Sage Journals: Discover world-class research

Abstract

Theoretically, futures prices are unbiased predictors of subsequent cash prices only if a market is efficient and there is no risk premium. This research empirically tests both market efficiency hypothesis and unbiasedness of futures price hypothesis in the context of Thai stock index futures (SET50 futures). This article also investigates whether any long-run or short-run inefficiencies or pricing biases exist by identifying and estimating a risk premium. This article finds that in the long run, futures and subsequent cash prices move together and are cointegrated with one cointegrating vector. The statistical test could not reject the null hypothesis of futures unbiasedness. The results do not support the existence of a constant risk premium. The error terms are also free from autocorrelation as required by market efficiency. In the short run, this research could not detect a constant or a time-varying risk premium. This result does not support either normal backwardation hypothesis (futures price < average subsequent cash price) or contango hypothesis (futures price > average subsequent cash price). The overall results support the unbiasedness hypothesis.

Keywords

unbiasedness hypothesis market efficiency SET50 futures futures pricing

Introduction

The futures prices are forecasts of the subsequent cash prices at the maturity of those particular futures contracts. Market efficiency requires that futures prices will equal expected cash prices at futures contract maturity plus or minus a risk premium, which can be either constant or time varying. In other words, futures prices will be unbiased predictors of subsequent cash prices only if both market efficiency condition and no risk premium condition hold. In other words, it is a joint hypothesis.

Even in the case that futures prices are biased, it is difficult to distinguish empirically whether the forecasting bias is due to the failure of the markets to incorporate all relevant information leading to market inefficiency, or the existence of a risk premium. Markets may even be efficient and futures prices are unbiased in the long run, but in the short run, there could be inefficiencies and pricing biases. In addition, a risk premium itself may be time varying. The reason is that the amount that hedgers are willing to pay to speculators to trade in futures contracts depends on time-varying uncertainty in the underlying cash prices.

This article empirically investigates both market efficiency hypothesis and unbiasedness of futures price hypothesis in the context of Thai stock index futures (SET50 futures¹) to determine whether any long-run or short-run inefficiencies or pricing biases exist. This study also identifies and investigates whether the risk premium, if exists, behaves as predicted by normal backwardation hypothesis (futures price < average subsequent cash price) or contango hypothesis (futures price > average subsequent cash price). This important point is usually ignored in previous studies of stock index futures (e.g., Antoniou & Holmes, 1996; Kenourgios, 2005).

Previous studies on the unbiasedness hypothesis are previously done mostly in the mature markets environment and focus mostly on commodity futures. There are few papers that study financial futures prices in emerging markets. The Thai SET50 futures contract is an interesting case in the sense that it has a lower liquidity and is more subject to manipulation from participants than more mature markets. The question is then whether in this context a futures price can still efficiently incorporate all information in predicting a subsequent cash price.

In terms of methodology, this article shares similar techniques, namely, the cointegration and the error correction model (ECM), with previous studies such as Wahab and Lashgari (1993), So and Tse (2004), and Bohl, Salm, and Schuppli (2011) in investigating relationships between cash prices and futures prices. However, former studies overlooked the importance of a stationarity of the net cost of carry. Brenner and Kroner (1995) show that if the net cost of carry has a unit root, then cash and futures prices could not be cointegrated. In addition, former studies do not use a fixed time to maturity. Again, Brenner and Kroner (1995) show that cointegration can only exist when studying futures prices with a fixed time to maturity.

To overcome shortcomings of previous papers, this study does explicitly test for a stationarity of the net cost of carry before testing for a cointegration between cash and futures prices. Moreover, futures price data in this study have a constant time to maturity at either 1 or 3 months. This makes the empirical models to be consistent with the theory (Brenner & Kroner, 1995).

The article applies the Johansen cointegration test to investigate the issue of long-run market efficiency and unbiasedness. For the short run, this article uses an ECM with an Autoregressive Conditional Heteroskedasticity-in-Mean (ARCH-in-mean) to study futures price dynamics. The advantage of this model specification is that it allows for both a constant and a time-varying risk premium. Testing is conducted over two forecasting horizons, namely, 1 month and 3 months.

In the long-run analysis, this article finds that both futures and subsequent cash prices have unit roots but they move together as they are cointegrated. In fact, the cointegrating vector between futures and subsequent cash prices is close to [1, −1] as suggested by the unbiasedness hypothesis. In addition, the constant term that represents a constant risk premium is not statistically significant. The error terms are also white noise as suggested by market efficiency. In short, the hypothesis of futures unbiasedness could not be rejected.

In the short-run analysis, there is no support for the existence of a constant or a time-varying risk premium. The result does not support either normal backwardation hypothesis, which posits that futures price will be lower than expected subsequent cash price, or contango hypothesis, which posits the opposite. In summary, a futures price of SET50 is an efficient and unbiased forecast.

The organization of this article is as follows. The “Introduction” section is followed by “Literature Review” section. The “Method” section discusses research methodology, which is followed by “Data” section that provides a discussion about data. The final sections are “Empirical Results” and “Conclusion.”

Literature Review

Theory

Brenner and Kroner (1995) use a cost-of-carry model to show that whether the existence of cointegration between cash and futures prices depends on the time-series properties of the net cost of carry, called henceforth the “differential.” More specifically, if the differential is stationary, then cash and futures prices are tied together and they would be cointegrated. On the contrary, if the differential is not stationary, then cash and futures prices will move apart, and they will not be cointegrated. In this case, cash price, futures price, and differential will form a three-variable cointegration with a cointegrating vector of (1, −1, 1).

They also caution that cointegration can only exist when studying time series of futures prices with fixed time to maturity. Therefore, any regression of cash and future prices on a fixed expiry date would have an error term that moves toward zero as futures contracts are about to expire. This means that the variance of this residual decreases through time, implying nonstationarity. As a result, cointegration cannot exist.

