· Contents. 1 Introduction 5 2 The TIIE rate and its futures contracts 11 2.1 Tasa de Inter es...

Trading patterns in the Mexican TIIE futures

market

GVELASCO

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 Interes 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

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.1 Day-of-the-week effects . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 Expiration day effects . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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

4.4 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4.1 Estimates of time-to-maturity effects on volatility . . . . . . . . . . . 49

1

4.4.2 Effect of controlling for variation in information flow . . . . . . . . . 54

4.4.3 Estimation of maturity effect on the basis. . . . . . . . . . . . . . . . 59

4.5 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Number of 28-day TIIE futures contracts traded per week relative to contract ex-

piration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . 6

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

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 Panel A. Conditional Mean Equation Estimates . . . . . . . . . . . . . . . . 28

3.5 Descriptive statistics for the estimated standardized residuals ut/√ht . . . . 32

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.9 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 fu-

tures 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,


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-of-

the-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 volatil-

ity 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 informa-

tion 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 Samuel-

son hypothesis empirically, yielding mixed results. In general, the maturity effect has been


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-of-

the-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 trad-

ing 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 con-

sideration 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


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.


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 strate-

gies. 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 be-

tween 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 Interes Interbancaria de Equilibrio (spot

TIIE)

Since March 1995, Banco de Mexico determines and publishes the short-term interest rate

benchmark known as Tasa de Interes 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 in-

formational 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

4.0

5.0

6.0

7.0

8.0

9.0

10.0

11.0

Dec-02 Jun-03 Dec-03 Jun-04 Dec-04 Jun-05 Dec-05 Jun-06 Dec-06

Daily

TIIE

(%)

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

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 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 (ap-

proximately 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 Eu-

ronext.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


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

0

50

100

150

200

250

300

1999 2000 2001 2002 2003 2004 2005 2006

Total

volum

e (mi

llions

of con

tracts

)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Daily

avera

ge (m

illion

s of c

ontra

cts)

Total volume 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.


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

volume

Broker

1 Santander2 ING Bank3 BBVA Bancomer4 Stock & Price5 Valmex Casa de Bolsa6 JP Morgan7 Banorte8 Grupo Financiero Scotiabank Inverlat9 HSBC

10 Invex11 Nacional Financiera12 Finamex13 GBM Casa de Bolsa14 IXE Banco15 Multivalores16 Grupo Financiero Banamex17 Monex18 Serafi Derivados19 Derfin20 Gamma Derivados21 Deutsche22 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-the-

week, 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 instru-

ments, 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 func-

tions. 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 af-

fect 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.


Seasonal patterns in futures price volatility have also been reported. Most studies at-

tribute 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 dif-

ferent 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 Data and methodology

3.3.1 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.


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 autore-

gressive 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.


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∗ 9.8719 −0.00038 0.0100 0.306 2.529 238.41∗ 19.83∗

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 autore-

gressive conditional heteroscedasticity effect with 5 lags.

* indicates significance at 5% level.


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

ration

-

10.0

20.0

30.0

40.0

50.0

60.0

0 6 12 18 24 30 36Months to expiration

Volum

e (m

illion

s of c

ontra

cts)

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

rt = µ+ φrt−1 +∑

k

δkDkt + ut, ut ∼ N (0, ht) (1)

where, for each of the series considered, µ is a constant for the mean equation, rt is the loga-

rithmic 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


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

tion.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0 2 4 6 8 10 12 14 16 18 20 22 24Weeks to expiration

Volum

e (mi

llions

of con

tracts

)

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:

ht = α0 + α1u2t−1 + β1ht−1 +

∑

k

γkDkt (2)

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 Results

3.4.1 Day-of-the-week effects

In testing for seasonality, a preliminary statistical analysis is performed using the stan-

dard 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 pro-

cedure 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. Fail-

ure 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.


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

Mon Tues Wed Thurs Fri All days t-stat F5

All 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

2003 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

2004 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

2005 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

2006 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.


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 ex-

plained 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 (Alegrıa, 2006).

The maximum-likelihood parameter estimates for the GARCH model with all the dum-

mies 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.


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

2003 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

2004 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

2005 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

2006 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

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).


Table 3.4: Panel A. Conditional Mean Equation Estimates

Series µ× 103 φ δM × 103 δT × 103 δH × 103 δF × 103 δZ × 103

1 −0.0289 0.1479∗ 0.4282 −0.8227∗ −0.2157 −0.6915 −0.3566

2 0.6116 0.1946∗ 0.2012 −1.1532 −1.833∗ −1.6175∗ −1.1068

3 0.7353 0.2167∗ 0.1971 −1.8535∗ −2.0438∗ −2.2670∗ −1.2414

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

6 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

9 0.1555 0.1580∗ 1.0509 −0.8342 −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∗

22 −0.2480 0.1332∗ 2.0021∗ −0.0286 −1.1202 −1.1698 −1.9709∗

23 0.1368 0.1246∗ 1.6228 −0.4532 −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

32 0.0938 0.1495∗ 1.6696 0.3327 −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 0.1351∗ −0.6433 0.0420 0.9731∗ −2.2720∗ 0.2375

Note. The table reports the conditional mean coefficients under the following GARCH specification:

rt = µ+ φrt−1 +∑

k

δkDkt + ut, ht = αo + α1u2t−1 + β1ht−1 +

∑

k

γkDkt

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.


