Prediction of spot prices of potatoes and wheat with futures contracts

Prediction of spot prices of potatoes and wheat with

futures contracts

Master thesis

Reinoud Metselaar May 27, 2008

Student number 1594753

University of Groningen

Faculty of Economics and Business

Master of Science in Business Administration Specialisation Finance

Profile Risk and Portfolio Management

__________________________________________________________________________

Key words: potato and wheat futures contract, prediction of spot prices and GARCH. JEL classification: G13; Q1; Q21 ; Q11; Q13

Introduction

The main purpose of the agricultural futures markets is to hedge risks. These futures markets are important for the price of many agricultural commodities, since cash prices are often settled by using nearby futures contract prices. Futures contracts are used to ensure that a producer or the processing industry receives or pays a certain price in the future. The earliest record of trade in a commodity futures contract was in Japan. The modern futures contract trade started in the early 1800’s in Chicago (Bernstein [1996]). This is not surprising since the area around Chicago is famous for their farmlands and cattle. Chicago has the biggest commodity exchange of the world. In recent years the variance in yields of agricultural commodities increased, probably due to the changing climate. An increasing variance in yields means an increasing variance in prices and higher risks. Because of these higher risks producers of agricultural products, traders, processing industry, the government and retailers are more and more interested in prediction of agricultural commodity spot prices.

A possible basis for price forecasting is the current price of a contract for a future delivery. When it is possible to test and accept a prediction model with the use of futures contracts it will be easier to make decisions about production and trading. Agricultural commodity futures contracts differ from futures contracts with an underlying stock or index. The most important variable that influence the agricultural futures contract price is the price of the underlying. This variable is also the most import variable for stock and index futures contract prices. However, agricultural commodity futures contract prices are also influenced by storage costs and the convenience yield, which is the value of the benefit of holding the commodity. In general these two variables have a low variance compared to the variance of the underlying.

In this study the potato and wheat futures contract market will be investigated. The planting time for “winter wheat”1 is from September till February. “Summer wheat”2 is normally planted from February till May. In Western Europe potatoes are generally planted in April. Another difference besides planting time is that the potato has a discontinuous inventory between season while wheat is a continuous storable commodity (storable for over a year). This means that potatoes are difficult to store for over a year, the quality decreases after time passes by. Consequently, the price depends strongly on the yield of the new harvest. The differences between the commodities could influence the power of a potential prediction model.

1

Winter wheat is planted in the autumn, before the winter 2

The objective of this research is to find out if it is possible to predict spot prices of potatoes and wheat with RMX (Risk Management Exchange) futures contract prices. Therefore the research question is:

Do RMX potato and wheat futures contract prices have prediction power for spot prices?

The central question can be answered with several sub-questions.

1. Are there stationarity problems with the spot and futures contract prices? 2. Is there a difference in prediction accuracy if you change the time to maturity?

3. Is there a difference in the prediction accuracy between potato and wheat futures contract prices? 4. Which prediction model fit best (OLS, ARCH, GARCH3)?

5. If a risk premium exist, is it constant or time varying?

The Dutch commodity futures market in Amsterdam closed in 2006, due to the low liquidity of futures contracts. After 47 years traders had to move to the commodity futures market of Hannover. In this master thesis the futures contract market of Hannover (RMX), will be investigated. The data used in this study consist of weekly spot prices of three indexes and the corresponding futures contract prices of Table potatoes (consumption), Processing potatoes (processing industry), London potatoes (process industry, trade in the UK) and Wheat for the period covering 1998-2007. The results of this thesis can give the opportunity to predict the spot price of processing, table and London potatoes and wheat listed at the RMX over the forecasting horizons of 1 week and 1, 2 and 3 months4. Furthermore, it investigates which time series model fits best. Former studies (Fama and French [1987], McKenzie and Holt [2002], Wang and Ke [2006], a.o.) have investigated different agriculture futures contact markets with different forecast horizons and different models.

In section 1 relevant literature will be discussed, including potential problems. In section 2 the methodology and the methods and solutions for potential problems will be explained. After the methodology section, a discussion of the data and descriptive statistics will be presented in section 3. Finally in section 4 the results of this study will be summarised and discussed.

3

Generalised Autoregressive Conditional Heteroscedasticity 4

1. Literature review

1.1 Theory

There are many researchers who investigated commodity futures markets. The main topics of their articles are market efficiency, risk premiums, variance, seasonality and prediction models. In this literature review several reports are discussed including the theme prediction model or a related subject. Fama and French [1987] studied the commodity futures contract price in two ways. The theory of storage which explains that the return from purchasing the futures contract at time t and sell it at time T equals the interest forgone plus the marginal storage costs minus the convenience yield (equation 1). The convenience yield is the productive value of the inventory. Fama and French [1987] and Hull [2006] discussed that the futures contract price of a consumption asset can in general be determined by equation 1.

Ft = St e (c-y) (T-t) [1]

Where Ft is the futures contract price at time t and St is the commodity spot price at time t. C and Y are

the cost of carry (the measure of the storage cost plus the interest less the income earned on the asset) and the convenience yield, they are multiplied by the time to maturity T-t. The convenience yield is high when inventories are low, holding stocks is then more valuable.

Ft = Êt (ST) [2]

Equation 2 Êt (ST) describes the expected future spot price at time T (maturity time of futures contract Ft),

under the assumption that the required return equals the risk free return. The difference between the required return and the risk free return is called the risk premium5. This risk premium can change considerably over time. The expected premium in absolute amount is defined as the bias of the futures contract price as a forecast of the spot price at maturity. Both models (equation 1 and 2) are alternative perspectives for the same economic phenomena.

For an investment asset it should be expected that the futures contract price is higher than the spot price in the real world (Ft >St). This is called contango, the futures contract price is higher than the spot price. For

commodities that are physical delivered there could be backwardation, which means that the futures contract price is below the spot price (Ft <St). This happens when the convenience yield from holding the

physical commodity exceeds the cost of physical storage plus the interest forgone. There is also a possibility that the market is in “normal” backwardation which means that the futures contract price is

5

In real returns Ft = Et (ST) - RPt. Where Et is the expectation of the future spot price with the information

lower than the “expected” spot price (Ft < Et (ST)). “Normal” backwardation or contango is not known

when the futures contract is traded. It is proven or disproved over time.

The futures contract price contains information about the expected spot price and there is no risk premium under the assumptions that the agents are risk neutral. Fama and French [1987] used the regression model of Fama [1984a, 1984b] to test for time varying expected premiums and price forecasts. The simple regression analysis to forecast spot prices at maturity with futures contracts prices is the basis of many studies including Mc Kenzie and Holt [2002] and Laws and Thompson [2004].

S T = α + ß Ft,T + ε(t,T) [3]

Equation 3 is a time series forecast to test the relation between the variable ST and Ft,T. Where ST is the

spot price at time T (the forecast) and Ft,T is the price of the futures contract at 1 week or 1, 2, 3 months

before maturity. The condition that the futures contract price contain information about the future spot price would imply that α = 0 and ß = 1 under the assumption that the required return equals the risk-free

return (risk neutrality). If these conditions are not fulfilled, it means the futures contract market does not contain information about the future spot price, or there is a constant risk premium, or a time varying risk premium which means that α is non-zero. Wang and Ke [2002] as well as Bowman and Husain [2004]

discussed that if α is non-zero there exist a risk premium.

