Pollcy, Roesrch, and ExterNal Afflrs WORKING PAPERS uSO%5 International Commodity Markets International Economics Department The World Bank June 1990 WPS 436 Commodity Price Forecasts and Futures Prices Boum-Jong Choe Commodity futures prices have biases due to risk premia and expectational errors, thus limiting their usefulness as a short- term price forecasting tool. Also, futures prices are more adaptive to spot price movements than price expectations, but not necessarily more rational. x ,t The Policy, Reearch, and External Arfairs Complex disrnbuLcs PRI- Working Papers todissumninatethe rudLnp of wozt in pm and to mtnouage the exchange of ideas among lank staff and aU others interested in developnnent issuCs. These papes carry the names of the authors, flect only thcir views, and should he used and cited accordingly lbe rindings, intapractuons, and conclusiomi are the authors' own, They should rot be &attrbutd to the World lank its loard or Directors. iL managernint, or any of its menber countries. Policy, Research, and External Affairs International Commodity Markets WPS 436 This paper - a product ol' the International Commodity Markets Division, International Economics Department- is part ol' a larger effort in PRE to understand the short- and long-run behavior of primary commodity prices and the implications of movements in these prices for the developing countries. Copies are available free from the World Bank. 1818 H Street NW, Washington DC 20433. Please contact Sarah Lipscomb, room S7-062, extcnsion 33718 (23 pages with charts and tables). Choc investigates the relationship between Choc investigates whether commodity commodity futures prices and price expectations, futures prices exhibit similar characteristics. He to determine the usefulness of futures prices as a also estimates a relationship between futures short-term price forccasting tool. prices and price expectations. His main findings include: Previous studies of forecasts from thc B;.;ik's Intcrnational Commodity Markets Division * The rational expectations hypothesis is morc found evidence that commodity specialists' widely rejected with futures prices than with the forecasts are outperfomied by "naive" (static) division's forecasts. l'orecasts, which will hold if commodity markets are efficient. On the other hand, lutures prices * Risk premia and expectational errors are also have shown biases, typically underforecast- equally impornilt in explaining the futurcs ing subsequent spot prices. Some researchers forecast bias --- SO l'utures priccs have to be attribute this futures discount bias to time- adjusted for risk prcmia to be uscful for short- varying risk premia. Others assumc that aoenis temi forecasting. are risk-neutral and that biases reflect market inreficiencs and the failure of rational expecla- * TIhe varitance of risk premia is not larger tions. than that of cxpected price changes for most comImIoditics. The infomiational value of' futums lrrices for lorecasting depcnds on thc sizc of' thc risk * The risk premium appears to be correlated prcnlium relative to the expectational error. A iit tihe l'utures discounit lor at Icast half the recent studv found that expeclational errors cominmo(ities. dominae the lforu ard discounit bias o' the foreign cxchange rate and that thc risk premiunm is small. Futures prices are rmorc heavily influenced relatively stable, and uncorrelated Mith the bT current spot prices than by expected future expectational crror. pnces. The PRE Working Palpr Series disseninates the fitdings of sAork under A i in the Bank s Pohlic, Research. and External Affairs Complex. An objictis c of the seri.s i to gCt Lhese findings gOut quckl\ , Ie en if presentations arc less than fullt pv1ished. The findings. interpretaItions. auid .owl. hlsixins iI the.e papers do neot necessarild re present official Bank polic. P!odt1,cdti h licR Pk ) em1TaJ;1 ('eTIter Commodity Price Forecasts and Futures Prices by Boum-Jong Choe Table of Contents I. Introduction 1 II. Data and A Comparisu.t of Performance 2 A. Data 2 B. Forecast Performance: A First Comparison 3 III. Rationalitv of Futures Prices 6 A. Rationality Test 6 B. Decomposition of Futures Forecast Bias 10 C. Variances of Risk Premium and Expected Price Change 11 IV. Tests of Risk Premia and Expectational Errors 12 A. Tests of Risk Premium 13 B. Tests of Expectational Error 14 V. Futures Prices and Price Expectations 19 VI. Conclusions 20 References 22 I. INTRODUCTION The International Commodity Markets Division (CM) of the World Bank started forecasting primary commodity prices more than two decades ago. The forecasts have been prepared mainly for the Bank's internal use in project