Their results suggest that the appropriate cointegrating vector for stock index futures would involve stock indices, futures prices, and the differentials, which, in this case, are the differences between interest rates and dividend yields. If interest rates are nonstationary, then cash and futures prices will not be cointegrated by themselves as the differential is required in the cointegrating vector. This same result also holds for dividend yields.

They also prove that if the differential is nonstationary, then the unbiasedness hypothesis cannot hold. Intuitively, the difference between the cash and futures price contains the differential. If this differential is nonstationary, or in other words, has a unit root, then it must be strongly serially correlated. However, serial correlation in the forecast error violates the unbiasedness hypothesis.

Empirical Studies

Unbiasedness in the case of commodity futures

McKenzie and Holt (2002) investigate four agricultural commodity futures contracts by applying cointegration and ECMs with generalized quadratic ARCH-in-mean processes. The results indicate that a futures price is unbiased in the long run, but may have short-run pricing biases. The results support the notion that there are short-run time-varying risk premiums.

Carter and Mohapatra (2008) study a hog futures contract traded at the Chicago Mercantile Exchange. This is the largest nonstorable commodity futures contract. They conclude that a hog futures price was an unbiased predictor of a subsequent cash price during 1998 to 2004. The empirical method utilizes a cointegration method and an ECM.

Armah (2008) studies the cocoa futures market and finds no evidence of a constant or a time-varying risk premium. However, the author still concludes that the futures price is a biased predictor.

Liu (2011) investigates market efficiency of crude palm oil futures traded in the Bursa Malaysia. Johansen cointegration test and vector error correction model (VECM) are used to test market efficiency of both cash and futures markets. The author finds a cointegrating relationship between the futures and cash prices for all studied horizons. In short, the unbiasedness hypothesis of crude palm oil futures price cannot be rejected.

In summary, overall results are still mixed but most studies tend to find that at least in the long run, a futures price is unbiased. In contrast, there may be pricing biases resulting from market inefficiency or a risk premium in the short run.

Unbiasedness in the case of stock index futures

Antoniou and Holmes (1996) test both market efficiency and unbiasedness hypothesis in the context of the FTSE-100 futures contract. They apply an ECM and cointegration to examine the short-run and long-run relationships between cash and futures prices, respectively. They show that a futures price is an unbiased predictor of subsequent cash prices for 1- and 2-month maturities but not longer.

Kenourgios (2005) uses a cointegration technique to test the unbiasedness hypothesis of the FTSE Athens Stock Exchange (ASE). The authors find the joint hypothesis of market efficiency and unbiasedness is rejected.

The roles of stock index futures in price discovery

Early study like Wahab and Lashgari (1993) find that both the S&P 500 and FTSE 100 cash and futures prices are cointegrated as they move together. This suggests that both cash and futures markets are important in terms of price discovery. In a more recent work, So and Tse (2004) analyze minute-by-minute data of the Hang Seng index, the Hang Seng index futures, and the Hang Seng tracker fund. They show that these prices all contribute in terms of price discovery as they are cointegrated with just one cointegrating vector. Their methods include the common factor model (Hasbrouck-and-Gonzalom method and Granger method) and the multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model.

Their results also confirm the trading cost hypothesis which states that stock index futures prices reflect more timely information than the underlying stock index due to lower transaction costs. As such, a futures market will have a more dominant role in terms of price discovery than the cash market. They also find volatility spillovers across cash and futures markets.

Bohl et al. (2011) extend the literature further by examining whether the existence of large individual investors, who are presumably act as noise traders, in the futures market would reduce the “informational contribution” of futures trading in terms of price discovery. They hypothesize that trading of unsophisticated individual investors may bring more noises and lower the price signal quality. Their sample covers the Polish WIG20 index futures market once dominated by individual investors. A change in mutual fund regulation in 2004 caused the share of individual investors to decline and the share of sophisticated institutional investors to rise. They apply a VECM with a multivariate Dynamic Conditional Correlation GARCH (DCC-GARCH) extension (VECM-DCC-GARCH) to the Polish data.

They find that during the period that the market was dominated by individual investors, the futures market had limited roles in terms of price discovery. However, as institutional investors’ share in futures trading increases, they notice that future prices provide more information to the cash market as the correlation between both markets increases significantly. They conclude that more trading share from institutional investors has made the futures market to be more efficient and better reflect relevant information.

In a related paper, Antoniou, Koutmos, and Pericli (2005) investigate whether the existence of stock index futures has increased positive feedback trading in the six stock market indexes, namely, Toronto 300 Composite (Canada), the Continuous Assisted Quotation (CAC) industrial price index (France), Frankfurt Commerzbank index (Germany), Nikkei 225 (Japan), the FT All Share index (United Kingdom), and the S&P 500 (United States). The data cover from 1969 to 1996. Their model assumes two different groups of investors: “risk averse expected utility maximizing investors” and “positive feedback traders.”

They find evidence of positive feedback trading in the cash markets only before but not after the introduction of stock index futures. The results find no evidence of positive feedback trading in the futures markets. They conclude that futures markets seem to limit the impact of the positive feedback trading and help stabilize the markets by making them more efficient.

However, in a later work, Hou and Li (2014) find that the above conclusions do not necessarily hold for an emerging market, particularly in China. More specifically, they examine whether the introduction of the CSI 300 (China Securities Index) futures stabilizes the underlying stock market. They utilize a univariate Autoregressive Glosten-Jagannathan-Runkle GARCH-in-Mean (AR-GJR-GARCH-M) model and a bivariate VECM-GARCH-M model to analyze high-frequency (10-min intervals) data.

They detect the positive feedback trading in the Chinese stock market only after the introduction of CSI 300 futures. In addition, they find that futures trading also attracts positive feedback trading in itself. They surmise that institutional investors may use positive feedback trading in their investment strategies. They conclude that the introduction of CSI 300 futures destabilizes its own spot market and thus lowers market efficiency.

