Trading patterns in the Mexican TIIE futures market

Trading patterns in the Mexican TIIE futures market

Renata Herrerías Franco Pedro Gurrola Pérez Instituto Tecnológico Autónomo de México

Contents 1 Introduction

5

2 The TIIE rate and its futures contracts

11

2.1

Tasa de Interés Interbancaria de Equilibrio (spot TIIE) . . . . . . . . . . . .

11

2.2

TIIE futures contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3 Day of the week and expiration effects

16

3.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

3.2

Previous studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3.3

Data and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.3.1

Sample data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.3.2

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.4.1

Day-of-the-week effects . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.4.2

Expiration day effects

. . . . . . . . . . . . . . . . . . . . . . . . . .

31

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

3.4

3.5

4 Maturity effects

35

4.1

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

4.2

Previous studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

4.3

Data and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.3.1

Sample data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.3.2

Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.4.1

49

4.4

Estimates of time-to-maturity effects on volatility . . . . . . . . . . .

1

4.5

4.4.2

Effect of controlling for variation in information flow . . . . . . . . .

54

4.4.3

Estimation of maturity effect on the basis. . . . . . . . . . . . . . . .

59

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5 Final conclusions

65

2

List of Figures 2.1

TIIE spot rate during the period 2003-2006 . . . . . . . . . . . . . . . . . . . .

12

2.2

Total number and daily average of TIIE futures contracts traded per year

. . . .

14

3.1

Number of 28-day TIIE futures contracts traded per month relative to contract expiration

3.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

Number of 28-day TIIE futures contracts traded per week relative to contract expiration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

4.1

Average log-basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4.2

Volume of TIIE Futures contracts traded during the whole period . . . . . . . . .

46

3

List of Tables 1.1

Number of futures contracts traded in 2006 by contract type . . . . . . . . .

2.1

Ranking of brokers trading TIIE futures contracts in 2006 according to trading

6

volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.1

Summary Statistics of 28-day TIIE Futures Daily Rate Changes. . . . . . . .

21

3.2

Statistics of Daily Rate Changes According to the Day of the Week. . . . . .

25

3.3

Trading Volume Statistics According to the Day of the Week. . . . . . . . .

27

3.4

28

3.5

Panel A. Conditional Mean Equation Estimates . . . . . . . . . . . . . . . . √ Descriptive statistics for the estimated standardized residuals ut / ht . . . .

4.1

Descriptive statistics for TIIE futures contracts daily logarithmic changes . .

41

4.2

Average log-basis by month to expiration . . . . . . . . . . . . . . . . . . . .

43

4.3

Descriptive statistics for daily basis changes . . . . . . . . . . . . . . . . . .

45

4.4

Regression of daily volatility on days to expiration . . . . . . . . . . . . . . .

51

4.5

Test for individual and time effects in futures volatility series . . . . . . . . .

52

4.6

Panel regression of daily volatility on time to expiration . . . . . . . . . . . .

53

4.7

Regression of daily volatility on days to expiration and spot volatility . . . .

56

4.8

Test for individual and time effects in futures volatility series with TIIE spot variance as control variable . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

32

57

Panel regression of daily volatility on time to expiration and spot rate volatility 58

4.10 Regression of basis changes volatility on days to expiration . . . . . . . . . .

60

4.11 Test for individual and time effects in basis changes volatility series. . . . . .

61

4.12 Panel regression of basis changes volatility on time to expiration

62

4

. . . . . .

Chapter 1 Introduction The Mexican Derivatives Market (MexDer) has been successfully trading over the last seven years. At the end of 2006 it became the eighth largest derivatives exchange in the world behind Korea Exchange, Eurex, Euronext.liffe and Chicago exchanges (Holz, 2007). MerDer offers futures contracts for currencies (euro and US dollar), interest rates (five products), one equity index (IPC) and five individual stocks. It also trades with options on one equity index (IPC), two individual stocks (America Movil and Naftrac), two ETFs (Nasdaq 100-Index and iShares S&P 500 index) and on the US dollar. Among the future contracts traded in the MexDer, the 28-day TIIE futures contract is an interest rate futures contract whose underlying consists of 28-day deposits that produce yield at the 28-day Interbank Equilibrium Interest Rate (Tasa de Interes Interbancaria de Equilibrio, or TIIE). This is the rate that serves as a measure of the average cost of funds in the Mexican interbank money market. Table 1.1 reports the number of futures contracts traded during 2006. It is evident that the 28-day TIIE contract is the leading derivatives contract traded in the MexDer, representing an astonishing 96% of total futures contracts traded. From a global perspective, the numbers are also impressive: the TIIE futures contract was the third most actively traded futures contract in the world in 2006, only after the CME’s Eurodollar and Eurex’ Eurobond, and experienced during that year the largest increase in volume in any futures contract (Holz, 2007). Despite the growing importance of the MexDer and the key role that 28-day TIIE futures play within the derivatives markets around the world, there are very few empirical

5

CHAPTER 1. INTRODUCTION

6

Table 1.1: Number of futures contracts traded in 2006 by contract type Contract

Volume

(%) Participation

US Dollar

6,026,940

2.19

Euro

50,469

0.02

Currencies

6,077,409

2.21

IPC

620,557

0.23

Equity indices

620,557

0.23

Cete 91

3,290,100

1.2

TIIE 28

264,160,131

96.18

3 years Bond (M3)

28,600

0.01

10 years Bond (M10)

471,879

0.17

UDI

0

0

Interest rates

267,950,710

97.56

Individual equities

3,000

0

Total

274,651,676

100

Figures are the number of contract traded during 2006 and the percentage of participation respect the total. Source: MexDer, December 2006

studies that analyze its behavior and characteristics. Research in this direction is certainly important for participants, including non-Mexican investors, as well as for regulators and clearinghouses. The aim of this work is to study the presence of trading and nonstationary patterns in the TIIE futures market. In particular, this work focuses on the presence of day-of-the-week, expiration day and maturity effects in the 28-day TIIE futures contracts. The intention is to investigate whether rate changes and their volatility are systematically different in some days of the week, in the week when the next-to-expiration contract matures, or as contracts approach their own expiration. The day-of-the-week effect refers to the evidence that asset returns present different distributions in some of the days of the week. It has been extensively reported in equity,


7

foreign exchange, commodities and T-Bill markets around the world (Aggarwal & Rivoli, 1989; Agrawal & Tandon, 1994; Berument & Kiymaz, 2001; French, 1980; Harvey & Huang, 1991; Jaffe & Westerfield , 1985; Lakonishok & Levi, 1982). However, research on day-ofthe-week effects on futures markets is less abundant. Among the different day-of-the-week effects occurring in different markets perhaps the most persistent is the weekend effect: Friday returns are reported to be abnormally high and Monday returns abnormally low and, on average, negative. Most of the studies on the existence of day-of-the-week patterns have found evidence of the weekend effect. With respect to the existence of patterns linked to expiration dates, it should be noted that in the last decades a great number of studies have been published regarding possible effects of stock indexes derivatives on the underlying. Evidence has been found of abnormal price behavior, higher trading volume or price reversals in the underlying assets around the expiration dates. This effect, known as expiration effect, arises primarily from a combination of factors including the existence of index arbitrage opportunities, the cash settlement feature of index options and futures, the unwinding of arbitrage positions in the underlying index stocks, and attempts to manipulate prices as explained, for example, in Stoll & Whaley (1997). In the case of interest rate futures a different but similar question arises: at the dates of expiration of short term contracts, are there persistent changes, upward or downward, on longer term contracts rates, in their volatility, or in both? A priori, one should expect price movements consistent with the term structure determined by the forward rate curve. However, such a behavior may also reveal seasonal patterns induced by trading activity. Hence, in this study the use of the term expiration effect will refer to the abnormal behavior of futures contracts with different maturities on the days around the expiration dates. Samuelson (1965) was the first to propose a theoretical model postulating that the volatility of futures prices should increase as the contract approaches expiration. This effect, more commonly known as Samuelson hypothesis or maturity effect, occurs because price changes are larger when more information is being revealed. Early in a contract’s life, little information is known about the future spot price for the underlying. Later, as the contract nears maturity, the rate of information acquisition increases, more relevant information arrives and participants are more sensitive to information arrival which affects the futures price. In consequence, price volatility increases. Numerous studies have investigated the Samuelson hypothesis empirically, yielding mixed results. In general, the maturity effect has been


8

supported for commodities, while it has not appeared to be significant for financial assets. In order to investigate the presence of these patterns in the 28-day TIIE futures market, our research considers a data set which includes all the contracts maturing between January 2003 and December 2006. Concerning the methodology employed, expiration and day-ofthe-week effects on futures daily rate changes are investigated using a set of 36 rollover time series and applying a GARCH(1,1) model specification that includes daily dummies and a dummy for expiration day effects, in both the conditional mean and the conditional volatility functions. On the other hand, the relation between volatility and time to expiration (maturity effect) is assessed for rate changes and basis changes using 48 time series grouped in a panel where observations are arranged not according to calendar day, but according to days to maturity. This permits to apply panel data estimation techniques in addition to the usual time series methods. Relative to previous literature, the contribution of this study is as follows. First, it documents the existence of day-of-the-week, expiration day and maturity effects in a market for which, in spite of its increasing importance, there are almost no previous studies. Usually these anomalies are attributed to the arrival of new information; however, the rationale behind the anomalies in the Mexican market may be different considering, for example, that the TIIE futures market is a very liquid market but with only few participants. Furthermore, day-of-the-week and expiration effects are investigated using a whole set of 36 rollover time series, ranging from the next-to-expiration contract to the contract with expiration in 35 months. This data set permits to assess the existence of nonstationarity and to identify trading patterns not only for next-to expiration contracts but also for long term contracts. The use of 36 time series allows distinguishing between the effects of trading activity and those of information arrival. For example, under the assumption that new information does not necessarily equally affect short and long run contracts, a monotonic behavior across futures contracts will denote anomalies highly influenced by trading activity patterns, and to a lesser extent by new information arrival. On the other hand, the consideration of long term contracts also leads to study the possible effect of expiration days on the whole forward curve. To the best of our knowledge, this effect on long term futures contracts has not been previously studied. Finally, this study also expands upon previous research on maturity effects by considering a panel of contracts with the same underlying but differing in its expiration date, where


9

observations are arranged not according to calendar day, but according to days to maturity. This data arrangement permits to apply panel data techniques to assess the existence of cross-sectional individual effects. The main findings can be summarized as follows, TIIE futures rate changes are strongly heteroscedastic. There is a weekend pattern consistent with the Monday effect observed in other interest

rate futures markets: On Mondays rates (prices) tend to increase (decrease) while on Fridays they tend to decrease (increase). This effect seems to be idiosyncratic, a consequence of particular trading activities. There are expiration effects on short-term TIIE futures contracts: on the expiration

dates (usually every month’s third Wednesday), the volatility of contracts expiring in seven month or less increases. Maturity effects are present in 2003 and 2004, an inverse maturity effect appears in

2005 and 2006, and there is not evidence of maturity effect once all contracts are considered (2003-2006). Maturity effects appear in the whole set of contracts when the spot volatility is included

as a proxy for information flow. The expected maturity effect is present in basis changes for contracts between Septem-

ber 2004 and March 2006, while panel analysis indicates an inverted effect in 2003 and the expected maturity effect in every year from 2004 and when the whole sample is considered. The study of the behavior of rate changes and volatility of futures prices has important implications for market participants, for derivatives pricing and for risk management. First, it gives information about market efficiency. A systematic price behavior on a specific day of the week would permit to create profitable trading strategies based on historic patterns. If this the case even when transaction costs are taken into account, then this behavior clearly contradicts the efficient market hypothesis.


10

Clearinghouses set margin requirements on the basis of futures price volatility. Therefore, if there is any relation between volatility and time to maturity the margin should be adjusted accordingly as the futures approaches its expiration date. The relation between volatility and maturity also has implications for hedging strategies. To minimize risk, hedgers must choose between futures contracts with different time to maturity according to the positive or negative relation between volatility and maturity. Furthermore, the fact that once the next-to-expiration contract matures, roll-over is done with contracts expiring in seven months or less indicates that there is a lack of balance between basis risk and transaction costs. Traders may prefer short-term contracts to minimize basis risk and, given that margin requirements are equal regardless of maturity, there is no incentive to use long term contracts for hedging. Volatility and time to maturity relation is essential too for speculators in the futures markets as they bet on the futures price movements of the assets. If maturity effect holds, then speculators may find advantageous to trade in futures contracts close to expiry, as greater volatility implies greater short time profit opportunities. Finally, since volatility is central to derivatives pricing, the relation between maturity and volatility should also be taken into consideration when pricing derivatives on futures. The rest of the document is organized as follows. The next chapter describes how 28-TIIE rate is calculated and its behavior over the studied period. It also explains the mechanics of operation of TIIE futures contracts and gives some general statistics. The third chapter explores day-of-the-week and expiration effects, providing a review of previous studies, the data and methodology used, and the main findings. In a similar fashion, Chapter Four investigates the presence of maturity effects in TIIE futures. General concluding remarks are given in the last chapter.

Chapter 2 The TIIE rate and its futures contracts 2.1

Tasa de Inter´ es Interbancaria de Equilibrio (spot TIIE)

Since March 1995, Banco de Mexico determines and publishes the short-term interest rate benchmark known as Tasa de Interés Interbancaria de Equilibrio, or TIIE. There are two variants for the TIIE: 28- and 91-day. The 28-day TIIE rate is based on quotations submitted daily by full-service banks using a mechanism designed to reflect conditions in the Mexican Peso money market. The participating institutions submit their quotes to Banco de Mexico by 12:00 p.m. Mexico City time and in the case that less than six quotes are received, Banco de Mexico will determine TIIE rate according to the prevailing money market conditions in that day. Following the receipt of the quotes, Banco de Mexico determines the TIIE with the average between bid and ask quotes weighted by its corresponding amount of money and by the differential between lower and higher quotes. Rates quoted by institutions participating in the survey are not indicative rates for informational purposes only; they are actual bids and offers by which these institutions are committed to borrow from or lend to Banco de Mexico. In case Banco de Mexico detects any collusion among participating institutions or any other irregularity, it may deviate from the stated procedure for determination of the TIIE rates. As mentioned previously, this study considers the period between January 2003 and

11

CHAPTER 2. THE TIIE RATE AND ITS FUTURES CONTRACTS

12

Figure 2.1: TIIE spot rate during the period 2003-2006 11.0 10.0

Daily TIIE (%)

9.0 8.0 7.0 6.0 5.0 4.0 Dec-02

Jun-03

Dec-03

Jun-04

Dec-04

Jun-05

Dec-05

Jun-06

Dec-06

Daily TIIE spot rate from January 2003 to December 2006. Source: Banco de México.

December 2006. In order to set the macroeconomic context in which the study was developed, Figure 2.1 graphs the daily TIIE quotes in the spot market over that period. This data is provided by Banco de Mexico. Between 2003 and 2006 the highest level was reached in March 2003, declining monotonically after that and until August 2003 when it reached the historic minimum (4.745%). A period of uncertainty started after September that year and it prevailed throughout the first half of 2004 where movements of almost 150 bps within very short periods (2 weeks) were present. A stable pattern is present in 2005 until August when the rate declined again to settle between 7.0 and 7.5 percent during the second half of 2006. Later in the document it will be seen that the behavior of TIIE futures rates highly reflect the movements in the spot market and that particular patterns can only be explained by the volatility and movements in the spot rate.


2.2

13

TIIE futures contracts

The TIIE futures contracts are traded in the Mexican Derivatives Exchange (MexDer). Each 28-day TIIE Futures Contract covers a face value of 100,000.00 Mexican Pesos (approximately 9,100 U.S. dollars). MexDer lists and makes available for trading different series of the 28-day TIIE futures contracts on monthly basis for up to ten years. It is important to observe that, in contrast with analogous instruments like CME’s Eurodollar futures or Euronext.liffe’s Short Sterling futures, TIIE futures are quoted by annualized future yields and not by prices. The relation between the quoted future yield on day t and the corresponding futures price Ft is determined by MexDer by the formula Ft =

100, 000 1 + Yt (28/360)

where Yt is the quoted yield divided by 100. The last trading day and the maturity date for each series of 28-day TIIE futures contracts is the bank business day after Banco de Mexico holds the primary auction of government securities in the week corresponding to the third Wednesday of the maturity month. Since these primary auctions are usually held every Tuesday, in general expiration day for TIIE futures corresponds to the third Wednesday of every month. As mentioned before, the study considers contracts traded from January 2003 since before that date TIIE futures trading volume was not enough to evaluate the statistic significance of the results. Figure 2.2 shows total and daily average trading volume for TIIE contracts per annum. From modest 156 thousand contracts traded in 1999, 264 millions contracts were traded in 2006, becoming the third most actively traded futures contract in the world and also the one with the largest growth in volume during 2006 (Holz, 2007). Nowadays there is an average of 1.06 millions of TIIE contracts traded every day, representing a daily average notional amount of approximately $9.5 billions of US dollars. The TIIE futures market is a very liquid market but with only few participants. For example, in 2006 there were on average seven operations per day per type of contract, each of them for an amount of around 20 million U.S. Dollars. The contrast between the large size of the market and the small number of participants suggests the market could behave differently in comparison to other more mature markets. It may be the case that the reduced number of participants promotes some collusion among them, and this collusion


14

300

1.20

250

1.00

200

0.80

150

0.60

100

0.40

50

0.20

Daily average (millions of contracts)

Total volume (millions of contracts)

Figure 2.2: Total number and daily average of TIIE futures contracts traded per year

0

0.00 1999

2000

2001

2002

2003

Total volume

2004

2005

2006

Daily average

Numbers are total and average millions of TIIE futures contracts traded per year. Source: Mexder.

could originate nonstationary patterns in prices. Table 2.1 presents the brokers trading with TIIE futures during 2006. It can be seen that MexDer only reports 22 brokers and that there are several foreign institutions among these.