evaluation and balance-of-payments forecasting. As such, the forecast accuracy, or forecast biases and informational efficiency, has been a major concern and the subject of occas_.onal retrospective studies. Two recent examples are Castelli et al. (1985) and Warr (1988). Their main conclusions were that the forecasts have been biased and also informationally inefficient; that the forecasters have tended to adhere too long to their previous forecasts. Furthermore, these studies found some evidence that commodity specialists' forecasts have been outperformed by "naive" (static) forecasts,1 prompting the suggestion that for the purpose of short- term (up to one year) forecasting it may be better to rely on current spot prices or futures prices. Theoretically, this proposition derives its rationale from the belief that the commodity futures markets are efficient and that the futures prices incorporate all available information about the expected future spot prices. If so, commodity specialists would not be able to outperform the futures prices. However, a large and growing number of empirical studies of futures market efficiency have mostly rejected the unbiasedness hypothesis of futures prices as a predictor of the future spot rates. Typically, futures prices underforecast the subsequent spot prices. Although there has been a consensus regarding the existence and direction of the futures price bias, opinions are divided as to its causes. Hansen and Hodrick (1980), for example, assume that the investors are risk averse and the biases represent time-varying risk premia. The underprediction therefore, should not be interpreted as evidence of market inefficiency or irrational expectations. Other researchers have assumed that agents are risk neural and, therefcre, attribute the biases to market inefficiency and the failure of rational expectations .2 Carrying the Hansen and Hodrick results a step further, Fama (1984) and Hodrick and Srivastava (1986) advanced the proposition that the variance of the risk premium is greater than the variance of expected changes in prices and that the two are correlated. In its extreme form, under the conditions of truly efficient markets, the proposition implies that the futures price bias consists only 1 In fact, evidence on this point is mixed as will be shown in the next section. 2For a survey of literature on the efficiency of the foreign exchange futures market, see Hodrick (1988). 1 of a risk premium. Thus, the use of futures prices for short-term price forecasting wLll produce biased forecasts, with the biases being the risk premium. Under such conditions, since prices are martingales, the naive forecasts will do better than futures prices. Therefore, to give credit to futures prices as predictors of subsequent price changes is to admit market inefficiency, nullifying the very rationale for their use. Thus, the informational value of futures prices for forecasting the future depends on the size of the risk premium relative to the expectational error. The main purpose of this paper is to explore the relationship between commodity futures prices and price expectations. It focuses on the usefulness of futures prices as a short-term price forecasting tool. Recently, Froot and Frankel (1989) used survey data on exchange rate expectations to estimate the relative importance of risk premium and expectational error in explaining the forward discount bias in foreign exchange rates. They found that, contrary to the claims of Fama and of Hodrick and Srivastava, expectational errors dominate the forward discount bias and that the risk premium is small, relatively stable, and not correlated with the expectetional error. This paper follows the Froot and Frankel analysis to see if commodity prices exhibit similar characteristics. It goes a step further and estimates a relationship between futures prices and price expectations, a relationship derived from an explicit model of spot and futures price determination developed by Turnovsky (1983). The paper is organized as follows. The next section summarizes the characteristics of the forecast and futures price data. Section III tests the rationality of futures prices and decomposes the futures price bias. Section IV conducts direct statistical tests of the relative importance of risk premium and expectational error. Finally, a relationship between futures prices and price expectations is estimated in Section V. The paper ends with a summary and conclusions. II. DATA AND A COMPARISON OF PERFORMANCE A. Data The International Commodity Markets Division of the World Bank has been forecasting the prices of many primary commodities over the last two decades. From the late 