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

Series α0 × 103 α1 β1 γM × 103 γT × 103 γH × 103 γF × 103 γZ × 103

1 0.0005 0.1157∗ 0.8872∗ −0.0093∗ −0.0006 −0.0002 0.0067∗ 0.0040∗

2 −0.0080∗ 0.0398∗ 0.9569∗ 0.0026 0.0052 0.0147∗ 0.0174∗ 0.0031∗

3 −0.0075∗ 0.0528∗ 0.9453∗ 0.0012 0.0112∗ 0.0139∗ 0.0103∗ 0.0039∗

4 −0.0162∗ 0.0670∗ 0.9332∗ 0.0037 0.0297∗ 0.0256∗ 0.0225∗ 0.0036∗

5 −0.0204∗ 0.0693∗ 0.9316∗ 0.0035 0.0341∗ 0.0296∗ 0.0334∗ 0.0059∗

6 −0.0195∗ 0.1657∗ 0.8353∗ 0.0006 0.0389∗ 0.0299∗ 0.0366∗ 0.0002

7 −0.0187∗ 0.0644∗ 0.9370∗ −0.0012 0.0355∗ 0.0326∗ 0.0238∗ 0.0088∗

8 −0.0069∗ 0.0720∗ 0.9280∗ −0.0266∗ 0.0239∗ 0.0039 0.0307∗ 0.0054

9 −0.0189∗ 0.0620∗ 0.9365∗ 0.0013 0.0394∗ 0.0206∗ 0.0319∗ 0.0042

10 −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 −0.0063 0.0065

16 0.0075 0.0866∗ 0.8868∗ −0.0113 0.0035 −0.0317∗ 0.0145∗ −0.0060

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∗

28 −0.0016 0.0465∗ 0.9529∗ −0.0121 0.0170 0.0025 −0.0045 0.0087∗

29 −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∗ 0.0004

36 0.0013 0.3408∗ 0.7040∗ 0.0339∗ −0.0199 0.0187 0 −0.0008

TIIE −0.0020 0.4543∗ 0.6598∗ 0.0123∗ −0.0060∗ 0.0056∗ 0.0079∗ 0.0010

Note. The table reports the conditional variance coefficients under the following GARCH specification:

rt = µ+ φrt−1 +∑

k

δkDkt + ut, ht = αo + α1u2t−1 + β1ht−1 +

∑

k

γkDkt

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


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 men-

tioning. 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 con-

tracts 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-


nouncements and trading activities. Given that on Tuesdays new information concerning

interest rates arrives, higher volatility should be related with these events, supporting Har-

vey & 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 Mon-

days, 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 con-

tract 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


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

Standarized residuals Squared standardized residuals

Series Skewness Kurtosis BJ LB(8) p-value LB(16) p-value LB(8) p-value LB(16) p-value

1 0.047 2.90 296.4 8.62 0.281 14.73 0.471 4.35 0.738 10.03 0.818

2 0.240 2.71 266.7 3.90 0.792 9.25 0.864 4.35 0.739 6.56 0.969

3 0.195 1.33 67.7 4.30 0.744 10.38 0.795 5.93 0.548 10.43 0.792

4 0.044 2.20 170.7 4.05 0.774 7.64 0.937 5.05 0.654 12.25 0.660

5 0.407 2.44 232.9 3.12 0.874 9.86 0.828 4.38 0.736 10.90 0.760

6 0.258 1.29 68.0 3.98 0.783 12.44 0.646 4.38 0.735 11.82 0.693

7 0.016 0.96 32.5 2.09 0.955 8.60 0.897 5.27 0.627 8.69 0.893

8 -0.036 1.12 44.4 4.59 0.710 11.00 0.752 3.51 0.834 5.46 0.988

9 0.119 0.57 13.4 5.29 0.624 10.04 0.817 7.75 0.355 12.55 0.637

10 0.143 0.79 24.8 3.11 0.874 6.41 0.972 3.41 0.845 8.09 0.920

11 0.194 1.33 67.6 1.01 0.995 6.65 0.967 6.76 0.454 12.86 0.613

12 0.249 1.81 124.1 0.66 0.999 4.96 0.992 3.52 0.833 11.73 0.699

13 0.148 1.30 62.6 1.35 0.987 8.09 0.920 3.57 0.828 13.82 0.539

14 0.177 1.08 45.5 3.17 0.869 11.90 0.686 2.42 0.933 10.67 0.776

15 0.178 1.06 44.0 2.62 0.918 10.11 0.813 3.94 0.787 7.61 0.939

16 0.393 1.78 133.2 3.32 0.854 8.91 0.882 4.30 0.744 6.25 0.975

17 0.154 0.86 29.4 9.87 0.196 16.71 0.336 4.85 0.678 7.23 0.951

18 0.153 1.88 127.8 8.22 0.313 17.06 0.316 1.99 0.960 5.29 0.989

19 0.193 1.07 45.6 6.73 0.458 12.91 0.609 2.03 0.958 4.41 0.996

20 0.104 1.05 40.4 5.48 0.602 12.08 0.673 1.81 0.969 5.25 0.990

21 -0.021 1.16 47.4 5.82 0.561 10.46 0.790 2.69 0.912 6.71 0.965

22 -0.061 1.11 43.9 4.31 0.743 8.21 0.915 2.47 0.929 8.57 0.899

23 -0.010 1.00 35.2 5.45 0.605 9.71 0.837 4.44 0.728 13.07 0.597

24 -0.083 1.27 57.8 8.63 0.280 14.07 0.520 7.29 0.399 14.17 0.513

25 -0.086 1.33 63.3 5.24 0.631 13.01 0.601 3.29 0.857 17.42 0.294

26 -0.018 1.46 75.1 4.01 0.779 11.98 0.680 3.68 0.816 14.57 0.483

27 0.086 1.21 52.6 3.58 0.827 9.24 0.865 6.26 0.510 12.88 0.611

28 0.097 1.32 62.7 2.43 0.933 5.62 0.985 4.01 0.779 11.37 0.726

29 0.131 1.37 68.5 2.15 0.951 6.22 0.976 6.62 0.470 12.96 0.606

30 0.266 1.61 101.2 2.86 0.897 5.33 0.989 2.59 0.920 8.23 0.914

31 0.295 1.73 117.7 5.38 0.614 7.58 0.940 1.94 0.963 7.91 0.927

32 0.327 1.99 154.4 5.16 0.641 9.18 0.868 7.27 0.401 12.79 0.619

33 0.839 4.58 837.6 9.92 0.193 15.93 0.386 28.88 0.000 32.63 0.005

34 0.131 4.20 623.5 7.42 0.386 21.71 0.116 20.31 0.005 22.35 0.099

35 0.156 4.25 639.4 6.51 0.482 22.59 0.093 14.05 0.050 33.56 0.004

36 0.077 3.27 377.3 5.94 0.546 9.85 0.829 4.12 0.766 14.69 0.474

Note. This table presents normality and correlation tests for standardized residuals and squared stan-

dardized residuals under the GARCH(1,1) model and for the estimated coefficients. LB(k) denotes the

Ljung-Box statistic with k lags.