In a more recent paper, Corredor, Ferrer, and Santamaria (2015) extend the literature to study the relationship between investor sentiment and price dynamics in the cash and futures markets. They analyze cash and futures markets of stock market indexes. Their sample includes S&P 500, CAC 40, DAX 30, FTSE 100, IBEX 35, and Euro Stoxx 50.

They show that the correlation between spot and futures prices decreases significantly during periods of high investor sentiment and volatility shocks tend to have less impact during these periods. Their results support behavioral finance theories, which predict that there would be an increase in noise trading during a high investor sentiment period. This would cause a lower arbitrage activity from rational investors who attempt to limit their risk exposure during this period.

The case of SET50 futures

Previous studies on SET50 futures focus mostly on detecting arbitrage opportunities or testing pricing models of futures contracts, and there is none on testing unbiasedness hypothesis or detecting any risk premium in futures contracts. Important works are reviewed below.

Thongthip (2010) investigates whether there is any arbitrage opportunity between SET50 cash and futures markets. He finds that there was mispricing of SET50 futures as the futures prices were outside the constructed no-arbitrage bounds. Nevertheless, he also points out that the arbitrage is not profitable for individual investors due to transaction costs but probably profitable for proprietary traders with lower transaction costs. In addition, he finds that there is no lead–lag relationship between SET50 cash and futures prices.

Tungsong and Srijuntongsiri (2011) find that SET50 futures is generally underpriced relative to a theoretical value based on the cost-of-carry model. They argue that low liquidity and thin trading volume explain the persistence of mispricing. Interestingly, they also find that SET50 futures is mispriced against the SET50 index more frequently and the average mispricing magnitude increases after the SET50-ETF (ThaiDEX SET50 Exchange Traded Fund which is also known as TDEX) was launched.

Tharavanij (2012) utilizes VECM to test the cost-of-carry model in explaining SET50 futures prices. He finds that the model works extremely well. The cash and futures prices form a cointegrating vector, and the elasticity coefficient is approximately 1 as suggested by the model. Furthermore, he finds that a futures price may differ from a forward price due to a future’s daily settlement. This fact can be noticed from a nonzero intercept in an estimated cointegrating vector. The article also finds that cash prices help predict futures prices and not the other way around.

In term of prediction accuracy, there is only one previous work on SET50 futures. Chaninta, Sriya, Apimuntakul, and Tharavanij (2012) compare prediction accuracy of a subsequent cash price by a futures price, a predictor from a random walk model, and a predictor from an Autoregressive Moving Average (ARMA) model. They find that predictors from an ARMA model and a random walk model are the best predictors for short- and long-term forecasts, respectively. However, futures price performs nearly as well as the best predictor in the short-term forecast and beats an ARMA model in the long-term forecast. Interestingly, futures prices tend to underpredict subsequent cash prices.

Method

This research would examine market efficiency and unbiasedness hypothesis both in the short run and long run. The 1-month and 3-month forecasting horizons would be analyzed. Concerning unbiasedness hypothesis, this research would investigate all three cases—zero risk premium, a constant risk premium, and a time-varying risk premium.

The “Method” section is organized into (a) unit root test, (b) long-run analysis, and (c) short-run analysis. Unit root test and response to Brenner and Kroner (1995) are discussed in the first part. In the last two parts, each analysis section contains the empirical model, testing of market efficiency and testing of unbiasedness hypothesis.

Please note that all variables are in natural log form.

Unit Root Test

Stationarity of the data (in natural logs) is evaluated by the augmented Dickey–Fuller (ADF) test, the modified Dickey–Fuller test by transforming time-series data via a generalized least squares regression (DF-GLS), the Phillips–Perron (PP) test, and the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test for stationarity. Although the ADF, the DF-GLS, and the PP tests state the null hypothesis of nonstationarity (a unit root), the KPSS test defines stationarity as the null.

Brenner and Kroner (1995) show that if the “differential” (net cost of carry) has a unit root, then cash and futures prices will tend to drift apart, and they would not be cointegrated. In this case, the unbiasedness hypothesis cannot hold. On the contrary, if the differential is stationary, then cash and futures prices are tied together and they would be cointegrated.

This study investigates statistical properties of cash prices, futures prices, and differentials as they would impact the interpretation from the empirical models. The above unit root tests are performed over all variables.

Long-Run Analysis

Empirical model for long-run analysis

If futures price is an efficient predictor of a subsequent cash price, then both series cannot move too far apart in the long term. Therefore, both series should form a cointegrating relationship. The Johansen procedure would be used to test for cointegration. The cointegrating vector would have the following form.

$u_{t} = S_{t} - α - δ . F_{t - 1} .$

u_t is the forecasting error (or in time-series parlance, the error correction term). If future price is an efficient predictor, then this error term should be stationary and white noise. F_t₋₁ is the price of the futures contract that would expire at period t. The subscript means that this futures price is observed one period before the subsequent cash price. This one period is basically the forecasting horizon. The futures price can be interpreted as the market forecast of the subsequent cash price at the expiry date of that particular futures contract. S_t is the cash price of an underlying at maturity date.

Testing of market efficiency in the long run

Market efficiency simply means that the futures market incorporate all available relevant information, concerning subsequent cash price of the underlying, in its futures price. Basically, futures price can be interpreted as the market predictor of the subsequent cash price. So, there should be no other variables which help to predict subsequent cash price after we control for the futures price.

To test for market efficiency in the long run is basically to test that the forecasting error is stationary and white noise. Forecasting error will be stationary only if subsequent cash price and futures price are cointegrated.

Testing unbiasedness hypothesis in the long run

The above market efficiency conditions are necessary conditions but not sufficient ones for unbiasedness of futures price. There are three cases of unbiasedness hypothesis: a zero risk premium, a constant risk premium, and a time-varying risk premium. In fact, however, it is the first case with a zero risk premium that an unbiasedness hypothesis truly holds. Cases with a constant or a time-varying risk premium are also called unbiasedness only in a sense that a futures price is still an unbiased predictor of a subsequent cash price after taking into account these risk premiums. The Johansen procedure would be used to test all hypotheses.