15

Table 2.1: Ranking of brokers trading TIIE futures contracts in 2006 according to trading volume Broker 1

Santander

2

ING Bank

3

BBVA Bancomer

4

Stock & Price

5

Valmex Casa de Bolsa

6

JP Morgan

7

Banorte

8

Grupo Financiero Scotiabank Inverlat

9

HSBC

10

Invex

11

Nacional Financiera

12

Finamex

13

GBM Casa de Bolsa

14

IXE Banco

15

Multivalores

16

Grupo Financiero Banamex

17

Monex

18

Serafi Derivados

19

Derfin

20

Gamma Derivados

21

Deutsche

22

GFD

Source: MexDer (Trading volume by broker is confidential)

Chapter 3 Day of the week and expiration effects 3.1

Introduction.

The existence of nonstationary patterns in futures contracts prices has been documented extensively in the finance literature. For example, contract month volatility, day-of-theweek, year, and calendar month effects, have been identified for equity, stock indexes and commodities futures (Crato & Ray, 2000; Galloway & Kolb, 1996; Kenyon, Kenneth, Jordan, Seale & McCabe, 1987; Khoury & Yourougou , 1993; Milonas & Vora, 1985). However, for interest rates futures the number of studies about the existence of prices anomalies is still reduced and frequently limited to short-term contracts. Interest rate futures are highly liquid traded financial assets mainly used for hedging purposes. The lower transactions costs, their ability to expand risk management capabilities and their flexibility, among other reasons, have boosted their popularity over the last decades not only in mature markets, but also in emerging economies. Like other derivative instruments, interest rates futures are supposed to increase price efficiency of financial markets and to improve risk sharing among economic agents. The aim of this chapter is to study the presence of day-of-the-week and expiration day effects in the 28-day TIIE futures contracts. The effects on futures daily rate changes are tested using a GARCH(1,1) model specification that includes daily dummies and a dummy for expiration day effects, in both the conditional mean and the conditional volatility functions. Moreover, by using not only next-to-expiration contracts but a whole set of 36 rollover time series, ranging from the next-to-expiration contract to the contract with expiration in

16

CHAPTER 3. DAY OF THE WEEK AND EXPIRATION EFFECTS

17

35 months, the study will permit to assess the existence of nonstationarity and to identify trading patterns not only for next-to-expiration contracts but also for long term contracts. This will allow to distinguish between the effects of trading activity and those of information arrival. Hence, under the assumption that new information does not necessarily equally affect short and long run contracts, a monotonic behavior across futures contracts will denote a day-of-the-week anomaly highly influenced by trading activity patterns, and to a lesser extent by new information arrival. Finally, the consideration of long term contracts also leads to study the possible effect of expiration days on the whole forward curve. The results show that TIIE futures rate changes are strongly heteroscedastic and that there is a weekend pattern consistent with the Monday effect observed in other interest rate futures markets. On the other hand, there are expiration effects on short-term TIIE futures contracts: on the expiration dates (usually every month’s third Wednesday), the volatility of contracts expiring in six months or less increases. The rest of this chapter is organized as follows. The next two sections review the previous studies and describe the data and the methodology employed. In section 3.4 the results are reported. Concluding remarks concerning these results are given in 3.5.

3.2

Previous studies

The day-of-the-week effects, i.e. evidence that asset returns present different distributions in some of the days of the week, have been extensively reported in equity, foreign exchange, commodities and T-Bill markets around the world (Aggarwal & Rivoli, 1989; Agrawal & Tandon, 1994; Berument & Kiymaz, 2001; French, 1980; Harvey & Huang, 1991; Jaffe & Westerfield , 1985; Lakonishok & Levi, 1982). In most of these studies there is evidence of a weekend effect: Friday returns are reported to be abnormally high and Monday returns abnormally low and, on average, negative. Literature on day-of-the-week and futures markets is more limited. Chiang and Tapley (1983) found weekly patterns, including Monday effect, on a variety of future contracts. Studies of Dyl and Maberly (Dyl & Maberly, 1986a,b) found evidence about the existence of day-of-the-week effect on the S&P500 stock index futures rejecting the hypothesis of equal mean returns across days of the week. Similar results were obtained by Gay and Kim (1987) for commodity futures.


18

Seasonal patterns in futures price volatility have also been reported. Most studies attribute seasonal changes in volatility mainly to scheduled macroeconomic announcements and to other public information releases. This conclusion is in line with efficient market hypothesis where asset prices should change only with the arrival of new information. For example, Harvey & Huang (1991) found higher volatility of price returns of major currencies futures on Thursdays and Fridays. They attribute this phenomenon to the concentration of scheduled announcements of macroeconomic indicators on those days of the week. Also, Ederington and Lee (1993) reported higher volatility of currency futures and interest rates futures immediately after macroeconomic announcements. They show that volatility is different across days of the week on announcements days only. In contrast, Han, Kling & Sell (1999), after controlling for the announcement effect and maturity effect, found a strong day-of-the-week effect in Deutsche Mark and Japanese Yen futures. Their results suggest that currency futures are not moved by announcements of macroeconomics indicators, but by factors such as trading process and market microstructure. In the case of interest rates futures, Johnston, Kracaw & McConnell (1991) identified Monday effects on T-bond future contracts, but found no significant seasonal patterns on T-bill contracts. Lee and Mathur (1999) found Monday and Thursday effects using data of futures contracts listed in the Spanish derivative market. On average, Monday returns were negative while on Thursday they were positive for all studied contracts. In addition, for MIBOR90 and MIBOR360 contracts volatility was found to be higher on Mondays. Also, Buckle, ap Gwilym, Thomas, and Woodhams (1998), analyzing intraday empirical regularities in the Short Sterling interest rate futures, report a Monday effect in which returns, volatility and trading volume tend to be lower on Mondays than across the rest of the week. As mentioned previously, in this study the use of the term expiration effect will refer to the abnormal behavior of futures contracts with different maturities on the days around the expiration dates, which in the case of the 28-day TIIE futures correspond to the Wednesdays on the third week of every month. To the best of our knowledge, there are no previous studies on this behavior.


3.3 3.3.1

19

Data and methodology Sample data

The data used in this study are obtained from the MexDer. In particular, the analysis uses daily settlement rates for 28-day TIIE futures contracts from January 2nd, 2003 to June 30th, 2006 (a total of 888 daily observations), for contracts expiring every month from January 2003 to June 2009. Using these daily observations, a panel is created by rolling over contracts: for each series, once the most immediate contract is close to maturity, we rollover each of the series to the contract that is next according to maturity. In applying this kind of rolling over methods there is no generally accepted procedure on the choice of rollover date. The most common choices include switching at the expiration date, at the time of volume crossover or at some arbitrary number of days before the expiry of the front month contract. Considering that the shortest TIIE futures contract has only three weeks to maturity, and that abnormal rate variability may arise at the expiration date (Ma, Mercer & Walker, 1992), the switching is done 5 trading days before the contract expires. The result of this procedure is a panel consisting of 36 rollover series according to time to maturity. The first series contains rates for the most immediate contract, the second one contains rates for the contract that will be delivered in one month, the third one rates for the contract with delivery date in two months, and so on. In other words, for every trading day between January 2nd 2003 and June 30th 2006 there are settlement yields for 36 futures contracts expiring from 3 weeks to the next 35 consecutive months. For each of these series, plus the series of TIIE spot rates, the analysis considers the series of logarithmic rate changes rt = ln(St /St−1 ), where St is the settlement rate on day t. We will sometimes refer to these rt simply as rate changes. There is evidence that the choice of rollover date and linking method can potentially generate biases on the statistical properties of the series (Geiss, 1995; Ma et al., 1992; Rougier, 1996). In order to minimize the impact that the splicing procedure may have on the statistical tests, increments across the splicing points are not included in the statistical calculations, resulting in a data set of 37 series of daily yield changes (including the one corresponding to the spot rate) with 845 observations each one.


20

Table 3.1 provides summary statistics of each of the series of rate changes. Almost no mean is statistically different from zero and the standard deviation tends to increase when contracts approach expiration. Most of the contracts show positive skewness and all series, including the spot rate, are leptokurtic. For all series the Bera-Jarque statistic rejects the hypotheses of normality. With the exception of only one series (No. 18) , the Engle (1982) LM-test for an autoregressive conditional heteroscedasticity (ARCH) effect clearly rejects the null of no ARCH effect in both the futures and TIIE rate changes. Further evidence that rate changes are not independently drawn from a normal distribution is provided by the autocorrelation of the series. The Ljung-Box test for autocorrelation of rate changes and squared rate changes (not reported in the Table) indicates that there is evidence of dependence. With respect to trading volume, another panel is constructed that contains volume data aligned by days to maturity instead of calendar day. Taking all the contracts that mature from January 2003 until June 2006, daily volume is tracked since the day the contract first appeared. Then the average traded volume across the contracts and relative to the days to expiration is obtained. Since 2005, contracts with maturity up to 10 years are available; however trading volume is almost negligible for contracts with expiration longer than 3 years. Figure 3.1 presents the number of contracts traded according to months before expiration. The results show that the traded volume increases monotonically as the contract approaches expiration. As in other futures market, contracts with the shortest maturity are far more liquid than contracts with maturities longer than three months. A weekly analysis over the last 6 months, as shown in Figure 3.2, indicates that the peak in trading volume is reached around four to ten weeks before expiration while in the last four weeks volume declines.


21

Table 3.1: Summary Statistics of 28-day TIIE Futures Daily Rate Changes. Series

Mean

Std. Dev. ∗

Skewness

Excess Kurtosis

Bera-Jarque

ARCH-LM

1

−0.00112

0.0141

−0.323

9.509

3198.23

79.44∗

2

−0.00085

0.0138

0.360

6.349

1437.60∗

49.87∗

∗

∗

∗

3

−0.00084

0.0122

0.161

3.591

457.68

92.08∗

4

−0.00068

0.0122

−0.117

5.411

1032.59∗

73.81∗

5

−0.00071

0.0118

0.094

4.453

699.24∗

94.19∗

6

−0.00070

0.0114

0.006

3.908

537.84∗

81.31∗

∗

7

−0.00060

0.0110

−0.044

3.633

464.90

38.54∗

8

−0.00054

0.0110

−0.022

2.594

237.03∗

40.61∗

9

−0.00049

0.0110

0.196

2.080

157.71∗

61.88∗

10

−0.00044

0.0105

0.227

2.184

175.18∗

43.37∗

11

−0.00046

0.0100

0.211

2.291

191.01∗

31.59∗

∗

12

−0.00041

0.0105

0.226

3.494

437.12

37.56∗

13

−0.00041

0.0110

0.042

2.904

297.10∗

43.05∗

14

−0.00043

0.0105

0.151

2.615

244.03∗

33.34∗

15

−0.00042

0.0105

0.197

2.506

226.51∗

34.50∗

∗

16

−0.00040

0.0105

0.335

2.741

280.41

43.92∗

17

−0.00036

0.0100

0.185

1.713

108.10∗

44.25∗

18

−0.00037

0.0105

0.321

4.337

676.65∗

9.87

2.529

∗

19.83∗

∗

19

−0.00038

0.0100

0.306

238.41

20

−0.00035

0.0100

0.275

2.050

158.63

24.49∗

21

−0.00042

0.0100

0.126

2.171

168.15∗

17.03∗

22

−0.00042

0.0100

0.005

2.125

158.94∗

14.77∗

23

−0.00037

0.0100

0.042

1.996

140.56∗

19.88∗

∗

24

−0.00042

0.0100

0.015

2.591

236.41

32.76∗

25

−0.00037

0.0100

−0.100

2.944

306.47∗

28.88∗

26

−0.00035

0.0100

−0.063

3.055

329.14∗

21.11∗

27

−0.00035

0.0095

0.124

2.574

235.39∗

25.19∗

28

−0.00036

0.0095

0.123

2.746

267.71∗

17.52∗

∗

29

−0.00030

0.0095

0.302

3.205

374.50

25.57∗

30

−0.00030

0.0089

0.444

3.235

396.14∗

29.83∗

31

−0.00031

0.0089

0.484

3.548

476.16∗

22.55∗

32

−0.00036

0.0089

0.415

3.587

477.28∗

29.13∗

∗

33

−0.00022

0.0100

0.773

6.117

1401.43

55.15∗

34

−0.00036

0.0126

0.171

10.176

3649.93∗

55.15∗

35

−0.00018

0.0158

−0.076

9.456

3148.70∗

208.33∗

36

−0.00041

0.0184

−0.523

15.837

8868.85∗

98.62∗

TIIE

−0.00032

0.0151

0.928

7.054

1873.26∗

140.12∗

Note. Each series consists of 845 observations. Series number corresponds to the months to expiration. The 1% critical value of the Bera-Jarque statistic is 9.21. The ARCH-LM is the LM -statistic of autoregressive conditional heteroscedasticity effect with 5 lags. * indicates significance at 5% level.


22

Figure 3.1: Number of 28-day TIIE futures contracts traded per month relative to contract expiration 60.0

Volume (millions of contracts)

50.0 40.0 30.0 20.0 10.0 -

0

6

12

18

Months to expiration

24

30

36

Numbers are millions of contracts traded during each month before the expiration date.

3.3.2

Methodology

The statistical significance of expiration and day-of-the-week effects is examined using the following regressions for each of the series. To address the autocorrelation the equation of the conditional mean is set as an AR(1) process with exogenous variables X rt = µ + φrt−1 + δk Dkt + ut , ut ∼ N (0, ht )

(1)

k

where, for each of the series considered, µ is a constant for the mean equation, rt is the logarithmic change of settlement rates on day t, and the residuals, ut , are assumed to be normally distributed with mean zero and variance ht . The variables Dkt , with k ∈ {M, T, H, F, Z}, are binary dummies representing the day of the week or the maturity: M stands for Monday, T for Tuesday, H for Thursday, F for Friday and Z for the last three days of the contract, that is, Monday, Tuesday and Wednesday of the expiration week (approximately every four


23

Figure 3.2: Number of 28-day TIIE futures contracts traded per week relative to contract expiration. 16.0

Volume (millions of contracts)

14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0

0

2

4

6

8

10

12

14

Weeks to expiration

16

18

20

22

24

Numbers are millions of contracts traded during the last 24 weeks before the expiration date.

weeks). Given that a constant term is allowed in the regression equation, Wednesdays dummy is omitted since this is the usual expiration day for all contracts. Additionally, the variance of TIIE futures contracts is examined using a GARCH(1,1) model with day of the week and maturity days as exogenous variables: X γk Dkt ht = α0 + α1 u2t−1 + β1 ht−1 +

(2)

k

where ht is the conditional variance for the series on day t, and Dkt represent the exogenous variables mentioned before. The maximum likelihood estimates were obtained with RATS (v.5) software package using the Berndt-Hall-Hall-Hausman algorithm. Since the accuracy of GARCH model estimation and of the associated t-statistics may depend on the software employed, the maximum likelihood estimation was also performed under EViews package using the Marquardt optimization algorithm. Although the coefficient estimates and their standard errors differ slightly, the reported results are qualitatively the same.