A futures market is efficient if al information available in the market is reflected in the futures contract price, including the expectations of the traders in the market. As discussed by McKenzie and Holt [2002] as well as Bowman and Husain [2004] the hypothesis that futures contract prices provide unbiased forecasts of spot prices is a joint hypothesis. McKenzie and Holt [2002] suggested that market efficiency implies that futures market prices will equal expected future spot prices plus or minus a constant or, possibly a time varying risk premium. If futures markets are unbiased predictors of spot prices at maturity they have to be efficient and have no risk premium. Bowman and Husain [2004] discussed that a market efficiency test with equation 3 would require careful matching of futures contract horizons and expiration dates with actual spot prices. The data in this research is not accurate enough for efficiency tests. Instead, equation 3 will be used to see if futures contract prices have prediction power.

Stationarity (fixed mean and variance)

for different commodity spot and futures contract prices this can give biased results with the wrong tests. The first difference of equation 3 can “remove” non-stationarity of the spot and futures contract prices (equation 4a). Equation 4b is another popular transformation, it regresses the cash prices on the lagged basis (Fama and French, 1987; McKenzie and Holt, 2002, Zulauf et al., 1999).

ST -St = α + ß (Ft - Ft-1) + ε(t,T) [4a]

ST -St = α + ß (Ft,T - St) + ε(t,T) [4b]

Where ST is the spot price at expiration time T (for Ft,T), Ft,T is the price of the futures contract at time t

with expiration date T, St is the spot price at time t (1 week, 1, 2 or 3 months before maturity T of Ft,T) and

ε(t,T) is the error term. Brooks [2000] discussed that the first difference is statistically valid but has the

problem that pure first difference models have no long-run solution.

A potential problem with equation 4a and 4b is cointegration. If two or more series are of themselves non-stationary, but a linear combination of them is stationary, than the series are said to be cointegrated. With cointegration problems futures contract prices tend to exhibit less variability than spot prices. McKenzie and Holt [2002] as well as Bowman and Husain [2004] discussed the problem of cointegrated commodity spot and futures contract prices. In both studies the hypothesis of no cointegration was rejected. To see if there are cointegration relationships between the stock price and the futures contract price a Johansen test was applied in both papers6. This test can also be used to test if futures contract prices are unbiasedness predictors of spot prices in the long run. McKenzie and Holt [2002] mentioned that an Eagle-Granger two-step cointegration procedure should not be used, since the test procedure does not have well-defined limiting distributions.

If cointegration exist there is a relation between the futures contract prices and spot prices. But cointegration does not rule out short-run in-efficiency and pricing bias. As McKenzie and Holt [2002] discuss, this bias can be observed due to the problem that markets take time to adjust to new information.

Cointegration problems can be evaded through applying an error-correction model (ECM). The ECM uses combinations of first differenced and lagged levels of cointegrated variables and can be used to test the short-run relation between the futures contract prices and spot prices. An ECM provides the series stationary. A problem is that an ECM does not allow risk premiums varying in nature.

A similar model to test the short run relation between the spot prices and futures contract prices is a Vector Autoregression model (VAR). In a VAR model the current values of the underlying and futures contract depends on different combinations of the previous values of the two variables. All the variables

6 _ε

in a VAR are treated symmetrically by including (for each variable) an equation explaining its evolution based on its own lags and the lags of all the other variables in the model

A second problem with an ECM model is that it assumes that the distribution of the spot price change is characterized by a constant variance. Kenyon, Jones and McGuirk [1993] and McKenzie and Matthew [2002] concluded that the error variance of equation 3 often changes considerably over time. Consequently, an ARCH (Autoregressive Conditional Heteroscedasticity) test could be applied. An ARCH model is appropriate when the data has periods of swings followed by periods of relative calmness.

In a GARCH (Generalised Autoregressive Conditional Heteroscedasticity) model the current volatility depends on the previous volatility. (G)ARCH is an auto regression model which allows the error variance to change over time on the lagged value of itself. Wen and Wang [2002] also state that an ARCH model succeeds in modelling the stochastic time varying volatility and excess kurtosis of commodity price changes.

1.2 Empirical results

Tomek [1979] was one of the first who discussed that agricultural futures contract prices can provide a poor forecast. But this forecast could still be superior to other techniques such as econometric models. As discussed in the introduction efficiency tests are closely related to the literature of forecasting with futures contracts. The results of former studies about the relationship between futures contract prices and spot prices and the prediction power of futures contracts will be discussed in this section. The different studies are summarized in table 1.

Previous studies testing the relationship between futures contract prices and spot prices include Haigh [2000], Laws and Thompson [2004] and Mckenzie and Holt [2002]. These studies are not directly related to potato and wheat futures contract markets7 and results of stationarity tests showed mixed results. As discussed in literature section, if stationarity problems are found a Johansen cointegration test should be applied. All three studies found non-stationarity and cointegration problems for some but not all futures contract prices and spot prices. Results of stationarity tests in the wheat futures markets are mixed. Wang and Ke [2006] as well as Wen and Wang [2004] could not reject the hypothesis that the Chinese wheat futures prices had one unit root. Bowman and Husain [2004] rejected the hypothesis that wheat futures prices and spot prices listed at the Chicago Board of Trade are non-stationary.

Haigh [2002] as well as Laws and Thompson [2004] and McKenzie and Holt [2002] applied a Johansen cointegration test to analysis the long-run relation between the futures contract prices and spot prices. For most futures contract prices they could not reject the individual hypotheses α = 0 and ß = 1. This implies

7

that the Baltic International Freight Futures Exchange (BIFFEX), different currency futures contract markets and different commodities futures markets do contain significant information of the expected spot price in the long-run. The results of the Johansen cointegration test showed that most futures contract prices and spot prices are cointegrated. Because of the cointegration problems an ECM was used to test the short run relation. Results of the short-run relation between futures contract prices and spot prices will be discussed later on.

Wang and Ke [2006] investigated the Chinese wheat and soybean futures market. They also applied a Johansen cointegration test. Results of Johansen cointegration tested showed that wheat futures contract prices are not cointegrated with spot prices. This suggest that no long-run equilibrium relation between the Chinese wheat futures contract prices and spot prices exists. The inefficiency is probably caused by market manipulation of large traders and government regulation. Only the soybean market was further investigated because of the lack of cointegration in the wheat futures contract market. For most maturities and markets the hypothesis that ß = 1 could not be rejected. This implies that the soybean futures market contains information of the future spot price. Bowman and Husain [2004] found cointegration for the 3 months to maturity wheat futures contract, but not for the 6 months to maturity futures contract. This means for the 3 months wheat futures contract there exist a relation between the futures contract price and spot price.

