Unbiased Estimation, Price Discovery, and Market Efficiency: Futures Prices and Spot Prices

Systems Engineering — Theory & Practice Volume 28, Issue 8, August 2008 Online English edition of the Chinese language journal Cite this article as: SETP, 2008, 28(8): 2–11

Unbiased Estimation, Price Discovery, and Market Efficiency: Futures Prices and Spot Prices CHEN Rong, ZHENG Zhen-long Department of Finance, Xiamen University, Xiamen 361005, China

Abstract: In most cases, futures prices are not unbiased estimates of future spot prices. The price discovery function of futures markets should be defined as the lead-lag relationship between current futures prices and current spot prices, not the unbiased estimation of future spot prices. The pricing efficiency and information efficiency of futures markets are different. Three empirical models appropriate to investigate the relationship between futures prices and spot prices were discussed. As an application, these models were used to test the pricing efficiency, the lead-lag relationship, and the information efficiency of the S&P 500 index spot and futures markets from 21 September 1990 to 20 December 2007. Key Words: futures markets; unbiased estimates; price discovery; market efficiency

Futures prices1 are naturally highly related to spot prices2 because futures are derivatives of spot assets. However, there are certain long-existing misunderstandings about the relationship between these two prices in academics and in financial industry. The purpose of this article is to clarify these misunderstandings and illustrate the real relationship between futures prices and spot prices.

1 Futures prices and spot prices: a review There exist two types of researches on the relationship between futures prices and spot prices. 1.1 Current futures prices and the expected future spot prices3 Because futures prices represent the prices at which market participants agree to transact on a set date in the future, a conclusion seems natural that current futures prices Ft should be the prediction of future spot prices ST in an efficient market, which is defined as the function of “price discovery” of futures. In futures markets literature, this argument is expressed as “futures prices are unbiased estimates of future spot prices”, that is Ft = E (ST |It )4 . This view is broadly believed. In academics, one example is the Pure Expectation Theory of the interest rate term structure proposed by Fisher in 1896. In modern financial language, this theory actually

regards forward rates as unbiased estimates of future spot rates5 . Another example is the well-known Unbiased Forward Exchange Rate Hypothesis (UFER), which argues that forward exchange rates are unbiased estimates of future spot exchange rates. Similar opinions exist extensively in many other futures markets. Accordingly, many researchers consider the failure of unbiased estimation as evidence of market inefficiency, such as Leuthold[2] , Martin & Garcia[3] , Hokkio & Rush[4] and Bhattacharya & Singh[5] , etc. In their articles, they all relate the issue of Ft = E (ST |It ) to the efficiency of futures market. The development of “theories” brings a mass of empirical studies. To our knowledge, at least hundreds of articles try to use different kinds of econometrics techniques to investigate the efficiency of currency futures markets, commodity futures markets, or other futures markets by testing Ft = E (ST |It ). However, so far no convincing evidence is obtained so that Ft = E (ST |It ) could not be either supported or rejected. For example, the Unbiased Forward Exchange Rate Hypothesis is regarded as a puzzle in finance (literatures [6] and [7].). In industry, the opinion that futures prices are unbiased estimates of future spot prices is also deeply ingrained in peoples mind. One typical example is people keep using the fed funds futures rates for forecasting future spot fed-

Received date: March 1, 2007 ∗ Corresponding author: Tel: +86-592-2180762; E-mail: [email protected] Foundation item: Supported by the The Ministry of Education of the Peoples Republic of China (No.05JJD790026, No.07JA790077) c 2008, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright 

1

It could be proved that when the risk-free interest rate is constant and the same for all maturities, the forward price for a contract with a certain delivery date in theory is the same as the futures price for a contract with that delivery date. When interest rates vary unpredictably (as they do in the real world), they are no longer the same in theory. However, the theoretical differences are in most circumstances sufficiently small to be ignored in reality. In this article, we assume that futures and forward contracts are just the same. Anything we discuss about futures will be the same for forwards. 2 Hereafter, we use spot assets to denote the underlying assets corresponding to the futures contracts we discuss. Underlying assets in this article are all investment assets, not consumption assets. 3 In this article, we refer to the markets average opinion about what the spot price of an asset will be at maturity of futures contracts discussed as the expected spot price of the asset. 4 Here, T is the time until delivery date in a futures contract, t denotes the current time point, which is prior to the delivery date. It is the information set at time t and E(·|It ) is the conditional expectation operator at time t. 5 There exist three different versions of this theory. Cox, Ingersoll and Ross[1] give a detailed discussion.