3.4 3.4.1

24

Results Day-of-the-week effects

In testing for seasonality, a preliminary statistical analysis is performed using the standard methodology. Considering the 36 series, rate changes are classified by day of the week, year by year and for the entire period. Mean changes and other statistics are computed for each day of the week, and t-tests are performed for comparing two means. Since this procedure implies dividing the sample in multiple subsamples, a standard F -test is performed to test the null hypothesis that means across all days of the week are jointly equal. Failure to reject the null would suggest that any apparent patterns observed when performing significant tests in isolation are not robust and are probably due to the effect of multiple subsamples. The results of this analysis are presented in Table 3.2. It can be seen that, for the entire period and all the subperiods, Monday means are always positive while Friday means are always negative. Moreover, the highest mean rate change for the entire sample occurs on Mondays (0.00144) and the lowest occurs on Fridays (-0.00180). This pattern is repeated when the sample is divided by calendar year, except in 2003 when the lowest mean change is on Thursdays (-0.00331). To test if the observed difference between Mondays and Fridays mean changes is significant, a t-test is performed. For the entire period and all the subperiods, the t-test rejects the null that Monday and Friday means are equal while the F -test confirms in all cases that the means across days of the week are significantly different. Concerning volatility there is not any noticeable pattern across the days of the week, although Table 3.2 shows that on annual basis the standard deviation has been gradually decreasing.


25

Table 3.2: Statistics of Daily Rate Changes According to the Day of the Week.

All

2003

2004

2005

2006

Mon

Tues

Wed

Thurs

Fri

All days

t-stat

F5

Mean

0.00144

-0.00055

0.00004

-0.00165

-0.00180

-0.00045

16.19*

88.19*

Std. Error

0.00015

0.00014

0.00014

0.00015

0.00013

0.00006

Std. Dev.

0.01178

0.01139

0.01073

0.01034

0.01073

0.01110

Max

0.10862

0.07032

0.08281

0.12758

0.08837

0.12758

Min

-0.15101

-0.10862

-0.06754

-0.09970

-0.12009

-0.15101

Sample

6372

6444

6300

5004

6336

30456

Mean

0.00139

-0.00266

-0.00047

-0.00331

-0.00184

-0.00131

6.48*

28.00*

Std. Error

0.00038

0.00034

0.00034

0.00038

0.00032

0.00016

Std. Dev.

0.01649

0.01477

0.01401

0.01479

0.01330

0.01482

Max

0.10862

0.06287

0.08281

0.12758

0.08837

0.12758

Min

-0.15101

-0.10862

-0.06754

-0.09970

-0.12009

-0.15101

Sample

1836

1836

1692

1476

1764

8604

Mean

0.00166

0.00126

0.00075

-0.00078

-0.00214

0.00020

9.52*

36.34*

Std. Error

0.00029

0.00026

0.00025

0.00022

0.00028

0.00012

Std. Dev.

0.01240

0.01132

0.01080

0.00830

0.01205

0.01130

Max

0.05977

0.07032

0.04625

0.05946

0.06812

0.07032

Min

-0.09407

-0.04699

-0.05560

-0.03692

-0.05946

-0.09407

Sample

1872

1872

1836

1368

1872

8820

Mean

0.00044

-0.00058

-0.00073

-0.00058

-0.00127

-0.00054

8.22*

16.97*

Std. Error

0.00013

0.00016

0.00014

0.00017

0.00016

0.00007

Std. Dev.

0.00554

0.00692

0.00603

0.00628

0.00683

0.00637

Max

0.02204

0.02367

0.02272

0.03414

0.01912

0.03414

Min

-0.01709

-0.02222

-0.02350

-0.02608

-0.02757

-0.02757

Sample

1800

1872

1836

1440

1800

8748

Mean

0.00312

0.00007

0.00108

-0.00203

-0.00207

0.00010

14.64*

49.33*

Std. Error

0.00022

0.00035

0.00035

0.00032

0.00027

0.00014

Std. Dev.

0.00658

0.01028

0.01080

0.00870

0.00825

0.00929

Max

0.02538

0.04039

0.03335

0.02382

0.02368

0.04039

Min

-0.01326

-0.03023

-0.04472

-0.03568

-0.03727

-0.04472

Sample

864

864

936

720

900

4284

Note. Summary statistics of 28-day TIIE futures contracts, considered all together, and classified by day of the week, year by year and for the whole period (January 2nd. 2003 to June 30th., 2006). t-stat tests the null hypothesis that Monday mean is different from Friday’s using a two tailed t-test. F5 is the F -statistic testing the null hypothesis that mean changes are equal across all five days of the week. The critical 0.05 value for the F5 -test is 2.76 (aprox.). * indicates significance at 5% level.


26

To reinforce the above analysis, Table 3.3 presents summary statistics for trading volume by day of the week, year by year and for the entire period. Consistently, either Tuesdays or Thursdays are the days with higher trading activity, suggesting there is no relation between rate changes on Mondays and Fridays and higher trading volume. Tuesdays and Thursdays volume coincides with trading activities in the Treasury Certificates market as will be explained later. It is worth mentioning that, according to MexDer, the lower trading volume in 2005 is explained by tax issues that increased the OTC trading on TIIE Swaps, provoking local banks to move their books offshore (Alegr´ıa, 2006). The maximum-likelihood parameter estimates for the GARCH model with all the dummies are reported in Panels A and B of Table 3.4. Table 3.5 reports the analysis of residuals, confirming the adequacy of the model for all the series considered, with the exception of series 33 and 35, which appear to still have significant serial correlation, according to the Ljung-Box statistics. In line with the trading pattern shown in Figure 3.1 these exceptions could be attributed to low trading volume.


27

Table 3.3: Trading Volume Statistics According to the Day of the Week. Mon

Tues

Wed

Thurs

Fri

All Days

Whole period

Mean

579,153

769,805

711,997

752,576

594,477

681,798

(2003-2006)

Std. Error

56,043

65,737

62,593

98,740

51,536

30,737

Max

5,087,510

6,594,200

7,856,000

14,360,000

6,945,000

14,360,000

Min

13,000

90,400

60,500

47,900

61,000

13,000

Std. Deviation

747,711

881,949

842,105

1,298,721

681,755

915,413

Sample

178

180

181

173

175

887

Mean

547,320

680,054

766,559

667,132

567,695

645,726

Std. Error

60,716

51,327

74,432

64,642

59,238

28,208

Max

1,962,353

1,935,860

2,180,300

1,950,000

1,932,000

2,180,300

Min

41,000

90,400

108,500

105,000

62,000

41,000

Std. Deviation

437,830

366,550

531,551

447,854

414,663

446,903

Sample

52

51

51

48

49

251

Mean

680,133

898,899

836,717

756,589

708,818

776,460

Std. Error

120,286

155,702

150,015

105,051

136,770

60,350

Max

5,087,510

6,594,200

7,856,000

4,005,500

6,945,000

7,856,000

Min

132,000

192,000

182,000

142,300

61,000

61,000

Std. Deviation

867,398

1,122,786

1,081,777

735,360

986,263

967,478

Sample

52

52

52

49

52

257

Mean

309,243

502,492

357,481

451,517

339,242

393,035

Std. Error

73,840

67,720

47,174

131,942

43,297

35,630

Max

3,780,000

2,544,652

1,923,500

6,755,200

1,675,000

6,755,200

Min

13,000

125,010

60,500

47,900

65,050

13,000

Std. Deviation

522,127

488,335

340,177

942,257

303,080

567,846

Sample

50

52

52

51

49

254

Mean

991,646

1,240,394

1,064,560

1,522,926

909,403

1,146,369

Std. Error

229,763

275,468

239,249

563,506

142,687

143,804

Max

4,636,244

6,194,500

5,286,244

14,360,000

2,660,000

14,360,000

Min

59,000

106,070

166,504

174,010

129,000

59,000

Std. Deviation

1,125,605

1,377,342

1,219,937

2,817,528

713,437

1,607,778

Sample

24

25

26

25

25

125

2003

2004

2005

2006

Note. 28-day TIIE futures trading volume statistics grouped by day of the week, for each year and for the whole analyzed period (January 2nd. 2003 to June 30th., 2006).


28

Table 3.4: Panel A. Conditional Mean Equation Estimates Series

µ × 103

δM × 103

φ

δT × 103

δH × 103

δF × 103

δZ × 103

0.4282

−0.8227∗

−0.2157

−0.6915

−0.3566

0.2012

−1.1532

−1.833∗

−1.6175∗

−1.1068

−2.0438∗

−2.2670∗

−1.2414

1

−0.0289

0.1479∗

2

0.6116

0.1946∗

3

0.7353

0.2167∗

0.1971

−1.8535∗

4

0.1134

0.1760∗

1.3282∗

−1.2615

−1.2838

−1.9581∗

−1.8690∗

5

−0.2039

0.1643∗

1.5081∗

−1.1512

−0.9495

−1.4258∗

−0.9732

0.2849

0.1882∗

0.8163

−2.0550∗

−1.1520

−2.5535∗

−1.7116∗

7

−0.0799

0.1544∗

1.3230

−1.1070

−0.9264

−1.8045∗

−1.2403

8

−0.5383

0.1652∗

2.0695∗

−0.5468

−0.2762

−1.9414∗

−0.6787

0.1555

0.1580∗

−0.7098

−2.3557∗

−1.8814∗

10

0.0003

0.1863∗

1.4309

−0.8182

−0.5963

−2.0995∗

−1.3279

11

0.2053

0.1641∗

1.4385

−1.1199

−0.8981

−2.1125∗

−1.5371

12

0.2684

0.1107∗

1.3113

−0.8637

−1.3282

−1.9237∗

−1.6937

13

−0.0623

0.1011∗

1.6022

−0.5077

−0.9202

−1.7263

−1.7647

14

−0.3985

0.0971∗

2.0151∗

−0.2318

−0.0910

−1.6033

−1.7096

15

0.2663

0.1355∗

1.5674

−0.9408

−1.0773

−2.0938∗

−1.5592

16

0.3902

0.1413∗

1.2057

−1.2357

−1.5006

−2.0721∗

−1.5846

17

0.2923

0.1455∗

1.8130

−1.2632

−1.1519

−1.8791∗

−1.6565

18

0.1119

0.1387∗

2.1922∗

−0.6668

−1.1851

−1.3971

−1.8677

19

0.0995

0.1493∗

2.1960∗

−0.7561

−1.2514

−1.1721

−1.9101

20

−0.3204

0.1453∗

2.4831∗

−0.3457

−0.9294

−0.7241

−1.7216

21

−0.1443

0.1484∗

2.2199∗

−0.5075

−1.2023

−1.1292

−1.8864∗

−0.2480

0.1332∗

2.0021∗

−0.0286

−1.1202

−1.1698

−1.9709∗

0.1368

0.1246∗

−1.4132

−1.8515∗

−1.9441∗

24

−0.0812

0.1171∗

1.6206

−0.1899

−0.9182

−1.7976∗

−1.7860∗

25

0.0641

0.1065∗

1.1966

−0.6524

−0.4936

−2.1515∗

−1.5686

26

−0.2630

0.1217∗

1.4047

−0.4738

−0.2791

−2.0491∗

−1.0801

27

−0.3946

0.1314∗

1.6077∗

−0.5924

−0.1711

−1.9076∗

−0.9144

28

−0.1540

0.1391∗

1.8456∗

−0.4548

−0.2637

−2.1599∗

−1.1987

29

−0.0601

0.1282∗

1.7089∗

−0.2279

−0.4647

−2.2596∗

−1.4409

30

0.1056

0.1418∗

1.7781∗

−0.1242

−0.8041

−2.2251∗

−1.6976

31

0.0844

0.1447∗

1.6850

0.1862

−1.0338

−2.2106∗

−1.7390

0.0938

0.1495∗

1.6696

−0.9764

−2.2938∗

−1.9103∗

33

−0.1814

0.1070∗

1.8918∗

0.8671

−0.8349

−2.0734∗

−1.9081∗

34

0.1476

0.0004

1.6233

0.2399

−1.4606

−2.6770∗

−2.0714

35

0.2967

−0.1009∗

2.5275∗

−0.2925

−0.8258

−2.3204∗

−2.8998∗

36

−0.2431

−0.0539

0.7328

−0.3188

−1.2519

−1.9510

−2.3775∗

TIIE

−0.0192

−0.6433

0.0420

−2.2720∗

0.2375

6

9

22 23

32

0.1351∗

1.0509

1.6228

−0.8342

−0.4532

0.3327

0.9731∗

Note. The table reports the conditional mean coefficients under the following GARCH specification: rt = µ + φrt−1 +

X

δk Dkt + ut ,

ht = αo + α1 u2t−1 + β1 ht−1 +

k

X

γk Dkt

k

where Dkt are day of the week and maturity dummy variables (k ∈ {M, T, H, F, Z}). M stands for Monday, T for Tuesday, H for Thursday, F for Friday and Z for the last three days of the contract, that is, Monday, Tuesday and Wednesday of the expiration week (approximately every four weeks). * indicates significance at the 5% level.


29

Table IV (continued). Panel B: Conditional Variance Equation Estimates Series

α0 × 103

α1

β1

γM × 103

1

0.0005

0.1157∗

0.8872∗

2

−0.0080∗

0.0398∗

0.9569∗

0.0026

3

−0.0075∗

0.0528∗

0.9453∗

4

−0.0162∗

0.0670∗

0.9332∗

5

−0.0204∗

0.0693∗

0.9316∗

6

−0.0195∗

0.1657∗

0.8353∗

7

−0.0187∗

0.0644∗

0.9370∗

8

−0.0069∗

0.0720∗

9

−0.0189∗

10

γH × 103

−0.0006

−0.0002

γF × 103

γZ × 103

0.0067∗

0.0040∗

0.0052

0.0147∗

0.0174∗

0.0031∗

0.0012

0.0112∗

0.0139∗

0.0103∗

0.0039∗

0.0037

0.0297∗

0.0256∗

0.0225∗

0.0036∗

0.0035

0.0341∗

0.0296∗

0.0334∗

0.0059∗

0.0006

0.0389∗

0.0299∗

0.0366∗

0.0002

−0.0012

0.0355∗

0.0326∗

0.0238∗

0.0088∗

0.9280∗

−0.0266∗

0.0239∗

0.0039

0.0307∗

0.0054

0.0620∗

0.9365∗

0.0013

0.0394∗

0.0206∗

0.0319∗

0.0042

−0.0129

0.0645∗

0.9322∗

−0.0043

0.0323∗

0.0120

0.0242∗

0.0039

11

−0.0064

0.0464∗

0.9484∗

0.0037

0.0115

0.0044

0.0105

0.0055

12

−0.0029

0.0501∗

0.9445∗

0.0036

0.0061

−0.0010

0.0049

0.0052

13

−0.0013

0.0880∗

0.9011∗

−0.0005

0.0107

−0.0065

0.0072

0.0022

14

0.0111

0.1209∗

0.8629∗

−0.0221

0.0022

−0.0316∗

0.0076

−0.0048

15

0.0001

0.0472∗

0.9488∗

−0.0046

0.0111

−0.0037

16

0.0075

0.0866∗

0.8868∗

−0.0113

0.0035

−0.0317∗

0.0145∗

17

−0.0036

0.0527∗

0.9348∗

0.0005

0.0189

−0.0057

0.0065

0.0017

18

0.0065

0.0376∗

0.9571∗

−0.0077

0.0007

−0.0161

−0.0124

0.0050

19

−0.0017

0.0468∗

0.9442∗

0.0039

0.0106

−0.0064

0

0.0034

20

−0.0025

0.0763∗

0.9064∗

0.0076

0.0124

−0.0085

0.0049

0.0028

21

−0.0061

0.0488∗

0.9440∗

0.0017

0.0196

0.0007

0.0074

0.0046

22

−0.0074

0.0499∗

0.9474∗

−0.0033

0.0236

0.0012

0.0124

0.0047

23

−0.0079

0.0504∗

0.9463∗

−0.0013

0.0260∗

0.0014

0.0117

0.0034

24

−0.0071

0.0548∗

0.9387∗

0.0075

0.0195

0.0010

0.0086

0.0020

25

−0.0069

0.0543∗

0.9465∗

−0.0094

0.0299∗

0.0066

0.0049

0.0049

26

−0.0007

0.0563∗

0.9490∗

−0.0243∗

0.0252∗

0.0058

−0.0117∗

0.0131∗

27

−0.0053

0.0525∗

0.9525∗

−0.0184∗

0.0304∗

0.0125∗

−0.0075

0.0153∗

−0.0016

0.0465∗

0.9529∗

−0.0121

0.0170

0.0025

−0.0045

0.0087∗

−0.0018

0.0486∗

0.9491∗

−0.0121

0.0167

0.0008

0.0018

0.0045

30

−0.0008

0.0422∗

0.9543∗

−0.0095

0.0112

0.0021

−0.0004

0.0040

31

−0.0031

0.0424∗

0.9526∗

−0.0018

0.0111

0.0038

0.0015

0.0048

32

−0.0021

0.0412∗

0.9516∗

0.0030

0.0027

0.0050

0.0013

0.0030

33

−0.0002

0.1196∗

0.8061∗

−0.0128

0.0240∗

0.0123

0.0186

−0.0018

34

−0.0247∗

0.0845∗

0.9043∗

0.0251∗

0.0322∗

0.0213

0.0565∗

−0.0010

35

−0.0172∗

0.1969∗

0.7785∗

0.0975∗

−0.0237

0.0304∗

0.0289∗

36

0.0013

0.3408∗

0.7040∗

0.0339∗

−0.0199

0.0187

0

−0.0020

0.4543∗

0.6598∗

0.0123∗

−0.0060∗

0.0056∗

0.0079∗

28 29

TIIE

−0.0093∗

γT × 103

−0.0063

0.0065 −0.0060

0.0004 −0.0008 0.0010

Note. The table reports the conditional variance coefficients under the following GARCH specification: rt = µ + φrt−1 +

