The rollover algorithm must follow an essential economic constraint imposing that the value of the portfolio changes only when the market prices of the individual bond futures move, with no change arising solely from the rollover procedure. When a rollover occurs, the number of new contracts to be bought is calculated so that the total amount of capital invested is constant, that is, the number of new contracts is given by the number of old contracts being sold multiplied by their middle price, divided by the middle price of the new contracts being bought. For bid-ask data, middle prices are defined as means of bid and ask prices.21 Middle price are used because traders can go long or short; the continuous series should not only reflect one of the two directions.

Two different schemes are presented:

A simple scheme involving a conversion factor to render continuous the transition from one contract to the next one, at a fixed date before the first contracts expiry.

The construction of bond futures portfolios with "constant mean time-to-expiry," through a daily partial rollover, whereas the constituent bond futures have a fixed calendar date expiry.

The timing of the rollover determines the character of the obtained continuous time series. It is possible to make a "first position series" or a "second position series" or something in between. Typically there is not sufficient data in the third position to allow for a serious study.

The proposed simple rollover scheme has just one contract in the reference portfolio at once. At a fixed delay D\ before the expiry of the contract in the portfolio, we roll over the entire holdings to the next expiry. If we start with one contract and the price at the rollover time is F\,T\-Di, we can afford to buy a number aj2 of new contracts at a price Fi\-d\?2 where

a12 = b7iza (2.10)

Clearly the number of contracts we hold at any time can be calculated as the product of the a factors of all past rollovers.

The execution of rollover procedures (even if it happens only in theory, not in real transactions) requires the approximate simultaneous availability of reliable market prices for the two involved contracts. This is not always guaranteed, but it is likely if the rollover time is chosen when both contracts are liquid. The a factors computed by Equation 2.10 are slowly varying over time, often slower than the prices themselves. This fact can be used to take a mean a from those daytimes where simultaneous quotes of both contracts are found, rather than an a determined at only one fixed daytime.

21 In the case of transaction data, the transaction prices take the role of middle prices.

22 In reality, we can only buy an integer number of contracts. This does not matter for our theoretical rollover formula: a may bc a fraction.

In empirical studies, however, few discontinuities of a factors over time are detected. Ballocchi and Hopman (1997) explain these discontinuities by asyn-chronicities in the underlying bonds, the "cheapest-to-deliver." If the underlying bonds of two successive contracts do not change exactly at the same time, the a factor is affected by the difference. When analyzing a continuous series, the exact knowledge of the underlying bonds (cheapest-to-deliver, or benchmark bonds) is helpful.

The second scheme based on a "constant time-to-expiry" is to have a portfolio that does not expire at a fixed calendar date, but keeps a constant mean time-to-expiry as time moves on, by means of an appropriate daily rollover procedure.23 The time-to-expiry (or horizon) h for a portfolio is defined as a weighted average of the time-to-expiry of the constituents. We consider a constant time-to-expiry portfolio consisting at time t of two contract expiries, with a number Bu of contracts in the first expiry, corresponding to time 7} and a number Yi+\,t of contracts in the second expiry, corresponding to time 7}+i. The time-to-cxpiry of this reference portfolio is then

PU -(7} - 0+ „ Y+]J (71+1-0 (2.11)

6,,, + Yi+\J Pi.t + Yi+l,t

Each day we arrange a partial rollover procedure, selling a proportion of the holdings in the first expiry and buying the second one, in order to keep h constant.

Some tools for generating long samples from several contracts through rollover schemes are commercially available, such as the Liffestyle program from LIFFE, see Gwilym and Sutcliffe (1999). This software also offers volume-dependent timing of the rollover (e.g., rolling over when the volume of a new contract overtakes that of an old contract).

2.6 COMMODITY FUTURES

Commodity futures are similar to the futures contracts presented in Section 2.1.2. The settlement at expiry means physical delivery of the underlying commodity. Commodities such as raw materials or agricultural products often exist in different variations and quality levels. Therefore a typical holder of a long position in commodity futures does not want to receive the commodity exactly in the form delivered at expiry. Most commodity futures traders offset their contracts (or roll them over) before expiry, in some markets so early that the second position (the contract with the second next expiry) has a higher liquidity than the first position.

A purpose of commodity futures trading is hedging. A manufacturer confronted with rapidly rising raw material prices is protected by holding some corresponding futures of simultaneously increasing value. Some investors use commodity futures as a vehicle for portfolio diversification.

23 The related high transaction costs are irrelevant, since this rollover procedure does not need to be executed in practice.

Futures of agricultural commodities may have an irregular schedule of expiry dates due to the seasonality of agricultural production. The cocoa futures market of New York, for example, has five irregularly spaced expiry dates per year, in March, May, July, September, and December.

As in other futures markets, contracts with different expiry dates are not independent. A contract with distant expiry, for instance, cannot be much more expensive than a near contract; otherwise traders would buy the near contract and profitably store the commodity afterward in a warehouse. This condition is similar to the condition that forward interest rates cannot be far below zero.

Commodity futures markets are often much smaller than FX or money markets. They are not liquid enough for huge transactions. Large orders often cause considerable slippage with immediate price movements to the unfavorable direction.

High-frequency commodity futures data are available from the exchanges and from data vendors. Rollover schemes similar to those of Section 2.5.2 are needed to build long time series from different contracts.

2.7 EQUITY MARKETS

Equity markets are a major source of high-frequency data. The authors of this book have only casually investigated time series from equity markets. Therefore, only a brief description is given.

Equities are traded at stock exchanges of different kinds. Also the instruments derived from equities are exchange-traded. High-frequency data are mainly produced during the opening hours of the exchanges. In some main markets, there is also some electronic trading outside the normal opening hours, which yields some sparse additional data.

High-frequency data are available from the following markets:

Equity of individual companies as traded by stock exchanges. This data type is strongly determined by the specific behavior of an individual firm and some general trends of the market and the economy. Stock splits (e.g. five new equity units replacing one old unit) and dividend payments affect the equity prices, and price series can only be understood with a full account of all these events. Only the most traded individual equities have a data frequency high enough to be called high-frequency.

Equity indices, also called stock indices. These are weighted sums of individual equity prices according to a formula. The basket of equities includes important equities of specific countries or industry sectors. The basket and the weights are adapted from time to time, according to the changing size of the companies. A performance index reflects the value of a realistic portfolio of investments according to the basket, including all dividend payments and reinvested profits. It is thus possible to replicate the behavior of an equity index by a real portfolio (it is a better approximation if the index is a performance index). Equity indices represent large segments