CHEN Rong, ZHENG Zhenlong/Systems Engineering — Theory & Practice, 2008, 28(8): 2–11

eral funds rates, just as we can see in articles on The Wall Street Journal, Financial Times, or in the reports of some Fed Watchers (such as Altig[8] and Hamilton[9] ) and some central banks6 . 1.2 Current futures prices and current spot prices After 90’s, cointegration tests and Granger causality tests were quite popular in financial researches, and empirical studies of the relationship between current futures prices Ft and current spot prices St became hot topics. Some researchers conduct cointegration analysis on Ft and St to investigate the pricing efficiency of futures markets, such as Brooks, et al.[12] and Crowder & Phengpis[13] . Some other researchers use Granger causality tests and vector error correction models (VECMs) to examine the lead-lag relationship and information flows between futures and spots markets, such as Wahab & Lashgari[14] , Abhyankar[15] and Chiang & Fong[16] . Accordingly, some researchers define the lead-lag relationship and information flows between futures and spots markets as “price discovery”. The market absorbing and reflecting new information more rapidly is said to have the function of price discovery. As we can see, the above two types of researches are both concerning the efficiency of futures markets and the function of price discovery. Which one captures the real relationship between futures prices and spot prices ? Are they related to each other? Is the efficiency of futures market the same concept as the classical Efficient Market Hypothesis (EMH) in finance? How should we test the efficiency of futures markets? What on earth is the function of price discovery of futures markets? The purpose of this article is to answer these questions. In brief, our conclusions include: 1) In most cases, current futures prices are not unbiased estimates of future spot prices. Futures do not have the function of “discovering” future spot prices. 2) However, current futures prices in one well-developed futures market would lead the change of current spot prices. It is more reasonable to define “price discovery” as the lead-lag relationship and information flows between two markets. 3) Different from the information efficiency of stock market defined in the classical EMH, the efficiency of futures market includes pricing efficiency and information efficiency. The former should be tested by investigating the relationship between Ft and St , whereas the latter should be tested by examining the residuals of futures log returns. 4) Anyhow, it is almost impossible to obtain reliable evidence about the market efficiency and the function of price discovery of futures markets by investigating the relationship between Ft and E (ST |It ).

2 Futures prices and spot prices: A theoretical analysis 2.1 Pricing formulas of futures Supposing that the no-arbitrage condition holds7 , futures prices satisfy Ft = St e(rt −qt )(T −t) , 6

(1)

where rt is the continuously compounded risk-free interest rate, qt is the average yield per annum on the spots during the life of the futures contracts with continuous compounding8 . This equation is also called the cost of carry model. If Eq.(1) does not hold, arbitrageurs will execute arbitrage strategies until Eq.(1) holds again and the market reaches equilibrium. However, if arbitrage is limited (for example, borrowings, lendings, and short sales are restricted, futures and spots are not good substitutions for each other.), the above no-arbitrage pricing formula will not hold any more. In such situations, theoretical futures prices like Eq.(1) are hard to obtain. 2.2

Current futures prices and current spot prices

From the above discussion, a natural conclusion is whether the relationship between Ft and St that satisfies Eq.(1) could be used to tell whether futures markets are efficient in pricing. Consider the natural logarithm on both sides of Eq.(1), we have ft = st + (rt − qt ) (T − t) ,