X

δk Dkt + ut ,

k

ht = αo + α1 u2t−1 + β1 ht−1 +

X

γk Dkt

k

where Dkt are day of the week and maturity dummy variables (k ∈ {M, T, H, F, Z}). M stands for Monday, T for Tuesday, H for Thursday, F for Friday and Z for the last three days of the contract, that is, Monday, Tuesday and Wednesday of the expiration week (approximately every four weeks). * indicates significance at the 5% level


30

The results in Panel A of Table 3.4 show that, in accord to the statistics obtained previously (Table 3.2), in the conditional mean equation, Monday’s coefficients (δM ) are always positive and frequently significant while Friday’s (δF ) are always negative and almost always significant. This indicates that changes on the TIIE futures rates tend to be positive on Mondays (from Friday close to Monday close) and negative on Fridays. Since futures yields and futures prices have an inverse relation, this Monday pattern is consistent with the Monday effect reported in other interest rate futures markets, like in Buckle et al. (1998) for the Short Sterling futures, in Johnston et al. (1991) for T-bond future contracts, or in Lee and Mathur (1999) for the Spanish MIBOR-futures market. However, the significant low rates on Fridays seem to be idiosyncratic. Since there is no scheduled macroeconomic announcement or other public information release occurring on those days of the week, this anomaly seems to be produced by the particular characteristics of the trading activity in the Mexican futures market. The last line of Table 3.4 reports the coefficients for the spot rate, showing that TIIE rate changes on Friday are also significant and negative. The fact that on Fridays the spot rate also tends to decrease leads to suspect that the weekend abnormal behavior on future contracts could be a consequence of the positions on the TIIE spot rate presented by market participants on Fridays. On Mondays participants may then bring back rates to match market conditions inducing, on average, positive changes. The rest of the days of the week do not appear to have any significant effect on the conditional mean. Related with day-of-the-week effect and volatility, several observations are worth mentioning. On Table 3.4 Panel B it can be seen that coefficients for Tuesdays, Thursdays and Fridays dummies in the conditional variance equation are significant for short run contracts but not for longer terms. There are also some significant coefficients in estimations for contracts expiring around two or three years, but not for contracts in between. For example, contracts expiring in two years present significant coefficients for Tuesdays’ dummies. Higher volatility on Tuesdays should exist for any term contract as this is the day when the Central Bank carries out the auction of Treasury Certificates (CETES) in the primary market. This is the leading interest rate in money market. Even though there are important announcements on Tuesdays, and on Thursdays the market is more liquid because Treasury Certificates are settled, the presence of significant coefficients on Fridays does not help to discriminate between the reaction to public an-


31

nouncements and trading activities. Given that on Tuesdays new information concerning interest rates arrives, higher volatility should be related with these events, supporting Harvey & Huang (1991). Alternatively, if the market is more liquid on Thursdays and market participants may manipulate rates on Fridays, then volatility should be explained by trading activities and market microstructure consistently with the results of Andersen and Bollerslev (1998) for spot rates. Friday effect may be attributable to some collusion among participants to lower their margin requirements. In general, even if the day-of-the week effect on volatility is not as unambiguous as it is for mean rate changes, the results provide some indication that on Mondays the TIIE futures market shows no structural change in volatility. Also there is evidence that, as a whole, short term contracts are more volatile than longer term contracts. This is further demonstrated by the magnitude of the dummies coefficients, that progressively decrease as the term of the contract increases, and by the results on volume presented in Figures 3.1 and 3.2.

3.4.2

Expiration day effects

In this section the expiration day effects on rates changes and volatility are investigated. This analysis is performed considering a dummy variable that takes the value one on Mondays, Tuesdays and Wednesdays of the expiration week and zero otherwise. The estimated coefficients are reported in the last column of Table 3.4, Panels A and B. Results for the conditional mean indicate that the coefficients for this dummy are always negative, although only in eleven cases they appear to be significant. With respect to the estimates for expiration day effect dummy in the GARCH process, the null hypothesis of no structural change cannot be rejected for contracts maturing in seven months or less. In these cases coefficients are positive and different from zero at the 5% significance level, meaning that the conditional volatility of those contracts increases when the next-to-expiration contract matures. On the other hand, there are no significant alterations in the spot rate near expiration days. Apparently, on the days prior to expiration, market participants change their hedging positions to contracts expiring one to six months ahead, while longer term contracts are not considered by investors for their rollover strategies. Since short term contracts involve lower


32

√ Table 3.5: Descriptive statistics for the estimated standardized residuals ut / ht

Series

Skewness

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

0.047 0.240 0.195 0.044 0.407 0.258 0.016 -0.036 0.119 0.143 0.194 0.249 0.148 0.177 0.178 0.393 0.154 0.153 0.193 0.104 -0.021 -0.061 -0.010 -0.083 -0.086 -0.018 0.086 0.097 0.131 0.266 0.295 0.327 0.839 0.131 0.156 0.077

Standarized residuals Kurtosis BJ LB(8) p-value 2.90 2.71 1.33 2.20 2.44 1.29 0.96 1.12 0.57 0.79 1.33 1.81 1.30 1.08 1.06 1.78 0.86 1.88 1.07 1.05 1.16 1.11 1.00 1.27 1.33 1.46 1.21 1.32 1.37 1.61 1.73 1.99 4.58 4.20 4.25 3.27

296.4 266.7 67.7 170.7 232.9 68.0 32.5 44.4 13.4 24.8 67.6 124.1 62.6 45.5 44.0 133.2 29.4 127.8 45.6 40.4 47.4 43.9 35.2 57.8 63.3 75.1 52.6 62.7 68.5 101.2 117.7 154.4 837.6 623.5 639.4 377.3

8.62 3.90 4.30 4.05 3.12 3.98 2.09 4.59 5.29 3.11 1.01 0.66 1.35 3.17 2.62 3.32 9.87 8.22 6.73 5.48 5.82 4.31 5.45 8.63 5.24 4.01 3.58 2.43 2.15 2.86 5.38 5.16 9.92 7.42 6.51 5.94

0.281 0.792 0.744 0.774 0.874 0.783 0.955 0.710 0.624 0.874 0.995 0.999 0.987 0.869 0.918 0.854 0.196 0.313 0.458 0.602 0.561 0.743 0.605 0.280 0.631 0.779 0.827 0.933 0.951 0.897 0.614 0.641 0.193 0.386 0.482 0.546

LB(16) 14.73 9.25 10.38 7.64 9.86 12.44 8.60 11.00 10.04 6.41 6.65 4.96 8.09 11.90 10.11 8.91 16.71 17.06 12.91 12.08 10.46 8.21 9.71 14.07 13.01 11.98 9.24 5.62 6.22 5.33 7.58 9.18 15.93 21.71 22.59 9.85

p-value 0.471 0.864 0.795 0.937 0.828 0.646 0.897 0.752 0.817 0.972 0.967 0.992 0.920 0.686 0.813 0.882 0.336 0.316 0.609 0.673 0.790 0.915 0.837 0.520 0.601 0.680 0.865 0.985 0.976 0.989 0.940 0.868 0.386 0.116 0.093 0.829

Squared standardized residuals LB(8) p-value LB(16) p-value 4.35 4.35 5.93 5.05 4.38 4.38 5.27 3.51 7.75 3.41 6.76 3.52 3.57 2.42 3.94 4.30 4.85 1.99 2.03 1.81 2.69 2.47 4.44 7.29 3.29 3.68 6.26 4.01 6.62 2.59 1.94 7.27 28.88 20.31 14.05 4.12

0.738 0.739 0.548 0.654 0.736 0.735 0.627 0.834 0.355 0.845 0.454 0.833 0.828 0.933 0.787 0.744 0.678 0.960 0.958 0.969 0.912 0.929 0.728 0.399 0.857 0.816 0.510 0.779 0.470 0.920 0.963 0.401 0.000 0.005 0.050 0.766

10.03 6.56 10.43 12.25 10.90 11.82 8.69 5.46 12.55 8.09 12.86 11.73 13.82 10.67 7.61 6.25 7.23 5.29 4.41 5.25 6.71 8.57 13.07 14.17 17.42 14.57 12.88 11.37 12.96 8.23 7.91 12.79 32.63 22.35 33.56 14.69

0.818 0.969 0.792 0.660 0.760 0.693 0.893 0.988 0.637 0.920 0.613 0.699 0.539 0.776 0.939 0.975 0.951 0.989 0.996 0.990 0.965 0.899 0.597 0.513 0.294 0.483 0.611 0.726 0.606 0.914 0.927 0.619 0.005 0.099 0.004 0.474

Note. This table presents normality and correlation tests for standardized residuals and squared standardized residuals under the GARCH(1,1) model and for the estimated coefficients. LB(k) denotes the Ljung-Box statistic with k lags.


33

basis risk, this preference for short term contracts can be due to hedgers preferring to assume frequent rollover transaction costs than the risk of future mispricing.

3.5

Conclusions

This chapter investigates sources of nonstationarity in the 28-day futures contracts, searching for day-of-the-week and expiration day effects. The presence of these effects, both in the rate changes and in their volatility, is tested in the context of GARCH models. The results show that there is a Monday effect similar to the one observed in other interest rate futures markets: rates (prices) tend to increase (decrease) on Mondays. In addition to this, rates tend to decrease on Fridays. Since there is no scheduled macroeconomic announcement or other public information release occurring on those days of the week, this anomaly seems to be produced by the particular characteristics of the trading activity in the market. The fact that on Fridays the spot rate also tends to decrease leads to suspect that the anomaly could be attributable to the need of market participants to lower their margin requirements during the weekend and to other reporting necessities. That is, given that TIIE spot rate is determined by the bid-ask positions set by a few participants (usually six or seven major banks), it may happen that on Fridays those participants set positions with lower values than the rest of the week to diminish the cost of money during the weekend. If this is the case, it indicates that the fact that only few participants trade these contracts makes it easy to induce nonstationarity patterns and, in consequence, market inefficiencies. A priori, ignoring the impact of market frictions, the existence of such patterns opens the possibility of abnormal profits by taking short positions on Fridays and closing them on Mondays. Concerning volatility, event though it is not possible to accurately assess the cause of a day-of-the-week effect, it has been shown on Mondays there is no structural change in volatility. On the other hand, the difference in volatility between short and long term contracts has also implications in the adequate specification of margin requirements. Since low margins promote investment and high margins tend to diminish it, it may be important for the clearinghouse to establish a margin policy that distinguishes between contracts with high or low volatility in order to optimize the relation between investment and risk control. With respect to a possible abnormal behavior during the expiration days, there is evidence


34

of significant changes in conditional volatility around days previous to expiration in contracts with seven months or less to maturity. Apparently, on the days prior to expiration market participants roll their hedging positions to contracts expiring one to six months ahead, while longer term contracts are not considered by investors for their rollover strategies. Since short term contracts involve lower basis risk, this preference for short term contracts can be due to hedgers preferring to assume frequent rollover transaction costs instead of the risk of future mispricing. This indicates fees and margin requirement policies that make no distinction between contracts with different maturities are not an adequate incentive for using long-term contracts in roll-over strategies.

Chapter 4 Maturity effects 4.1

Introduction.

Understanding the dynamics of futures price volatility is important for all market participants. In this chapter the study focuses on a specific aspect of futures price volatility: the relation between volatility and time to expiration. Samuelson (1965) was the first to investigate theoretically this relation, providing a model that postulates the volatility of futures prices should increase as the contract approaches expiration. This effect, more commonly known as Samuelson hypothesis or maturity effect, occurs because price changes are larger when more information is being revealed. The study of the behavior of volatility of futures prices near the maturity date has important implications for risk management, for hedging strategies, and for derivatives pricing, among others. Clearinghouses set margin requirements on the basis of futures price volatility. Therefore, if there is any relation between volatility and time to maturity, the margin should be adjusted accordingly as the futures approaches its expiration date. The relation between volatility and maturity also has implication for hedging strategies. Depending on the positive or negative relation between volatility and maturity, hedgers should choose between futures contracts with different time to maturity to minimize the price volatility. For example, Low et al. (2001) propose a multiperiod hedging model that incorporates the maturity effect. Their empirical results show that the model outperforms other hedging strategies that do not account for maturity. Thirdly, volatility and time to maturity relation is also essential for speculators in the futures markets. Speculators bet on the futures price movements of

35

CHAPTER 4. MATURITY EFFECTS

36

the assets. If maturity effect holds then speculator may find beneficial to trade in futures contracts close to expiry as greater volatility implies greater short time profit opportunities. Finally, since volatility is central to derivatives pricing, the relation between maturity and volatility should also be taken into consideration when pricing derivatives on futures. Numerous studies have investigated the Samuelson hypothesis empirically, yielding mixed results. In general, the maturity effect has been supported for commodities, while it appears to be insignificant for financial assets. The aim of this chapter is to study the presence of maturity effects in the Mexican interest rate futures market. For this purpose 48 time series are used, consisting of the settlement prices of the contract with maturities from January 2003 to December 2006. With these series a panel is constructed arranging observations not according to calendar day, but according to days to maturity. This permits to apply panel data estimation techniques in addition to the usual time series methods, and thus to assess the existence of cross-sectional individual effects. Our findings show that maturity effects are present in 2003 and 2004, inverse maturity effect appears in 2005 and 2006, and it indicates that there is not evidence of maturity effect once all contracts are considered (2003-2006). When spot volatility is included as a proxy for information flow, results are qualitatively the same on each separate period, indicating information flow does not affect the presence of maturity effects. However, when considering the whole period the inclusion of spot volatility yields stronger significance to the observed maturity effect. With respect to the basis, results show the expected maturity effect in contracts between September 2004 and March 2006, while panel analysis indicates an inverted effect in 2003 and the expected maturity effect in every year from 2004 and in the whole sample. In the final section we discuss some possible explanations of this behavior. The rest of the chapter is organized as follows. The next section briefly reviews the existing literature. In section 4.3 we describe the data and the methodology employed. In section 4.4 we report the results. Concluding remarks are given in section 4.5.

4.2

Previous studies

Samuelson (1965) was the first to provide a theoretical model for the relation between


37

the futures price volatility and time to maturity. The theoretical hypotheses introduced by Samuelson, known as the Samuelson hypothesis or the maturity effect, predicts volatility of futures prices rises as maturity approaches. The intuition is that when there is a long time to the maturity date, little is known about the future spot price for the underlying. Therefore, futures prices react weakly to the arrival of new information since our view of the future will not change much with it. As time passes and we approach maturity, the futures price is forced to converge to the spot price and so it tends to respond more strongly to new information. The example used by Samuelson to present the hypothesis relies on the assumptions that 1) futures price equals the expectation of the delivery date spot price, and 2) spot prices follow a stationary, first-order autoregressive process. This specification implies that the spot price reverts in the long run to a mean of zero. However, Rutledge (1976) argued that alternative specifications of the generation of spot prices are equally plausible and may lead to predict futures price variation decreases as maturity approaches (an ”inverse” maturity effect). Later, Samuelson (1976) showed that a spot generating process that includes higher order autoregressive terms can result in temporary decreases in a generally increasing pattern of price variability. Hence a weaker result is obtained: if delivery is sufficiently distant then variance of futures prices will necessary be less than the variance very near to delivery. Numerous studies have investigated the Samuelson hypothesis empirically, with different sets of data and different methodologies, and have obtained mixed results. In general, the effect appears to be stronger for commodities futures, while for financial futures the effect is frequently statistically nonsignificant or non existent at all. For commodity markets, early empirical work by Rutledge (1976) finds support for the maturity effect in silver and cocoa but not for wheat or soybean oil. Milonas (1986) derives, in line with Samuelson’s arguments, a theoretical model for the maturity effect and provides empirical evidence. He calculates price variability as variances over daily price changes within a month and adjusts these variances for month, year and contract month effects. He tests for significant differences in variability among the different time to maturity groups of variances and finds general support for the maturity effect in ten out of the eleven future markets examined, which included agricultural, financial and metal commodities. Grammatikos & Saunders (1986), investigating five currency futures, find no relation between time to maturity and volatility for currency futures prices.


