,689 23,320 21, , ,539 98,572 03/15/ ,927 1,237 7,876 8,008 4,763 25, ,970 06/14/ , ,068 7,003 10,700 30, ,430 09/13/96 2, , , ,008 12/13/96 10,894 5, ,107 11,440 35, ,190 03/14/97 25,184 14,627 1,332 1,391 14,749 16,509 43, ,614 06/13/97 3,920 5,193 2,849 29, , , ,912 09/12/97 22,756 4,030 2,583 6,588 10,374 3,878 24, ,609 12/12/97 5,230 1,613 4,135 14,257 14,124 21,085 38, ,884 03/13/98 2,638 4, , ,231 8, ,876 06/12/98 22, ,766 22,117 20,720 4,428 51, ,889 09/11/98 4, ,096 4,988 23, , , , ,840 12/11/98 2,623 23,906 3,176 25, ,401 26, , ,502 03/12/99 25,496 14,292 21,849 23,970 5,513 17,282 25, ,275 06/11/99 21,166 6,635 27,112 27, ,732 21, ,920 09/10/ ,959 1,728 3, ,827 5,794 3, ,945 12/10/ , ,241 28,230 4,836 6, ,171 (continued)

15 Contract month British pound Euro Canadian dollar Australian dollar Japanese yen Swiss franc Portfolio profit/loss Cumulative profit/loss 03/10/00 4,644 9,126 21,719 5,733 8,400 10,689 36, ,043 06/16/00 6, ,140 2,302 2,733 24,083 9, ,273 09/15/00 13,003 19,823 2,732 16,998 4,404 17,006 73, ,239 12/15/00 28,826 27,151 5, ,192 28,787 28, ,544 03/16/01 5, , ,377 16,262 4,363 16, ,988 06/15/01 22,524 26,359 23,340 10,868 2,247 6,768 7, ,648 09/14/01 8,477 12,594 24,605 3,304 26, ,835 23, ,612 12/14/01 21,251 3, ,179 14, , ,651 03/15/02 22,696 23,532 22,602 3,034 3,041 2, ,774 06/14/02 7,133 12,590 3,988 12,089 25, ,211 19, ,520 09/13/02 9,534 5,469 23,255 21,619 22,470 25,375 2, ,804 12/13/02 4,900 9,326 2,966 5, ,973 14, ,627 03/14/ ,129 9,684 11,405 22,521 29,251 18, ,558 06/13/03 10,329 18,152 17,807 21, ,313 58, ,931 09/12/03 25,778 27,675 22, ,041 25, ,933 12/12/03 15,966 15,368 7,030 22, , ,087 31, ,060 Total 235,560 87,207 91, ,903 10, , ,060 14,869,684 Notes: Quarterly profits in each of the six currencies based on equal-dollar-valued positions. The random walk strategy adopts a contrarian stance, selling strong and purchasing weak currencies. Foreign currency futures selling at discounts were always purchased, and those selling at a premium were always sold. Equally weighted currency strategy is implemented by investing equally in all six currencies. The resulting portfolio positions are shown for each quarter, Cumulative profit and loss is profit and loss accumulated over time Random walk currency futures 277 Table VI.

16 IJMF 3,3 Currency Number of trades Number of profitable trades Number of contracts traded Cumulative profits 278 Table VII. Square-dollar portfolio profit summary Euro , British pound , Canadian dollar , Swiss franc , Japanese yen , Australian dollar , All , Notes: Number of trades, number of profitable trades, number of contracts traded, and cumulative profits in Euro, British pound, Canadian dollar, Swiss franc, Japanese yen, and Australian dollar from 1984 to 2003 based on square-dollar strategy. The random walk strategy adopts a contrarian stance, selling strong and purchasing weak currencies. Foreign currency futures selling at discounts were always purchased, and those selling at a premium were always sold. Square-dollar strategy is implemented by investing resources equally divided between long and short dollar positions discounts or premia in futures-market cross exchange rates is also profitable from As can be seen, both trading rules produce cumulative profits during the approximately 20-year period that has elapsed since the trading rules were published. The first trading rule produces an information ratio (profit divided by standard deviation) of 26.0 percent. The second produced an information ratio of 17.6 percent. During the same period, a buy and hold position in the US stock market (the S&P 500 futures contracts) produced an information ratio of 16.0 percent, after allowing for $20 round trip commissions each quarter ($80/year per contract). The final topic considered is the risk of the currency futures trading strategies. The discussion is complicated by two factors. First, the gross variability of the profit streams simulated above depends on the degree of leverage employed. The apparent risk of the strategies can be reduced by posting more than $150,000 per period in margin. More substantially, the analysis of the strategies is plagued by the difficulty of partitioning risking into their systematic and non-systematic components. Measuring systematic risk requires first identifying the market portfolio. US equity prices are commonly used in the finance literature as a proxy for the market portfolio. Despite the theoretical inadequacies of this approach, that practice is adopted in the following. Total returns to currency portfolios have been computed by adding 80 percent of the 30-day Treasury bill yield (applied to the assumed investment of $150,000 per period) to the trading profits. The return to holding US common stocks is measured by changes in the Standard and Poor s Index of 500 stock prices plus dividend yield. All returns are before taxes. Table IX compares summary statistics on the returns and risks of four investments the equally weighted and square-dollar currency futures portfolios, common stocks, and the 30-day US Treasury Bills from the first quarter of 1984 through the fourth quarter of The equally weighted currency futures portfolio is the riskiest investment of those considered, registering a standard deviation of quarterly returns of almost 20 percent. The square dollar procedure reduces risk to 15 percent, approximately twice that of common stock. However, as Table IX and Figure 1 show, the investor is amply rewarded for bearing these risks. A line connecting the