Fama and French [1987] investigated different commodity markets in the USA between 1967-1984. To avoid stationarity problems the first difference of equation 3 was used. They concluded that the Chicago wheat futures contracts in spring are weak predictors for harvest spot prices. For the wheat futures contracts price 10 months to maturity the ß (equation 4a) was - 0.78 which means the futures contract price 10 month before maturity does not contain information about the change in spot prices from time t to T. The R2 was 0.05 which implies that the model does not fit well. For 2 months to maturity wheat futures contract prices the ß was 0.18 which means that with this time to maturity the futures contract prices contains information. Kofi [1973] found, with equation 3, similar results for the Chicago wheat futures market between 1953-1969.

The results of the different studies summarized in table 1 are mixed, the ß for wheat is not for all markets significant different from unity. But in general the futures contract price contains information about the spot prices, especially for shorter periods to maturity.

respectively significant different from zero and unity. Kofi [1973] concluded that the R2 of equation 3 for potatoes in the USA is 0.16 between 1953-1969 (time to maturity of 8 months). This means that equation 3 does not fit well, probably caused by the discontinuous inventory of potatoes. This reasoning is confirmed by a R2 of 0.70 for the potato futures contract with 2 months before maturity. After planting, the potato market receives new information and there are no stocks left of the prior period. French [1986] and Tomek and Gray [1970] discussed that from year to year this should result in a low variation of potato futures contract (maturity November) prices in spring.

Wang and Ke [2006] tested for ARCH effects in the Chinese wheat futures market. The null hypothesis of no ARCH effect could not be rejected in general. Some studies used a GARCH model to test if current spot prices contain information about future spot prices. Wen and Wang [2004] used a GARCH model to test if prices of the Chinese wheat market in the last period are related to the current price. They found that prices in the last period contained information on the current spot price. The ARCH and GARCH properties showed that large price changes tend to be followed by other large changes and small changes by smaller ones.

Wen and Wang [2004] used continuous data of futures contract prices and spot prices to estimate the parameters of an AR, ARCH and GARCH model. All three models showed that there is a significant relation between the current price and the price in the last period. They found the ARCH and GARCH models outperformed as forecasting model, this was probably caused by the ARCH and GARCH specifications. Morana [2001] applied a GARCH model to forecast the 1 month forward price of oil. The GARCH model fitted better than a random walk process.

Table 1 Overview literature

Researchers Subject Market Model Results Tomek and

Gray [1970]

Relationship spot and futures contracts prices on (non)-inventory commodities

Corn, soybean and potato futures contracts USA (1952-1968) Equation 3 (OLS) Relation between the futures contracts and spot prices - Potatoes - m = 7 months - ß =3.94 (rejected sign. 5%) -α =- 662.99 (rejectedsign5%) - R2 = 0.57 Kofi [1973] Framework, comparing the efficiency of futures markets

Potatoes, cacao, corn, soybeans and coffee Chicago (1953-1969)

Equation 3 (OLS) Relation between the futures contracts and spot prices

- Wheat / Potatoes - m = 3 months - ß= 0.92 / 0.96

Sign. at 5%/rejected sign.5%

-α = 12.40 / 19

Sign. at 5% / Not sign. at 5%

- R2

= 0.78 / 0.61 Fama and

French [1987]

Forecast power, theory of storage and premiums. Different commodities New York-Chicago (1967-1984) Equation 3 and 4 (OLS) - Wheat - m = 2 months - ß = 0.18 (not sign. 10%) Haigh [2000] Freight market Baltic International

Freight Futures Exchange (1985-1999) Johansen cointegration test. Short-run. Extra seasonal dummy and error correction term. (OLS, VAR ECM) - Freight market - m = 2 months - ß = 1.110 (sign. at 1%)c - α = 0.769 (sign. at 1%) - RMSEb: ECM McKenzie and Holt [2002] Market efficiency in agricultural futures market Cattle, hogs, soybeans and corn Chicago (1959-2000) Johansen cointegration test. + GARCH conditional variance. ((G)ARH, ECM) - Corna - m= 2-3 months - ß = 0.998 (sign. at 5%)c - α = 0.485 (sign. at 5%) - RMSE = ARCH-ECM Bowman and Husain [2004]

Forecast with futures contracts or judgement 15 commodities Wheat (IMF) (1976-2003) Johansen cointegration test (Judgement, Regression, ECM) - Wheat - m = 3 months - ß = 0.42 (sign. at 5%) - RMSE = ECM Wen and Wang [2004] Price behaviour in China’s wheat futures market Chinese wheat market (2000-2004) AR, (G)ARCH (with switching dummy which is zero for forecasting model) - Wheat - m = 10 days - ß = 0.16 (sign. at 5%) - RMSE: GARCH Laws and Thompson [2004] The efficiency of financial futures markets Euro/dollar -Yen/dollar Sterling/dollar (1987 – 2000) Johansen cointegration test, OLS, ARIMA (rolling 260 days), VECM - Euro/dollar

-1 and 3 months rejectedc

ß = 1 and α = 0

- 2 months not rejected

ß = 1 and α = 0 - m = 3 months - RMS = OLS Wang and Ke [2006] Efficiency tests of agricultural commodity

Chinese wheat and soybean (1998-2002) Johansen cointegration test - Soybean - m = 1 month - ß = 1.46 (sign. at 1%)c - α = -1.18 (sign. at 5%) a. Corn has almost the same characteristics as wheat

b. The best forecasting model based on the Root Mean Squared Error.

c. Reported intercepts and coefficients are estimated with a Johansen cointegration test

Law and Thompson found a risk premium for some futures contracts in the long run. Haigh [2000] as well as McKenzie and Holt [2004] rejected the hypothesis of a risk premium in the long run, respectively for the BIFFEX and Corn, Soybeans, Hogs and Cattle futures contract markets.

As discussed above there can exist a risk premium in equation 3 and 4. McKenzie and Holt [2002] and Laws and Thompson [2004] rejected the hypothesis that α = 0 in the short run which means that a (time varying) risk premium in equation 4 exists. Jalali-Naini and Kazemi [2006] investigated the crude oil futures market in Texas from 1999 to 2003 and used an ARCH and GARCH model to calculate the hedge ratio. They found a non-zero risk premium and the variability of the hedge ratio increased as maturity increased.

Bowman and Husain [2004] as well as Wang and Ke [2006] do discuss the risk premium sideways in their literature review but do not relate it to their own empirical results. Wen and Wang [2004] do not discuss the risk premium at all. This is odd if you consider that the risk premium is closely related to forecasting power with futures contracts. Fama and French (1987) tested for risk premiums by regressing the futures contract price at time t minus the spot price at time T against the difference between the futures contract price at time t minus the spot price at time t. They found a time varying risk premium in the wheat futures contracts market. Fama and French [1987] found evidence of variation in the basis Ft-St. This was

probably due to seasonally in production and/or demand. Another method to test for a time-varying risk premium is the ARCH-M model. Beck [1993] used an ARCH-M model, in his study the risk premium was hypothesized as a function of the conditional standard deviation of the change in spot price.