(2)

where ft = ln (Ft ) and st = ln (St ). We know that log prices series are usually nonstationary. Thus, the step before we examine whether Eq.(2) holds is to test the stationarity of these three time series ft , st , and (rt − qt ) (T − t). If they are all nonstationary,  cointegration tests needto be conducted for {Xt } = T (ft , st , (rt − qt ) (T − t)) . If these three series are cointegrated with the cointegration vector (1, −1, −1), it could be thought that there exists long-term equilibrium among ft , st , and (rt − qt ) (T − t) and the futures markets could be thought efficient in pricing. If (rt − qt ) (T − t) is stationary, we only need to perform cointegration tests on ft and st to tell whether there is long-term equilibrium between these two prices. If ft and st are cointegrated, the Granger causality test could be conducted on the basis of a VECM to examine the information flow and the lead-lag relationship between futures markets and spot markets. Or we could rearrange Eq.(2) and establish the following econometrics model ft − st = α + β (rt − qt ) (T − t) + εt .

(3)

We could then use regressions to investigate whether the relationship between the basis ft − st and the cost of carry satisfies the pricing efficiency (that is, to test whether α = 0, β = 1 given εt is white noise.) In one sentence, the pricing formula of futures prices explicitly shows that the investigation of the relationship between Ft and St could provide us evidence not only about the pricing efficiency of futures market but also about the information flow and the lead-lag relationship between markets. Consequently, it is quite reasonable to define “price discovery” as the information flow mechanism and the leadlag relationship between futures markets and spot markets.

For example, similar opinions could be found in European Central Bank Monthly Bulletin[10] and Federal Reserve Monetary Policy Report to Congress[11] . 7 In one perfect market, risk-free arbitrage is described as one trading strategy that requires no initial investment but guarantees a riskless profit higher than the riskless return. 8 During the life of one futures contracts, a positive, negative, and zero average yield means qt > 0, qt < 0, and qt = 0, respectively.

CHEN Rong, ZHENG Zhenlong/Systems Engineering — Theory & Practice, 2008, 28(8): 2–11

If Ft changes more rapidly than St when new information arrives and St changes after Ft , we would say the futures market has the function of “price discovery”. 2.3 Futures prices and the expected future spot prices The relationship between Ft and E (ST |It ) remains a controversial and puzzling issue. The pricing formula of futures prices, Eq.(1), does not explicitly give answer to this question. However, we could still find the answer with some basic ideas in finance. 2.3.1 Are current futures prices unbiased estimates of future spot prices? According to the CAPM, in an efficient market, we always have9 E (ST |It ) = St e(yt −qt )(T −t) .

(4)

Here, yt is the expected return for the asset with continuous compounding and yt = rt + βt (rmt − rt ) , where rmt is the market return, βt is the beta coefficient of this asset. It is evident that the greater the systematic risk, the higher the expected return yt . Then, we could discuss the relationship between Ft and E (ST |It ). First, we consider the situation that Eq.(1) holds. Compare Eq.(1) and Eq.(4)  Ft = St e(rt −qt )(T −t) . E (ST |It ) = St e(yt −qt )(T −t) Obviously, only when yt = rt , that is, only when the expected return of the spot asset is equal to the risk-free interest rate, Ft = E (ST |It ) holds. There are only two situations in which yt = rt holds. One is investors are risk-neutral, and the other is the systematic risk of the asset is zero. Because most investors are not risk-neutral and the systematic risk of most assets is not zero in reality, in most cases, current futures prices are NOT unbiased estimates of future spot prices. Next, we consider the situation that the no-arbitrage condition does not hold. It follows that Eq.(1) does not hold and theoretical futures prices are not available. It seems impossible to examine the relationship between Ft and E (ST |It ). However, it is not the case. In financial markets, three broad categories of traders can be identified: arbitrageurs, hedgers, and speculators. When the no-arbitrage condition holds, it is arbitrageurs who push futures prices back to the level of no-arbitrage prices. When arbitrage opportunities could not be utilized, there will only be hedgers and speculators left in the market. Because hedgers only care about how to hedge their risk, speculators become the market force to determine futures prices. Then, how will speculators influence futures prices? Rational speculators always require risk premium when they invest. For them, futures are just a part of their portfolio. The more systematic risk they assumes when they invest on futures, the higher the risk premium they will ask for. If the risk premium is too small, which means the price are too 9