17 Cumulative profit/loss Portfolio profit/loss Swiss franc Japanese yen Australian dollar Canadian dollar Euro dollar British pound Contract month 03/16/ ,424 1, , ,549 26,549 06/15/84 6,373 6, , ,901 8,636 15,076 8,527 09/14/84 8,835 10,961 1, ,354 10,742 37,893 46,420 12/14/84 6,958 3, ,547 3,923 17,282 63,702 03/15/85 223,365 16, , ,674 21,033 12,457 76,159 06/14/85 49, ,124 4, , ,847 12,385 88,543 09/13/85 12,618 28, , , ,760 77,783 12/13/85 21, ,071 22, , , ,142 21,641 03/14/86 7, , , , , ,865 06/13/86 12,126 22,428 3, ,444 25,398 22, ,575 09/12/86 25, , , , , ,646 12/12/86 25,893 23,417 3, ,180 2,824 4, ,742 03/13/87 28, ,689 11, , ,870 2, ,328 06/12/87 9,716 22,636 22,052 13,881 28,777 24,939 5, ,136 09/11/ ,784 3,251 5,212 1,640 1,437 13, ,268 12/11/87 20, ,359 2,318 21, , , , ,719 03/11/88 1,770 4,837 6,783 6, ,835 26, ,821 06/10/88 22,618 7,244 5,786 18,952 22,003 9,884 37, ,575 09/16/88 212,502 15, ,859 17,053 32,461 16,886 12/16/88 15,646 29,190 3,647 13, ,112 29,145 2,005 18,890 03/10/89 27,907 10,437 1,127 23,182 8,710 12,698 21,882 40,773 06/16/89 216,149 11,521 1, ,945 19,784 13,767 18,042 58,815 09/15/89 5, ,977 9,583 2,577 21,503 18,405 77,220 12/15/89 5, ,222 5,101 4, , ,577 55,643 03/16/90 3, ,244 21,750 30,197 3,332 33,718 89,361 06/15/90 9,130 21,485 2,707 5,955 2,553 6,896 25, ,116 09/14/90 21, ,825 2,966 13, , , , ,731 12/14/90 3, ,756 1,685 26,107 2,918 1, ,180 77,550 03/15/91 23,630 25,969 1,044 1,188 22,675 26, ,838 60,712 (continued) Random walk currency futures 279 Table VIII. Quarterly profits: square-dollar portfolio strategy

18 IJMF 3,3 280 Table VIII. Cumulative profit/loss Portfolio profit/loss Swiss franc Japanese yen Australian dollar Canadian dollar Euro dollar British pound Contract month 06/14/91 27,615 28,149 1, ,486 27, ,435 37,277 09/13/91 6,200 6,009 1,006 5,378 4,581 3,931 27,107 64,383 12/13/91 4,971 5, ,081 3,413 4,932 17,237 81,621 03/13/92 23,754 22,924 23, ,622 25, ,465 63,155 06/12/92 8,352 6,016 1,106 1,067 4,705 5,693 26,938 90,093 09/11/92 4,478 8,773 21,316 23,099 1,691 10,358 20, ,977 12/11/92 214,532 25,654 23,371 23, , ,086 77,892 03/12/93 27,176 23,762 3,478 3, ,671 26, ,145 42,746 06/11/93 5,772 2,357 21,773 23,352 9,431 4,144 16,579 59,325 09/10/93 2,672 3,623 21,983 23,432 1,008 5,007 6,895 66,220 12/10/93 22,561 24, ,232 22,238 23,327 29,738 56,482 03/11/ ,452 22,171 6, ,510 2,448 28,413 48,068 06/10/ , ,189 24,919 1,324 1,204 49,273 09/16/94 6,172 10,199 2,813 1, , , ,550 35,723 12/16/94 21,921 3,035 24,932 7,083 3,226 6,827 13,316 49,039 03/10/ ,453 22,161 28, , , ,155 22,116 06/16/95 4,474 21,318 3,651 22, ,783 21,899 28, ,734 09/15/95 25,713 8,744 2,426 7,688 32,625 7,400 53,170 42,436 12/15/ ,689 23,320 21, , ,539 27,897 03/15/ ,927 1,237 7,876 8,008 4,763 25,398 53,295 06/14/96 1,112 5, ,602 5,252 8,025 27,203 80,498 09/13/96 3, , ,914 87,411 12/13/96 16,341 3, ,015 5,330 8,580 35, ,475 03/14/97 27,776 10, ,086 11,062 12,382 29, ,198 06/13/97 5,879 3,894 2, ,495 27, , ,114 09/12/97 28,268 2,418 1,550 3,953 6,224 2,327 8, ,318 12/12/97 15, ,481 8,554 8, , ,834 03/13/98 7,913 2, , ,738 11, ,560 06/12/98 28, ,060 13,270 12,432 2,657 23, ,139 (continued)