If the individual hypotheses can not be rejected, it is still possible that the joint hypothesis will be rejected. Wang and Ke [2006] rejected the joint hypothesis of α = 0 and ß = 1 with the Johansen cointegration test for some but not all markets. This hypothesis was probably rejected due to the existence of a risk premium, as the individual hypothesis α = 0 was rejected more often than the hypothesis ß = 1.

Other studies gave mixed results.

Mckenzie and Holt [2002] found that all markets are unbiased in the long run. For corn the ARCH-ECM model for forecasting outperformed. Wen and Wang [2004] tested the Chinese wheat market with the ARMA, ARCH and GARCH model. A 10 days forecast horizon was evaluated. They found that the GARCH (1,1) model outperformed the other models. In general the ECM outperformed as a forecasting model. The difference appeared due to the cointegrated relationships between the futures contract prices and spot prices. These results were conclusive with the results of Bowman and Husain [2004] who found that the ECM model outperformed as a forecasting model for the International Monetary Fund Index wheat market.

In this literature review the relevant theoretical background of commodity futures contract pricing and spot price forecasting were discussed. Almost all papers discussed used equation 3 as a basis for their study. The market efficiency in commodity markets implies that futures market prices will equal expected future spot prices plus or minus a constant or, possibly a time varying risk premium. As discussed in the theory section, market efficiency tests with equation 3 would require careful matching of futures contract horizons and expiration dates with actual spot prices.

2. Methodology

First of all the data sets with futures contract prices and index prices will be tested if they are normally distributed. The descriptive statistics, including the Jarque-Bera test, will be applied. To reduce heteroscedasticity problems and the problem of non-normality the logarithm of the variables are investigated. As mentioned in the literature review, a potential problem with the OLS procedure in equation 3 is stationarity of the variables spot price and futures contract price. To test if there are problems with stationarity a Dicky-Fuller unit root test will be applied on the logarithm of the spot and futures contract prices. Rejection of the H0 DF test is taken as evidence of stationarity.

t t t y u y =Φ ₋₁ + [5a] t t t y u y =Ω + ∆ ₋₁ [5b]

Where in equation 5a yt is the spot price or futures contract price and yt-1 is the lagged futures contract or

spot price. The null hypothesis is that the series contain unit root (Ф ≥ 1) and the alternative hypothesis is that the series are stationary (Ф < 1). With a unit root the variance of y increases with time. The DF test of equation 5a is sensitive for autocorrelation, the alternative is adding lagged values of yt-i until

autocorrelation is eliminated. Equation 5b is an alternative regression which is called the Augmunted Dickey Fuller test. The test of Ф ≥ 0 in equation 5a is equivalent to the test of Ώ = 0 (Ф-1 = Ώ) in equation 5b. The Augmunted Dickey Fuller test in Eviews 5.0 automatically adds lagged values till autocorrelation is eliminated.

If non-stationarity is found, equation 4 will be applied. This equation removes the non-stationarity (integrated I(1)) by taking the first difference of the general OLS model in equation 3. An ECM uses combinations of first differenced and lagged levels of cointegrated variables. The advantage of an ECM is that it has a long-run solution which the first difference has not. If the data is stationary, it is not necessary to apply an ECM to test the short term relation between the spot prices and futures contract prices. In this research, if necessary, the first difference will be used.

The OLS model will be tested for autocorrelation with the Durbin-Watson test which measures the serial correlation in the residuals. A potential problem with above models is that it assumes a constant variance of the spot price. If there is heteroscedasticity found in the residuals an ARCH model will be applied. The ARCH model was introduced by Engle [1982]. To test for heteroscedasticity (conditional variance is not constant) the ARCH-LM (Lagrange Multiplier) test will be applied. If the hypothesis of no ARCH effects is rejected a conditional heteroscedasticity model will be applied. An ARCH model allows the conditional variance of a variable to be time varying. Equation 6a is the general ARCH model, were ∆ST is ST - St. The

variance of the error term (σ2ε,t) is related to the previous squared error. The variance is obtained from the

lagged regression of the spot price and futures contract price. The ARCH term will be further discussed in the data section.

) , ( ,T tT t T F

S =

α

β

ε

) , ( 1 tT t T F S =

α

β

ε

υ

σ

ε

=

υ

=

ω

+

γ

ε

σ

The hypothesis that can be tested with equation 6c is y1 = 0. In this research we are especially interested in

the change of the parameters in equation 6a due to the change of the error term. A GARCH (1,1) model allows the previous volatilities to effect current volatility. The lag structure of this model is much more flexible. A GARCH model is better than an ARCH model because it avoids overfitting. The past residuals can affect the current variance either directly or indirectly through the lagged variance terms. Equation 6c is replaced by equation 6d in the GARCH (1,1) model.

2 1 2 2 1 1 2 ,t

=

ω

+

γ

ε

+

γ

σ

σ

The ω represents the constant term, γ1 represents the parameter of the lag of the squared residual from

equation 6a (ARCH term) and γ2 represents the parameter of the last periods forecast variance. An extra

extension in the GARCH model is the term γ2 σ2t-1 which is the function of past variances. As discussed in

the literature, Wang and Ke [2004] and Moreno [2001] used high frequency data of spot prices to calculate the parameters of the (G)ARCH model. In this study the residuals and conditional variance will be obtained from the nearest regression. In the data section this will be discussed in detail.

To test whether there exist a risk premium equation 7 can be used. The assumptions of an absolute constant risk aversion utility function and uncertainty of the interest rate must be made to test for a variable risk premium. The difference between the futures contract price at time t and the spot price at time T must equal the difference between the futures contract price at time t minus the spot price at time t. If ßr is significant different from zero and positive it means that the basis at time t contains information

about the risk premium realized at maturity T. Fama and French [1987] used the same regression analysis to test for time varying risk premiums and found mixed results among different commodities.

To compare the results of the different tests a Root Mean Squared Error (RMSE) will be calculated. The RMSE takes the average deviation of the forecast to the actual spot price and gives an overview of the prediction power in one number. An out of sample data set will be used to calculate the RMSE.

∑

−

=

FC

S

n

RMSE

)

(

/

1

where ST is the actual spot price and FCt is the forecast price. The lower the RMSE the better the

prediction.

2.1 Hypotheses

There are 2 hypotheses of importance that will be tested in this thesis. First the hypotheses that examines the relation between the futures contract price and spot price will be tested. Secondly, the hypothesis for a time-varying risk premium will be applied.

The hypotheses that will be tested with equation 3, 4 or 6 (depending on the model) are:

H0 α = 0 and H1 α ≠ 0

H0 ß = 1 and H1 ß ≠ 1

If hypothesis α = 0 cannot be rejected it means that there is no significant prove of the existence of a risk

premium in the market. The null hypothesis that the coefficient ß = 1 will be rejected if the market does not contain significant information of the spot price. Both hypotheses will also be tested jointly, α = 0 and ß = 1. If α ≠ 0 and ß = 1 it would imply that the futures contract has forecasting power in combination

with a constant risk premium.