Similar discussions could also be found in [17] and [18].

high, rational speculators will not buy or they would short futures; if the risk premium is too high, which means the prices are too low, speculators will buy as much as possible. All these transactions will drive futures prices converging to an equilibrium level at which speculators could gain a reasonable risk premium for the risk they take. Note that as redundant securities of spot assets, futures do not bring new risk to investors and the risk premium of futures is just the systematic risk premium of the spot assets. Consequently, in efficient markets, the difference between Ft and E (FT |It ) should be the systematic risk premium of the spot assets. As we have FT = ST on the expiration date of futures, the difference between Ft and E (ST |It ) should also be the systematic risk premium of the spot assets in efficient markets. So far, all the above analyses point to one important conclusion: in an efficient market, no matter whether no arbitrage condition holds or not, the difference between Ft and E (ST |It ) should always be the systematic risk premium of the spot assets and current futures prices are NOT unbiased estimates of future spot prices unless investors are riskneutral or the systematic risk of the assets is zero. Therefore, in most cases, futures do not have the function of “discovering” future spot prices. Now, one may ask a question. In reality, when the market expects the future prices would rise (fall), we do observe the increase (decrease) of futures prices, reflecting these expectations. How could we say current futures prices are not unbiased estimates of future spot prices? The reason lies in that when there are some expectations, not only futures prices but also spot prices will change to reflect these expectations. Consequently, in an efficient market, although the change of futures prices do reflect market expectations, the difference between current futures prices and the expected future spot prices remains the systematic risk premium because spot prices also react to those expectations. Anyhow, current futures prices are NOT unbiased estimates of future spot prices unless investors are risk-neutral or the systematic risk of the assets is zero. 2.3.2

Unbiased estimates and the pricing efficiency of futures markets We have mentioned that there are quite a few researches trying to study the efficiency of futures markets by testing whether Ft = E (ST |It ) holds. However, we have proved that in most cases, Ft is not equal to E (ST |It ). This means that whatever our empirical studies reveals about the relationship between Ft and E (ST |It ) could not be used as evidence to support or reject the efficiency of futures markets. It appears attractive to utilize the equation of the difference between Ft and E (ST |It ), E (ST |It ) = Ft eρt (T −t) ,

(5)

to examine the pricing efficiency of futures markets, where ρt = yt − rt , which is the systematic risk premium per annum of the spot assets during the period of (T − t) with continuous compounding. In theory, it is reasonable. But the problem is E (ST |It ), and ρt are usually not available. Substituting Eq.(4) into Eq.(5), we obtain Eq.(1) Ft = St e(rt −qt )(T −t)

CHEN Rong, ZHENG Zhenlong/Systems Engineering — Theory & Practice, 2008, 28(8): 2–11

again. The equivalence between Eq.(5) and Eq.(1) in efficient markets shows that we can explore evidence of the pricing efficiency of futures market by tesing Eq.(1) instead of Eq.(5). This indicates again a better way to investigate the pricing efficiency of futures market is to test the relationship between current futures prices and current spot prices just as mentioned in Section 2.2. 2.3.3 Pricing efficiency and the information efficiency of futures markets So far, we discussed about the pricing efficiency of futures markets. However, in financial literature, information efficiency seems to be paid much more attention. Actually, most of the articles on the relationship between Ft and E (ST |It ) argue that what they are testing is the information efficiency of futures markets. In this section, we illustrate that the pricing efficiency and the information efficiency of futures markets are two different issues and many researchers confuse them. We will also show that it is hard to examine the information efficiency of futures markets by investigating the relationship between Ft and E (ST |It ). Fama[19] is the first one who develops the concept of information efficiency in stock markets as an academic concept of study. The basic idea is if stock prices always rapidly change to reflect new information so that it is impossible to consistently outperform the market by using any information that the market already knows, except through luck, this market is “informationally efficient”. It is usually expressed as (6) st − st−1 = μt + εt , where st = ln St , st−1 = ln St−1 , μt is the daily expected return from t − 1 to t with continuous compounding and εt is a martingale difference sequence10 with the property of E(εt |It−1 ) = 0. Daily expected returns are usually quite small that researchers assume them to be zero. Thus, we have (7) st − st−1 = εt . Consider conditional expectation on both sides of Eq.(7), we get (8) E (st |It−1 ) = st−1 . Eq.(8) could be explained as follows. If today’s stock price is the conditional unbiased estimate of tomorrow’s stock price, the stock market is efficient in weak form. It seems that the idea of unbiased estimation in the test of information efficiency of stock markets in Eq.(8) is very similar to the idea of unbiased estimation in futures markets. This is the reason why many researchers apply Eq.(8) directly in the futures markets and get E (fT |It ) = ft . Because fT is equal to sT when the futures contracts mature, the above equation could be rewritten as E (sT |It ) = ft .