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


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 partic-

ipants. 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 inves-

tigate 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 im-

portant 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


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.


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 ordi-

nary 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 hypothe-

sis, 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


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 in-

formation flow or time to expiry, but on production and demand inflexibilities arising from


decision making. Their results on CME commodity futures support of the maturity effect.

More recently, Arago & Fernandez (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 Data and methodology

4.3.1 Sample data

The study considers daily TIIE spot and futures rates between January 2003 and De-

cember 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


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

Standard ARCH

Contract Mean t-stat Deviaton Skewness Kurtosis BJ p-value (LM) p-value

Jan03 -0.2720 −0.86 0.3109 1.99 15.85 1824.63∗ 0.0000 2.59 0.7621

Feb03 -0.0052 −0.02 0.3210 1.86 15.50 1714.39∗ 0.0000 0.80 0.9770

Mar03 -0.0510 −0.16 0.3070 1.61 12.94 1100.25∗ 0.0000 0.74 0.9810

Apr03 -0.0216 −0.07 0.3114 0.95 10.02 532.22∗ 0.0000 2.89 0.7163

May03 -0.1014 −0.34 0.2929 1.34 13.86 1261.49∗ 0.0000 3.40 0.6379

June03 -0.2991 −1.02 0.2880 0.05 7.01 161.84∗ 0.0000 14.21∗ 0.0143

July03 -0.4714 −1.63 0.2839 0.44 6.96 165.52∗ 0.0000 13.61∗ 0.0183

Aug03 -0.5384 −1.87 0.2829 0.20 6.04 95.04∗ 0.0000 18.63∗ 0.0023

Sept03 -0.5568 −2.04∗ 0.2687 0.41 6.17 108.22∗ 0.0000 6.01 0.3054

Oct03 -0.7011 −3.12∗ 0.2212 0.32 4.03 14.91∗ 0.0006 3.53 0.6195

Nov03 -0.5935 −2.62∗ 0.2227 0.40 3.83 13.40∗ 0.0012 8.71 0.1212

Dec03 -0.6151 −2.58∗ 0.2345 0.36 4.15 18.65∗ 0.0001 2.69 0.7478

Jan04 -0.3708 −1.52 0.2399 1.00 8.47 342.19∗ 0.0000 2.74 0.7400

Feb04 -0.6025 −2.45∗ 0.2415 0.47 4.83 42.66∗ 0.0000 8.8 0.1173

Mar04 -0.5225 −2.08∗ 0.2472 0.39 5.15 52.82∗ 0.0000 11.25∗ 0.0467

Apr04 -0.3620 −1.39 0.2572 0.39 4.64 33.02∗ 0.0000 16.04∗ 0.0067

May04 -0.2959 −1.16 0.2514 0.12 3.72 5.83 0.0541 9.27 0.0986

June04 -0.0930 −0.37 0.2443 0.39 6.39 121.91∗ 0.0000 5.67 0.3400

July04 -0.0294 −0.13 0.2304 -0.02 5.42 59.08∗ 0.0000 18.35∗ 0.0025

Aug04 -0.0091 −0.04 0.2126 0.18 4.27 17.44∗ 0.0002 9.29 0.0981

Sept04 0.0337 0.16 0.2106 0.11 3.88 8.34∗ 0.0154 6.01 0.3056

Oct04 0.0916 0.43 0.2115 0.33 4.30 21.40∗ 0.0000 6.67 0.2462