1970s, the forecast interval has been well established; every two years, forecasts are made for both the short and long term (10-15 years ahead), after extensive reassessments of global supply and demand. The forecasts are updated every six months, primarily focusing on the short-term outlook. For the purpose of this study, only the short-term (one year) forecasts made durinr. the 1979-88 period are used. At around 2 the mniddle of each year, the average prices of commodities expected to prevail in the following year are forecast. The forecast horizon, therefore, is about one year. The choice of the sample period and the forecast horizon were constrained by the availability of corresponding futures prices, which were retrieved from the commodity database (DRICOM) of Data Resources Inc. For each CM forecast, the matching futures price is the average price of all futures contracts maturing in the target year of the price forecast and observed during the week when the forecast was made. Because of the limited availability of flutures prices, only eight commodities are included in this study; i.e., copper, sugar, coffee, cocoa, maize, cotton, wheat, and soybeans. B. Forecast Performance: A First Comparison Table 1 lists the data in termr of percentage changes in prices forecast by the CM and futures markets. Summary statistics of the data are shown in Table 2. Percentage changes are measured in logarithmic termTs. Let Pt denctt. the logarithm of the spot price at time t, Etpt+1 be the logarithr of the mathematical expectation of the spot price in t+l conditJonal on the information available in t, and ft+,t be the logarithm of the price in t of the futures contract maturing in t+l. Then, the percentage changes in the spot, expected, and futures prices are measured, respectively, by: APt+1l = Pt+1 - Pt; Apet+,, = Etpt+l - Pt; Apft+1i = ft+it Pt- A feature that stands out from Table 1 is that the CM forecasts and futures prices generally agree on the future dire tion of price changes. In 50 out of the 72 cases, the sign of price changes is the same for both CM forecasts and futures prices. A notable exception is coffee where the signs were more often different than the same. Secondly, the percentage changes forecast by CM are much larger than those implied by the futures prices. In 38 of 50 cases where CM and futures markets forecasted prices to move in the same direction, CM forecast larger percentage price changes than the futures prices. This is also shown in Table 2 where the average percentage changes forecast by CM are often more than twice those of the futures prices. Thirdly, the standard deviations of percentage price changes shown in Table 2 are hiuh compared with their means, indicating extreme volatility of commodity prices. It is seen that the actual prices fluctuated much more widely than those forecasted by CM or the futures prices. In fact, the futures prices had the least variance, confirming the underestimation bias mentioned above. 3 Table 1: CM For :asts vs. Futures Prices (percent per annum) 1980 1981 1982 1983 1984 1985 1986 1987 1988 CM Forecasts Copper -11.4 7.5 40.0 52.8 3C.6 14.7 4.5 10.9 -15.8 Sugar 35.4 -68.6 22.0 68.3 11.4 45.3 40.5 42.7 62.6 Coffee -11.5 15.6 16.2 0.3 5.8 2.8 6.5 -8.2 25.6 Cocoa 11.3 30.7 18.6 15.0 -4.4 -10.8 -9.9 3.9 -0.9 Maize 35.0 15.2 22.3 11.5 5.5 -2.9 -2.6 34.3 17.8 Cotton 18.8 -0.9 19.1 20.5 7.4 13.5 14.6 28.2 -10.5 Wheat 29.5 7.1 7.6 13.9 11.3 10.2 -8.6 -14.8 21.0 Soybeans -12.4 4.3 24.0 16.6 13.8 8.8 9.F 0.5 10.1 Futures Prices Copper -0.1 -11.5 -20.7 14.7 10.3 12.0 -1.4 4.0 -5.1 Sugar 20.4 1.1 10.3 24.5 15.2 34.0 15.8 29.6 18.3 Coffee 3.1 -1.9 -11.1 -22.0 -8.3 -9.0 0.1 6.8 9.3 Cocoa 6.0 4.4 14.4 7.4 5.3 -14.0 1.3 7.0 2.6 Maize 10.3 18.9 20.4 15.7 -6.4 -5.0 1.0 2.5 15.4 Cotton 2.9 -1.0 -1.1 8.6 5.9 2.6 -5.) 11.0 -4.0 Wheat -9.6 7.7 26.9 14.0 3.6 -2.2 -3.4 -9.4 4.8 Soybeans -3.3 8.4 9.9 8.3 -0.1 2.3 8.1 1.4 -1.9 Note: The years shown are the target years of forecasts. Source: International Economics Department, World Bank. Table 2: Mean and Standard Deviations of Forecasted, Futures, and Actual Price Changes &petl,+, APf c+1, I APt+i,. iMean Errors Mean S.D. Mean S.D. Mean S.D. (1) (2) (3) 1-3 2-3 Copper 14.9 22.7 7.4 8.5 4.4 21.1 10.5 2.9 Sugar 28.9 40.6 18.8 9.9 8.1 72.3 20.8 10.7 Coffee 5.7 11.5 -3.7 9.9 6.9 23.1 -1.2 -1.1 Cocoa 5.9 14.0 3.8 7.6 -4.1 16.8 10.0 7.9 Maize 15.1 34.0 8.1 10.3 -2.5 21.6 17.6 10.5 Cotton 12.3 11.9 2.2 5.5 -2.0 31.5 14.3 -1.6 Wheat 8.6 13.6 3.6 11.7 3.5 18.5 5.1 -1.6 Soybeans 8.4 10.4 3.7 5.0 -0.2 16.7 8.6 3.9 Source: International Economics Department, World Bank. 4 Forecast errors of the ~M forecasts and futures prices are measured, respectively, by: ret+l = Etpt+l pt+- ; rft+lt = ft+-t Pt+.i Means of these forecast errors are provided in the last two columns of Table 2. Since the naive forecasts have zero forecast changes in prices, their mean forecast error is the negative of the mean actual price changes (column 3). It is shown that the mean forecast errors are large for both CM forecasts and the futures prices, but the CM forecasts have larger absolute mean errors than the futures prices for all commodities. In terms of mean forecast errors, the CM forecasts also have done worse than naive forecasts except for coffee. However, they all are not statistically different from zero because of large standard deviations. The CM forecasts on average showed an overestimation bias compared with the futures prices. The mean forecast error of CM forecasts is positive in nine out of ten commodities, compared with three out of ten for futures prices. The mean forecast error could be a misleading indi-ator of forecast accuracy in the sense that large positive and negative forecast errors do not matter as long as they cancel out. A more useful indicator of the closeness of the forecasts to the actuals is the mean of absolute forecast error, which is shown in Table 3. In terms of mean absolute forecast errors, the CM forecasts beat the futures prices and naive forecasts for 3 of the 8 commodities. In any case, the differences between them are r as pronounced as those of the simple mean forecast errors shown -iove. Table 3: Mean Absolute Errors of CM, Futures, and Naive Forecasts CM Futures Naive Aret+,, Arf t+l, A Pt+1. Copper 30.0 21.0 17.3 Sugar 47.1 55.0 57.8 Coffee 14.4 21.3 17.7 Cocoa 16.3 13.6 13.0 Maize 23.6 16.9 17.4 Cotton 24.1 20.3 24.5 Wheat 9.5 15.8 14.0 Soybeans 17.2 14.9 12.5 Source: International Economics Department, World Bank. 5 To illustrate the nature of the forecast errors, Charts 1-4 plot the expected (both CM and futures) and actual percentage changes in prices of copper, coffee, wheat and cotton -- each commodity representing one of the four major commodity groups (metals, beverages, cereals, and agricultural raw materials). It is clear that the large positive bias in CM fovecasts resulted from the failure to anticipate the severity of the commodity price depression in the 1981-86 period. The 1980s started with relatively high commodity prices. With the economic recession in 1982, commodity prices generally collapsed in 1981/82 and remained at dismal levels until 1987, except for a minor recovery in 1983/84. The severity of the commodity price depression was largely unforeseen by most experts. This failure can be explained by the agents' inability to immediately recognize and adapt to changes in the market regime, which Lewis (1989) cites as a cause for forecast bias that is consistent with rational expectations. Secondly, whenever the spot price changes significantly, the CM forecasts tend to come closer to the full extent of the price change than futures prices. Possible explanations for this could be that market expectations incorporated in futures prices are more conservative than experts' expectations or that futures prices contain risk premia. III. RATIONALITY OF FUTURES PRICES A. Rationality Test To test the rationality of futures prices as predictors of future spot prices, previous studies relied heavily on the regression of changes in spot prices on the futures discount: Pt+lt ~= a + ) Apft+1t + Et+lt, (1) where the notations are the same as before and ct+lt is white noise. The null hypothesis of rationality implies that a=o and 0=1, i.e., the actual change in spot price is the same as anticipated by the futures price plus a random error term. Table 4 shows the results of estimating (1) using ordinary least squares (OLS). None of the F-test statistics are significant at the 5% level, but they are significant in four of the eight commodities at the 10% level. For these commodities, the evidence against the rationality of futures prices seems quite strong when 3 Coffee (and cocoa to a lesser extent) has been a major exception to this price pattern, maintaining relatively high prices during 1980-86 before collapsing in 1987-88. 