Unbiasedness with a zero risk premium hypothesis in the long run

This hypothesis states that future price is an unbiased predictor of the subsequent cash price.

$\begin{array}{l} u_{t} = S_{t} - α - δ . F_{t - 1} \\ If α = 0 and δ = 1, then \\ S_{t} = F_{t - 1} + u_{t} \end{array}$

The Johansen multivariate cointegrating procedure can be used to formally test the long-run unbiased hypothesis (α = 0 and δ = 1).

Unbiasedness with a constant risk premium hypothesis in the long run

This hypothesis states that on average, futures price is different from the subsequent cash price with a fixed proportion equal to the risk premium. If the Johansen test could reject the null of α = 0, but could not reject the null of δ = 1, this would imply unbiasedness with a constant risk premium.

$S_{t} = α + F_{t - 1} + u_{t}$

Unbiasedness with a time-varying risk premium hypothesis in the long run

This hypothesis states that on average, futures price is different from the subsequent cash price with a nonconstant risk premium, which varies over time. Basically, the parameter α is not a constant over time, whereas δ still equals 1. This hypothesis cannot be tested directly in the long run because the Johansen procedure does not allow a nonconstant α in the cointegrating relationship. However, this hypothesis would be tested in the short-run.

Short-Run Analysis

Empirical model for short-run analysis

The existence of a cointegrating relationship between futures price and subsequent cash price is a necessary but not a sufficient condition for short-run market efficiency and unbiasedness. In fact, it only guarantees market efficiency in the long run. However, there could still be short-run pricing biases and market inefficiencies, where past price data may help forecast subsequent cash prices.

This work modifies the model used in McKenzie and Holt (2002) and uses an ARCH-in-Mean Error Correction Model (ARCH-M-ECM) to test the short-run relationship between cash and futures prices.

$\begin{array}{l} Δ S_{t} = - ρ . u_{t - 1} + β . Δ F_{t - 1} + \sum_{i = 2}^{m} β_{i} . Δ F_{t - 1} + \\ \sum_{j = 1}^{k} ψ_{j} . Δ S_{t - j} + θ . σ_{t} + v_{t} \\ u_{t - 1} = S_{t - 1} - α - δ . F_{t - 2} \\ v_{t} | v_{t - 1} ~ N (0, σ_{t}^{2}) \\ σ_{t}^{2} = w + a_{1} . v_{t - 1}^{2} \end{array}$

u_t₋₁ is the previous period error correction term. σ_t² is the time-varying conditional variance of a residual term (v_t). v_t is a stationary white-noise residual term. Cointegration between cash and future prices would imply that ρ > 0 as cash price changes in response to disequilibrium from the previous period.

This article uses a simpler ARCH model as a workhorse instead of a more sophisticated model to conserve precious degrees of freedom. In addition, a more advanced model like a GARCH one can be written in the form of the summation of infinite ARCH terms as it is just simply an exponential smoothing of past shocks. The weight given to each past shocks would decline over time. As such, the highest weight would be on the first ARCH term. In other words, an ARCH(1) can be thought of as an approximation.

Nevertheless, a GARCH model and even a more advanced model like an asymmetric GARCH would also be applied as a robustness check. The asymmetric GARCH would take into account two stylized facts that bad news cause more volatility than good news and volatility is highly persistent (Koutmos & Tucker, 1996).

We can rewrite Equation 2 to get the following equivalent equation:

$\begin{array}{l} S_{t} = (1 - ρ) . S_{t - 1} + β . F_{t - 1} + \\ (ρ . δ - β) . F_{t - 2} + ρ . α + \sum_{i = 2}^{m} β_{i} . Δ F_{t - i} + \\ \sum_{j = 1}^{k} ψ_{j} . Δ S_{t - j} + θ . σ_{t} + v_{t} \end{array}$

Testing of market efficiency in the short run

Short-run market efficiency requires the following restrictions on the parameters of Equation 3:

$ρ = 1, β \neq 0, ρ . δ = β, β_{i} = ψ_{j} = 0$

If the above restrictions in Equation 4 do not hold, then past futures or cash prices would contain relevant information in predicting subsequent cash prices (S_t). As this information is not completely incorporated in the futures price at time t−1, it means that the futures market is inefficient. The efficient market hypothesis would require that all past information is incorporated in the futures price at time t−1 (F_t₋₁), and therefore, the lagged futures prices or lagged cash prices should not help forecasting subsequent cash prices. The above restriction allows for the existence of both a constant risk premium (α) and a time-varying risk premium ( $θ . σ_{t}$ ).

Testing of unbiasedness hypothesis in the short run

The above market efficiency condition in Equation 4 and the long-run unbiasedness condition are necessary conditions but not sufficient ones for short-run unbiasedness of the futures price. There are three cases of unbiasedness hypothesis: a zero risk premium, a constant risk premium, and a time-varying risk premium. Only the first case with a zero risk premium that an unbiasedness hypothesis truly holds. Other cases are called unbiasedness only in a sense that a futures price is an unbiased predictor of a subsequent cash price after taking into account a risk premium.