Nov04 0.1948 0.91 0.2118 0.35 4.28 21.56∗ 0.0000 8.71 0.1211

Dec04 0.2563 1.29 0.1958 0.32 4.05 15.20∗ 0.0005 14.81∗ 0.0112

Jan05 0.2616 1.45 0.1772 0.25 4.13 15.32∗ 0.0005 12.99∗ 0.0235

Feb05 0.2416 1.38 0.1721 0.28 4.35 21.56∗ 0.0000 14.71∗ 0.0117

Mar05 0.3104 1.83 0.1673 0.29 4.27 19.70∗ 0.0001 12.39∗ 0.0298

Apr05 0.3393 2.04∗ 0.1634 0.34 4.44 25.42∗ 0.0000 16.58∗ 0.0054

May05 0.1717 1.16 0.1451 -0.17 3.31 2.17 0.3386 23.54∗ 0.0003

June05 0.1145 0.82 0.1372 -0.08 3.48 2.51 0.2849 27.35∗ 0.0000

July05 0.1426 1.21 0.1164 -0.03 3.24 0.62 0.7344 8.97 0.1103

Aug05 0.0732 0.68 0.1067 0.02 3.39 1.57 0.4563 4.32 0.5049

Sept05 0.0783 0.77 0.1003 0.07 3.67 4.76 0.0928 8.29 0.1410

Oct05 0.0376 0.39 0.0962 0.10 3.83 7.32∗ 0.0258 9.93 0.0771

Nov05 -0.0634 −0.66 0.0940 -0.11 3.84 7.48∗ 0.0237 9.58 0.0882

Dec05 -0.0734 −0.80 0.0906 -0.05 4.34 18.19∗ 0.0001 11.55∗ 0.0414

Jan06 -0.0687 −0.73 0.0924 0.00 4.03 10.70∗ 0.0047 3.99 0.5505

Feb06 -0.1602 −1.79 0.0880 0.09 3.93 9.03∗ 0.0110 4.13 0.5308

Mar06 -0.2365 −2.62∗ 0.0887 -0.11 4.04 11.32∗ 0.0035 5.70 0.3368

Apr06 -0.3270 −3.67∗ 0.0877 -0.27 4.15 16.29∗ 0.0003 8.43 0.1341

May06 -0.3231 −3.65∗ 0.0871 -0.38 4.57 30.55∗ 0.0000 3.07 0.6898

June06 -0.2917 −3.21∗ 0.0895 -0.30 4.25 19.40∗ 0.0001 5.74 0.3323

July06 -0.2496 −2.77∗ 0.0887 -0.15 4.36 19.64∗ 0.0001 6.25 0.2828

Aug06 -0.2722 −2.83∗ 0.0948 0.01 3.90 8.23∗ 0.0163 8.25 0.1432

Sept06 -0.2243 −2.37∗ 0.0931 0.24 4.42 22.56∗ 0.0000 10.10 0.0724

Oct06 -0.2221 −2.27∗ 0.0963 0.57 6.28 121.44∗ 0.0000 15.62∗ 0.0080

Nov06 -0.2058 −1.91 0.1060 0.36 6.40 121.75∗ 0.0000 21.59∗ 0.0006

Dec06 -0.1574 −1.42 0.1090 0.79 10.63 613.09∗ 0.0000 17.30∗ 0.0040

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 hy-

pothesis of normal distribution. The ARCH-LM is the LM-statistic of autoregressive conditional

heteroscedasticity effect with 5 lags. ∗ indicates 5% significance.


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. Stan-

dard 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 statis-

tic 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 = lnYT, t − lnSt.

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

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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

∆BTt = [lnYT, t+1 − lnSt+1]− [lnYT,t − lnSt]

= ∆YTt −∆St.


Table 4.3 presents summary statistics for the basis changes ∆BTt of each contract. Most

of the means are negative and tend to increase over time, although none of them is sta-

tistically 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 mono-

tonically 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 (Alegrıa, 2006).