38

Galloway & Kolb (1996) examined a set of 45 commodities futures contracts, including twelve financial contracts. Using monthly variances, they investigated the maturity effect both in an univariate setting, searching for maturity effect patterns, and performing ordinary least squares (OLS) regressions. They found strong maturity effect in agricultural and energy commodities, concluding that time to maturity is an important source of volatility in contracts with seasonal demand or supply, but they did not found the effect in commodities for which the cost-of-carry model works well (precious metals and financial). In particular, T-Bill, T-bond and Eurodollar futures showed no evidence of any significant maturity effect. A similar result for currency futures was reported in Han, Kling & Sell (1999). Anderson & Danthine (1983) offer an alternative explanation of the time pattern of futures price volatility by incorporating time-varying rate of information flow. The hypothesis, named state variable hypothesis, states that variability of futures prices is systematically higher in those periods when relatively large amounts of supply and demand uncertainty are resolved, i.e. during periods in which the resolution of uncertainty is high. Within this context, Samuelson’s hypothesis is a special case in which the resolution of uncertainty is systematically greater as the contract nears maturity. Under this perspective, the maturity effect reflects a greater rate of information flow near maturity, as more traders spend time and resources to uncover new information. Some studies have applied the state variable hypothesis to test the existence of maturity effect. Anderson (1985) studies volatility in nine commodity futures for the period 1966 to 1980. Using both nonparametric and parametric tests he finds that on six of these markets (oats, soybean, soybean oil, live cattle and cocoa) there is strong evidence of maturity effects but no such effect for wheat, corn or silver. However, he also reports that seasonality is more important in explaining the patterns in the variance of futures price changes. Barnhill, Jordan & Seale (1987) apply the state variable hypothesis to the Treasury bond futures market during the period 1977-1984 and find evidence supportive of the maturity effect. The effects of time to maturity have also been studied on the futures basis (defined as the futures price less the spot price). Castelino & Francis (1982), based on Samuelson’s analysis of futures prices, study the effect of time to maturity on the basis over the life of commodity futures contracts. Assuming a first-order autoregressive price process, they show that the volatility of changes in the basis must decline as contract maturity approaches. The rationale behind this is that the arrival of new information is more likely to affect spot and


39

futures prices in the same manner if it arrives closer to maturity than further away. As a corollary, it follows that hedging in a nearer contract involves less basis risk than hedging in a more distant contract. Using daily data for futures on wheat, soybeans, soybeans meal and soybean oil they provide empirical evidence of this maturity effect on the basis. Beaulieu (1998) studies the basis in two stock market equity indices. The paper utilizes GARCH model to estimate the volatility of the basis since there is heteroscedasticity and leptokurtosis present. The results indicate that the size of the variance of the basis decreases as the futures contracts approach expiration, in line with the previous results of Castelino & Francis (1982). Chen, Duan & Hung (1999) focus on index futures and propose a bivariate GARCH model to describe the joint dynamics of the spot index and the futures basis. They use the Nikkei-225 index spot and futures prices to examine empirically the Samuelson effect and study the hedging implications under both stochastic volatility and time-varying futures maturities. Their finding of decreasing volatility as maturity approaches contradicts the Samuelson hypothesis. Bessembinder, Coughenour, Seguin & Monroe Smoller (1996) present a different analysis of the economic issues underlying the maturity effect. With respect to the state variable hypothesis, they note that there is an absence of satisfactory explanations of why information should cluster towards a contract expiration date. According to their model, neither the clustering of information flow near delivery dates nor the assumption of that each futures price is an unbiased forecast of the delivery date spot price is a necessary condition for the success of the hypothesis. Instead they focus on the stationarity of prices. They show that Samuelson hypothesis is generally supported in markets where spot price changes include a predictable temporary component, a condition which is more likely to be met in markets for real assets than for financial assets. Their analysis predicts that the Samuelson hypothesis will be empirically supported in those markets that exhibit negative covariation between spot price changes and the futures term slope. Since financial assets do not provide service flows, they predict that the Samuelson hypothesis will not hold for financial futures. To test their predictions they consider data from agricultural, crude oil, metals and financial futures. Performing regressions on days to expiration, spot volatility and monthly indicators they obtain supportive evidence for their model. Hennessy & Wahl (1996) propose an explanation of futures volatility based not on information flow or time to expiry, but on production and demand inflexibilities arising from


40

decision making. Their results on CME commodity futures support of the maturity effect. More recently, Aragó & Fernández (2002) study the expiration and maturity effects in the Spanish market index using a bivariate error correction GARCH model (ECM-GARCH). Their results show that during the week of expiration conditional variance increases for the spot and futures prices, according to Samuelson hypothesis.

4.3 4.3.1

Data and methodology Sample data

The study considers daily TIIE spot and futures rates between January 2003 and December 2006. Volatility patterns are assessed using logarithm changes. For the spot rate St those are defined as ∆St = ln(St+1 /St )

(1)

Futures data includes daily settlement yields and trading volume data for all 28-day TIIE futures contracts with maturities between months mentioned above. These data were obtained from the Mexican Derivatives Exchange (MexDer). Since, for the majority of contracts, open interest is low and trading volume is thin in periods long before maturity, the sample used for each futures contract includes only the thirteen months preceding its expiration. The result is a data set of 12,624 observations corresponding to 48 TIIE futures contracts with 263 daily settlement rates each. Logarithmic rate changes for futures rates are defined as ∆YT t = ln(YT, t+1 /YT, t )

(2)

where YT, t denotes the settlement yield on calendar day t for the contract with maturity T . We will refer to these logarithmic rate changes ∆YT t simply as rate changes. As for the expiration month itself, it will be excluded from the analysis, considering that trading volume decreases as the contract enters the expiration month inducing abnormal price variability. Hence, we have a set of 48 series of logarithmic rate changes, corresponding to contracts with expiration dates ranging from January 2003 to December 2006, and with 242 observations each. Table 4.1 presents summary statistics for the rate changes ∆YT t . Mean rate changes are predominantly negative with the exception of contracts that matured between September


41

Table 4.1: Descriptive statistics for TIIE futures contracts daily logarithmic changes Standard Deviaton

Skewness

Kurtosis

−0.86 −0.02 −0.16 −0.07 −0.34 −1.02 −1.63 −1.87 −2.04∗ −3.12∗ −2.62∗ −2.58∗

0.3109 0.3210 0.3070 0.3114 0.2929 0.2880 0.2839 0.2829 0.2687 0.2212 0.2227 0.2345

1.99 1.86 1.61 0.95 1.34 0.05 0.44 0.20 0.41 0.32 0.40 0.36

15.85 15.50 12.94 10.02 13.86 7.01 6.96 6.04 6.17 4.03 3.83 4.15

-0.3708 -0.6025 -0.5225 -0.3620 -0.2959 -0.0930 -0.0294 -0.0091 0.0337 0.0916 0.1948 0.2563

−1.52 −2.45∗ −2.08∗ −1.39 −1.16 −0.37 −0.13 −0.04 0.16 0.43 0.91 1.29

0.2399 0.2415 0.2472 0.2572 0.2514 0.2443 0.2304 0.2126 0.2106 0.2115 0.2118 0.1958

1.00 0.47 0.39 0.39 0.12 0.39 -0.02 0.18 0.11 0.33 0.35 0.32

Jan05 Feb05 Mar05 Apr05 May05 June05 July05 Aug05 Sept05 Oct05 Nov05 Dec05

0.2616 0.2416 0.3104 0.3393 0.1717 0.1145 0.1426 0.0732 0.0783 0.0376 -0.0634 -0.0734

1.45 1.38 1.83 2.04∗ 1.16 0.82 1.21 0.68 0.77 0.39 −0.66 −0.80

0.1772 0.1721 0.1673 0.1634 0.1451 0.1372 0.1164 0.1067 0.1003 0.0962 0.0940 0.0906


-0.0687 -0.1602 -0.2365 -0.3270 -0.3231 -0.2917 -0.2496 -0.2722 -0.2243 -0.2221 -0.2058 -0.1574

−0.73 −1.79 −2.62∗ −3.67∗ −3.65∗ −3.21∗ −2.77∗ −2.83∗ −2.37∗ −2.27∗ −1.91 −1.42

0.0924 0.0880 0.0887 0.0877 0.0871 0.0895 0.0887 0.0948 0.0931 0.0963 0.1060 0.1090

Contract

Mean


-0.2720 -0.0052 -0.0510 -0.0216 -0.1014 -0.2991 -0.4714 -0.5384 -0.5568 -0.7011 -0.5935 -0.6151


t-stat

p-value

ARCH (LM)

p-value

1824.63∗ 1714.39∗ 1100.25∗ 532.22∗ 1261.49∗ 161.84∗ 165.52∗ 95.04∗ 108.22∗ 14.91∗ 13.40∗ 18.65∗

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0006 0.0012 0.0001

2.59 0.80 0.74 2.89 3.40 14.21∗ 13.61∗ 18.63∗ 6.01 3.53 8.71 2.69

0.7621 0.9770 0.9810 0.7163 0.6379 0.0143 0.0183 0.0023 0.3054 0.6195 0.1212 0.7478

8.47 4.83 5.15 4.64 3.72 6.39 5.42 4.27 3.88 4.30 4.28 4.05

342.19∗ 42.66∗ 52.82∗ 33.02∗ 5.83 121.91∗ 59.08∗ 17.44∗ 8.34∗ 21.40∗ 21.56∗ 15.20∗

0.0000 0.0000 0.0000 0.0000 0.0541 0.0000 0.0000 0.0002 0.0154 0.0000 0.0000 0.0005

2.74 8.8 11.25∗ 16.04∗ 9.27 5.67 18.35∗ 9.29 6.01 6.67 8.71 14.81∗

0.7400 0.1173 0.0467 0.0067 0.0986 0.3400 0.0025 0.0981 0.3056 0.2462 0.1211 0.0112

0.25 0.28 0.29 0.34 -0.17 -0.08 -0.03 0.02 0.07 0.10 -0.11 -0.05

4.13 4.35 4.27 4.44 3.31 3.48 3.24 3.39 3.67 3.83 3.84 4.34

15.32∗ 21.56∗ 19.70∗ 25.42∗ 2.17 2.51 0.62 1.57 4.76 7.32∗ 7.48∗ 18.19∗

0.0005 0.0000 0.0001 0.0000 0.3386 0.2849 0.7344 0.4563 0.0928 0.0258 0.0237 0.0001

12.99∗ 14.71∗ 12.39∗ 16.58∗ 23.54∗ 27.35∗ 8.97 4.32 8.29 9.93 9.58 11.55∗

0.0235 0.0117 0.0298 0.0054 0.0003 0.0000 0.1103 0.5049 0.1410 0.0771 0.0882 0.0414

0.00 0.09 -0.11 -0.27 -0.38 -0.30 -0.15 0.01 0.24 0.57 0.36 0.79

4.03 3.93 4.04 4.15 4.57 4.25 4.36 3.90 4.42 6.28 6.40 10.63

10.70∗ 9.03∗ 11.32∗ 16.29∗ 30.55∗ 19.40∗ 19.64∗ 8.23∗ 22.56∗ 121.44∗ 121.75∗ 613.09∗

0.0047 0.0110 0.0035 0.0003 0.0000 0.0001 0.0001 0.0163 0.0000 0.0000 0.0000 0.0000

3.99 4.13 5.70 8.43 3.07 5.74 6.25 8.25 10.10 15.62∗ 21.59∗ 17.30∗

0.5505 0.5308 0.3368 0.1341 0.6898 0.3323 0.2828 0.1432 0.0724 0.0080 0.0006 0.0040

BJ

This table reports the statistics of the daily logarithmic changes of each of the futures contracts along 242 days before expiration month. BJ is the Bera-Jarque statistic for testing the null hypothesis of normal distribution. The ARCH-LM is the LM-statistic of autoregressive conditional heteroscedasticity effect with 5 lags.

∗

indicates 5% significance.


42

2004 and October 2005. However, few of these mean estimates are significantly different from zero. Most contracts are leptokurtic (kurtosis greater than 3) and positively skewed, although these departures from normality tend to diminish for more recent contracts. Standard deviation also diminishes over time, with contracts expiring in 2006 being the less volatile. With the exception of contracts maturing in 2005, in all cases Bera-Jarque statistic rejects the hypothesis of normality. The table includes the results for the Engle (1982) LM-test for an autoregressive conditional heteroscedasticity (ARCH) effect. In most of the series, ARCH effects are not significant. The basis at time t for a contract i with maturity in T will be measured by the log-basis, that is, by the difference between the futures log-rate and the spot log-rate, BT t = ln YT, t − ln St . Figure 4.1 shows the average log basis for each contract in the sample. From the highest point in the graph for the contract that matured in May 2004, average log basis declined progressively until it became negative in contracts with expiration between November 2005 and September 2006. Figure 4.1: Average log-basis $# # " !


43

Table 4.2: Average log-basis by month to expiration Months to Expiration

Semester defining contract expiration S1’2003

S2’2003

S1’2004

S2’2004

S1’2005

S2’2005

S1’2006

S2’2006

2

0.0258

0.1158

0.0228

0.0713

0.0063

-0.0235

-0.0132

0.0240

3

0.0305

0.1660

0.0430

0.0892

0.0146

-0.0278

-0.0223

0.0375

4

0.0608

0.1759

0.0904

0.0970

0.0227

-0.0256

-0.0303

0.0489

5

0.1005

0.1490

0.1511

0.0921

0.0403

-0.0172

-0.0410

0.0572

6

0.1442

0.0991

0.2064

0.0964

0.0694

-0.0116

-0.0478

0.0571

7

0.1691

0.0813

0.2497

0.1199

0.1048

-0.0031

-0.0471

0.0395

8

0.1770

0.0801

0.2831

0.1190

0.1427

0.0051

-0.0454

0.0162

9

0.1679

0.0635

0.3020

0.1409

0.1556

0.0234

-0.0387

-0.0031

10

0.1815

0.0953

0.2847

0.1720

0.1731

0.0384

-0.0267

-0.0171

11

0.1790

0.1378

0.2225

0.2352

0.1716

0.0567

-0.0091

-0.0329

12

0.1618

0.1761

0.1569

0.2770

0.1694

0.0924

0.0043

-0.0409

13

0.1835

0.2032

0.1118

0.3034

0.1916

0.1290

0.0152

-0.0379

This table shows the average log-basis (the difference between log-changes of futures and spot rates) according to the month of expiration of each contract.

Furthermore Table 4.2 presents the monthly average log basis across contracts grouped by semesters according to their expiration date. For contracts that expired in the first semester of 2005 negative basis appeared 7 months before expiration. Negative basis are also present from 11 months before expiration in first semester of 2006 contracts and from 13 to 9 months before maturity in contracts of the second semester of 2006. This effect seems to be related with the declining patterns of the TIIE after the second half of 2005. Considering that when spot rates are high it is very likely that the term structure of interest rates will show negative slope, the presence of negative log-basis in 2006 is not surprising since the spot rate reached its highest point in mid 2005. The log-basis change between t and t + 1 is ∆BT t = [ln YT, t+1 − ln St+1 ] − [ln YT,t − ln St ] = ∆YT t − ∆St .


44

Table 4.3 presents summary statistics for the basis changes ∆BT t of each contract. Most of the means are negative and tend to increase over time, although none of them is statistically different from zero. For the majority of contracts basis changes are leptokurtic. Standard deviation of these changes diminish over time, with contracts expiring in 2006 having the less volatile basis. With the exception of two, in all cases Bera-Jarque statistic rejects the hypothesis of normality of the basis changes. The results for the LM-test show that, in most of the series, the hypothesis of no ARCH effects cannot be rejected. As we have noted in Section 3.3 of the previous chapter, traded volume increases monotonically as the contract approaches expiration. The peak in trading volume is reached around four to ten weeks before expiration while in the last four weeks volume declines (see Figures 3.1 and 3.2). This justifies the decision of considering for the analysis only the thirteen months previous to the expiration of the contract. Finally, Figure 4.2 reports the number of TIIE futures contracts traded every month from January 2003 to December 2006. It is noticeable the significant drop in volume during 2005 as compared to previous years. According to MexDer, this fact is explained by some tax issues that induced participants to switch their hedge positions to swaps traded over the counter (Alegr´ıa, 2006).