The second set of hypotheses that will be tested with equation 7 are:

H0 ßr = 0 and H1 ßr ≠ 0

3. Data

The data used in this study are weekly observations of the RMX potato futures contracts and RMX wheat futures contracts. The different RMX potato futures contracts consist of three different categories: European processing potatoes, Table potatoes and London potatoes. All three categories will be used for testing equation 3. For the wheat futures contract there is one type of futures contract traded on the RMX futures market. All potato futures contracts are cash settlement futures contracts, the wheat futures contract is a physical delivery contract. The contract specifications are presented in appendix A (table A1 and A2). Potato futures contract prices and wheat futures contract prices are available on the website of RMX Hannover.

Figure 1 Weekly spot prices of the different potato indexes

€ -€ ₅ € ₁₀ € 15 € ₂₀ € ₂₅ € ₃₀ € ₃₅ 1999 2000 2001 2002 2003 2004 2005 2006 2007 Year P ri c e p e r 1 0 0 K g . Table potato London potato Processing potato

For details see appendix A

The underlying values of potato futures contracts are not quoted during the entire year (figure 1). If there is no underlying data available the closing futures contract price, which is in the maturity month, will be used as underlying (if available). The data for this thesis is not accurate enough for a market efficiency test. Because the values of the variables are not available for the entire year.

The underlying of a RMX wheat futures contract have to be delivered physical. Unfortunately there is no underlying index available for wheat futures contracts. In this research futures contract prices that are in their maturity month will be used as spot prices. In appendix B the September wheat futures contracts settlement prices are presented. It is clear that in 2007 the prices of the September wheat futures contract rose tremendously compared to previous years. From 1999 till 2006 the price of the September wheat futures contract was between 100 and 150 euro. The wheat stocks in Europe decreased the past years and in 2007 the stocks were at the lowest level since 19828.

Table 2 Contracts and available data

Futures contract

Data available

Underlying a Settlement Expiration monthsb Number of contracts Processing potatoes 1998c -2007 POTXd Cash Bid – ask 2, 3, 4 ,10, 11, 12 39 Table potatoes 2002-2007 TAPX Cash Bid – ask 2, 4, 6 and 11 14 London potatoes 2003-2007 WTBBPC Cash Bid – ask 4,5, and 11 14 Wheat 1999-2007

Futures contract prices maturity month Physical Bid – ask 2, 5, 8 and 11 (12 for 2007) 36

a. POTX=Index of processing potatoes, TAPX=Index of table potatoes and WTBBPC=Index London potatoes b. 1 = January, 2= February, etc. Some expiration months are not available in every year.

c. 1999 prices are converted into euros, exchange rate 1 Euro = 1,99583 D-mark.

d. Before 2002 no underlying index available, prices of futures contracts in there maturity month are used. Source www.rmx-online.de

A potential method to increase the sample size is pooling. All potatoes contracts can be pooled which increases the sample size to 67 futures contracts. The total number of wheat futures contracts that can be investigated is 36. In appendix C the volumes and open interest of the different futures contracts are shown. The high liquidity of the processing potatoes in 2006 and 2007 is probably due to the closed Dutch commodity futures market. In appendix C it is shown that the volumes of wheat futures contracts increased in 2007, this was probably due to the lower stocks available.

The last trading days of the Table, Processing and London potatoes futures contracts are respectively the last Wednesday, Thursday and Friday of the expiration month. For wheat the last trading day is the exchange day before the first day of the delivery month. In this study the weekly settlement prices of Monday will be used as input for the different models.

To calculate the conditional variance and past residuals in equation 6c and 6d the nearest past regression is used. For example the conditional variance and past residuals for the regression of the month 3 (March) processing potato futures contract price will be based on the variance and residuals of the regression of the month 2 (February) futures contract price.

To evaluate the prediction power of the different models a new regression will be made with in-sample data. Due to the small data set the parameters of the OLS, ARCH and GARCH tests presented in the main text will be calculated with the total data set. The in-sample data for potatoes will start at 1998 and end in June 2006 (52 observations), the out-of sample data set will contain the futures contract prices from June 2006 till December 2007 (15 observations). For wheat the in-sample data will contain all contracts from 1999 till May 2006 (29 observations) and the out-of sample data will contain all futures contract prices from May 2006 till December 2007 (7 observations). When the different parameters are calculated with the in-sample data they will be evaluated with equation 8 (out-of sample data).

3.1 Descriptive statistics

The potatoes as well as wheat futures contract prices with 1 week (5 trading days) and 1, 2 and 3 months (respectively 20, 40 and 60 trading days) to maturity will be investigated. In table 3 and 5 the descriptive statistics of the observed real futures contract and spot prices are presented. In table 4 the log basis (log

Ft,T – log St ) for potatoes are shown. The t represents the weeks before maturity, so t(1) are the futures

contract prices 1 week before maturity. For wheat is was not possible to present the basis, this was not possible due to the missing prices of an underlying index.

When the time to maturity increases the maximum, mean and standard deviation increases, this corresponds with theory. The skewness of the spot price and futures prices for all maturities are positive, which means that the distribution has a long right tail. The kurtosis for the maturities 2 and 3 months are above 3 which means these series show leptokurtosis. This means that the distribution exhibits fat tails and peakness around the mean. Therefore the past futures contract prices have a relatively low variance, because the futures contract prices are close to the mean.

Table 3 Descriptive statistics of potatoes spot and futures price obs. week 40-1998 till week 12-2007

Potato STa Ft (1) b Ft (4) Ft (8) Ft (12) Mean 13.110 13.604 14.196 14.631 15.151 Maximum 29.900 29.900 32.500 50.000 51.600 Minimum 1.000 3.200 3.000 1.000 1.000 Std. Dev. 8.333 8.264 8.677 9.921 10.900 Skewness 0.478 0.508 0.591 1.094 1.283 Kurtosis 2.118 2.032 2.238 4.050 4.430 Jarque-Bera 4.721 5.4938 5.524 16.443 24.082 Probability 0.094 0.064 0.063 0.000 0.000 JB of log - - - 2.880 2.253 ADF t-stat.c -4.223* -3.429** -3.294** -4.018* -4.281* Observations 67 67 67 67 67

Eviews output (absolute values) a. Spot price

b. Futures contract price 1 week for maturity, 4 weeks for maturity, etc.

c. ADF critical values *1% = - 3.633, **5% = - 2.948 and ***10% = - 2.613 rejection H0 unit root

This implies that the H0 of a normal distribution can not be rejected with a significance level of 5%. For

the maturities of 2 and 3 months the logarithmic values are tested. The ADF t-statistic for the index as well as the futures contract prices with different horizons are not significant. This means that the unit root hypothesis is rejected with a significance level of 1% and 5% (Fuller, 1979). For Potatoes it is valid to apply an OLS model, the data are normal distributed and stationary. As discussed in the literature review, Bowman and Husain [2004] could not reject stationarity for soybeans, soybean meal, and soybean oil spot prices and for the futures contract prices of tin, wheat, maize, and soybeans. Haigh [2000] could not rejected the hypothesis of a unit root in the Biffex freight futures contract market.

