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Introduction to Zero Curve Construction

The term structure of interest rates describes the relationship between the yield on a particular instrument and its time to maturity. Graphically, we view the term structure of interest rates through the yield curve. A typical yield curve is based on the yield to maturity of a variety of coupon bearing bonds, where the yield to maturity is defined as the rate that equates the present value of the cashflows from the bond to the current market price. As such, the yield to maturity represents a weighted average of the required rates of return demanded by investors for each of the cashflows that they expect to receive over the life of the bond.

This information is of limited value when one wishes to price some other 'package' of cashflows, even when these cashflows are assumed to have a similar risk profile. For example, assume we wish to value a security that will provide only one cashflow of $100 in 3 years time. On the face of it, one might determine the appropriate discount rate for this cashflow with reference to the current required yield on a 3-year bond as described in the yield curve. However, if this yield relates to a bond that will provide periodic coupon payments, then the measured yield does not accurately reflect the required return on a single cashflow that occurs in 3 years. Instead it represents the average required return on all of the coupon payments as well as the redemption of the bond's face value at maturity.

What we really want to use however is the required rate of return on a security that only pays a single cashflow at the same time as the security under consideration (in 3 years time). Such a return is called the zero-coupon yield and the graphical representation of a collection of these rates is called the Zero-Coupon Yield Curve, or more simply, the Zero Curve. Unfortunately, in most cases we are only able to directly observe zero rates for a limited number of maturities, and the remainder of the points in the curve must be constructed using a combination of bootstrapping and interpolation techniques. These techniques are used to consistently merge data from one source with that from another, as well as to fill in the many gaps that will exist between the rates that are actually observed in the market.

A zero curve can be constructed for any class of interest rate and denominated in any type of currency. The particular curve under consideration will dictate what kind of base data is available to be used in the building process. For example, a zero curve describing returns from Government issued securities might be constructed using a small number of observed yields on discount bonds (which typically have relatively short-dated maturities) and a much larger set of coupon-bearing bonds. Perhaps the most commonly used zero curve is based on LIBOR rates and the curve is therefore constructed using LIBOR-based components. These components or inputs will typically include money market deposit rates, Eurodollar futures rates, and par swap rates.

The nature of the calculations that are used to combine and smooth all of the input rates in order to generate the zero curve differ depending on the type of curve that is being constructed, and therefore the type of input data that will be used. The zero curve routines supported by Resolution are only suitable for building curves like the LIBOR-based curve, where the input rates are drawn from cash and swap data.

 

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