45

Table 4.3: Descriptive statistics for daily basis changes Contract

Mean

tstat

Standard Deviation

Skewness

Kurtosis

p-value

ARCH (LM)


-0.3243 0.0175 -0.1981 -0.3283 -0.2093 0.0011 0.0475 -0.0899 -0.0907 -0.0636 -0.2081 -0.1844

-0.65 0.04 -0.40 -0.69 -0.46 0.00 0.10 -0.19 -0.20 -0.15 -0.49 -0.43

0.4949 0.4993 0.4845 0.4663 0.4495 0.4662 0.4502 0.4603 0.4564 0.4173 0.4171 0.4226

-0.11 -0.24 -0.04 -0.28 -0.30 -0.71 -1.03 -0.77 -0.75 -0.69 -0.66 -0.72

8.33 8.65 7.94 8.12 10.45 8.49 7.60 5.79 5.84 5.12 4.56 4.88

p-value

286.50∗ 324.53∗ 246.00∗ 267.68∗ 563.42∗ 324.00∗ 256.04∗ 102.73∗ 103.48∗ 64.21∗ 42.01∗ 56.40∗

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

34.68∗ 32.53∗ 30.75∗ 36.03∗ 42.69∗ 18.32∗ 14.95∗ 14.77∗ 12.39∗ 15.85∗ 17.62∗ 17.01∗

0.0000 0.0000 0.0000 0.0000 0.0000 0.0026 0.0106 0.0114 0.0298 0.0073 0.0035 0.0045


-0.0498 0.0659 0.0340 -0.0090 -0.2548 -0.3877 -0.2788 -0.3407 -0.3126 -0.2403 -0.3221 -0.0205

-0.12 0.15 0.08 -0.02 -0.56 -0.90 -0.71 -0.90 -0.84 -0.67 -0.93 -0.06

0.4265 0.4237 0.4396 0.4632 0.4510 0.4253 0.3859 0.3726 0.3656 0.3533 0.3400 0.3150

-0.59 -0.49 -0.64 -0.61 -0.43 -0.39 -0.32 -0.51 -0.50 -0.56 -0.57 -0.55

4.22 3.94 4.02 3.88 3.76 4.00 3.68 4.20 4.14 4.38 4.58 5.35

28.92∗ 18.42∗ 27.10∗ 22.66∗ 13.29∗ 16.29∗ 8.74∗ 25.15∗ 23.05∗ 31.68∗ 38.30∗ 67.91∗

0.0000 0.0001 0.0000 0.0000 0.0013 0.0003 0.0127 0.0000 0.0000 0.0000 0.0000 0.0000

9.26 16.64∗ 16.76∗ 15.87∗ 21.25∗ 10.72 10.44 9.42 10.12 10.67 14.07∗ 17.58∗

0.0992 0.0052 0.0050 0.0072 0.0007 0.0572 0.0637 0.0934 0.0720 0.0582 0.0152 0.0035


-0.3033 -0.2163 -0.1229 -0.1433 -0.1745 -0.2499 -0.2282 -0.2284 -0.1837 -0.1583 -0.1686 -0.0928

-1.02 -0.80 -0.50 -0.61 -0.90 -1.40 -1.48 -1.62 -1.41 -1.29 -1.40 -0.82

0.2928 0.2656 0.2441 0.2317 0.1899 0.1757 0.1514 0.1385 0.1280 0.1204 0.1181 0.1108

-0.69 -0.58 -0.39 -0.40 -0.13 -0.18 -0.40 -0.47 -0.58 -0.56 -0.56 -0.34

6.02 6.73 7.39 8.71 3.19 3.19 3.34 3.73 4.43 4.98 5.06 4.85

110.97∗ 153.79∗ 200.36∗ 334.68∗ 1.06 1.72 7.56∗ 14.07∗ 34.10∗ 52.04∗ 55.36∗ 39.13∗

0.0000 0.0000 0.0000 0.0000 0.5897 0.4226 0.0228 0.0009 0.0000 0.0000 0.0000 0.0000

22.07∗ 29.23∗ 58.42∗ 69.86∗ 20.17∗ 26.15∗ 13.69∗ 34.14∗ 41.56∗ 49.96∗ 56.20∗ 39.80∗

0.0005 0.0000 0.0000 0.0000 0.0012 0.0001 0.0177 0.0000 0.0000 0.0000 0.0000 0.0000


-0.0304 -0.0505 -0.0248 -0.0524 -0.0127 0.0300 0.0718 0.0521 0.0752 0.0461 0.0404 0.0572

-0.27 -0.49 -0.23 -0.50 -0.12 0.29 0.69 0.48 0.70 0.42 0.34 0.48

0.1090 0.1004 0.1042 0.1038 0.1022 0.1037 0.1026 0.1069 0.1063 0.1072 0.1160 0.1176

0.08 0.27 0.31 0.28 0.32 0.19 0.29 0.30 0.58 0.74 0.45 0.80

3.90 3.71 4.48 4.78 5.26 4.79 4.93 4.35 5.05 6.00 6.10 9.10

8.39∗ 8.11∗ 26.09∗ 34.93∗ 55.89∗ 33.87∗ 41.10∗ 21.94∗ 56.13∗ 112.70∗ 105.28∗ 401.23∗

0.0151 0.0174 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

27.16∗ 12.27∗ 21.80∗ 16.68∗ 12.04∗ 17.09∗ 19.88∗ 12.07∗ 11.35∗ 12.14∗ 18.24∗ 18.16∗

0.0001 0.0312 0.0006 0.0052 0.0342 0.0043 0.0013 0.0338 0.0449 0.0329 0.0027 0.0028

BJ

This table reports the statistics of the daily basis changes along 242 days before expiration month. BJ is the Bera-Jarque statistic for testing the null hypothesis of normal distribution. The ARCHLM is the LM-statistic of autoregressive conditional heteroscedasticity effect with 5 lags. 5% significance.

∗

indicates


46

Figure 4.2: Volume of TIIE Futures contracts traded during the whole period

)%" (' %&! $ # "!

Numbers are millions of contracts traded monthly during the period.

4.3.2

Methodology

Different studies have employed different approaches to test Samuelson hypothesis. Some studies calculate price variability as variances over daily price changes within a month, record the number of months left to maturity of the contract and then perform OLS regressions using these monthly variances, like in Milonas (1986) or Galloway & Kolb (1996). In Bessembinder et al. (1996) daily volatility is estimated as the absolute value of future returns and regressions are performed on days to expiration, spot volatility and monthly indicators. Other studies build long term future series by rolling over contracts and apply different GARCH models with time to maturity as an exogenous variable. In this study, the focus is on extending the usual OLS regressions by applying panel estimation techniques. Hence, from the 48 series of rate changes a panel data set is constructed by aligning the data by days to expiration instead of calendar day. This implies rearranging subindexes to express the cross-sectional and time dimensions. Specifically, if the contracts are labelled with the variable i (i = 1, ..., 48), T (i) is the


47

maturity date defining the i-th contract, and τ = T (i) − t is the number of days to maturity, then all data can be defined in terms of the pair (i, τ ) instead of the previous (T, t). For example, in terms of time to maturity, rate changes for contract i are expressed as ∆Yiτ = ln(Yi,τ /Yi,τ +1 )

(3)

Recall that the expiration month has been excluded from the analysis. Hence, the time variable τ ranges from the 20-th day before the contract expires (τ = 20) to 262 days before expiration (τ = 262). For each futures series i there is a corresponding series of contemporaneous spot rates, which will be denoted Si,t . To maintain coherence with the panel data structure, each of these series is subsequently aligned in terms of the days to maturity τ defined by contract i. Then, as with the future series, the study considers the logarithmic returns ∆Siτ = ln(Si,τ /Si,τ +1 )

(4)

where, for the i-th contract, Si,τ is the value St of the TIIE spot rate on calendar day t = T (i) − τ . Finally, in terms of time to expiration, the basis for contract i at time τ is ∆Biτ = ∆Yiτ − ∆Si τ .

(5)

As in Rutledge (1976) or Bessembinder et al. (1996), daily variability will be measured by the absolute value of the logarithmic rate changes. That is, σ(Y )iτ = | ln(Yi,τ /Yi,τ +1 )|

(6)

for the case of futures variability. Analogous expressions hold for spot changes volatility σ(S)iτ and basis changes volatility σ(B)iτ . The maturity effect will first be investigated by performing individual OLS regressions for each contract. This amounts to considering the unrestricted model, σ(Y )iτ = αi + βi τ + uiτ

(7)

corresponding to linear regressions the futures volatility on time to expiration. The hypothesis is that if maturity effect is present, the coefficient βi should be negative.


48

Next, imposing the restrictions αi = α,

βi = β

∀i = 1, ..., N

yields the restricted model σ(Y )iτ = α + βτ + uiτ

(8)

After estimating coefficients, analysis of variance tests of the residuals will give information on the presence of individual effects, time effects or both. To perform a panel data analysis, the two-way error component regression model disturbances are decomposed as uiτ = µi + λτ + ηiτ

(9)

where µi denotes the unobservable individual effect, λτ the unobservable time effect, and ηiτ is the remainder stochastic disturbance term. λτ is individual invariant and accounts for any time specific effect that is not included in the regression. In the last stage of the analysis, consists of panel regressions both with fixed and random effects. The fixed and random effects estimators are designed to handle the systematic tendency of uiτ to be higher for some individuals that for others (individual effects) and possibly higher for some periods than for others (time effects). The fixed effects estimator does this by (in effect) using a separate intercept for each individual or time period. When considering a fixed-effects model, µi and λτ are treated as constants and are swept out. Under a random effects model, they are treated as part of the error term and β is estimated by GLS. There are advantages and disadvantages to each treatment of the individual effects. A fixed effects model cannot estimate a coefficient on any time-invariant regressor since the individual intercepts are free to take any value. By contrast, the individual effect in a random effects model is part of the error term, so it must be uncorrelated with the regressors. On the flip side, because the random effects model treats the individual effect as part of the error term, it suffers from the possibility of bias due to a correlation between it and regressors In order to test for the effects of information flow, the above analysis is performed including spot volatility as a regressor. Finally, to test the hypothesis of decreasing volatility of the basis as maturity approaches, the same analysis is performed with basis volatility as dependent variable.


49

All the coefficient estimates were obtained using Rats v.5.0 software package.

4.4 4.4.1

Empirical results Estimates of time-to-maturity effects on volatility

Table 4.4 reports results of individual regressions of the daily volatility estimates on the number of days until the contract expires1 . Estimated coefficients on the time to expiration variable are negative, as predicted by Samuelson hypothesis, but only for contracts that matured in 2003 and 2004. In 2005 all coefficients are positive and significant, contrary to Samuelson hypothesis. In 2006 all the coefficients are still positive, although only a few are significant. This particular behavior of contracts maturing in 2005 is also evident in the estimated mean coefficients. In contrast with all the other periods, in 2005 no mean coefficient is significantly different from zero. The last two columns of Table 4.4 report the adjusted R2 and the Durbin-Watson statistics. The adjusted R2 values show the model has little explanatory power. On the other hand, Durbin-Watson test results indicate there is no significant first order autocorrelation of the residuals. The results of the test for individual and time effects in volatility series are presented in Table 4.5. The first columns depict the results of the restricted model regression σ(Y )iτ = α + βτ + uiτ The estimated regression coefficients are negative and significant for contracts expiring in 2003 and 2004 but is positive and significant for contracts expiring in 2005 and 2006. Moreover, when the whole period is considered, β is not significantly different of zero. These results indicate a maturity effect was present but either disappeared in contracts expiring from 2005 onwards or turned into an inverted effect. The analysis for the presence of individual effects, time effects or both shows the presence of individual effects in contracts expiring in 2005 and in the whole set of contracts. Table 4.6 reports the results of panel regression of daily volatility on days to expiration. Estimation is done either by fixed effects or by random effects. The results support Samuleson 1

Similar regressions were performed considering, instead of days to maturity, the squared root of days to

maturity. The results obtained are qualitatively the same.


50

hypothesis for contracts with expiration in 2003 and 2004. However, the results for contracts with expiration in 2005 and 2006 is against the hypothesis. In fact, for these contracts volatility appears to decrease as maturity approaches. When the whole set of contracts is considered, the β coefficient is not significant, indicating there is no evidence of relation between volatility and time to maturity.


51

Table 4.4: Regression of daily volatility on days to expiration Contract Jan03 Feb03 Mar03 Apr03 May03 June03 July03 Aug03 Sept03 Oct03 Nov03 Dec03 Jan04 Feb04 Mar04 Apr04 May04 June04 July04 Aug04 Sept04 Oct04 Nov04 Dec04 Jan05 Feb05 Mar05 Apr05 May05 June05 July05 Aug05 Sept05 Oct05 Nov05 Dec05 Jan06 Feb06 Mar06 Apr06 May06 June06 July06 Aug06 Sept06 Oct06 Nov06 Dec06

αi 0.01636 0.01725 0.01617 0.01497 0.01200 0.01522 0.01647 0.01760 0.01400 0.01080 0.01152 0.01195 0.01400 0.01463 0.01531 0.01588 0.01362 0.01451 0.01359 0.01352 0.01260 0.01038 0.00828 0.00751 0.00654 0.00512 0.00450 0.00256 0.00303 0.00240 0.00209 0.00240 0.00174 0.00145 0.00179 0.00184 0.00378 0.00305 0.00299 0.00328 0.00363 0.00368 0.00386 0.00435 0.00364 0.00353 0.00429 0.00358

t-stats 7.58∗ 8.40∗ 8.87∗ 7.96∗ 7.38∗ 6.89∗ 7.80∗ 8.99∗ 7.95∗ 8.98∗ 8.83∗ 9.03∗ 7.39∗ 8.17∗ 8.72∗ 9.36∗ 8.96∗ 8.18∗ 9.09∗ 10.11∗ 9.51∗ 8.40∗ 7.29∗ 7.72∗ 8.28∗ 6.67∗ 5.64∗ 3.11∗ 4.81∗ 4.05∗ 4.44∗ 5.44∗ 4.49∗ 3.18∗ 3.69∗ 3.73∗ 6.56∗ 6.07∗ 5.73∗ 6.50∗ 6.94∗ 6.64∗ 7.12∗ 7.08∗ 6.32∗ 5.72∗ 6.95∗ 6.41∗

βi -0.00003 -0.00004 -0.00003 -0.00001 -0.00001 -0.00002 -0.00003 -0.00003 -0.00001 0.00000 0.00000 -0.00001 -0.00002 -0.00003 -0.00003 -0.00003 -0.00001 -0.00003 -0.00002 -0.00003 -0.00002 0.00000 0.00001 0.00001 0.00001 0.00002 0.00002 0.00004 0.00003 0.00003 0.00003 0.00002 0.00002 0.00002 0.00002 0.00002 0.00001 0.00001 0.00001 0.00001 0.00000 0.00000 0.00000 0.00000 0.00000 0.00001 0.00000 0.00001

t-stats −2.61∗ −3.08∗ −2.40∗ −1.29 −0.51 −1.67 −2.38∗ −2.94∗ −1.08 −0.08 −0.59 −0.69 −2.04∗ −2.41∗ −2.71∗ −2.82∗ −1.29 −2.38∗ −2.11∗ −3.28∗ −2.41∗ −0.39 1.63 1.99∗ 2.45∗ 3.48∗ 3.75∗ 5.29∗ 5.71∗ 5.99∗ 6.52∗ 5.24∗ 6.25∗ 5.73∗ 4.70∗ 4.19∗ 1.41 2.49∗ 2.62∗ 2.05∗ 1.16 1.24 0.67 0.26 1.32 1.37 0.87 2.44∗

¯2 R 0.014 0.019 0.011 0.001 -0.003 0.010 0.022 0.033 0.001 -0.004 -0.003 -0.002 0.018 0.023 0.028 0.027 0.002 0.022 0.018 0.035 0.018 -0.004 0.005 0.007 0.012 0.035 0.054 0.125 0.117 0.136 0.129 0.092 0.117 0.127 0.089 0.080 0.005 0.021 0.022 0.013 0.002 0.002 -0.003 -0.004 0.002 0.002 -0.002 0.009

D-W 1.73 1.84 1.89 1.92 1.89 1.74 1.69 1.78 1.97 1.81 1.64 1.83 1.59 1.75 1.63 1.53 1.68 1.75 1.62 1.84 1.92 1.82 1.99 2.02 2.01 1.90 2.01 2.07 2.17 2.17 2.12 2.08 1.93 1.81 1.87 1.90 1.90 1.90 1.92 1.93 1.97 1.81 1.80 1.77 1.68 1.62 1.60 1.65

The table reports the estimates of the regression model σ(Y )iτ = αi + βi τ + εiτ ¯ 2 is the adjusted R2 . DW is the Durbin-Watson test for where τ represents days to maturity. R first-order serial correlation of the residuals. There are 242 observations. * indicates significance at 5%. Estimation with heteroscedasticity-consistent standard errors.