6 CHART 1: PRICE EXPECTATIONS VS ACTUALS CPER -- PERCENT PER ANN8I 60 30 - 20 - 30 20 -10 -10 -20 1980 1982 1994 1996198 o CM FORECAST + FUTURES PRICE o ACTUAL PPICE CHART 2: PRICE EXPECTATIONS VS ACTUALS COFFE -- PERCENT PER ANNNl 40 - 30- 20- 10 -10 -20- -30- -40- -50 - W lll 19B0 1982 1994 1996 1989 CM FORECAST t FUTURES PRICE o ACTUAL PICE 7 CHART 3: EXPECTED VS. ACTUAL PRICES WrEAT - PERCBr FM ANNL 40 - 30 - 20 - -10 -120 t980 1982 1994 e988 ISI cm C FORECACST + FUTURES PRICE o ACTUAL PICE CHART 4: EXPECTED VS. ACTUAL PRICES CWTON -- PERCENT PEA ANNUM 60- so 40 - 30 - 20 *10 -0 -20- -30 1980 1982 1994 1968 IgSo CM FORECAST + FUTUPES PRICE o ACTUAL PPICE 8 the F-statistics are looked at in connection with the t-tests of the coefficients. These four commodities have estimates of / that are significantly differ.nt from one at the 5% level; in three of the four, the estimates of / are negative. Although not statistically significant, sugar and coffee have estimates of / that deviate widely from unity. On the other hand, the hypothesis that a-0 cannot be rejected for all of the commodities. Table 4: Rationality of Futures Prices F-test Commodities a a t.:P=0 t:P=1 R**2 a-0,0-1 Copper 0.137 -1.235 -1.44 -2.61** 0.12 3.50* (0.094) (0.857) Sugar -0.565 3.435 1.42 1.01 0.11 0.62 (0.509) (2.423) Coffee 0.073 0.105 0.12 1.25 -0.14 1.34 (0.088) (0.883) Cocoa -0.071 0.789 1.02 -0.27 0.01 1.04 (0.063) (0.772) Cotton -0.063 4.640 3.18*** 2.49** 0.53 3.14* (0.083) (1.459) Maize -0.089 0.793 1.08 -0.28 0.02 1.14 (0.093) (0.735) Wheat 0.061 -0.256 -0.56 2.73** -0.09 3.78* (0.053) (0.460) Soybean 0.064 -1 806 -1.72 2.67** 0.20 3.89* (0.063) (1.049) * Significant at 10% level. Significant at 5% level. *** Significant at 1% level. Source: International Economics Department, World Bank. The above result broadly agrees with the findings of previous studies that overwhelmingly rejected the rationality hypothesis.4 Estimates of / are often found to be significantly less than one, and sometimes even negative. However, there have been two diametrically opposite interpretations of the biases. One school takes it as evidence of irrational expectations; the other adheres 4 Essentially the same results emerged from studies using different futures price data. These include the Froot and Frankel study using foreign exchange futures prices; Warr used coiamodity futures prices. 9 to the assumption of dtionality of expectations, which then leaves time-varying risk premia as the only explanation of the biases. B. Decom osition of Futures Forecast Bias If the marke, expectations incorporated in futures prices were the same as the CM forecasts, it is possible to calculate the size of the time-varying risk premium from its definition: rpt+1L = Apft+lt - Apet+it, (2) which states that the futures price falls short of the expected future spot price by the amount of the risk premium.5 In the absence of direct observations of market expectations, the CM forecasts are used for 6pet.i,t to cormpute the risk premium. Froot and Frankel showed that the probability limit of 1 in the regression of (1) can be decomposed into two parts, one due to expectational error and the other arising from the risk premium: 3 - 1 - bre - brPF (3) where bre = ccv(ret+,t,Apft+t)/var(;pft+,,t) brp = £var(rpt+1,,t)+COV(rpt+1 t,Apet+1) ]/var(Apft+1,). If exnoctations are rational, the expectational errors will not be correlated with the Intormation set, including the futures prices, and thus bre=O. A zero risk premium implies brp=O. The available data allows computation of both bre and brp -- the contributions, respectively, of expectational errors and risk premia to the departure of p from unity. Table 5 shows the computation results. Generally, it appears that neither expectational errors not risk premia has been the consistently dominant cause of the futures discount bias. In half of the commodities (copper, cocoa, cotton, and soybeans), expectational errors appear to have been the main cause of the forecast biases of futures prices, while in the other half it has been the risk premia. This result is in sharp contrast to the Froot and Frankel findings that expectational e-rrors explained most of the foreign exchange futures discount bias and risk premia were only of minor importance. A possible explanation for the different results could be that commodity prices are much more volatile than 5Since the seller of a futures contract transfers the price risk to its buyer, it is argued that a premium should be paid from the seller to the buyer if the agents are risk averse. Thus, rptf+,t in (2) will be a negative number. 