(9)

Now, it seems that we could test the information efficiency of futures market by examining whether ft is equal to E (sT |It ) in Eq.(9), that is, examining whether futures (log) prices are unbiased estimates of future spot (log) prices. Such understandings could be seen everywhere in the futures markets literature. However, two important difference between Eq.(8) and Eq.(9) are ignored. First, in the test of the information efficiency of stock markets, the hypothesis that μt = 0 is acceptable since those are all daily data. However, in the study of futures markets, such a hypothesis is no longer acceptable since the time interval is the period of T − t instead of one day. As we discussed in previous sections, the expected return of futures prices is the systematic risk premium, which in most cases is not zero. Second, in the test of the information efficiency of stock markets, researchers usually study the property of the residual εt in Eq.(7). However, in the study of futures markets, researchers conduct regression analysis of Eq.(9)11 and conclude by observing whether the coefficient of ft is equal to 1. In fact, when the time interval is quite long, μt is not only nonzero but also time-varying. The expected return in a long run is not the simple sum of daily return. An adjustment of − 12 σ 2 is needed12 . Besides, because the data of E (sT |It ) is not available, the real spot prices sT have to be used as substitutions of E (sT |It ), which implies another important hypothesis of rational expectation. Anyway, the attempt to test the information efficiency of futures markets is doomed to failure. Even if the coefficient of ft is not equal to 1, it is not convincing evidence that the futures market studied is informationally inefficient. Consequently, the right way to test the information efficiency of futures markets is to examine whether εt in the following equation ft − ft−1 = εt

(10)

is a martingale difference sequence. Note that Eq.(10) shares the same idea with Eq.(7) and daily data need to be used. Therefore, the issue of market efficiency is different for stock markets and futures markets. For stock markets, only information efficiency is considered because we do not have a reliable pricing model for stocks so far. For futures markets, however, both pricing efficiency and information efficiency need to be considered. The pricing efficiency is about the no-arbitrage prices while information efficiency is about the reaction of futures prices to new information. The pricing efficiency and the information efficiency are not necessarily consistent with each other. The former depends on whether arbitrage strategies could be utilized while the latter rests with the market development and some other factors

10

In early literature about the EMH, εt is usually assumed to be an iid normal process. However, more and more evidence indicates the volatilities of prices are time-varying and log returns are not normally distributed. Nowadays, the condition of the weak-form efficiency is believed to be that εt is a martingale difference sequence. A detailed discussion could be found in [20]. 11 There are three forms of regression models. (1) sT = α + βft + εt , (2) ΔsT = α + βΔft + εt , and (3) sT − st = α + β (ft − st ) + εt . The null hypotheses are all (α, β) = (0, 1).   12

Suppose the price of the underlying asset follows geometric Brownian motion where there is a convexity adjustment term.

dS S

= μdt + σdZ . Then, we have E (ST |It ) = St e

μ− 1 σ 2 2

(T −t)

