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16 edge points and 7mat, where the forward period meets the forward periods of the neighbor contracts. The value of q at the meeting point 7exp is determined from the four implied forward rates nearest to 7exp (in a regular sequence of 3month futures contracts) by polynomial interpolation , 9 (q2 + q3) q\ 04 ,07. e(Te)tp) =  C27) where q is computed by Equation 2.5 and its index (1,2,3,4) indicates the position in the series of the four nearest forwards. For instance, qt, refers to the forward from rcxp to Tmat. Equation 2.7 can be interpreted as the interpolation of a cubic polynomial going through the values Qi ... qa, located at the midpoints of the corresponding forward periods. The same equation if shifted by one forward period leads to e(7mat). If a contract at the edge (q\ or 04 in Equation 2.7) is not available from the data source, we extrapolate 3 q2  q3  c Qi = (28) and analogously for Q4. The numerical impact of an extrapolation error is small, as qi and Q4 have little impact in Equation 2.7. By fulfilling all the mentioned requirements, the coefficients of the polynomial q(T) of Equation 2.6 can be formulated as a = [<?(7eXp) + e(7mat)  2 Q) (2.9) b = \ leTmat)  e(7exp)]  [6 Q  e(7exp)  0(Tmat)] where q follows from Equation 2.5 (q  , using the indexing of Equation 2.7). Now we can compute the annualized forward rate r for a given forward period by substituting Equations 2.5 through 2.9 into Equation 2.4. The integration is simple because the integrand, Equation 2.6, is a simple polynomial. The forward period to be considered (from rslart to 7enu) may typically extend over many original forward periods of the futures contracts. In this case, we need to integrate over several piecewise polynomials. A consequence of the polynomial interpolation is the potential overshooting of the yield curve. If the forward rates implied by a series of IR futures have a maximum somewhere around mediumterm expiries, an interpolated forward rate (for a period close to that maximum) may exceed all the implied forward rates corresponding to original futures quotes. The modest overshooting of polynomial interpolation is not undesirable as it leads to a strong reduction of the 3month seasonalities obtained for a forward IR series generated by linear interpolation (which has no overshooting). We conclude that polynomials with their smooth but
sometimes overshooting behavior represent true yield curves better than piecewise linear interpolation with hard corners at the nodes. In our method, overshooting is a local effectdistant contracts cannot affect the behavior of q(T). This is better than some methods based on spline interpolation, where even very distant contracts have an influence on the local behavior. A special case of overshooting might be "undershooting" of (forward) interest rates. Some parts of q{T) might have values below zero. The method should include an element to avoid strongly negative q{T), although interest rates can move slightly below zero under extreme circumstances. The polynomial method relies on a regular quarterly sequence of expiries. It can be adapted to include also irregular contracts such as contracts based on serial months. The period from the current time to the first expiry cannot be covered by IR futures. We need spot IRs to fill that gap, which is a method described by Muller (1996). After filling the gap, the described methods to compute implied forward interest rates can also be used to compute implied spot rates, simply by choosing rstart = 0, which implies the current time. The yield curve consists of a set of implied spot rates with different 1 There is an interesting application for the yield curve of implied interest rates by comparing the curve derived from IR futures at any point in time with the curve derived from other instruments (such as deposits or overthecounter forward rate agreements), which allows an investigation of arbitrage opportunities. Convexity corrections of the yield curve, which represent the difference between futures and forward contracts due to the presence of margining arrangements for futures (collateral accounts, see Burghardt and Hoskins, 1995), are negligibly small in our case, because the futures contracts under consideration are never more than 18 months from expiry. Time series of forward rates share many properties of FX time series, as studied in later chapters of this book. 2.5 BOND FUTURES MARKETS 2.5.1 Bonds and Bond Futures Bonds are the dominating financial instrument related to longterm interest rates.19 There is a wealth of bonds issued by governments or individual companies with different credit ratings. Bonds are interestpaying contracts. After a lifetime of several years, the capital is paid back to the holder of the contract. Some bonds are complicated financial constructions including special option contracts. The world of bonds is not simple enough to be studied in the form of few, long, consistent time series. The bond futures market, on the other hand, is more standardized. Similar to the shortterm interest rate futures discussed in Section 2.4, bond futures are a 19 There is also a market for interest rate swap transactions with maturities of few years (with a focus on shorter maturities than those of bonds). The IR swap rates of this market constitute yet another source of highfrequency IR data.
liquid financial instrument in the area of interest rate markets, traded at the same exchanges. Highfrequency, highquality intraday data are available in the form of transaction prices, sometimes bid and ask prices, and volume figures. Bond futures markets supply more accurate and more frequent information on bonds than the cash market for bonds. In spite of this, there is little published research on the intraday behavior of bond futures, Ballocchi and Hopman (1997). Bond futures are futures contracts as discussed in Section 2.1.2. As for most shortterm IR futures, there are four settlement months in a year (March, June, September, and December) known as quarterly expiries. The exact settlement and delivery rules can be obtained from the exchanges (or their web sites). A practical introduction to the bond futures markets can be found in LIFFE (1995a,b). Active trading of bond futures is focused on the first two positions which are the contracts with the nearest expiries. The underlying instruments of bond futures are often government bonds with maturity periods of years, for example the 30year U.S. Treasury bond futures traded at Chicago Board of Trade (CBOT). This long duration is the main difference from shortterm IR futures where the underlying instruments are notional 3month deposits. Threemonth interest rates are strongly influenced by monopolistic players such as the central banks with powers to set shortterm rates. In this respect, the bond market and thus the bond futures market is "freer" than the shortterm IR futures market. The underlying instrument of some typical bond futures is the cheapest bond(s) available to the exchange under certain conditions, the "cheapesttodeliver." Unfortunately, the choice of the cheapesttodeliver can change sometimes, leading to a disruptive behavior of bond futures prices. For studying long samples, we need to create continuous time series.20 A theoretically appealing method to construct such a series is to use arbitrage formulas such as in Hull (1993), with input from both shortterm and longterm interest rates, as well as from the underlying deliverable instrument, to find a relationship between two successive bond futures contracts. Such an approach, which could be seen as an extended variation of the approach of Section 2.4.2 based on forward IRs, would require data input from several sources and imply considerable methodological efforts. Instead of this, we suggest using rollover schemes, which allow for an analysis based on bond futures data only. 2.5.2 Rollover Schemes As mentioned in Section 2.1.2, a rollover scheme is a general way to create a continuous time series from the time series of futures contracts with different expiries. The proposed schemes have been mainly applied to bond futures in Ballocchi and Hopman (1997). The schemes are recommended to researchers but not necessarily to traders or investors who pay transaction costs. Investment strategies may also include rollovers, but the different optimization goal leads to different schemes. 20 As before, the word "continuous" means a consistent behavior as close to stationary as possible. The series should not suffer shocks unrelated to market movements when crossing the contract expiries.
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