10 Table 5: Decomposition of Futures Discount Bias Commodities bre brp P (1) (2)1-)(2 Copper 3.47 -1.27 -1.20 Sugar -0.36 -2.08 3.44 Coffee -0.07 0.96 0.11 Cocoa 0.32 -0.11 0.79 Cotton -3.22 -0.32 4.54 Maize -0.18 0.39 0.79 Wheat 0.52 0.83 -0.35 Soybean 3.06 -0.25 -1.81 Note: Not all the values of i here exactly match those in Table 3 because of rounding errors and adjustments for the differences in the prices quoted in the spot and futures trades. Source: International Economics Department, World Bank. foreign exchange rates and, therefore, the risk premium is accordingly larger. Like Froot and Frankel, most of the brp's are negative, meaning that the risk premium tends to push P above one. However, unlike Froot and Frankel, who found the bre 's to be all positive, our estimates are equally divided between positive and negative values, suggesting that the CM commodity specialists, in forming price expectations, have used different weights for different commodities on the relative importance of the current spot prices vis-a-vis futures prices. C. Variances of Risk Premium and Expected Price Change Fama, and Hodrick and Srivastava claim that in the foreign exchange market the variance of the risk premium is greater than the variance of the expected change in prices: var(6pet+,Jt) < var(rpt+i,t). (4) Froot and Frankel showed that, if (4) holds and Apft,+t and ret+1,t are uncorrelated (i.e., if expectation is rational or br.=O), then /<1/2 (or equivalently brp>l/2). Since it was revealed in Tables 4 and 5 that /<1/2 because brp>l/2 and br.=0 only for coffee and wheat, the variance of risk premium may not be greater than that of expected price changes for the majority of the commodities. However, this test is valid only under the condition of zero expectational errors. Another way of directly confirming (4) is to calculate the variances involved, which is done in Table 6. Among the four 11 Table 6: Variances of Risk Premium and Forecasted Price Changes Variances of Commodities APt+11 Q Aft+X1 &pet+,1 rpt+,1 (3)-(4) (1) (. (3)- (4) Copper 0.0480 0.0072 0.0516 0.0260 0.0256 Sugar 0.5224 0.0099 0.1650 0.1140 0.0510 Coffee 0.0533 0.0098 0.0132 0.0223 -0.0091 Cocoa 0.0281 0.0059 0.0197 0.0126 0.0071 Cotton 0.1116 0.0031 0.0142 0.0091 0.0051 Maize 0.0465 0.0106 0.0196 0.0172 0.0024 Wheat 0.0212 0.0137 0.0184 0.0276 -0.0092 Soybean 0.0278 0.0025 0.0107 0.0069 0.0038 Source: International Economics Department, World Bank. variances shown, that of actual price changes (column 1) is the greatest (the exceptions are copper and wheat); and that of the futures premium (column 2) is substantially smaller than the rest. The variance of risk premium (column 4) is larger than that of expected price change (column 3) only for coffee and wheat, which were also shown to be the case under the assumption of rational expectations. For the rest of the commodities, the variance of risk premium is either substantially smaller than that of expected price change (copper, sugar, cocoa, cotton and soybeans) or is roughly the same (maize). The result in general disputes Fama, and Hodrick and Srivastava proposition contained in the inequality (4). However, Table 6 confirms their claim of covariance between risk premium and expected price change. This can be seen by the relatively large negative covariance between the two implied by the variance shown in Table 6.6 In any case, the variances of risk premium and expected price changes are both sizable, which deviates considerably from the Froot and Frankel finding that risk premium in the exchange markets has been relatively constant, has smaller variance than expected changes in exchange rates, and probably not correlated with it. IV. TESTS OF RISK PREMIA AND EXPECTATIONAL ERRORS The point estimates in Table 4 of the relative importance of the risk premia and expectational errors do not lend themselves to statistical tests of their significance. Fortunately, there are direct statistical tests available for the estimates, br, and brp. 6 The covariance between risk primium and expected price change is equal to ((2)-(3)-(4)]/2, where the numbers in the parentheses are the column numbers in Table 6. 12 A. Tests of Risk Premium To test the importance of risk premium, we run the OLS regression of the following equation: pet+, = C2 + 2 APft+lt + eL1t, (5) where the null hypothesis to test is that futures discount is perfectly correlated with expected price changes or 0,=1. By virtue of the identity (2), if (2=1 holds, then futures discount is not correlated with risk premium. Since 02=1-b,p,' 2=1 implies bp=O. Thus, a test of 02=1 is equivalent to a test of brl=0 in Table 4. Regressions of (5) can also be used to test the hypothesis that the risk premium has a mean of zero, i.e., 2=°0. The error term in (5) represents the random error in measuring the market expectation. Table 7 provides the OLS estimates of (5). In four of the commodities (cocoa, cotton, maize, and soybeans), the null hypothesis, 132=1, cannot be rejected, while in the other four, it is rejected. It should be noted that the commodities for which the null hypothesis cannot be rejected are the same ones that showed larger variances for forecasted price changes than for risk premia Table 7: Tests of Risk Premium F-test Commodities 92 t:32=0.5 t:(32=1 R**2 