,

CHEN Rong, ZHENG Zhenlong/Systems Engineering — Theory & Practice, 2008, 28(8): 2–11

is, the most important function of futures market, i.e., risk management. It also indicates that to make futures markets a good place to manage and hedge risk, the pricing efficiency or an unrestricted arbitrage mechanism are fundamental. Overall, in previous sections, we reach many important conclusions. First, in most cases, no theories or evidence support the argument that futures prices are unbiased estimates of future spot prices. Futures prices do not have the function of “discovering” future spot prices as many people have taken for granted. The failure of such defined “price discovery” in empirical studies could not be used as evidence of the inefficiency of futures markets. Second, a welldeveloped futures market might have the function of “discovering” current spot prices. Third, there exist two forms of market efficiency in futures markets: the pricing efficiency and the information efficiency. They are not necessarily consistent with each other. Finally, a futures market which is efficient in pricing will have a strong function of risk management. Based on these conclusions, we argue that researchers could test the pricing efficiency of futures markets by running regressions of ft − st = α + β (rt − qt ) (T − t) + εt Figure 1. Futures prices, spot prices, and the expected future spot prices

like transactin costs. As shown in previous discussion, the pricing efficiency could be tested by investigating the relationship between Ft and St , whereas the information efficiency could be tested by examining whether the residuals of futures log returns, εt in Eq.(10), is a martingale difference sequence. The study of the relationship between Ft and E (ST |It ) could not provide us any convincing evidence about the market efficiency. 2.3.4 Risk premium and the risk management function of futures Then, is the study of the relationship between Ft and E (ST |It ) meaningless? No. Suppose the underlying asset provides no yield during the life of one futures contract, that is, qt is zero13 . From Eq.(1) and Eq.(5), we have  Ft = St ert (T −t) . E (ST |It ) = E (FT |It ) = Ft eρt (T −t) It could be seen that in a futures market, which is efficient in pricing, the part futures prices over current spot prices is the risk-free interest rate; at the same time, the expected return of futures is just the systematic risk premium of the underlying asset. Figure 1 illustrates this idea. What does it mean? The fact that the expected return of futures is just the systematic risk premium of the underlying asset means hedgers could use opposite futures positions to hedge the systematic risk they take in the spot market. That

13

When qt is not zero, similar conclusions will be reached.

(11)

= or conducting cointegration analysis on {Xt }  T  ft , st , (rt − qt )(T − t) . If researchers want to test the information efficiency of futures markets, researchers should examine whether the residual of futures log returns, εt in Eq.(10) ft − ft−1 = εt is a martingale difference sequence. Besides, researchers could use causality tests to investigate the lead-lag relationship between the futures prices and spot prices. These three are appropriate empirical models to investigate the relationship patterns between futures prices and spot prices.

3

Empirical study on S&P 500 index futures

In this section, we would like to conduct an empirical study on daily closing prices of S&P500 index spot and futures. The reason we choose S&P500 index futures is S&P500 index futures market is generally acknowledged to be one of the most developed, liquid and efficient futures markets. This empirical study is rather an illustration of the reasonable relationship between futures markets and spot markets than a test of market efficiency of the S&P500 index futures. The S&P 500 spot and futures closing prices were retrieved for each trading day from 21 September 1990 to 20 December 2007 from the Bloomberg for a total of 4307 observations. The futures prices are those from the nearest contracts. Contracts are rolled over to the next nearby contracts one month prior to expiration. The T-bill data were obtained from the the United States Department of the Treasury. The dividend yields were collected from the website of Aswath Damodaran who is a professor in finance at New York University and the website of Standard & Poor’s.

CHEN Rong, ZHENG Zhenlong/Systems Engineering — Theory & Practice, 2008, 28(8): 2–11

Table 1. Augmented dickey-fuller (ADF) unit root tests

ft

Δft

−1.706079

−67.51094∗∗

Trend and intercept

−1.628570

−67.52329

∗∗

No trend and no intercept

2.1387384

−67.43363∗∗

Intercept

st

Δst

−1.748424

−66.62816∗∗

−7.044060∗∗

−7.630526∗∗

−1.600765

∗∗

∗∗

−7.040139

−7.649243∗∗

−4.631692∗∗

−5.216830∗∗

2.218953

ft − st

−66.64248

−66.54693∗∗

(rt − qt )(T − t)

** denotes significance at 1% level. Table 2. Johansen cointegration tests between ft and st

Hypothesized No. of CE(s)

Eigenvalue

Trace statistic ∗

Max-eigen statistic

None

0.015491

70.63451

67.16339∗

At most 1

0.000807

3.471124

3.471124

* denotes significance at 5% level.