Unbiasedness with a zero risk premium in the short run

This hypothesis states that future price is an unbiased predictor of the subsequent cash price. With regard to Equation 3, short-run unbiasedness with zero risk premium implies the following conditions: (a) long-run unbiasedness with zero risk premium (α = 0, δ = 1) and (b) short-run restrictions for zero or constant risk premium (ρ = β = 1, β_i = ψ_j = 0, θ = 0). If all above conditions hold, then we can reduce Equation 3 to the following:

$S_{t} = F_{t - 1} + u_{t}$

Unbiasedness with a constant risk premium in the short run

This hypothesis states that on average, futures price is different from the subsequent cash price with a fixed proportion equal to the risk premium. With regard to Equation 3, short-run unbiasedness with a constant risk premium implies the following conditions: (a) long-run unbiasedness with a constant risk premium (α ≠ 0, δ = 1) and (b) short-run restrictions for zero or constant risk premium (ρ = β = 1, β_i = ψ_j = 0, θ = 0). If all above conditions hold, then we can reduce Equation 3 to the following equation, where α is interpreted as a constant risk premium:

$S_{t} = α + F_{t - 1} + v_{t}$

Unbiasedness with a time-varying risk premium in the short run

This hypothesis states that on average, futures price is different from the subsequent cash price with a nonconstant risk premium. Theoretically, the risk premium will vary with a cash price risk as measured by the conditional standard deviation of percentage changes of cash prices. With regard to Equation 3, short-run unbiasedness with a time-varying risk premium implies the following conditions: (a) long-run unbiasedness with a constant risk premium (α ≠ 0, δ = 1) and (b) short-run restrictions for time-varying risk premium (ρ = β = 1, β_i = ψ_j = 0, θ ≠ 0). If all above conditions hold, then we can reduce Equation 3 to the following:

$S_{t} = α + F_{t - 1} + θ . σ_{t} + v_{t}$

“α + θ.σ_t” is a total risk premium, which is composed of α, a constant risk premium part, and “θ.σ_t,” a time-varying risk premium part.

A positive sign of θ implies normal backwardation (futures price < average subsequent cash price) where short hedgers pay a risk premium to long speculators to bear the risk. The implication is that a futures price tends to underpredict a subsequent cash price. In contrast, a negative sign implies contango (futures price > average subsequent cash price) where long hedgers pay a risk premium to short speculators to bear the risk. The implication is that futures price tends to overpredict a subsequent cash price.

Data

Data Sources

SET50 index data were collected from the SETSMART database provided by the Stock Exchange of Thailand (SET). SET50 futures data were collected from the Thailand Futures Exchange (TFEX).

The samples include only already expired nonoverlapping futures contracts with a constant time-to-maturity. For 1-month time to maturity, the sample covers 39 matured futures contracts with expiry dates from the end of November 2012 to the end of January 2016. For 3-month time to maturity, the sample covers 38 matured futures contracts with expiry dates from the end of September 2006 to December 2015.

The net cost of carry or “differential” (d) is calculated from the following equation based on the cost-of-carry model:

$F = S . e^{d . T T M} \Rightarrow d = \frac{\ln (F / S)}{TTM}$

“F” and “S” are current futures and cash (or spot) price, respectively. “TTM” is the time to maturity of the futures contract. “d” is the net cost-of-carry term in a unit of percentage point per year.

Data Description

Table 1 reports forecasting accuracy of futures prices with 1- and 3-month maturities. As expected, forecasting errors, as measured by mean error (ME) and root mean square error (RMSE), increase with longer forecasting horizons. Interestingly, ME and mean percentage error (MPE) are negative in all horizons. This means that futures prices are lower than subsequent cash prices on average. This seems to imply that investors could make money by longing SET50 futures contracts and holding them until maturity.

Table 1.

Forecasting Accuracy of Futures Prices.

Forecasting accuracy statistics	f1m-level	f3m-level
Number of observations	39	38
ME	−1.22	−14.22
MPE	−0.02%	−1.51%
MAE	33.15	53.33
MAPE	3.48%	8.43%
RMSE	39.61	66.48
RMSPE	4.17%	11.75%
Theil’s inequality coefficient	0.02	0.04
Bias proportion	0.09%	4.58%
Variance proportion	0.25%	0.07%
Covariance proportion	99.66%	95.35%
Correlation with subsequent cash prices	.83	.96

Note. f1m-level and f3m-level are the levels (not a natural log) of 1-month maturity and 3-month maturity futures prices, respectively. The cash price is also a level variable (not a natural log). ME = mean error; MPE = mean percentage error; MAE = mean absolute error; MAPE = mean absolute percentage error; RMSE = root mean square error; RMSPE = root mean square percentage error.

The reported Theil’s coefficients of forecasting errors are quite low and not higher than 5%. This shows that futures price is quite an accurate predictor. However, bias is a significant proportion (around 4%) for 3-month maturity. This is a result of an underprediction bias. The correlations between futures prices and subsequent cash prices are high as they vary from .83 to .96.

Figure 1 plots SET50 futures prices at 1 and 3 months before maturity and subsequent SET50 cash prices. Overall, the two series move together. The futures prices at 1 month before maturity have the tightest relationship with cash prices. Tables A1 and A2 in the appendix provide descriptive statistics.

Figure 1.

SET50 futures prices (level) at 1 and 3 months before maturity and subsequent SET50 cash prices (level).

Empirical Results

Unit Root Test

Tables 2 and 3 show results from unit root tests. The null hypotheses of a unit root for (log of) cash (spot) price ( s ) and (log of) futures price of both maturities (f1m and f3m) cannot be rejected at 5% significant level in the ADF, the DF-GLS, and the PP tests. Similarly, the null hypotheses of stationarity are rejected at 5% significant level by the KPSS test except for the 3 month to maturity sample.

Table 2.

Unit Root Tests of (Log of) Cash (Spot) Price and Its First Difference, (Log of) 1-Month Maturity Futures Price and Its First Difference, and the Net Cost of Carry.

Variables	ADF	DF-GLS	PP	KPSS
S	−1.240	−1.363	−7.027	0.189**
f1m	−1.494	−1.605	−6.068	0.161**
Δ.s	−4.449***	−2.642***	−36.297***	0.305
Δ.f1m	−4.178***	−4.070***	−30.861***	0.298
d1m	−3.802***	−3.423***	−18.838***	0.368*

Note. “s,” “f,” and “d” are (log of) cash (spot) price, (log of) futures price, and net cost of carry or “differential” term, respectively; 1m means 1 month to maturity of the futures contract. Δ.s and Δ.f1m are first differences of (log of) cash (spot) price and (log of) futures price at 1-month maturity, respectively. ADF = augmented Dickey–Fuller; DF-GLS = the modified Dickey–Fuller test by transforming time-series data via a generalized least squares regression; PP = Phillips–Perron; KPSS = Kwiatkowski–Phillips–Schmidt–Shin.