19 Cumulative profit/loss Portfolio profit/loss Swiss franc Japanese yen Australian dollar Canadian dollar Euro dollar British pound Contract month 09/11/98 14,711 26,058 2,993 22,040 29,077 26,789 26, ,878 12/11/98 7,870 22,344 1,905 23, ,041 23, , ,171 03/12/99 216,487 8,575 21,109 22,382 3,308 10,369 2, ,446 06/11/99 23,499 3,981 24,267 24, ,039 23, ,833 09/10/ , ,716 25,914 2,897 1, ,346 12/10/ , ,121 24,115 2,418 3, ,459 03/10/00 2,322 4, ,866 4,200 5,344 18, ,895 06/16/00 3, ,151 1,367 22,041 4, ,510 09/15/00 6,501 9,911 1,366 8,499 2,202 8,503 36, ,493 12/15/00 24,413 23,576 2, ,096 24,394 24, ,146 03/16/01 3, , ,131 9,757 2, , ,107 06/15/01 22,524 26,359 23,340 10,868 2,247 6,768 7, ,767 09/14/01 8,477 12,594 24,605 3,304 26, ,835 23, ,731 12/14/01 21,251 3, ,179 14, , ,770 03/15/02 22,022 22,649 21,952 2,276 4,561 4,316 4, ,300 06/14/02 5,350 9,443 2,991 9,067 28, ,317 2, ,069 09/13/02 7,151 4,102 22,441 21,214 23,705 28,062 24, ,899 12/13/02 3,675 6,995 2,225 4,119 21, ,459 5, ,120 03/14/ ,847 7,263 8,554 23, ,877 5, ,489 06/13/03 7,747 13,614 13,355 15,918 21, ,469 36, ,415 09/12/03 24,333 25,756 22, ,062 3, ,769 12/12/03 11,975 11,526 5,273 16, , ,630 1, ,066 Total 246,055 12,260 67, , , , ,066 8,442,525 Notes: Quarterly profits in each of the six currencies based on square-dollar-valued positions. The random walk strategy adopts a contrarian stance, selling strong and purchasing weak currencies. Foreign currency futures selling at discounts were always purchased, and those selling at a premium were always sold. Square-dollar strategy is implemented by investing resources equally divided between long and short dollar positions. The resulting portfolio positions are shown for each quarter, Cumulative profit and loss is profit and loss accumulated over time Random walk currency futures 281 Table VIII.

20 IJMF 3,3 Investment Number of quarters Mean quarterly return (percent) Standard deviation of quarterly return (percent) 282 Table IX. Summary return statistics Treasury Bills Common stock Equally weighted currency portfolio Square-dollar currency portfolio Notes: Number of quarters, mean quarterly returns, and standard deviation of quarterly returns of 30-day Treasury Bills, common stock, equally weighted currency, and square-dollar currency portfolios in British pound, Swiss franc, Euro, Canadian dollar, Australian dollar, and Japanese yen from 1984 to To compute equally weighted and square-dollar portfolios, the random walk strategy is adopted as a contrarian stance, selling strong and purchasing weak currencies. Foreign currency futures selling at discounts were always purchased, and those selling at a premium were always sold. Equally weighted currency strategy is implemented by investing equally in all six currencies and square-dollar currency strategy is implemented by investing resources equally divided between long and short dollar positions. Total returns to currency portfolios have been computed by adding 80 percent of the three-month Treasury Bill yield (applied to the assumed investment of $150,000 per period) to the trading profits. The return to holding US common stocks is measured by changes in the Standard and Poor s Index of 500 stock prices plus dividend yield. All returns are before taxes Figure 1. Portfolio performance statistics (quarterly, ) return/risk realization for investments in 30-days Treasury bills and the Standard and Poor s 500 stock index intersect a line connecting the return/risk for both currency portfolios at point A (Figure 1), which has a mean of 6.1 percent and standard deviation of percent. At this point, the amount of investing capital is percent in equally weighted and percent in square-dollar currency portfolios. The return/risk for both currency portfolios lies above the extension of a line connecting the return/risk realization for investments in 30-day Treasury bills and the Standard and