3.1 Unit root tests

Table 3. Granger causality tests between Δft and Δst

Table 1 reports the ADF unit root test results of the series of log futures prices (ft ), the first difference of log futures prices (Δft , the log returns of futures), log spot prices (st ), the first differences of log spot prices (Δst , the log returns of spots), the basis (ft − st ) and the cost of carry ((rt − qt ) (T − t)). It shows that at the 1% significance level, ft and st series could not reject the null hypothesis of nonstationary while Δft , Δst , ft − st , and (rt − qt ) (T − t) all reject the null hypothesis and are stationary series. 3.2 Pricing efficiency of the S&P 500 index futures market According to the results of section 3.1, the pricing efficiency of S&P 500 index futures could be tested by implementing cointegration tests on ft and st or running regression of ft − st on (rt − qt ) (T − t). 3.2.1 Cointegration test Johansen’s cointegration tests with the lag length of 714 are conducted on ft and st . The results in Table 2 shows that the null hypothesis of no cointegration, i.e., r ≤ 0, is rejected at the 5% level using Trace and Maximum Eigenvalue statistics while the hypothesis of r ≤ 1 is not rejected. It indicates that there exist long-term equilibrium between ft and st . 3.2.2 Regression analysis Since the series of ft −st and (rt − qt ) (T − t) are both stationary, we run regression of ft − st on (rt − qt ) (T − t) and obtain ft − st R 14

2

= 0.0002 + 1.095 (rt − qt ) (T − t) + εt (4.401)∗∗ (139.0598)∗∗ = 0.817871.

(12)

Chi-sq

df

Prob.

Regression of Δst on Δft

28.89817

7

0.0002

Regression of Δft on Δst

16.01993

7

0.0249

Here, the values in the parentheses are t-values and ** denotes significance at 1% level. Eq.(12) exhibits that almost 82% of the change of the basis could be explained by the cost of carry. The null hypothesis of zero coefficient is rejected at 1% level and the coefficient is 1.095. This result is consistence with the results of the cointegration test, which indicates the S&P500 futures market is efficient in pricing. 3.3

Granger causality test

The VECM model of Δft and Δst is established, and Granger causality tests are implemented on the VECM model. The results are given in Table 3. It could be observed that the null hypothesis Δft does not Granger cause Δst is rejected at 1% level with the lag length of 7. However, the null hypothesis Δst is the Granger cause of Δft is rejected at 5% level. In other words, S&P 500 index futures prices are in the leading position of price discovery when new information arrives. This is consistent with our intuition. As one of the most developed, liquid and efficient futures markets in the world, S&P index futures have the advantage of lower costs, higher leverage, and better liquidity over spot markets. Investors tend to trade first in the futures market instead of the spot market when new information arrives. The information flows from the futures market to the spot market. 3.4

Information efficiency of the S&P 500 index spot and futures markets

In this article, we consider the statistic of M1 (p) proposed by Hong and Lee in [21] to test whether the residuals of spot log returns and futures log returns are martingale difference sequences and, accordingly, examine the information efficiency of these two markets. Hong and Lee prove that if a ‘time series is a martingale, when p = cT λ for

The lag length is chosen by comparing different lag length selection methods including LR, FPE, AIC, SC, and HQ.