Table 4 Logarithm of the basis Ft – St for potato futures contracts

Potato Log Ft (1) – Log St Log Ft (4) –Log St Log Ft (8) –Log St Log Ft (12) -Log St

Mean -0.009 -0.002 0.065 0.077 Maximum 0.052 0.057 0.196 0.171 Minimum -0.502 -0.058 -0.028 -0.062 Std. Dev. 0.104 0.024 0.048 0.060 Skewness -4.198 0.037 0.506 -0.716 Kurtosis 20.449 3.704 3.185 2.578 Jarque-Bera 421.833 0.668 1.675 3.156 Probability 0.000 0.716 0.433 0.206 Observations 27 32 38 34 Eviews output

Table 4 describes the logarithm basis (Ft – St). The results can be compared with the results of Coppola

[2006] and Fama and French [1987]. They found that the variance rises when the time to maturity increases. This is also the case with the pooled potato spot prices and futures contract prices. The basis for potatoes for longer maturities are higher than for shorter time to maturities (mean). This is probably due to the difficulty to predict the expected spot price when time to maturity increases. Only the standard deviation of the 1 week basis is higher than the other standard deviations. As discussed in the literature review, the higher variance for longer maturities is probably caused by the season effects in production and/or demand.

The potato futures contract market is in general in backwardation 1 week and 1 month before maturity and in contango 2 and 3 months before maturity. Other commodity markets like the oil futures market and different other commodities markets are backwardation (Coppola, 2006 and Fama and French, 1987). As discussed in the introduction, backwardation (Ft < St) exists if the convenience yield from holding the

Figure 2 Closing prices 30 April for futures contracts with maturity at harvest time and spot price at maturity of the futures contract

Wheat 0 50 100 150 200 250 1999 2000 2001 2002 2003 2004 2005 2006 2007 E u ro s /T o n Futures contract prices 30 April (Exp. Sept.) Closing prices September contracts Potatoes 0 5 10 15 20 25 30 1998199920002001 20022003 20042005 20062007 C e n ts /K g Futures contract prices 30 April (Exp. Nov. or Oct.) Closing price October-November*

* October contracts not available for 2006 and 2007

Figure 2 shows that the prices of processing potato futures contracts in April deviate much more from their final settlement price than for the wheat futures contract prices. The wheat futures contract prices in April are more stable and close to the final settlement price. As discussed above this is probably due to the variable production yield in relation to the discontinuous inventory. The higher standard deviation of the basis for potatoes compared to wheat confirms this reasoning. Tomek and Gray [1970] presented the same graph with similar results.

The descriptive statistics of wheat futures contract prices and spot prices are presented in table 5. There is no clear relation between the mean, maximum, standard deviation and the time to maturities of the futures contracts. The skewness of the spot price and futures contract prices for all maturities are positive which means that the distribution has a long right tail. The kurtosis is for all maturities above 3 which means that these series are leptokurtosis. Leptokurtosis was also found for the 2 and 3 months potato futures contract prices. The Jarque-Bera value for the spot price and different futures contract prices are all above the critical value, which implies that the H0 of a normal distribution must be rejected with a significance level

of 1%.

Table 5 Descriptive statistics of wheat spot prices and futures contract prices 1998-2006 Wheat STa Ft (1) b Ft (4) Ft (8) Ft (12) Mean 133.644 133.572 132.306 130.447 129.949 Median 130.700 129.700 129.350 127.600 128.200 Maximum 243.000 235.000 263.500 261.100 229.700 Minimum 101.000 103.000 103.000 103.000 103.000 Std. Dev. 29.365 29.471 29.911 27.550 22.794 Skewness 2.200 2.217 2.865 3.153 2.486 Kurtosis 8.622 8.496 12.560 15.333 11.596 Jarque-Bera 76.466 74.792 184.885 287.777 143.824 Probability 0.000 0.000 0.000 0.000 0.000 JB of log 23.854 25.281 63.025 86.816 40.367 Probability 0.000007 0.000003 0.000000 0.000000 0.000000 ADF t-stat.a -5.829* -5.883* -5.848* -5.992* -5.964* Observations 36 36 36 36 35 Eviews output a. Spot price

b. Futures contract price 1 week for maturity, 4 weeks for maturity, etc.

c. ADF critical values rejection H0 unit root *1% = - 3.633, **5% = - 2.948 and ***10% = - 2.613

Unfortunately it is not possible to present the basis of wheat futures contracts because of the missing wheat index prices. In figure 2 it can be seen that the September wheat futures contract price in April is sometimes below and sometimes above the final settlement price. Which indicates that the wheat futures market is sometimes in contango and sometimes in backwardation. This is not consistent with the findings of Feldman and Till [2006] who found that the CBOT wheat futures market is contango.

As discussed the potato and wheat prices are stationary, therefore it is not necessary to use the first difference and/or apply a cointegration test. The standard deviation of the wheat futures contracts becomes lower when time to maturity increases. Opposite results are found for potatoes. This means that in general the variance of potato futures contract prices are higher for a longer time to maturity. As discussed earlier, this could be caused by the discontinuous inventory. Another interesting observation is that the potato futures market is in backwardation for shorter periods to maturity.

4. Empirical results

This section includes different time-series models that are tested empirically. First of all the traditional OLS model will be applied. Followed by an ARCH and GARCH model. Finally, the models shall be compared using their best specification.

OLS

The results of the traditional OLS model are presented in table 7. The OLS model, as expected, has a good fit. The R2 lowers when time to maturity becomes longer. Tomek and Gray [1970] as well as Kofi [1973] found lower R2’s for potatoes and wheat, the main difference between their studies is that they only used futures contracts that had a maturity around harvest time (October and November). Kofi [1973] found an

R2

for potato futures contracts 1 month before maturity of 0.71. This is lower than the results found in this study, probably due to the contract maturity month.

The R2 for potatoes and wheat for the other maturities (2 and 3 months) were also lower than the results of Tomek and Gray [1970] and Kofi [1973]. To test the individual hypothesis ß = 1 a Wald coefficient restriction test is applied in Eviews. The F-statistic for potato futures contracts 1 week, 1 and 2 months before maturity as well as all wheat futures contracts are significant which is evidence that the hypothesis of ß = 1 can not be rejected. The ß of the 3 months before maturity potato futures contract is significantly different from 1. As can be seen in the literature review this is conclusive with the results of Tomek and Gray [1970] and Kofi [1973]. They also rejected the hypothesis ß = 1 for longer time to maturities.

Table 7 General OLS model with logarithm prices log ( ST) = α + ß log (Ft,T)+ ε(t)

Eviews output

a. Probabilities are reported in the parenthesis. b. Wald test H0: ß = 1

The individual null hypotheses are: α = 0 and ß = 1

** H0 is rejected significance level of 5%

The p values of α are all above 0.05 which means that the hypothesis, α is zero, can not be rejected, this means that there is no significant proof of an existing risk premium. As discussed earlier the risk premium should rise if the time to maturity gets longer. This is implied by the increasing α, but it is not significant.

For the futures contract 3 months before maturity the fit of wheat futures contracts is better with an R2 of 0.754 against a 0.689 for potato futures contracts. This was expected due to the inventory problem of potatoes. If there were sufficient data available and pooling the potato futures contracts was not necessary, it could be expected that the difference between the two commodities would be much bigger, especially if the model was used to test for futures contracts that had a maturity during harvest.