*, **, and *** mean 10%, 5%, and 1% significance, respectively.

Table 3.

Unit Root Tests of (Log of) Cash (Spot) Price and Its First Difference, (Log of) 3-Month Maturity Futures Price and Its First Difference, and the Net Cost of Carry.

Variables	ADF	DF-GLS	PP	KPSS
s	−3.007	−3.103*	−9.773	0.117
f3m	−3.039	−3.088*	−10.250	0.123*
Δ.s	−4.443***	−4.385***	−19.886***	0.073
Δ.f3m	−4.318***	−4.387***	−21.089***	0.062
d3m	−4.272***	−4.405***	−23.831***	0.184

Note. “s,” “f,” and “d” are (log of) cash (spot) price, (log of) futures price, and net cost of carry or “differential” term, respectively; 3m means 3 months to maturity of the futures contract; Δ.s and Δ.f3m are first differences of (log of) cash (spot) price and (log of) futures price at 3-month maturity, respectively. ADF = augmented Dickey–Fuller; DF-GLS = the modified Dickey–Fuller test by transforming time-series data via a generalized least squares regression; PP = Phillips–Perron; KPSS = Kwiatkowski–Phillips–Schmidt–Shin.

*, **, and *** mean 10%, 5%, and 1% significance, respectively.

Nevertheless, first differences of cash and futures price are stationary. The ADF, DF-GLS, and PP tests all rejected a unit root hypothesis at 1% significant level. Likewise, KPSS test does not reject a stationary null hypothesis.

Regarding the net cost of carry or “differential” (d1m and d3m), all test statistics reject a unit root null hypothesis or cannot reject a stationary null hypothesis. The implication is that there is no need to include the net cost of carry in the cointegrating vector between cash and futures prices.

Cointegration Test

Table 4 shows results from the Johansen’s cointegration test. The test statistic clearly rejects the null hypothesis of no cointegration between cash and futures prices. However, it could not reject the null hypothesis of a maximum of one cointegrating vector. This implies that futures prices and subsequent cash prices at least move together overtime and will not drift apart indefinitely. This fact partially satisfies conditions for market efficiency and unbiasedness.

Table 4.

Johansen’s Cointegration Test Between Cash Price and Futures Price.

Variables	Johansen trace statistics (λ)
Variables	k = 0	5% critical value	1% critical value	k = 1	5% critical value	1% critical value
f1m	112.578***	15.41	20.04	2.055	3.76	6.65
f3m	179.167***	15.41	20.04	3.284	3.76	6.65

Note. “s,” “f,” and “d” are (log of) cash (spot) price, (log of) futures price, and net cost of carry, respectively; 1m and 6m mean 1-month and 6-month time to maturity of the futures contract, respectively. The trace test is used to test the null hypothesis that the number of cointegrating vectors is less than or equal to k, where k is either 0 (for no cointegration) or 1 (for a single cointegrating vector). If the null of both k = 0 and k = 1 are rejected, it would imply that both variables are stationary.

*, **, and *** mean 10%, 5%, and 1% significance, respectively.

Vector Error Correction (VEC) and Cointegrating Vector

Table 5 shows the VEC and cointegrating vector relationships between cash and futures prices. Changes in cash and futures price respond with statistical significance to last period error correction terms, EC(−1). The only exception is a change in cash price in the 1-month maturity sample. In the change in futures price equations (Δf), the coefficients of EC(−1) are positive as expected. This means that when a futures price is below an equilibrium level with a cash price, making an error correction term positive, it tends to rise toward equilibrium in subsequent periods. Interestingly, given the same positive error correction term (a cash price above an equilibrium), cash prices from the 3 month to maturity sample tend to move even higher. This can be seen from an unexpected positive sign of the coefficients of EC(−1) in the change in cash price equations (Δs). Nevertheless, the coefficient (0.396) is still lower than the 1 in the change in futures price equations (Δf; 1.034). This fact makes the system stable as indicated by the highest modulus of Eigenvalue being lower than 1.

Table 5.

Johansen’s VEC and CV of Cash (Spot) Price and Futures Price.

Variables	Coefficient symbol in Equation 2	1 month	3 months
Δs
EC(−1)		−0.013(0.171)	0.396** (0.156)
RMSE		0.044	0.113
R ²		.000	.156
χ²		0.006	6.447**
Jarque–Bera test		1.888	4.137
Δf
EC(−1)		0.993*** (0.041)	1.034*** (0.016)
RMSE		0.010	0.012
R ²		.943	.991
χ²		591.277***	4,029.677***
Jarque–Bera test		0.827	3.488
CV
s		1	1
f	−δ	−0.976*** (0.025)	−0.999*** (0.006)
constant	−α	−0.168(0.175)	−0.017(0.036)
χ²		1,468.275***	32,253.920***
t test b(f) = −δ = −1		0.945	0.234
Highest modulus of Eigenvalue		0.018	0.363
LM test of no autocorrelation at Lag 1		0.959	6.002
LM test of no autocorrelation at Lag 2		2.434	0.981
No. of observations		38	37

Note. “s,” “f,” and “d” are (log of) cash (spot) price, (log of) futures price, and net cost of carry, respectively; Δs and Δf are first (log) differences of cash and futures price, respectively. EC(−1) is a last period error correction term. CV is a cointegrating vector with a normalized coefficient of 1 for (log of) cash price. A number in parentheses is a standard error. VEC = vector error correction; RMSE = root mean square error; LM = Lagrange multiplier.