Table 4.3: Descriptive statistics for daily basis changes

Standard ARCH

Contract Mean tstat Deviation Skewness Kurtosis BJ p-value (LM) p-value

Jan03 -0.3243 -0.65 0.4949 -0.11 8.33 286.50∗ 0.0000 34.68∗ 0.0000

Feb03 0.0175 0.04 0.4993 -0.24 8.65 324.53∗ 0.0000 32.53∗ 0.0000

Mar03 -0.1981 -0.40 0.4845 -0.04 7.94 246.00∗ 0.0000 30.75∗ 0.0000

Apr03 -0.3283 -0.69 0.4663 -0.28 8.12 267.68∗ 0.0000 36.03∗ 0.0000

May03 -0.2093 -0.46 0.4495 -0.30 10.45 563.42∗ 0.0000 42.69∗ 0.0000

June03 0.0011 0.00 0.4662 -0.71 8.49 324.00∗ 0.0000 18.32∗ 0.0026

July03 0.0475 0.10 0.4502 -1.03 7.60 256.04∗ 0.0000 14.95∗ 0.0106

Aug03 -0.0899 -0.19 0.4603 -0.77 5.79 102.73∗ 0.0000 14.77∗ 0.0114

Sept03 -0.0907 -0.20 0.4564 -0.75 5.84 103.48∗ 0.0000 12.39∗ 0.0298

Oct03 -0.0636 -0.15 0.4173 -0.69 5.12 64.21∗ 0.0000 15.85∗ 0.0073

Nov03 -0.2081 -0.49 0.4171 -0.66 4.56 42.01∗ 0.0000 17.62∗ 0.0035

Dec03 -0.1844 -0.43 0.4226 -0.72 4.88 56.40∗ 0.0000 17.01∗ 0.0045

Jan04 -0.0498 -0.12 0.4265 -0.59 4.22 28.92∗ 0.0000 9.26 0.0992

Feb04 0.0659 0.15 0.4237 -0.49 3.94 18.42∗ 0.0001 16.64∗ 0.0052

Mar04 0.0340 0.08 0.4396 -0.64 4.02 27.10∗ 0.0000 16.76∗ 0.0050

Apr04 -0.0090 -0.02 0.4632 -0.61 3.88 22.66∗ 0.0000 15.87∗ 0.0072

May04 -0.2548 -0.56 0.4510 -0.43 3.76 13.29∗ 0.0013 21.25∗ 0.0007

June04 -0.3877 -0.90 0.4253 -0.39 4.00 16.29∗ 0.0003 10.72 0.0572

July04 -0.2788 -0.71 0.3859 -0.32 3.68 8.74∗ 0.0127 10.44 0.0637

Aug04 -0.3407 -0.90 0.3726 -0.51 4.20 25.15∗ 0.0000 9.42 0.0934

Sept04 -0.3126 -0.84 0.3656 -0.50 4.14 23.05∗ 0.0000 10.12 0.0720

Oct04 -0.2403 -0.67 0.3533 -0.56 4.38 31.68∗ 0.0000 10.67 0.0582

Nov04 -0.3221 -0.93 0.3400 -0.57 4.58 38.30∗ 0.0000 14.07∗ 0.0152

Dec04 -0.0205 -0.06 0.3150 -0.55 5.35 67.91∗ 0.0000 17.58∗ 0.0035

Jan05 -0.3033 -1.02 0.2928 -0.69 6.02 110.97∗ 0.0000 22.07∗ 0.0005

Feb05 -0.2163 -0.80 0.2656 -0.58 6.73 153.79∗ 0.0000 29.23∗ 0.0000

Mar05 -0.1229 -0.50 0.2441 -0.39 7.39 200.36∗ 0.0000 58.42∗ 0.0000

Apr05 -0.1433 -0.61 0.2317 -0.40 8.71 334.68∗ 0.0000 69.86∗ 0.0000

May05 -0.1745 -0.90 0.1899 -0.13 3.19 1.06 0.5897 20.17∗ 0.0012

June05 -0.2499 -1.40 0.1757 -0.18 3.19 1.72 0.4226 26.15∗ 0.0001

July05 -0.2282 -1.48 0.1514 -0.40 3.34 7.56∗ 0.0228 13.69∗ 0.0177

Aug05 -0.2284 -1.62 0.1385 -0.47 3.73 14.07∗ 0.0009 34.14∗ 0.0000

Sept05 -0.1837 -1.41 0.1280 -0.58 4.43 34.10∗ 0.0000 41.56∗ 0.0000

Oct05 -0.1583 -1.29 0.1204 -0.56 4.98 52.04∗ 0.0000 49.96∗ 0.0000

Nov05 -0.1686 -1.40 0.1181 -0.56 5.06 55.36∗ 0.0000 56.20∗ 0.0000

Dec05 -0.0928 -0.82 0.1108 -0.34 4.85 39.13∗ 0.0000 39.80∗ 0.0000

Jan06 -0.0304 -0.27 0.1090 0.08 3.90 8.39∗ 0.0151 27.16∗ 0.0001

Feb06 -0.0505 -0.49 0.1004 0.27 3.71 8.11∗ 0.0174 12.27∗ 0.0312

Mar06 -0.0248 -0.23 0.1042 0.31 4.48 26.09∗ 0.0000 21.80∗ 0.0006

Apr06 -0.0524 -0.50 0.1038 0.28 4.78 34.93∗ 0.0000 16.68∗ 0.0052

May06 -0.0127 -0.12 0.1022 0.32 5.26 55.89∗ 0.0000 12.04∗ 0.0342

June06 0.0300 0.29 0.1037 0.19 4.79 33.87∗ 0.0000 17.09∗ 0.0043

July06 0.0718 0.69 0.1026 0.29 4.93 41.10∗ 0.0000 19.88∗ 0.0013

Aug06 0.0521 0.48 0.1069 0.30 4.35 21.94∗ 0.0000 12.07∗ 0.0338

Sept06 0.0752 0.70 0.1063 0.58 5.05 56.13∗ 0.0000 11.35∗ 0.0449

Oct06 0.0461 0.42 0.1072 0.74 6.00 112.70∗ 0.0000 12.14∗ 0.0329

Nov06 0.0404 0.34 0.1160 0.45 6.10 105.28∗ 0.0000 18.24∗ 0.0027

Dec06 0.0572 0.48 0.1176 0.80 9.10 401.23∗ 0.0000 18.16∗ 0.0028

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 ARCH-

LM is the LM-statistic of autoregressive conditional heteroscedasticity effect with 5 lags. ∗ indicates

5% significance.


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

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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 esti-

mation 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


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 + βiτ + uiτ (7)

corresponding to linear regressions the futures volatility on time to expiration. The hypoth-

esis is that if maturity effect is present, the coefficient βi should be negative.


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 in-

cluding 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.


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

4.4 Empirical results

4.4.1 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 statis-

tics. 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 in-

dividual 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

1Similar regressions were performed considering, instead of days to maturity, the squared root of days to

maturity. The results obtained are qualitatively the same.


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.