52

Table 4.5: Test for individual and time effects in futures volatility series Equality Year 2003

2004

2005

2006

All

Regression coefficients

Analysis of variance

of variances

estimate

t-stat

p-value

Source

F-test

p-value

χ2

p-value

α

0.014500

26.42

0.0000

Individual

0.7458

0.6949

185.93

0.0000

β

-0.000019

-5.35

0.0000

Time

0.8826

0.8967

(df=11)

Joint

0.8766

0.9132

α

0.012800

31.04

0.0000

Individual

1.7542

0.0566

71.57

β

-0.000015

-5.85

0.0000

Time

0.9520

0.6868

(df=11)

Joint

0.9870

0.5455

α

0.002955

12.56

0.3144

Individual

21.6631

0.0000

609.11

β

0.000023

15.28

0.0000

Time

0.9869

0.5443

(df=11)

Joint

1.8895

0.0000

α

0.003637

21.34

0.0000

Individual

0.5384

0.8782

71.58

β

0.000005

4.76

0.0000

Time

1.0488

0.2980

(df=11)

Joint

1.0265

0.3794

α

0.008484

42.64

0.0000

Individual

31.953

0.0000

6430.03

β

-0.000002

-1.20

0.2313

Time

0.7779

0.9952

(df=47)

Joint

5.8655

0.0000

0.0000

0.0000

0.0000

0.0000

This table reports the coefficients of the restricted regression σ(Y )iτ = α + βτ + uiτ where α and β are assumed to be constant across contracts and τ represents days to maturity. Analysis of variance is an analysis of variance test for common means, across individuals, across time, or both. The last two columns report the results of a likelihood ratio test for equal variances across cross-sections. df = degrees of freedom.


53

Table 4.6: Panel regression of daily volatility on time to expiration Year 2003

Regression coefficients estimate

t-stat

p-value

¯2 R

α

0.014500

27.08

0.00000

0.00727

β

-0.000019

-5.35

0.00000

Panel Regression - Estimation by Random Effects 2004

α

0.012800

27.82

0.00000

β

-0.000015

-5.59

0.00000

0.02862


α

0.002955

5.27

0.00000

β

0.000023

11.13

0.00000

0.19787


α

0.003637

19.98

0.00000

β

0.000005

4.45

0.00000

0.02787

Panel Regression - Estimation by Random Effects All

α

0.008484

16.90

0.00000

β

-0.000002

-1.27

0.20404

0.11530

Panel Regression - Estimation by Random Effects This table reports the coefficients of the panel regression over absolute returns σ(Y )iτ = α + βτ + uiτ where τ is the variable for days to maturity. The error component descomposes as uiτ = µi + λτ + ηiτ , allowing for individual or time effects.


4.4.2

54

Effect of controlling for variation in information flow

As mentioned earlier, recent studies on the Samuelson hypothesis suggest that increased volatility prior to a contract expiring is directly due to the rate of information flow into the futures market. The significance of information effects is therefore investigated by following the testing procedure used in Bessembinder et al. (1996) which involves including spot price variability as an independent variable in the regression outlined above. If spot price stationarity is the most significant determinant of the Samuelson hypothesis, the coefficient on the days to expiry variable should remain negative and significant despite the inclusion of the spot volatility variable. Table 4.7 reports results of individual regressions of the daily volatility estimates on the number of days until the contract expires and on spot volatility. Compared with the results obtained previously, the inclusion of the spot volatility does not appear to have any significant effect. The TIIE spot volatility is only statistically significant in very few cases, showing in general, futures volatility is not being affected by spot volatility. The last two columns of Table 4.7 report the adjusted R2 and the Durbin-Watson statistics. The negative value of the adjusted R2 values for some of the series reveal a poor fit. On the other hand, Durbin-Watson test results indicate there is no significant first order autocorrelation of the residuals. When the spot volatility is introduced in the restricted regression as a control variable to account for the effects of information flow, the main change is that the maturity effect during the whole set of contracts becomes statistically significant, as we can see in Table 4.8. The first columns depict the results of the restricted model regression σ(Y )iτ = α + βτ + γσ(S)iτ + uiτ where σ(S)iτ is the spot rate volatility. Spot volatility is significant except in 2005 and 2006 contracts. The estimated β coefficients are negative and significant for 2003 and 2004 contracts and positive and significant for contracts expiring in 2005 and 2006. However for the whole set of contracts it appears to be negative. These results indicate that, when we account for the information flow effects, the maturity effect is present during the whole period, although it is not observable in the last two years. The analysis for the presence of individual effects, time effects or both shows the presence of individual effects is qualitatively the same as obtained without the spot volatility as


55

regressor. Table 4.9 reports the results of panel regression of daily volatility on days to expiration and spot volatility. Estimation is done either by fixed effects or by random effects. With the exception of 2006 contracts, the spot volatility appears to be significant. Again, 2005 seems to have a particular behavior, with the spot volatility coefficient being negative. As before, the results support Samuelson hypothesis for contracts with expiration in 2003 and 2004, while the results for contracts with expiration in 2005 and 2006 is against the hypothesis. As previous results showed (see Section 4.4.1), for these contracts volatility appears to decrease as maturity approaches. However, when the whole set of contracts is considered, the β coefficient becomes significant at 5%, indicating that, when we consider the effects of information flow, there is evidence of relation between volatility and time to maturity.


56

Table 4.7: Regression of daily volatility on days to expiration and spot volatility Contract Jan03 Feb03 Mar03 Apr03 May03 June03 July03 Aug03 Sept03 Oct03 Nov03 Dec03 Jan04 Feb04 Mar04 Apr04 May04 June04 July04 Aug04 Sept04 Oct04 Nov04 Dec04 Jan05 Feb05 Mar05 Apr05 May05 June05 July05 Aug05 Sept05 Oct05 Nov05 Dec05 Jan06 Feb06 Mar06 Apr06 May06 June06 July06 Aug06 Sept06 Oct06 Nov06 Dec06

α 0.015655 0.016236 0.014954 0.014357 0.010627 0.013193 0.014004 0.015863 0.012400 0.009167 0.010914 0.010945 0.013356 0.014505 0.014779 0.014596 0.013152 0.014311 0.013400 0.012597 0.011968 0.010237 0.008230 0.007487 0.006551 0.005167 0.004623 0.002688 0.003469 0.002809 0.002326 0.002513 0.001763 0.001481 0.001795 0.001847 0.003808 0.002895 0.002986 0.003186 0.003503 0.003437 0.003868 0.004366 0.003651 0.003543 0.004282 0.003552

t-stats 7.32∗ 7.89∗ 7.82∗ 6.94∗ 5.45∗ 5.70∗ 5.31∗ 6.35∗ 5.52∗ 5.95∗ 6.83∗ 6.74∗ 6.96∗ 8.03∗ 8.50∗ 8.45∗ 8.45∗ 7.74∗ 9.10∗ 9.11∗ 8.68∗ 8.11∗ 7.17∗ 7.65∗ 8.28∗ 6.71∗ 5.74∗ 3.22∗ 5.42∗ 4.54∗ 4.92∗ 5.70∗ 4.55∗ 3.23∗ 3.71∗ 3.69∗ 6.62∗ 5.70∗ 5.39∗ 5.82∗ 6.44∗ 5.88∗ 6.89∗ 7.00∗ 6.33∗ 5.73∗ 6.96∗ 6.42∗

β -0.000032 -0.000040 -0.000030 -0.000017 -0.000009 -0.000023 -0.000025 -0.000031 -0.000010 0.000003 -0.000004 -0.000005 -0.000022 -0.000025 -0.000028 -0.000031 -0.000014 -0.000026 -0.000021 -0.000030 -0.000024 -0.000006 0.000010 0.000011 0.000015 0.000022 0.000027 0.000040 0.000032 0.000033 0.000028 0.000021 0.000022 0.000022 0.000018 0.000017 0.000005 0.000008 0.000009 0.000007 0.000004 0.000005 0.000002 0.000001 0.000005 0.000006 0.000003 0.000009

t-stats −2.71∗ −3.50∗ −2.85∗ −1.46 −0.90 −1.72 −1.78 −2.39∗ −0.81 0.38 −0.45 −0.53 −2.02∗ −2.41∗ −2.72∗ −3.04∗ −1.45 −2.37∗ −2.05∗ −3.74∗ −3.06∗ −0.77 1.25 1.59 2.32∗ 3.5∗ 4.06∗ 5.72∗ 6.07∗ 6.53∗ 6.71∗ 5.17∗ 5.59∗ 5.34∗ 4.19∗ 3.89∗ 1.39 2.37∗ 2.62∗ 2.06∗ 1.20 1.33 0.67 0.27 1.32 1.44 0.92 2.44∗

γ 0.056243 0.108008 0.119048 0.057322 0.136344 0.130879 0.105744 0.072071 0.074178 0.072022 0.028368 0.048332 0.030458 0.006559 0.029454 0.084387 0.040226 0.013954 0.015833 0.102748 0.096045 0.042906 0.030395 0.021118 -0.01412 -0.0339 -0.07455 -0.09213 -0.16581 -0.1733 -0.15774 -0.09868 -0.05191 -0.03118 -0.02017 -0.00844 -0.0189 0.080446 0.001298 0.034588 0.048664 0.095035 -0.00539 -0.01072 -0.0159 -0.03956 -0.0227 -0.0372

t-stats 0.91∗ 1.68∗ 1.72∗ 0.81 1.59 1.99 1.66 1.20∗ 1.32 1.80 0.79 1.17 0.66∗ 0.17∗ 0.73∗ 1.84∗ 0.93 0.28∗ 0.34∗ 2.05∗ 2.16∗ 0.99 0.64 0.43 −0.30∗ −0.71∗ −1.42∗ −1.65∗ −2.57∗ −2.42∗ −2.27∗ −1.37∗ −0.63∗ −0.39∗ −0.22∗ −0.08∗ −0.24 0.61∗ 0.02∗ 0.51∗ 0.74 1.26 −0.09 −0.14 −0.20 −0.47 −0.29 −0.45∗

¯2 R 0.0148 0.0297 0.0266 0.0007 0.0141 0.0349 0.0336 0.0370 0.0077 0.0036 -0.0050 -0.0017 0.0161 0.0191 0.0259 0.0373 0.0014 0.0187 0.0142 0.0526 0.0327 -0.0046 0.0025 0.0039 0.0086 0.0326 0.0573 0.1308 0.1335 0.1523 0.1438 0.0959 0.1150 0.1242 0.0859 0.0757 0.0011 0.0211 0.0181 0.0093 -0.0013 0.0032 -0.0067 -0.0080 -0.0016 -0.0010 -0.0063 0.0048

DW 1.778 1.932 1.998 1.976 2.033 1.929 1.816 1.865 2.055 1.868 1.658 1.868 1.610 1.751 1.664 1.628 1.716 1.760 1.624 1.934 1.994 1.861 2.012 2.036 2.001 1.879 1.947 1.996 2.071 2.087 2.061 2.043 1.915 1.791 1.860 1.898 1.890 1.924 1.916 1.946 1.994 1.816 1.794 1.766 1.670 1.613 1.594 1.639

The table reports the estimates of the unrestricted regression model σ(Y )iτ = αi + βi τ + γi σ(S)iτ + uiτ ¯ 2 is the adjusted R2 . There where τ represents days to maturity and σ(S)iτ is the spot volatility. R are 242 observations. Expiration month is excluded. * indicates significance at 5%.


57

Table 4.8: Test for individual and time effects in futures volatility series with TIIE spot variance as control variable Equality Year 2003

2004

2005

2006

All



of variances

estimate

t-stat

p-value

Source

F-test

p-value

χ2

p-value

α

0.013100

21.91

0.00000

Individual

0.8003

0.6400

182.48

0.0000

β

-0.000018

-5.17

0.00000

Time

0.9299

0.7673

(df=11)

γ

0.081700

5.57

0.00000

Joint

0.9242

0.7912

α

0.012200

27.94

0.00000

Individual

1.1770

0.2974

67.13

β

-0.000017

-6.56

0.00000

Time

0.9513

0.6894

(df=11)

γ

0.057900

4.47

0.00001

Joint

0.9612

0.6542

α

0.002928

12.40

0.00000

Individual

20.3612

0.0000

597.19

β

0.000022

14.19

0.00000

Time

0.9885

0.5376

(df=11)

γ

0.021571

1.32

0.18805

Joint

1.8342

0.0000

α

0.003638

20.90

0.00000

Individual

0.5380

0.8786

71.57

β

0.000005

4.73

0.00000

Time

1.0488

0.2981

(df=11)

γ

-0.000466

-0.02

0.98541

Joint

1.0265

0.3795

α

0.007230

36.10

0.00000

Individual

15.0028

0.0000

5590.21

β

-0.000004

-3.24

0.00118

Time

0.8311

0.9727

(df=47)

γ

0.164300

24.94

0.00000

Joint

3.1439

0.0000

0.0000

0.0000

0.0000

0.0000

This table reports the coefficients of the restricted regression over the residuals of excess returns σ(Y )iτ = α + βτ + γσ(S)iτ + uiτ where α and β are assumed to be constant across contracts, τ is the variable for days to maturity and σ(S)iτ is spot volatility. Analysis of variance is an analysis of variance test for common means, across individuals, across time, or both. The last two columns report the results of a likelihood ratio test for equal variances across cross-sections. df = degrees of fredom.


58

Table 4.9: Panel regression of daily volatility on time to expiration and spot rate volatility Year 2003

Regression coefficients Estimate

t-stat

p-value

¯2 R

α

0.013200

22.28

0.00000

0.01796

β

-0.000018

-5.17

0.00000

γ

0.081500

5.56

0.00000


α

0.012200

27.37

0.00000

β

-0.000017

-6.55

0.00000

γ

0.056200

4.32

0.00002

0.01938


α

0.000000

0.14555

β

0.000025

16.31

0.00000

γ

-0.064700

-3.87

0.00011

Panel Regression - Estimation by Fixed Effects 2006

α

0.003642

21.90

0.00000

β

0.000005

4.74

0.00000

γ

-0.003401

-0.13

0.89328

0.00215


α

0.000000

0.11707

β

-0.000002

-2.05

0.04072

γ

0.061100

8.36

0.00000

Panel Regression - Estimation by Fixed Effects This table reports the estimated coefficients of the panel regression σ(Y )iτ = α + βτ + γσ(S)iτ + uiτ where τ is the variable for time to maturity and σ(S)iτ is the spot rate volatility.


4.4.3

59

Estimation of maturity effect on the basis.

Table 4.10 shows the results of the individual regressions of the basis volatility on time to maturity. The coefficients β are positive and significant for contracts with expiration between September 2004 and March 2006, indicating that basis volatility decreased as maturity approached. This is in agreement with the results of Castelino & Francis (1982) or Beaulieu (1998). However, for the rest of the contracts the results show some evidence against this effect, either because β is negative and significant, not significant at all or with very poor ¯ 2 ). fittings (negative R On Table 4.11 the results indicate the presence of individual effects on basis changes volatility during 2004 and 2005 and on the whole period. Positive betas confirm basis changes volatility decreases as maturity approaches, with the exception of 2003. Table 4.12 presents annual panel regressions for basis changes volatilities. Once again, 2003 coefficient for time to maturity is negative and significant, while coefficients are significant and positive from 2004 to 2006 and for the whole sample. Again with the exception of 2003, panel results indicate that as distance to maturity increases the volatility in the basis changes augment.