The DW statistics for all potato and wheat maturities are between du and 4-du 9

, except for the 1 week before maturity potato futures contract price. This means that the hypothesis of no evidence of autocorrelation cannot be rejected for all futures contracts except the 1 week potato futures contracts. For the 1 week potato futures contract the results are inconclusive10. In appendix D the Jarque-Bera values of the residuals are presented. The null hypothesis of a normal distribution cannot be rejected for the 1 week wheat OLS and 1 and 2 months potato OLS. The results of the OLS model can be misleading due to the assumption that the residuals are normally distributed. As discussed in the methodology section models that allow the residuals to change overtime are ARCH and GARCH models. Next, a test for heteroscedasticity will be applied.

Heteroscedasticity

When an OLS model is applied on data which is heteroscedastic it could give misleading results. Heteroscedasticity means that the variance of the errors are not constant. The standard errors will be too large for the intercept when the errors are heteroscedastic, this is called the ARCH effect. The ARCH model is a particular non-linear model often used for financial data. To test for ARCH effects an ARCH-LM test is used. In appendix E the results are presented. The ARCH-ARCH-LM test of the residuals from the 1 week and 3 months before maturity potato futures contracts regression are significant which suggest ARCH effects (different lags included). The hypothesis of no ARCH effects also has to be rejected for the 1 week and 2 and 3 months before maturity wheat futures contracts regressions (different lags included). Wen and Wang [2004] as well as Wang and Ke [2006] found evidence of ARCH effects for different commodities spot prices (a.o. wheat). The result suggest that the data of the 1 week and 1 month before maturity potato futures contract and 1 week and 2 and 3 months before maturity wheat futures contracts has periods of swings followed by periods of relative calmness. For the regression with these contracts it is appropriate to apply a conditional heteroscedasticity model.

The results for wheat futures contracts with the ARCH (1) model in table 8 are consistent with the results of the general OLS model except for the α and ß of the 1 month for maturity futures contract. The null hypothesis α = 0 of the 1 week to maturity wheat futures contract must also be rejected with a

significance level of 10%. This means that for shorter periods to maturity there is significant risk premium in the wheat futures contract market.

9

Potatoes (K = 65): dl = 4-1.41 = 2.59 du = 4-1.47 = 2.53 - Wheat (K = 36): dl = 4-1.21 = 2.79 du = 4-1.32 = 2. 68

The parameters for the other maturities for potatoes and wheat futures contracts are all close to the theoretical value and significant. Another point of attention, all ß’s for wheat futures contracts are above 1 and for potato futures contract below 1. When these parameters are above 1 and used to predict spot prices the market is in “normal” backwardation. The expected spot price is higher than the futures contract price (Ft < Et (ST)).

Table 8 ARCH (1) equation 6 a, b and c

Potatoes αααα z-statisticsa ß F-statisticsa,b R2 DW Week 1 0.019 1.443 (0.190) 0.986 1.034 (0.313) 0.990 2.349 Month 1 0.004 0.158 (0.874) 0.979 0.762 (0.386) 0.961 2.013 Months 2 0.015 0.332 (0.739) 0.957 1.093 (0.230) 0.906 2.054 Months 3 0.010 0.160 (0.873) 0.951 0.719 (0.400) 0.847 1.551 Wheat Week 1 -0.031 -1.166 (0.096)* 1.102 2.644 (0.114) 0.989 1.473 Month 1 -0.249 -5.264 (0.000)*** 1.118 27.531(0.000)*** 0.897 2.327 Months 2 -0.118 -0.858 (0.391) 1.060 0.835 (0.368) 0.811 2.238 Months 3 -0.184 -0.824(0.414) 1.092 0.754 (0.392) 0.754 1.913 Eviews output

a. Probabilities are reported in the parenthesis. b. Wald test H0: ß = 1

The individual null hypotheses are: α = 0 and ß = 1

* H0 is rejected significance level of 10% *** H0 is rejected significance level of 1%

In table 9 the results of the GARCH (1,1) model are presented. For potato futures contracts 2 months before maturity the hypothesis ß = 1 is rejected (significance level of 1%). This, as discussed earlier, is probably due to the discontinuous inventory. Another explanation of the difference between the ARCH and GARCH results could be that the GARCH model allows the current volatility to depend on previous volatility. As can be seen in appendix F, the GARCH terms are in general statistical significant. This means that the current volatility is influenced by the previous volatility. The other parameters in the variance equation 6d are in general not significant.

Table 9 GARCH (1,1) equation 6 a, b and d

Potatoes αααα z-statisticsa ß F-statisticsa,b R2 DW Week 1 0.019 1.369 (0.170) 0.984 1.062 (0.307) 0.990 2.350 Month 1 0.013 0.618 (0.536) 0.973 1.730 (0.193) 0.961 1.996 Months 2 0.044 1.113 (0.266) 0.941 3.169 (0.080)* 0.903 1.984 Months 3 0.012 0.239 (0.811) 0.961 0.808 (0.372) 0.843 1.524 Wheat Week 1 -0.006 -0.107 (0.915) 1.002 0.009 (0.927) 0.989 1.481 Month 1 0.036 0.654 (0.513) 0.984 0.390 (0.537) 0.916 2.219 Months 2 0.024 0.097 (0.923) 0.993 0.004 (0.951) 0.815 2.074 Months 3 -0.174 -0.673 (0.501) 1.085 0.483 (0.493) 0.752 1.884 Eviews output

a. Probabilities are reported in the parenthesis. b. Wald test H0: ß = 1

The individual null hypotheses are: α = 0 and ß = 1

The different α’s for the potato futures contracts are not significantly different from zero. The α’s for the 1 week and 3 months wheat futures contracts are negative in the GARCH (1,1) model which could imply a negative risk premium (not significant). Jalali-Naini and Kazemi Manesh [2006] as well as Wen and Wang [2006] found similar results in respectively the crude oil and wheat market. Neither did they find significant differences compared to the OLS model they applied.

To test for the joint hypothesisα = 0 and ß = 1 a Wald coefficient restriction test in Eviews is used. The

results are presented in table 10. The joint hypothesis must be rejected for all potato futures contracts except the 1 week before maturity futures contract (significance level of 10%). For wheat futures contracts the joint hypothesis is only rejected for the 1 month to maturity futures contract (ARCH) and 2 months to maturity (GARCH). Rejection of the null hypothesis means that the futures contract market does not contain information about the expected spot price, or that a risk premium exist. The GARCH model does not reject the joint hypothesis for the 1 week before maturity potato futures contracts and all wheat futures contracts except the for the 2 months to maturity wheat futures contract. Including the ARCH and GARCH terms does not, in general, change the hypotheses tests.