Table 4.4: Regression of daily volatility on days to expiration

Contract αi t-stats βi t-stats R2 D-W

Jan03 0.01636 7.58∗ -0.00003 −2.61∗ 0.014 1.73

Feb03 0.01725 8.40∗ -0.00004 −3.08∗ 0.019 1.84

Mar03 0.01617 8.87∗ -0.00003 −2.40∗ 0.011 1.89

Apr03 0.01497 7.96∗ -0.00001 −1.29 0.001 1.92

May03 0.01200 7.38∗ -0.00001 −0.51 -0.003 1.89

June03 0.01522 6.89∗ -0.00002 −1.67 0.010 1.74

July03 0.01647 7.80∗ -0.00003 −2.38∗ 0.022 1.69

Aug03 0.01760 8.99∗ -0.00003 −2.94∗ 0.033 1.78

Sept03 0.01400 7.95∗ -0.00001 −1.08 0.001 1.97

Oct03 0.01080 8.98∗ 0.00000 −0.08 -0.004 1.81

Nov03 0.01152 8.83∗ 0.00000 −0.59 -0.003 1.64

Dec03 0.01195 9.03∗ -0.00001 −0.69 -0.002 1.83

Jan04 0.01400 7.39∗ -0.00002 −2.04∗ 0.018 1.59

Feb04 0.01463 8.17∗ -0.00003 −2.41∗ 0.023 1.75

Mar04 0.01531 8.72∗ -0.00003 −2.71∗ 0.028 1.63

Apr04 0.01588 9.36∗ -0.00003 −2.82∗ 0.027 1.53

May04 0.01362 8.96∗ -0.00001 −1.29 0.002 1.68

June04 0.01451 8.18∗ -0.00003 −2.38∗ 0.022 1.75

July04 0.01359 9.09∗ -0.00002 −2.11∗ 0.018 1.62

Aug04 0.01352 10.11∗ -0.00003 −3.28∗ 0.035 1.84

Sept04 0.01260 9.51∗ -0.00002 −2.41∗ 0.018 1.92

Oct04 0.01038 8.40∗ 0.00000 −0.39 -0.004 1.82

Nov04 0.00828 7.29∗ 0.00001 1.63 0.005 1.99

Dec04 0.00751 7.72∗ 0.00001 1.99∗ 0.007 2.02

Jan05 0.00654 8.28∗ 0.00001 2.45∗ 0.012 2.01

Feb05 0.00512 6.67∗ 0.00002 3.48∗ 0.035 1.90

Mar05 0.00450 5.64∗ 0.00002 3.75∗ 0.054 2.01

Apr05 0.00256 3.11∗ 0.00004 5.29∗ 0.125 2.07

May05 0.00303 4.81∗ 0.00003 5.71∗ 0.117 2.17

June05 0.00240 4.05∗ 0.00003 5.99∗ 0.136 2.17

July05 0.00209 4.44∗ 0.00003 6.52∗ 0.129 2.12

Aug05 0.00240 5.44∗ 0.00002 5.24∗ 0.092 2.08

Sept05 0.00174 4.49∗ 0.00002 6.25∗ 0.117 1.93

Oct05 0.00145 3.18∗ 0.00002 5.73∗ 0.127 1.81

Nov05 0.00179 3.69∗ 0.00002 4.70∗ 0.089 1.87

Dec05 0.00184 3.73∗ 0.00002 4.19∗ 0.080 1.90

Jan06 0.00378 6.56∗ 0.00001 1.41 0.005 1.90

Feb06 0.00305 6.07∗ 0.00001 2.49∗ 0.021 1.90

Mar06 0.00299 5.73∗ 0.00001 2.62∗ 0.022 1.92

Apr06 0.00328 6.50∗ 0.00001 2.05∗ 0.013 1.93

May06 0.00363 6.94∗ 0.00000 1.16 0.002 1.97

June06 0.00368 6.64∗ 0.00000 1.24 0.002 1.81

July06 0.00386 7.12∗ 0.00000 0.67 -0.003 1.80

Aug06 0.00435 7.08∗ 0.00000 0.26 -0.004 1.77

Sept06 0.00364 6.32∗ 0.00000 1.32 0.002 1.68

Oct06 0.00353 5.72∗ 0.00001 1.37 0.002 1.62

Nov06 0.00429 6.95∗ 0.00000 0.87 -0.002 1.60

Dec06 0.00358 6.41∗ 0.00001 2.44∗ 0.009 1.65

The table reports the estimates of the regression model

σ(Y )iτ = αi + βiτ + εiτ

where τ represents days to maturity. R2 is the adjusted R2. DW is the Durbin-Watson test for

first-order serial correlation of the residuals. There are 242 observations. * indicates significance

at 5%. Estimation with heteroscedasticity-consistent standard errors.


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

Equality

Year Regression coefficients Analysis of variance of variances

estimate t-stat p-value Source F-test p-value χ2 p-value

2003 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

2004 α 0.012800 31.04 0.0000 Individual 1.7542 0.0566 71.57 0.0000

β -0.000015 -5.85 0.0000 Time 0.9520 0.6868 (df=11)

Joint 0.9870 0.5455

2005 α 0.002955 12.56 0.3144 Individual 21.6631 0.0000 609.11 0.0000

β 0.000023 15.28 0.0000 Time 0.9869 0.5443 (df=11)

Joint 1.8895 0.0000

2006 α 0.003637 21.34 0.0000 Individual 0.5384 0.8782 71.58 0.0000

β 0.000005 4.76 0.0000 Time 1.0488 0.2980 (df=11)

Joint 1.0265 0.3794

All α 0.008484 42.64 0.0000 Individual 31.953 0.0000 6430.03 0.0000

β -0.000002 -1.20 0.2313 Time 0.7779 0.9952 (df=47)

Joint 5.8655 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.


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

Year Regression coefficients

estimate t-stat p-value R2

2003 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 0.02862

β -0.000015 -5.59 0.00000


2005 α 0.002955 5.27 0.00000 0.19787

β 0.000023 11.13 0.00000


2006 α 0.003637 19.98 0.00000 0.02787

β 0.000005 4.45 0.00000


All α 0.008484 16.90 0.00000 0.11530

β -0.000002 -1.27 0.20404


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 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 station-

arity 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 statis-

tics. 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


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.


Table 4.7: Regression of daily volatility on days to expiration and spot volatility