D-W a=0, 12=1 Copper 2.270 3.31** 2.38** 0.68 1.65 4.35* (0.534) Sugar 3.079 2.54** 2.05** 0.51 2.53 2.65 (1.016) Coffee 0.035 -1.06 -2.20** -.14 2.14 5.03** (0.439) Cocoa 1.108 1.10 0.20 0.27 1.83 0.16 (0.553) Cotton 1.321 1.28 0.50 0.29 2.11 4.69* (0.064) Maize 0.614 0.25 -0.84 0.09 2.22 1.60 (0.460) Wheat 0.166 -0.77 -1.92** -0.12 1.53 2.39 (0.434) Soybean 1.250 1.21 0.40 0.28 1.31 1.38 (0.037) Notes: See Table 3. Source: International Economics Department, World Bank. 13 in Table 6. Three of these four had larger bre than brp in Table 5. Of the other four commodities for which the null hypothesis is rejected, three had larger brp than brs* Of these three, coffee and wheat showed larger variance of risk premium than expected price changes in Table 6. For coffee and wheat, therefore, the preponderance of risk premium seems quite strong. The only surprise is copper which showed a predominance of expectational errors but also a relatively large variance for risk premium. There is, however, no strong indication of the presence of non-zero risk premium; the estimates Of a2 (not reported in Table 7) are not significantly different from zero, except for cotton. Charts 5-8 show the time path of risk premia and forecast errors of futures prices for copper, coffee, cotton, and wheat. That the risk premium for these commodities fluctuated over time very much in line with the forecast errors suggests the presence of time- varying risk premium that is correlated with the futures discount. The F-test statistic is significant enough to reject a2=Q and /32=1 for copper, coffee and cotton; sugar and wheat also show relatively high F statistic. For these commodities, the risk premium either does not have mean zero or is correlated with the futures discount, or both. The regression of (5) also allows a direct test of the Fama, and Hodrick and Srivastava claim that the variance of the risk premium is greater than that of the futures discount, since the inequality (4) implies 32<1/2. Table 7 shows t-tests of /2=1/2. For all commodities except for copper and sugar, the null hypothesis of 32=1/2 cannot be rejected. Thus, even if (4) does not hold, the differences in variance of risk premium and expected price change are not likely to be large for most commodities. Overall, unlike the results obtained by Froot and Frankel for exchange rate futures, for commodity markets the importance of the risk premium in explaining the futures discount bias cannot be ruled out. Risk premia could be a significant part of the futures discount, even if not the dominant part, and may be correlated with price expectations. B. Tests of Expectational Error The flip side of the test of risk premium is the test of expectational error. Since a joint test of risk premium and expectational error is not available, the test of expectational error will provide independent evidence on its importance in explaining the futures discount bias. This will be done in two steps: first, the rationality of the CM forecasts will be tested and then a direct test of brO= will be made. 14 If the CM forecasts were rational, the biases in the futures premium would have to be attributed to risk premium. To test the rationality, we regress the forecast errors on forecasts: ret+,t = a3 + P3 Apett+1t + et+it, (6) where the null hypothesis of rationality implies a3=93=0. Results of estimating (6) with OLS are reported in Table 8. The F-test results indicate that the rationality of the CM forecasts is rejected only for copper. For cocoa and soybeans, the test statistics are not significant at the usual level, but are high enough to reject the null hypothesis at the 10% level. The results of t-tests of P3=0 are not much different; 03=0 is rejected only for copper, at the 5% significant level but at the 10% significance level it is rejected for cocoa and soybeans as well. The above results are similar to those found in a related study by Choe (1990). A problem in estimating (6) with OLS is that the same (expectation) variable appears on both sides of the equation. The error term in (6), therefore, is likely to be correlated with the independent variable and the OLS estimates of p3 may be biased toward unity. However, the hypothesis that P3=1 is rejected for at least three of the commodities, suggesting that the biasedness toward unity, if any, was not a factor for these commodities in the test results for rationality. To circumvent the estimation problem due to correlation of the error term with the independent variable in (6), we substitute the futures discount for price expectations as the explanatory variable: -re,+lt = a4 + P4 LPft+lt + et+l,t. (7) The futures discount can be viewed as an instrumental