Table 10 F statistic: Testing joint null hypothesis of α = 0 and ß = 1 with Wald test

Eviews output

a. Probabilities are reported in the parenthesis - The joint null hypothesis is: α = 0 and ß = 1

* H0 is rejected sign. level of 10% ** H0 is rejected sign. level of 5% ***H0 is rejected sign. Level of 1%

Risk premium

As discussed in the introduction, the expected spot price is the futures contract price plus or minus a risk premium. The hypothesis α = 0 in equation 3 and 6 was tested with and OLS and (G)ARCH model. The results were conclusive, the hypothesis α = 0 could not be rejected in the wheat futures contract and potato futures contract market. To prove that α = 0 equation 7 will be used. Unfortunately it is not possible to test equation 7 with wheat futures contracts due to the missing spot prices 1 week, 1, 2 and 3 months before maturity. The observed spot prices for wheat futures contract at time t are only 18, this is to low for testing equations 7. This is regrettable because with the ARCH model the hypothesis α = 0 had to be rejected for some maturities.

Potatoes OLS ARCH GARCH(1,1)

For all maturities ßr is not significantly different from zero, which means there is no evidence of a risk

premium. As was expected, the α’s in table 7, 8 and 9 were also not significantly different from zero. For the regression of the 1 week, 1 month and 2 months before maturity futures contract prices the ßr’s are

negative (not significant). The regression of the 3 months futures contract price is positive. This could mean that there exist a risk premium when the futures contract prices are further away from their maturity, but we must be careful about this because ßr is not significantly different from zero.

Table 11 Testing for time varying risk premium equation 7

Eviews output

a. Probabilities are reported in the parenthesis

** H0 is rejected significance level of 5%

As discussed earlier, Fama and French [1987] used equation 7 to test for time varying risk premiums. They found mixed results, for wheat they found a significant time varying risk premium for some maturities but not all of them. McKenzie and Holt [2002] found mixed results for the four commodities they investigated. For corn, that has similar characteristics as wheat and soy meal they did not find any evidence of time varying risk premium. They did find a time varying risk premium for live cattle and hogs.

RMSE

The Root Mean Squared Error of the different models with different times to maturity are presented in table 12. The parameters α and ß are re-estimated to evaluated the prediction power of the different models. In appendix G the values (calculated with in-sample data) of the parameters are presented.

Table 12 RMSE model

Potatoa OLS ARCH (1) GARCH (1,1)

1 week 0.013691 0.013606 0.013766 1 month 0.026528 0.026904 0.028194 2 months 0.062475 0.062394 0.061727 3 months 0.094513 0.088445 0.079972 Wheatb 1 week 0.016435 0.016552 0.016435 1 month 0.048833 0.048025 0.048632 2 months 0.067447 0.068216 0.065974 3 months 0.078927 0.079389 0.079289 Eviews output

a. In-sample observations1998 till June 2006 and out-of sample observations July 2006 till December 2007 b. In-sample observations 1999 till May 2006 and out-of sample observations June 2006 till December 2007

The differences between RMSE for the models are small. As discussed in the literature review Wen and Wang [2004] as well as McKenzie and Holt [2002] found evidence that the GARCH model outperformed

Potatoes _ααααr t-statisticsa ßr t-statistics

the other models. The results in table 12 are conclusive with these studies, the GARCH model fits best for most maturities.

5. Conclusion

Forecasting commodity prices with futures contracts is a subject that is often investigated by researchers. In this study equations 3, 4 and 6 is used to test if futures contracts are a good predictor for potato and wheat spot prices. A common problem with commodity futures contract prices is stationarity. For the RMX potato and wheat futures contract prices there are no stationarity problems found. These results are consistent with the results found by Bowman and Husain [2004] who found that wheat futures contracts and spot prices are stationary. The finding of stationary data was important for the process of finding the correct model. If data was non-stationary, applying an OLS model can give biased results.

The RMX potato futures contract market appeared to be a backwardation for short times to maturity where it is contango for longer times to maturity. If the convenience yield is lower than the cost of physical storage plus the interest foregone the market is in backwardation (Ft < St). The convenience yield

is high when inventories are low, holding stocks is than more valuable. A possible explanation of the differences between short and long maturities can be the storage cost which are in general higher for longer maturities. Most models show normal backwardation (Ft < Et (ST)) for the wheat futures contract

market, beta is above 1. Opposite results are found for the potato futures contract market. Further research should be done to explain the relation between backwardation/contango, storage cost and the convenience yield.

As discussed in the theory section, the prediction accuracy should decrease as time to maturity increases. There is a clear relation between the R2 and the time to maturity, as time to maturity increases the R2 decreases. This is the case for all models tested. As discussed in the literature this is theoretically expected. The longer the time to maturity, the higher the chance new information will appear in the market. This will change the expected spot price.

The null hypothesis ß = 1 for wheat futures contracts cannot be rejected for almost all contract maturities. The null hypothesis is only rejected for the 1 month to maturity futures contract in the ARCH model. This is probably caused by the varying conditional variance.

The hypothesis αis zero in equation 3 and 6 can not be rejected in the OLS, ARCH and GARCH model, except for the 1 week and 1 month wheat futures contracts in the ARCH model. Unfortunately the sample size of the wheat futures contract and spot prices were not large enough to test for a time varying risk premium in the wheat futures contract market. Earlier studies showed mixed results. The chance of finding a risk premium in the wheat futures market is bigger when there are futures contracts used that have a maturity in the same calendar month and around harvest time. For the other contracts there is no prove of a risk premium and a test for time varying risk premium is not by definition necessary. The existence of time varying risk premiums (only potatoes) was tested with an OLS model. As expected the test for time varying risk premiums with equation 7 did not show any significant premium in the potato futures contract market.

The joint null hypothesis of α = 0 and ß = 1 is also tested with the OLS, ARCH and GARCH model. The

results are mixed, the hypothesis is rejected in all models for potato futures contracts 2 and 3 months to maturity. For wheat futures contracts the joint hypothesis α = 0 and ß = 1 is not rejected for all maturities

where they are rejected for potato futures contracts. If the joint hypothesis is rejected it means that the futures contract market does not contain information about the future spot price, or that there exist a risk premium. Further research could explain the difference in results between the individual hypotheses and the joint hypothesis. The joint hypothesis is less frequently rejected for wheat futures contract prices. This as discussed throughout this paper, is probably caused by the discontinuous inventory of potatoes. In future research the variable production yield and stocks could be added to prove the influence of these variables.

The trading volumes at RMX Hannover are still growing, this is necessary to get a more efficient market. There is no significant difference between the prediction power of potato and wheat futures contracts. If there is sufficient data available, further research can focus on futures contracts that have a maturity in the same calendar month. Result will probably show a significant difference between the regression of wheat futures contracts and potato futures contracts.

In further research an ECM model could be applied, as discussed in the theory section an ECM uses combinations of first differenced and lagged levels of cointegrated variables. An ECM model captures the problem of lower variability of the futures contract prices and can be used to test the short term relation between the futures contract prices and spot prices. Further research could focus on short run market efficiency.

Future work should, if possible, focus on larger samples. Especially the small sample size of wheat futures contract prices and the individual potato futures contract prices might cause biased results. With the data used in this study it was not possible to test futures contracts that had a maturity in the same month for each year (harvest futures contracts). Another point of attention in further research could be the difference between the backwardation of the potato futures market in the short run and the contango market in the long run.