Contract α t-stats β t-stats γ t-stats R2 DW

Jan03 0.015655 7.32∗ -0.000032 −2.71∗ 0.056243 0.91∗ 0.0148 1.778

Feb03 0.016236 7.89∗ -0.000040 −3.50∗ 0.108008 1.68∗ 0.0297 1.932

Mar03 0.014954 7.82∗ -0.000030 −2.85∗ 0.119048 1.72∗ 0.0266 1.998

Apr03 0.014357 6.94∗ -0.000017 −1.46 0.057322 0.81 0.0007 1.976

May03 0.010627 5.45∗ -0.000009 −0.90 0.136344 1.59 0.0141 2.033

June03 0.013193 5.70∗ -0.000023 −1.72 0.130879 1.99 0.0349 1.929

July03 0.014004 5.31∗ -0.000025 −1.78 0.105744 1.66 0.0336 1.816

Aug03 0.015863 6.35∗ -0.000031 −2.39∗ 0.072071 1.20∗ 0.0370 1.865

Sept03 0.012400 5.52∗ -0.000010 −0.81 0.074178 1.32 0.0077 2.055

Oct03 0.009167 5.95∗ 0.000003 0.38 0.072022 1.80 0.0036 1.868

Nov03 0.010914 6.83∗ -0.000004 −0.45 0.028368 0.79 -0.0050 1.658

Dec03 0.010945 6.74∗ -0.000005 −0.53 0.048332 1.17 -0.0017 1.868

Jan04 0.013356 6.96∗ -0.000022 −2.02∗ 0.030458 0.66∗ 0.0161 1.610

Feb04 0.014505 8.03∗ -0.000025 −2.41∗ 0.006559 0.17∗ 0.0191 1.751

Mar04 0.014779 8.50∗ -0.000028 −2.72∗ 0.029454 0.73∗ 0.0259 1.664

Apr04 0.014596 8.45∗ -0.000031 −3.04∗ 0.084387 1.84∗ 0.0373 1.628

May04 0.013152 8.45∗ -0.000014 −1.45 0.040226 0.93 0.0014 1.716

June04 0.014311 7.74∗ -0.000026 −2.37∗ 0.013954 0.28∗ 0.0187 1.760

July04 0.013400 9.10∗ -0.000021 −2.05∗ 0.015833 0.34∗ 0.0142 1.624

Aug04 0.012597 9.11∗ -0.000030 −3.74∗ 0.102748 2.05∗ 0.0526 1.934

Sept04 0.011968 8.68∗ -0.000024 −3.06∗ 0.096045 2.16∗ 0.0327 1.994

Oct04 0.010237 8.11∗ -0.000006 −0.77 0.042906 0.99 -0.0046 1.861

Nov04 0.008230 7.17∗ 0.000010 1.25 0.030395 0.64 0.0025 2.012

Dec04 0.007487 7.65∗ 0.000011 1.59 0.021118 0.43 0.0039 2.036

Jan05 0.006551 8.28∗ 0.000015 2.32∗ -0.01412 −0.30∗ 0.0086 2.001

Feb05 0.005167 6.71∗ 0.000022 3.5∗ -0.0339 −0.71∗ 0.0326 1.879

Mar05 0.004623 5.74∗ 0.000027 4.06∗ -0.07455 −1.42∗ 0.0573 1.947

Apr05 0.002688 3.22∗ 0.000040 5.72∗ -0.09213 −1.65∗ 0.1308 1.996

May05 0.003469 5.42∗ 0.000032 6.07∗ -0.16581 −2.57∗ 0.1335 2.071

June05 0.002809 4.54∗ 0.000033 6.53∗ -0.1733 −2.42∗ 0.1523 2.087

July05 0.002326 4.92∗ 0.000028 6.71∗ -0.15774 −2.27∗ 0.1438 2.061

Aug05 0.002513 5.70∗ 0.000021 5.17∗ -0.09868 −1.37∗ 0.0959 2.043

Sept05 0.001763 4.55∗ 0.000022 5.59∗ -0.05191 −0.63∗ 0.1150 1.915

Oct05 0.001481 3.23∗ 0.000022 5.34∗ -0.03118 −0.39∗ 0.1242 1.791

Nov05 0.001795 3.71∗ 0.000018 4.19∗ -0.02017 −0.22∗ 0.0859 1.860

Dec05 0.001847 3.69∗ 0.000017 3.89∗ -0.00844 −0.08∗ 0.0757 1.898

Jan06 0.003808 6.62∗ 0.000005 1.39 -0.0189 −0.24 0.0011 1.890

Feb06 0.002895 5.70∗ 0.000008 2.37∗ 0.080446 0.61∗ 0.0211 1.924

Mar06 0.002986 5.39∗ 0.000009 2.62∗ 0.001298 0.02∗ 0.0181 1.916

Apr06 0.003186 5.82∗ 0.000007 2.06∗ 0.034588 0.51∗ 0.0093 1.946

May06 0.003503 6.44∗ 0.000004 1.20 0.048664 0.74 -0.0013 1.994

June06 0.003437 5.88∗ 0.000005 1.33 0.095035 1.26 0.0032 1.816

July06 0.003868 6.89∗ 0.000002 0.67 -0.00539 −0.09 -0.0067 1.794

Aug06 0.004366 7.00∗ 0.000001 0.27 -0.01072 −0.14 -0.0080 1.766

Sept06 0.003651 6.33∗ 0.000005 1.32 -0.0159 −0.20 -0.0016 1.670

Oct06 0.003543 5.73∗ 0.000006 1.44 -0.03956 −0.47 -0.0010 1.613

Nov06 0.004282 6.96∗ 0.000003 0.92 -0.0227 −0.29 -0.0063 1.594

Dec06 0.003552 6.42∗ 0.000009 2.44∗ -0.0372 −0.45∗ 0.0048 1.639

The table reports the estimates of the unrestricted regression model

σ(Y )iτ = αi + βiτ + γi σ(S)iτ + uiτ

where τ represents days to maturity and σ(S)iτ is the spot volatility. R2 is the adjusted R2. There

are 242 observations. Expiration month is excluded. * indicates significance at 5%.


Table 4.8: Test for individual and time effects in futures volatility series with TIIE spot

variance as control variable

Equality


estimate t-stat p-value Source F-test p-value χ2 p-value

2003 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

2004 α 0.012200 27.94 0.00000 Individual 1.1770 0.2974 67.13 0.0000

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

2005 α 0.002928 12.40 0.00000 Individual 20.3612 0.0000 597.19 0.0000

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

2006 α 0.003638 20.90 0.00000 Individual 0.5380 0.8786 71.57 0.0000

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

All α 0.007230 36.10 0.00000 Individual 15.0028 0.0000 5590.21 0.0000

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

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.


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


Estimate t-stat p-value R2

2003 α 0.013200 22.28 0.00000 0.01796

β -0.000018 -5.17 0.00000

γ 0.081500 5.56 0.00000


2004 α 0.012200 27.37 0.00000 0.01938

β -0.000017 -6.55 0.00000

γ 0.056200 4.32 0.00002


2005 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 0.00215

β 0.000005 4.74 0.00000

γ -0.003401 -0.13 0.89328


All 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 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 be-

tween 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

fittings (negative R2).

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 signif-

icant 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.