60

Table 4.10: Regression of basis changes volatility on days to expiration Contract Jan03 Feb03 Mar03 Apr03 May03 June03 July03 Aug03 Sept03 Oct03 Nov03 Dec03 Jan04 Feb04 Mar04 Apr04 May04 June04 July04 Aug04 Sept04 Oct04 Nov04 Dec04 Jan05 Feb05 Mar05 Apr05 May05 June05 July05 Aug05 Sept05 Oct05 Nov05 Dec05 Jan06 Feb06 Mar06 Apr06 May06 June06 July06 Aug06 Sept06 Oct06 Nov06 Dec06

α 0.01988 0.01949 0.01918 0.02052 0.01503 0.01976 0.02606 0.02942 0.02500 0.02451 0.02257 0.02147 0.02349 0.02095 0.02210 0.01984 0.01467 0.01749 0.01715 0.01602 0.01394 0.01071 0.00826 0.00711 0.00521 0.00558 0.00474 0.00320 0.00368 0.00285 0.00250 0.00214 0.00155 0.00178 0.00170 0.00156 0.00426 0.00341 0.00361 0.00372 0.00427 0.00421 0.00431 0.00451 0.00373 0.00366 0.00435 0.00332

t-stats 6.82∗ 6.84∗ 7.30∗ 8.10∗ 6.59∗ 5.56∗ 7.81∗ 9.62∗ 8.26∗ 10.08∗ 10.42∗ 11.11∗ 10.34∗ 9.56∗ 8.79∗ 7.44∗ 5.86∗ 6.32∗ 7.60∗ 7.87∗ 7.35∗ 6.76∗ 5.87∗ 5.43∗ 4.08∗ 4.74∗ 4.34∗ 2.70∗ 4.55∗ 4.05∗ 4.23∗ 4.08∗ 3.11∗ 3.14∗ 2.80∗ 2.55∗ 5.97∗ 6.02∗ 5.43∗ 5.96∗ 6.97∗ 6.73∗ 7.23∗ 6.91∗ 6.10∗ 5.80∗ 6.72∗ 5.58∗

β 0.00001 0.00001 0.00001 0.00000 0.00002 0.00000 -0.00005 -0.00006 -0.00003 -0.00004 -0.00002 -0.00001 -0.00002 0.00000 0.00000 0.00002 0.00005 0.00002 0.00001 0.00001 0.00003 0.00004 0.00006 0.00005 0.00006 0.00005 0.00005 0.00005 0.00004 0.00004 0.00003 0.00003 0.00003 0.00003 0.00003 0.00003 0.00001 0.00001 0.00001 0.00001 0.00000 0.00000 0.00000 0.00000 0.00001 0.00001 0.00000 0.00001

t-stats 0.54 0.54 0.62 −0.16 1.25 −0.11 −2.27∗ −3.29∗ −1.61 −2.66∗ −1.43 −0.72 −1.19 −0.03 −0.25 1.06 2.90∗ 1.32 0.89 0.94 2.10∗ 3.76∗ 5.00∗ 4.73∗ 5.02∗ 4.36∗ 4.72∗ 4.90∗ 6.13∗ 7.02∗ 6.93∗ 7.10∗ 6.90∗ 5.65∗ 5.44∗ 5.29∗ 1.59 2.68∗ 2.14∗ 1.92 0.84 0.98 0.71 0.64 1.83 1.70 1.17 2.93∗

¯2 R -0.003 -0.003 -0.003 -0.004 0.002 -0.004 0.020 0.041 0.008 0.017 0.002 -0.003 0.001 -0.004 -0.004 0.001 0.036 0.005 -0.001 -0.001 0.010 0.037 0.069 0.067 0.095 0.080 0.098 0.133 0.143 0.176 0.156 0.151 0.158 0.132 0.133 0.130 0.009 0.024 0.015 0.010 -0.002 -0.001 -0.002 -0.002 0.008 0.006 0.000 0.019

DW 1.49 1.65 1.56 1.53 1.52 1.53 1.63 1.67 1.72 1.63 1.67 1.57 1.66 1.68 1.97 2.05 2.08 2.00 2.04 2.04 1.97 1.98 2.03 1.94 2.05 1.88 1.73 1.74 1.78 1.77 1.82 1.63 1.57 1.57 1.50 1.54 1.60 1.71 1.59 1.65 1.72 1.64 1.64 1.69 1.64 1.61 1.59 1.58

The table reports the estimates of the unrestricted regression model σ(B)iτ = αi + βi τ + uiτ ¯ 2 is the adjusted R2 . where τ represents days to maturity and V Biτ is the basis volatility. R There are 242 observations. Expiration month is excluded. * indicates significance at 5%.


61

Table 4.11: Test for individual and time effects in basis changes volatility series. Equality Year 2003

2004

2005

2006

All



of variances

Estimate

t-stat

p-value

Source

F-test

p-value

χ2

p-value

α

0.021900

25.30

0.000

Individual

0.361

0.971

66.5

0.000

β

-0.000013

-2.35

0.019

Time

0.833

0.967

(df = 11)

Joint

0.813

0.984

α

0.015977

23.46

0.000

Individual

6.509

0.000

127.8

β

0.000023

5.22

0.000

Time

1.043

0.321

(df = 11)

Joint

1.281

0.003

α

0.003041

9.01

0.000

Individual

34.490

0.000

1077.4

β

0.000039

18.06

0.000

Time

0.988

0.541

(df = 11)

Joint

2.450

0.000

α

0.003946

20.30

0.000

Individual

0.345

0.975

42.0

β

0.000006

5.20

0.000

Time

0.956

0.671

(df = 11)

Joint

0.929

0.774

α

0.011218

34.61

0.000

Individual

59.694

0.000

9930.0

β

0.000014

6.66

0.000

Time

0.826

0.976

(df = 47)

Joint

10.433

0.000

0.000

0.000

0.000

0.000

This table reports the coefficients of the restricted regression over the residuals of the regression σ(B)iτ = α + βτ + γ + uiτ where α and β are assumed to be constant across contracts, τ is the variable for days to maturity and σ(B)iτ is basis changes volatility. Analysis of variance is an analysis of variance test for common means, across individuals, across time, or both. The last two columns report the results of a likelihood ratio test for equal variances across cross-sections. df = degrees of freedom.


62

Table 4.12: Panel regression of basis changes volatility on time to expiration Year 2003

Regression coefficients Estimate

t-stat

p-value

¯2 R

α

0.021900

27.03

0.00000

-0.00741

β

-0.000013

-2.35

0.01886


α

0.015977

16.41

0.00000

β

0.000023

5.28

0.00000

0.03227


α

0.003041

3.16

0.00158

β

0.000039

12.09

0.00000

0.26017


α

0.003946

19.59

0.00000

β

0.000006

5.01

0.00000

0.018933


α

0.011218

10.79

0.00000

β

0.000014

7.21

0.00000

Panel Regression - Estimation by Random Effects This table reports the estimated coefficients of the panel regression σ(B)iτ = α + βτ + uiτ where τ is the variable for time to maturity and τ is time to maturity.

0.20005


4.5

63

Conclusions

This chapter analyzed the volatility of TIIE futures contracts in relation with their maturity, i.e. the existence of maturity effect. The study complements and expands previous research using panel data techniques that permit cross-sectional and temporal analysis. In fact, descriptive statistics show that volatility has been consistently diminishing over time, indicating changes in return patterns and a possible reduction in information asymmetry in the Mexican futures markets. Results show that the common maturity effect in TIIE futures was present until 2004. Unexpectedly, volatility seems to be decreasing as time to maturity decreases in contracts expiring in 2005 and 2006, contrary to Samuelson hypothesis. Considering the performance of the spot TIIE during the analyzed period results for 2005 and some of the 2006 contracts may be reasonable. Particularly, the volatility of the spot rate registered during 2004 should be reflected between 13 to 7 trading months before expiration in contracts maturing in 2005. The TIIE reached its highest value around May 2005 and it was more or less stable until August, when it started to decrease. That is why volatility in 2005 contracts is higher in dates distant from maturity and lower when they approached to expiration. Panel analysis delivers the same conclusions, maturity effects are present in 2003 and 2004, inverse maturity effect appears in 2005 and 2005, and it indicates that there is not evidence of maturity effect once all contracts are considered (2003-2006). For individual series, results are qualitatively the same when the spot volatility is included as a proxy for information flow. In general, spot volatility does not explain futures volatility but only in 2005 contracts where there is an inverse relation. On the contrary, when panel data techniques are applied spot volatility explain futures volatility except for 2006 contracts and the maturity effect becomes statistically significant using the whole set of contracts. That is, panel analysis show that if information flow is controlled, evidence about the relation between volatility and maturity appears and results are contrary to Anderson & Danthine (1983). Finally, individual contract analysis of changes in the basis shows the expected maturity effect in contracts between September 2004 and March 2006, while panel analysis indicates an inverted effect in 2003 and the expected maturity effect in every year from 2004 and in the whole sample. In general it can be said that the volatility of the changes is decreasing


64

as contracts approach to expiration. The study of the behavior of volatility of futures prices near the maturity date has important implications for market participants, for derivatives pricing and for risk management. Hedging strategies that consider the effects of maturity normally outperform the strategies that do not. Clearinghouses set margin requirements on the basis of futures price volatility, in general, implying that the higher the volatility the higher the margin. Therefore, if there is any relation between volatility and time to maturity, the margin should be adjusted accordingly as the futures contract approaches its expiration date. Matching margins with price variability in an efficient way is the aim of an adequate margin policy. Although exchanges monitor price variability for different assets they do not usually consider differences among different contracts over the same underlying.

Chapter 5 Final conclusions Throughout the present work evidence has been provided about the existence of trading patterns and nonstationarity behavior in the TIIE futures contracts that in some cases are not present in similar futures contracts traded in other markets. The nonstationarity has been assessed not only in the next-to-expiration contract, but also in long term contracts. The main contribution of the study relies on the fact that previous empirical studies about TIIE futures contracts are limited and scarce. Also, to the best of our knowledge, nonstationary patterns in long term futures contracts has not been studied before and panel data techniques have not been used to asses maturity effects. The evidence provided in this study has several important implications for market participants, derivatives pricing and risk management. Firstly, considering that Mexican the TIIE futures contract is a highly liquid contract traded by very few participants, MexDer managers and regulators should be aware of the plausible collusion among institutions. This is important because it seems that the TIIE futures contract is not efficiently traded and that profitable strategies could be set taking short positions in Fridays and closing them on Mondays. Market efficiency is a key characteristic of a well functioning and mature market. Secondly, results about expiration effects show that market participants do not consider long term contracts for their hedging strategies. If there is not any price differentiation between short and long term contracts, there are no incentives to use long term contracts for hedging since basis risk is not compensated by lower fees and margin requirements. Finally, time to maturity should be considered in futures contracts pricing, speculation and hedging. If long term contracts are less volatile than short term contracts and margin requirements are

65

CHAPTER 5. FINAL CONCLUSIONS

66

based on volatility, then initial and maintenance margin should not be equal for different maturities. Hedging strategies that consider time to maturity are more effective than those that do not and, as a consequence, hedgers should be aware of the existence of maturity effects in TIIE futures contracts.

Bibliography Aggarwal, R., & Rivoli, P. (1989). “Seasonal and day-of-the-week effect in four emerging stock markets”. Financial Review, 24, 541-550. Agrawal, A., & Tandon, K. (1994). “Anomalies or illusions? Evidence of stock markets in eighteen countries”. Journal of International Money and Finance, 13, 83-106. Alegr´ıa, 2006”.

J.

(2006),

Presentation

“MexDer retrieved

en

la

Convención

May

15,

2007,

Anual from

de

la

FIA-

MexDer

Boca

webpage:

http://www.mexder.com.mx/MEX/Presentaciones.html. Andersen, T.G., & Bollerslev, T. (1998). “Deutsche mark-dollar volatility: Intraday activity patterns, macroeconomic announcements, and longer run dependencies”. The Journal of Finance, 53, 219-266. Anderson, R.W. (1985), “Some determinats of the volatility of futures prices”, The Journal of Futures Markets, 5, 331-348. Anderson, R.W., and Danthin, J. (1983), “The time pattern of hedging and the volatility of futures prices”, Review of Economic Studies, 50, 249-266. Aragó, F., and Fernández, A. (2002), “Expiration and maturity effect: empirical evidence from the Spanish spot and futures stock index”, Applied Economics, 34, 1617-1626. Baltagi, B.H. (1987), Econometric analysis of panel data, Chichester: John Wiley & Sons. Barnhill, T.M., Jordan, J.V., and Seale, W.E. (1987), “Maturity and Refunding Effects on Treasury-Bond Futures Price Variance”, Journal of Financial Research 10, 121-131. Beaulieu, M.-C., (1998), “Time to maturity in the basis of stock market indices: Evidence from the S&P500 and the MMI”, Journal of Empirical Finance, 5, 177-195.

67

BIBLIOGRAPHY

68

Berument, H., & Kiymaz, H. (2001). “The day of the week effect on stock market volatility”. Journal of Economics and Finance, 25, 181-193. Bessembinder, H., Coughenour, J.F., Seguin, P.J., Monroe Smoller, M., (1996), “Is there a term structure of futures volatilities? Reevaluating the Samuelson Hypothesis”, The Journal of Derivatives, 4, 45-57. Buckle, M., ap Gwilym O., Thomas, S.H., & Woodhams, M.S. (1998). “Intraday empirical regularities in interest Rate and equity index futures markets, and the effect of macroeconomic announcements”. Journal of Business, Finance & Accounting, 25 (7) & (8), 921-943. Castelino, M.G., and Francis, J.C. (1982), “Basis speculation in commoditiy futures: The maturity effect ”, The Journal of Futures Markets, 2, 195-206. Chen, Z., Daigler R. T. and Parhizgari, A.M. (2006), “Persistence of Volatility in Futures Markets”, The Journal of Futures Markets, 26, 571-594. Chen, Y.C., Duan J.C., and Hung, M.W.,(1999), “Volatility and maturity effects in the Nikkei index futures”, The Journal of Futures Markets, 19, 895-909. Chiang, R., & Tapley, C. (1983). “Day-of-the-week effects and the futures market”. Review of Research in Futures Markets, 2, 356-410. Crato, N., & Ray, B.K. (2000). “Memory in returns and volatilities of futures contracts”. The Journal of Futures Markets, 20, 525-543. Dyl, E.A., & Maberly, E.D. (1986a). “The daily ditribution of changes in the price of stock index futures”. The Journal of Futures Markets, 6, 513-521. Dyl, E.A., & Maberly, E.D. (1986b). “The weekly pattern in stock index futures: A further note”. The Journal of Finance, 61, 1149-1152. Ederington, L. H., & Lee, J. H. (1993). “How markets process information: News releases and volatility”. The Journal of Finance, 48, 1161-1191. Engle, R. (1982). “Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation”. Econometrica, 50, 987-1008.

BIBLIOGRAPHY

69

French, K. (1980). “Stock returns and the weekend effect”. Journal of Financial Economics, 8, 55-69. Galloway, T., & Kolb, R.W. (1996). Futures prices and the maturity effect. The Journal of Futures Markets, 16, 809-828. Gay, G.D., & Kim, T-H. (1987). “An investigation into seasonality in the futures market”. The Journal of Futures Markets, 7, 169-181. Geiss, C. (1995). “Distortion-free futures price series”. The Journal of Futures Markets, 15, 805-831. Grammatikos, T. and Saunders, A. (1986), “Futures price variability: A test of maturity and volume effects”, The Journal of Futures Markets 16 (7), 809-828. Han,L-M., Kling, J.L. and Sell, C.W. (1999), “Foreign exchange futures volatility: day-ofthe-week, intraday, and maturity patterns in the presence of macroeconomic announcements”, The Journal of Futures Markets 19, 665-693. Harvey, C.R. and Huang, R.D. (1991), “Volatility in the foreign currency futures market”, Review of Financial Studies, Vol. 3, 543-569. Hennessy, D. A. & Wahl, T. I. (1996).“The Effects of Decision Making on Futures Price Volatility”. American Journal of Agricultural Economics, 78, 591-603. Holz, R. (2007). “Trading volume: Nearly 10 billion contracts traded worldwide in 10 months”. Futures Industry Magazine, January-February. Retrieved March 5, 2007, from http://www.futuresindustry.org. Jaffe, J. and Westerfield, R. (1985), “The weekend effect in common stock returns: The international evidence”, The Journal of Finance, 40, 433-454. Johnston, E. T., Kracaw, W.A., & McConnell, J.J. (1991). “Day-of-the-week effects in financial futures: An analysis of GNMA, T-Bond, T-Note, and T-bill contracts”. Journal of Financial and Quantitative Analysis, 26, 23-44. Kenyon, D., Kenneth, K., Jordan, J., Seale, W., & McCabe, N. (1987). “Factors affecting agricultural futures price variance.” The Journal of Futures Markets, 7, 73-91.

BIBLIOGRAPHY

70

Khoury, N. and Yourougou, P. (1993), “Determinants of Agricultural Futures Price Volatilities: Evidence from Winnipeg Commodity Exchange”, The Journal of Futures Markets 13, 345-356. Lakonishok, J. & Levi, M. (1982). “Weekend effect in stock returns: A note”. The Journal of Finance, 37, 883-889. Lee, C.I., & Mathur, I. (1999). “Efficiency tests in the Spanish futures markets”. The Journal of Futures Markets, 19, 59-77. Low, A., Muthuswamy, J., Sajar, S. and Terry, E. (2001), “Multiperiod hedging with futures contracts”, The Journal of Futures Markets 22, 1179-1203. Ma, C., Mercer, J., & Walker, M. A. (1992). “Rolling over futures contracts: A note”. The Journal of Futures Markets, 12, 203-217. Milonas, N.T. (1986), “Price variability and the maturity effect in futures markets”, The Journal of Futures Markets 6, 443-460. Milonas, N.T., & Vora, A. (1985). “Sources of nonstationarity in cash and futures prices”. Review of Research in Futures Markets, 4, 315-326. Moosa, I.A., & Bollen, B. (2001), “Is there a maturity effect in the price of the S&P500 futures contract?”, Applied Economic Letters, 8, 693-695. Rougier, J. (1996). “An optimal price index for stock index futures contracts”. The Journal of Futures Markets, 16, 189-199. Rutledge, D.J.S. (1976), “A note on the variability of futures prices”, Review of Economics and Statistics, 58, 118-120. Samuelson, P. A. (1965), “Proof that Proper Anticipated Prices Fluctuate Randomly”, Industrial Management Review 6, 41-49. Samuelson, P. A. (1976), “Is real-world price a tale told by the idiot of chance?”, The Review of Economics and Statistics 58 (1), 120-123.

BIBLIOGRAPHY

71

Stoll, H.R., & Whaley, R.E. (1997). “Expiration-day effects of the All Ordinaries Share Price Index Futures: Empirical evidence and alternative settlement procedures”. Australian Journal of Management, 22, 139-174.