variable that is independent of the error term in (7). A test of /4=0 provides a test of rational expectations in the sense that, if P4 were found to be positive, for example, the CM forecasters could have improved the accuracy of their forecasts by betting against the futures market. In other words, the implication is that the forecasters have not used the available futures market information efficiently. Furthermore, regression of (7) provides a convenient direct test of the point estimate of bre in Table 4, because 0,=bre. Table 9 shows the OLS estimates of equation (7). The rationality hypothesis is rejected for more of the commodities than in regressions of equation (6); the F-test statistics are highly significant in rejecting the null hypothesis for copper, cotton, and soybeans; that for wheat is significant at 10%. The t-test results for p4=0 are similar to those of the F-tests. For copper, 15 CHART 5:FUTURES ERROR AND RISK PREMIUM COPPER: PERCENT PER ANNWU 0.4 0.3 0.2- 0.1 -0.1 -0.2 -0.3 -0.4 1980 1982 1984 1388 1988 a FUTrLRES ERROR + RI SK PR I WIN CHART 6: FUTURES ERROR AND RISK PREMIUM COFFEE: PERCENT PER ANNLN 0.5 - 0 4 - 0.3 0.2- 0.1 6 -0.1~~~~~~~~~~~4 -0.2- -0 3 1980 1982 1984 1988 Igoe a FL1TLSAes ERROR .+ AI SK PRG9d I IN 1 6 CHART 7: FUTURES ERROR AND RISK PREMIUM COrTTN: PERCENT PER ANNIN 0.2 0.1 -0.1 -0.2- -0.3 -0.4- -0.5 -0.6 1980 1982 1984 19B9 1988 a FUTLUES ERRO R I ISK PPJ I W CHART 8: FUTURES ERROR AND RISK PREMIUM WHEA r: PERCENr PER ANNW 0.2- 0.1 -0.1 -0.2- -0.3 1S9o 1982 1984 1988 19S8 a FuTLAES ERROR t PISK PRBEILW 17 Table 8: Tests of Rationality of CM Forecasts F-test Commodities f03 t: L3=0 t:03=1 R**2 D-W a3=63=° Copper 1.276 3.79** 0.82 0.63 1.83 14.37*** (0.337) Sugar -0.131 -0.25 -2.18** -0.13 1.08 0.06 (0.519) Coffee -0.120 -0.19 1.78* -0.14 2.95 0.04 (0.631) Cocoa 0.751 1.70* -0.56 0.19 1.09 2.90* (0.441) Cotton -0.840 -1.16 2.55** 0.04 1.96 1.36 (0.721) Maize 0.470 0.86 -0.97 -0.03 1.89 0.74 (0.546) Wheat -0.079 -0.25 3.44*** -0.13 0.80 0.06 (0.314) Soybean 1.035 1.70* 0.06 0.19 1.19 2.90* (0.608) Notes: See Table 3. Source: International Economics Department, World Bank. Table 9: Tests of Expectational Error F-test Commodities CZ /4 t:04=0 D-W R**2 C4=04=0 Copper -0.152 3.470 4.00*** 2.72 0.65 16.02*** (0.095) (0.867) Sugar 0.275 -0.355 -0.17 1.14 -0.14 0.03 (0.447) (2.128) Coffee -0.014 -0.070 -0.10 3.04 -0.14 0.01 (0.074) (0.735) Cocoa 0.088 0.319 0.33 0.87 -0.12 0.11 (0.078) (0.954) Cotton 0.213 -3.220 -2.74** 0.86 0.45 7.49** (0.067) (1.18) Maize 0.190 -0.178 -0.23 2.12 -0.13 0.05 (0.098) (0.780) Wheat 0.032 0.517 1.68* 1.50 0.18 2.81* (0.036) (0.309) Soybean -0.026 3.056 3.26*** 2.18 0.55 10.64*** (0.056) (0.937) Notes: See Table 3. Source: International Economics Department, World Bank. 18 cotton and soybeans (and possibly also for wheat), the bre's are significantly different trom zero -- in other words, expectational errors rather than risk premiums were the main cause of futures discount bias. For the other four commodities, however, the rationality of CM forecasts cannot be rejected, making risk premia rather than expectational errors a plausible cause of the futures forecast bias. V. FUTURES PRICES AND PRICE EXPECTATIONS So far, the analysis of futures discount bias has relied on the identity (2) that a futures price is the sum of the expected price and a risk premium. Thus, the risk premium is calculated as a residual, which in reality may include a variety of factors other than the lump-sum payment supposedly being paid for assuming the price risk. One such factor is the error of measuring the market expectation. Furthermore, since the risk premium may enter the futures price relationship in ways other than the simple additive form in equation (2), subsequent decomposition formulae may be inadequate to estimate its role. In this section, we utilize a result in Turnovsky (1983) that establishes a relationship between futures, spot and expected prices from an explicit model of spot and futures price determination. Under a standard set of assumptions, he showed that: F,,, = w, Pt + w2 EtPt+1. (8) It states that the future_ price of a commodity in period t for delivery in t+1, Ft+1 t, is a weighted average of the current spot price, Pt, and the price expected to prevail in t+l, E,P,+1. The weights, w, and W2, depend on the agents' coefficient of absolute risk aversion, the degree of price variability, and supply and demand parameters. If and only if the agents are risk neutral, then Ft+,,t = EtPt+1, or w1=O and W2=1. Thus, a test of w1=O and ,2=1 will provide a test of the null hypothesis that the agent s are risk neutral. Risk non-neutrality implies 1>0 and w2<1 and w1+w2