**, and *** mean 10%, 5%, and 1% significance, respectively.

The Lagrange multiplier (LM) test could not reject the null hypothesis of no autocorrelation among VEC residuals at least up to lag two periods. This fact satisfies conditions for market efficiency and unbiasedness.

The lag length selection of past change in cash price (Δs) and futures price (Δf) is based on the Schwarz’s Bayesian information criteria (SBIC). The underlying vector auto regression (VAR) with a lag length of one has the lowest SBIC. As the order of the corresponding VEC is always one less than that of the underlying VAR, there is no lagged changes in cash or futures prices in the estimated model.

The cointegrating relationship between cash and futures prices is reported in the “cointegrating vector” section of Table 5. The coefficients of cash price (s) are normalized to 1. There are constant terms in the cointegrating vector equation to measure a constant risk premium for each maturity. However, none of them are statistically significant. Therefore, the hypothesis of a constant risk premium in a long run is rejected.

The coefficients of futures prices in cointegrating vector equations are all highly significant with negative signs as expected. For a futures price to be an unbiased predictor of a subsequent cash price, the coefficient of cash price must equal “−1” and the constant term equals to zero in the cointegrating vector. The t tests cannot reject the null hypothesis of the coefficient of cash price being “−1” for both maturities. When combined with the fact that the constant terms are insignificant and there is no autocorrelation among VEC residuals, this result implies that unbiasedness hypothesis with zero risk premium and market efficiency hypothesis hold at least in a long run.

ARCH-in-Mean Error Correction Model

In this section, ECM is used to study short-term relationship between changes in cash prices (Δs) and changes in futures price (Δf). The ECM equation includes the last period error correction term, EC(−1), to take into account movement toward a long-run equilibrium. Given the fact that cash prices and futures prices are cointegrated, ordinary least squares (OLS) estimators and its residuals are super consistent. The error correction term terms are calculated as residuals of OLS regression of cash prices (s) on futures prices (f).

The lagged terms of changes in futures and cash prices are not included in the reported result. The reason is that they are not significant. This means β_i and ψ_j in Equation 3 are not statistically different from zero. Including them would consume precious degrees of freedom and lower the power of subsequent statistical tests.

The ARCH term (v²_t−1) is included in the variance equation to incorporate time-varying volatility in stock returns. The feedback effect of volatility on the expected stock return is captured by the ARCH-in-Mean (ARCH-M) term (σ_t) in the ECM (mean) equation.

Although financial theories argue that only nondiversifiable risks matter, the inclusion of total risk measured by standard deviation of returns like the ARCHM term in the mean equation is still appropriate in this case. The reason is that SET50 is a well-diversified portfolio. In fact, most movements in the broadest stock market index (SET index in this case) could be explained mainly by movements in the SET50 index. Therefore, its risk is mainly nondiversifiable.

This article uses an ARCH model as a workhorse for its simplicity. For robustness checks, however, estimations with a GARCH and an asymmetric GARCH are also performed. The GARCH model would better incorporate the stylized fact that return volatility tends to be persistent, whereas the asymmetric one would capture another stylized fact that bad news cause more volatility than good news (Koutmos & Tucker, 1996). The results are that coefficients of both GARCH and asymmetric terms are always not statistically different from 0. As such, the reporting table would be based on the ARCH model only.

Table 6 reveals that the error correction term is statistically significant at 1% for both maturities. The sign is negative as expected. It means that cash price tends to go down when it is above the last period equilibrium. The ARCHM terms (σ_t) in the ECM equation are not significant in both maturities. This simply indicates that a time-varying risk premium is not statistically significant, and both hypotheses of normal backwardation and contango can be rejected.

Table 6.

ARCH-M-ECM of Changes in Cash (Spot) Price.

Variables	Coefficient symbol in Equations 2 and 3	1 month	3 months
ECM: Δs
EC(−1)	−ρ	−1.577*** (0.566)	−1.439*** (0.498)
Δf	β	1.512*** (0.494)	1.382*** (0.472)
σ_t	θ	0.053(0.159)	0.199(0.186)
Variance equation: (σ_t²)
v²_t−1	a ₁	0.473(0.004)	0.914** (0.463)
Constant	w	0.001** (0.001)	0.003* (0.002)
No. of observations		38	37
χ²		10.33**	14.31***
Wald test
ρ = 1		1.04	0.78
β = 1		1.08	0.65
ρ = β = 1		1.09	0.78

Note. “s,” “f,” and “d” are (log of) cash (spot) price, (log of) futures price, and net cost of carry, respectively. Δs and Δf are first (log) differences of cash and futures price, respectively. EC(−1) is a last period error correction term. It is an OLS residual calculated from regressing (log of) cash (spot) price on (log of) futures price. “σ_t” is the autoregressive conditional heteroskedasticity in mean (ARCH-M) term. “v²_t−1” is the one-period lagged arch term in the variance equation. The χ² statistic tests the null hypothesis that coefficients of all explanatory variables are 0. A number in parentheses is a standard error. ARCH-M-ECM = ARCH-in-mean error correction model; ECM = error correction model; OLS = ordinary least squares.

*, **, *** mean 10%, 5%, and 1% significance, respectively.

The constant term (ρ.α in Equation 3) is a multiplication between the adjustment parameter (ρ) and the constant risk premium in the long-run cointegrating vector (α). As the constant risk premium is not significantly different from zero, this fact would imply a zero risk premium.

The Wald test could not reject the hypothesis that both ρ and β equal to 1 for both maturities. When combined this result with the fact that there is no evidence of a constant or a time-varying risk premium and the satisfaction of conditions of the long-run unbiasedness with zero risk premium, we can conclude that a futures price is also an unbiased and efficient predictor of a subsequent cash price in the short run.