CHEN Rong, ZHENG Zhenlong/Systems Engineering — Theory & Practice, 2008, 28(8): 2–11

Table 4. Empirical tests about the relationship between current futures prices and future spot prices

α ˆ

βˆ

¯2 R

sT − st = α + β (ft − st ) + εt

0.000061∗∗

−0.325977

0.000717

ΔsT = α + βΔft + εt

0.000343

0.015138

0.000061

Models

** denotes significance at 1% level.

8 7 6 5 4 3 2.33

2

1.65

1 0 1

3

5

7

9

11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 S&P 500 index

S&P 500 index futures

Figure 2. M1 (p) for S&P 500 index and S&P 500 index futures

0<λ<

1 3+ 4b − 2

4

−1 and 0 < c < ∞, we have

M1 (p) −→ N (0, 1) ,15 d

where p is the bandwidth in spectral tests and p ≡ p (T used to select initial bandwidth  ), c is the parameter 1/5 . Figure 2 exhibits the values of p¯ p¯ = c × (10T ) M1 (ˆ p0 ) for futures prices and spot prices. The horizontal axis is the value of c so that we could observe the stability p0 ) for different c. Two horizontal lines in the figure of M1 (ˆ are upper-tailed asymptotic N (0, 1) critical values at 5% level and 1% level, respectively. It could be seen that the null hypothesis of martingale difference sequences are rejected at 1% level for both the S&P500 spot market and the futures market. That is, neither market is efficient in weak form. This is consistent with the results in previous sections. In section 3.2, we find that the S&P 500 futures market is efficient in pricing. In section 3.3, there’s information flows between two markets, although the futures market plays a leading role in price discovery. These two markets fluctuate together. Therefore, if one market is not informationally efficient, the other will not be informationally efficient. Another finding is although the S&P 500 futures market is efficient in pricing, it is not informationally efficient. This proves that the pricing efficiency is not necessarily consistent with the information efficiency. 3.5 Relationship between current futures prices and future spot prices: a comparison We have obtained some evidence showing that the S&P 500 futures market is efficient in pricing but not informationally efficient with the data of current futures prices and current spot prices. Next, we would like to carry out empirical studies on two popular models about the relationship 15

between current futures prices and future spot prices. ¯2 Table 4 shows that in these two models, both the R values are both quite low and the null hypotheses of β = 0 could not be rejected at 1% level. That is, both models fail to capture the relationship between ft and st , not to mention testing the efficiency of futures market. As explained in previous sections, many factors including time-varying expected return, time-varying risk premium, time-varying volatility, and irrational expectations might be the causes . The results in Table 4 prove that it is difficult to obtain convincing conclusions with the data of current futures prices and future spot prices.

Conclusions

In this article, we clarify some misunderstandings about the relationship between futures prices and spot prices and illustrate the correct relationship patterns and the appropriate empirical models. We illustrate that in most cases, current futures prices are not unbiased estimates of future spot prices. We redefine the function of “price discovery” of futures markets as the leading role in the reactions to new information. We propose that the pricing efficiency and the information efficiency of futures market are different. Investigating the relationship between current futures prices and future spot prices will not provide us reliable evidence about the market efficiency and the function of price discovery of futures markets. In addition, three empirical models appropriate to investigate the relationship between futures prices and spot prices are discussed. Finally, an empirical study on the data of S&P 500 index spot and futures from 21 September 1990 to 20 December 2007 was carried out, and the results are consistent with the above arguments.

Acknowledgments We thank the referees and the editors for careful and insightful comments. We thank Yongmiao Hong at Cornell University for the programming codes of generalized spectral tests. We also thank seminar participants at the 2006 China Management Conference, the Fourth Conference on Banking and Finance and Financial Trend (2006), the Fourth International Conference on Risk Management & the Fifth International Conference on Financial System Engineering (2007) and the Fourth Chinese Finance Annul Meeting (2007) for helpful comments, discussions, and references, and the Ministry of Education of the People’s Republic of China for support via research grants 05JJD790026 and 07JA790077.

M1 (p) is a statistic that could be used to test model misspecification in mean robust to conditional heteroscedasticity and other higher order timevarying moments of unknown form. The tests enjoy the appealing “nuisance parameter free” property that parameter estimation uncertainty has no impact on the limit distribution of the tests. Further information could be found in [21].

CHEN Rong, ZHENG Zhenlong/Systems Engineering — Theory & Practice, 2008, 28(8): 2–11

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