Table 4.10: Regression of basis changes volatility on days to expiration

Contract α t-stats β t-stats R2 DW

Jan03 0.01988 6.82∗ 0.00001 0.54 -0.003 1.49

Feb03 0.01949 6.84∗ 0.00001 0.54 -0.003 1.65

Mar03 0.01918 7.30∗ 0.00001 0.62 -0.003 1.56

Apr03 0.02052 8.10∗ 0.00000 −0.16 -0.004 1.53

May03 0.01503 6.59∗ 0.00002 1.25 0.002 1.52

June03 0.01976 5.56∗ 0.00000 −0.11 -0.004 1.53

July03 0.02606 7.81∗ -0.00005 −2.27∗ 0.020 1.63

Aug03 0.02942 9.62∗ -0.00006 −3.29∗ 0.041 1.67

Sept03 0.02500 8.26∗ -0.00003 −1.61 0.008 1.72

Oct03 0.02451 10.08∗ -0.00004 −2.66∗ 0.017 1.63

Nov03 0.02257 10.42∗ -0.00002 −1.43 0.002 1.67

Dec03 0.02147 11.11∗ -0.00001 −0.72 -0.003 1.57

Jan04 0.02349 10.34∗ -0.00002 −1.19 0.001 1.66

Feb04 0.02095 9.56∗ 0.00000 −0.03 -0.004 1.68

Mar04 0.02210 8.79∗ 0.00000 −0.25 -0.004 1.97

Apr04 0.01984 7.44∗ 0.00002 1.06 0.001 2.05

May04 0.01467 5.86∗ 0.00005 2.90∗ 0.036 2.08

June04 0.01749 6.32∗ 0.00002 1.32 0.005 2.00

July04 0.01715 7.60∗ 0.00001 0.89 -0.001 2.04

Aug04 0.01602 7.87∗ 0.00001 0.94 -0.001 2.04

Sept04 0.01394 7.35∗ 0.00003 2.10∗ 0.010 1.97

Oct04 0.01071 6.76∗ 0.00004 3.76∗ 0.037 1.98

Nov04 0.00826 5.87∗ 0.00006 5.00∗ 0.069 2.03

Dec04 0.00711 5.43∗ 0.00005 4.73∗ 0.067 1.94

Jan05 0.00521 4.08∗ 0.00006 5.02∗ 0.095 2.05

Feb05 0.00558 4.74∗ 0.00005 4.36∗ 0.080 1.88

Mar05 0.00474 4.34∗ 0.00005 4.72∗ 0.098 1.73

Apr05 0.00320 2.70∗ 0.00005 4.90∗ 0.133 1.74

May05 0.00368 4.55∗ 0.00004 6.13∗ 0.143 1.78

June05 0.00285 4.05∗ 0.00004 7.02∗ 0.176 1.77

July05 0.00250 4.23∗ 0.00003 6.93∗ 0.156 1.82

Aug05 0.00214 4.08∗ 0.00003 7.10∗ 0.151 1.63

Sept05 0.00155 3.11∗ 0.00003 6.90∗ 0.158 1.57

Oct05 0.00178 3.14∗ 0.00003 5.65∗ 0.132 1.57

Nov05 0.00170 2.80∗ 0.00003 5.44∗ 0.133 1.50

Dec05 0.00156 2.55∗ 0.00003 5.29∗ 0.130 1.54

Jan06 0.00426 5.97∗ 0.00001 1.59 0.009 1.60

Feb06 0.00341 6.02∗ 0.00001 2.68∗ 0.024 1.71

Mar06 0.00361 5.43∗ 0.00001 2.14∗ 0.015 1.59

Apr06 0.00372 5.96∗ 0.00001 1.92 0.010 1.65

May06 0.00427 6.97∗ 0.00000 0.84 -0.002 1.72

June06 0.00421 6.73∗ 0.00000 0.98 -0.001 1.64

July06 0.00431 7.23∗ 0.00000 0.71 -0.002 1.64

Aug06 0.00451 6.91∗ 0.00000 0.64 -0.002 1.69

Sept06 0.00373 6.10∗ 0.00001 1.83 0.008 1.64

Oct06 0.00366 5.80∗ 0.00001 1.70 0.006 1.61

Nov06 0.00435 6.72∗ 0.00000 1.17 0.000 1.59

Dec06 0.00332 5.58∗ 0.00001 2.93∗ 0.019 1.58

The table reports the estimates of the unrestricted regression model

σ(B)iτ = αi + βiτ + uiτ

where τ represents days to maturity and V Biτ is the basis volatility. R2 is the adjusted R2.There are 242 observations. Expiration month is excluded. * indicates significance at 5%.


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

Equality


Estimate t-stat p-value Source F-test p-value χ2 p-value

2003 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

2004 α 0.015977 23.46 0.000 Individual 6.509 0.000 127.8 0.000

β 0.000023 5.22 0.000 Time 1.043 0.321 (df = 11)

Joint 1.281 0.003

2005 α 0.003041 9.01 0.000 Individual 34.490 0.000 1077.4 0.000

β 0.000039 18.06 0.000 Time 0.988 0.541 (df = 11)

Joint 2.450 0.000

2006 α 0.003946 20.30 0.000 Individual 0.345 0.975 42.0 0.000

β 0.000006 5.20 0.000 Time 0.956 0.671 (df = 11)

Joint 0.929 0.774

All α 0.011218 34.61 0.000 Individual 59.694 0.000 9930.0 0.000

β 0.000014 6.66 0.000 Time 0.826 0.976 (df = 47)

Joint 10.433 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.


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


Estimate t-stat p-value R2

2003 α 0.021900 27.03 0.00000 -0.00741

β -0.000013 -2.35 0.01886


2004 α 0.015977 16.41 0.00000 0.03227

β 0.000023 5.28 0.00000


2005 α 0.003041 3.16 0.00158 0.26017

β 0.000039 12.09 0.00000


2006 α 0.003946 19.59 0.00000 0.018933

β 0.000006 5.01 0.00000


All α 0.011218 10.79 0.00000 0.20005

β 0.000014 7.21 0.00000


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.


4.5 Conclusions

This chapter analyzed the volatility of TIIE futures contracts in relation with their ma-

turity, 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


as contracts approach to expiration.

The study of the behavior of volatility of futures prices near the maturity date has impor-

tant 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 volatil-

ity, 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 ex-

changes 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 par-

ticipants, 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.

Alegrıa, J. (2006), “MexDer en la Convencion Anual de la FIA- Boca

2006”. Presentation retrieved May 15, 2007, from MexDer 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.

Arago, F., and Fernandez, 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 macroeco-

nomic 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 vari-

ance 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-of-

the-week, intraday, and maturity patterns in the presence of macroeconomic announce-

ments”, 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 fi-

nancial 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 Volatil-

ities: 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


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


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”, In-

dustrial 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.

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