Conclusion

The futures prices can be interpreted as market forecasts of the subsequent cash prices at the maturity of those particular futures contracts. Market efficiency requires that futures prices will equal expected subsequent cash prices plus or minus a constant or a time-varying risk premium. In other words, futures prices will be unbiased predictors of future cash prices only if a market is efficient and there is no risk premium.

The unbiasedness property is important because a biased forecast of a subsequent cash price would complicate the use of futures contract for hedging operations as Lence (1995) shows that an unbiased futures price is a necessary condition for the utility-free optimal hedge ratio. Otherwise, the optimal hedge ratio would depend on the hedger’s degree of risk aversion.

This article empirically tests the two separate hypotheses of market efficiency and unbiasedness of the Thai stock index futures (SET50 futures) to determine whether any long-run or short-run inefficiencies or pricing biases exist. The study also attempts to identify and estimate a risk premium.

The Johansen cointegration procedure is used to test for long-run market efficiency and unbiasedness while still allowing for a constant risk premium. The short-run market efficiency and unbiasedness are analyzed by an ECM with an ARCH in-mean. This model specification allows for both a constant and a time-varying risk premium. Testing is conducted over two forecasting horizons, namely, 1 month and 3 months.

The results from the Johansen’s cointegration test suggest that futures prices and subsequent cash prices are cointegrated. As such, they move together over time and will not drift apart indefinitely. In the estimated cointegrating vector, constant terms in both maturities are not statistically significant. Therefore, the hypothesis of a constant risk premium in a long run is rejected. The overall result supports the idea that the market is efficient and the futures price is an unbiased and efficient predictor of a subsequent cash price at least in a long run.

In the short run, there is no evidence of the existence of a constant or a time-varying risk premium. Therefore, both normal backwardation and contango hypotheses are not supported by the data. The results reveal that a futures price is also an unbiased and efficient predictor of a subsequent cash price in the short run.

Future research would benefit more from longer time period. The study could be expanded to include other futures contracts like single stock futures or currency futures when there are enough observations of those futures. The methodology itself should be developed further to include several futures contracts in a form of panel time-series analysis.

Footnotes

The author thanks Mahidol University and the College of Management for providing a financial support for this research.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research and/or authorship of this article: This work was supported by the College of Management,Mahidol University (CMMU;faculty grant).

Author Biography

Piyapas Tharavanij is an assistant professor at College of Management,Mahidol University,Bangkok,Thailand. His research interests are in the fields of financial economics,investment and technical trading. His recent paper was published in the journal SpringerPlus titled “Performance of technical trading rules: evidence from Southeast Asian stock markets” in 2015.

References

Antoniou

Holmes

(1996). Futures market efficiency: The unbiasedness hypothesis and variance-bounds tests: The case of the FTSE-100 futures contract. Bulletin of Economic Research, 48, 115-128.

Antoniou

Koutmos

Pericli

(2005). Index futures and positive feedback trading: Evidence from major stock exchanges. Journal of Empirical Finance, 12, 219-238.

Armah

S. E.

(2008, July 27-29). Establishing the presence of a risk premium in the cocoa futures market: An econometric analysis. Paper presented at the American Agricultural Economics Association Annual Meeting, Orlando, FL.

Bohl

M. T.

Salm

C. A.

Schuppli

(2011). Price discovery and investor structure in stock index futures. Journal of Futures Markets, 31, 282-306.

Brenner

R. J.

Kroner

K. F.

(1995). Arbitrage, cointegration, and testing the unbiasedness hypothesis in financial markets. Journal of Financial and Quantitative Analysis, 30, 23-42.

Carter

C. A.

Mohapatra

(2008). How reliable are how futures as forecasts? American Journal of Agricultural Economics, 90, 367-378.

Chaninta

Sriya

Apimuntakul

Tharavanij

(2012). Prediction accuracy of futures contracts traded in Thailand Futures Exchange (TFEX). International Research Journal of Finance and Economics, (99), 169-186.

Corredor

Ferrer

Santamaria

(2015). Sentiment-prone investors and volatility dynamics between spot and futures markets. International Review of Economics & Finance, 35, 180-196.

Hou

(2014). The Impact of the CSI 300 stock index futures: Positive feedback trading and autocorrelation of stock returns. International Review of Economics & Finance, 33, 319-337.

10.

Kenourgios

D. F.

(2005). Testing efficiency and the unbiasedness hypothesis of the emerging Greek futures market. European Review of Economics & Finance, 4, 3-20.

11.

Koutmos

Tucker

(1996). Temporal relationships and dynamic interactions between spot and futures stock markets. Journal of Futures Markets, 16, 55-69.

12.

Lence

S. H.

(1995). On the optimal hedge under unbiased future prices. Economics Letters, 47, 385-388.

13.

Liu

(2011). Testing market efficiency of crude palm oil futures for European participants. In Baourakis

Mattas

(Eds.), A resilient European food industry in a challenging world (pp. 223-244). Hauppauge, NY, USA: Nova Science.

14.

McKenzie

A. M.

Holt

M. T.

(2002). Market efficiency in agricultural futures markets. Applied Economics, 34, 1519-1532.

15.

R. W.

Tse

(2004). Price discovery in the Hang Seng Index markets: Index, futures, and the tracker fund. Journal of Futures Markets, 24, 887-907.

16.

Tharavanij

(2012). Empirical test of the cost-of-carry model: A case of Thai stock index futures contract. International Research Journal of Finance and Economics, 101, 28-38.

17.

Thongthip

(2010). Lead-lag relationship and mispricing in SET50 index cash and futures markets (Master’s thesis). Thammasat University, Bangkok, Thailand.

18.

Tungsong

Srijuntongsiri

(2011, December). Index arbitrage and futures pricing efficiency: Evidence from Thailand. Paper presented at the 24th Australasian Finance and Banking Conference 2011, University of New South Wales, Sydney, Australia.

19.

Wahab

Lashgari

(1993). Price dynamics and error correction in stock index and stock index futures Markets: A cointegration approach. Journal of Futures Markets, 13, 711-742.