The term structure of interest rates (or term structure) is simply a price or yield relationship among a set of securities that differ only in the tim- ing of their cash flows or their term until maturity. These securities
Term Structure Modeling 95
invariably have a specified set of other attributes in common so that the study of the term relationship is meaningful.
It is common to think of the term structure as consisting of the current- coupon U.S. Treasury issues only. This restriction is not necessary since it is possible to define other term structures derived from other securi- ties. For example, it is meaningful to define the term structure of sets of coupon or principal Treasury strips. Other examples include off-the-run Treasury issues, agency debentures, interest-rate swaps or the notes of single-A rated banks and finance companies. The set of securities used to define a term structure is called the reference set. A market sector (sometimes referred to as a market or a sector) consists of all those instruments described by a specific term structure. There is the market sector of coupon or principal Treasury strips, off-the-run Treasuries, agency debentures, interest-rate swaps, and single-A rated banks and finance companies, and so forth. Very often, the reference set for a mar- ket sector may have restrictions on the structure (non-callables only), liquidity (recent issues only), or price (close to par only) of the securities that make up the set.
The relationship expressed by the term structure is traditionally the par-coupon yield relationship, hence the terminology: yield curve. This also is not a necessary restriction. In general, the term structure could be thediscount function, the spot-yield curve, or some other expression of the price or yield relationship between the securities. Given the wide- spread usage of the (par) yield curve for the Treasury market, it is not surprising that many market sectors are defined from a reference set derived from the Treasury market. For example, the reference set that defines the agency debenture market is a set of yield spreads to the on- the-run Treasuries, so that a 5-year debenture issued by an agency may be priced at par to yield 15 basis points more than the current 5-year Treasury issue. If the Treasury issue is trading at a 6.60% yield to matu- rity, the par priced agency issue has a 6.75% coupon. By inference, from the spread quote of 15 basis points, the reference yield for the 5-year term is 6.75%. Similar statements can be made for the interest-rate swap and the corporate-bond markets.
It needs to be emphasized that the reference set of bonds used to define the term structure of interest rates and the resulting term struc- ture itself are not one and the same. Indeed, the term structure, as a complete description of the entire yield curve, ultimately can be used to analyze all manner of option laden, index amortizing swaps or deben- tures that are in the same market sector. The “vanilla” reference set con- sists of individual bonds that are used mainly to define the term structure or to derive its defining relationships—spot-yield curve, spot- rate process, discount function, and the like.
5-Audley/Chin-TermStructModel Page 95 Thursday, August 29, 2002 10:00 AM
96 INTEREST RATE AND TERM STRUCTURE MODELING
Theories about the term structure of interest rates fall into two cate- gories:
■ Qualitative theories seek to explain the shape of the yield curve based on economic principles. Three theories attract the widest attention: the expectations, liquidity-preference, and preferred-habitat (or hedging pressure) theories.
■ Quantitative theories seek to mathematically characterize the term structure (often in harmony with one of the qualitative theories).
Usually, a quantitative theory about the term structure of interest rates culminates in a mathematical model, a term structure model, that exhibits useful properties. Specifically, a term structure model is the mathematical representation of the relationship among the securities in a market sector. This formalizes the distinction between the reference set used to define a market sector and a term structure model.
Term Structure Models
The simplest and most familiar term structure model is the (semi-logarithmic) graph of the U.S. Treasury yield curve found daily in the Wall Street Journal and in the business section of many newspapers. This model is useful mainly as a visualization of the yield relationship between the most recently issued shorter-term Treasury instruments and bonds. The graph can be characterized by a mathematical equation and is one example of the set of interpolation models of the term structure. These
“connect-the-dots” models can be useful in providing a quantitative way to price bonds outside the current-coupon Treasury issues, but their utility is rather limited. Bonds that are valued through a linear-interpo- lation technique may not be “fairly” valued in the sense that an average yield may not be equal to the “par-coupon” yield corresponding to the same date. Later in this chapter we provide a discussion of how the par- coupon curve is constructed to be fairly valued in comparison to the set of reference Treasury issues.
The term structure model as described above simply provides a snapshot of the relationship between the yields for selected Treasury maturities on a given day. It is often required that term structure models exhibit additional “analytic” properties. One such property is the con- sistency associated with the preclusion of riskless arbitrage when the term structure model is used for pricing. More will be said about this later in the chapter. For now, it is intended merely to indicate that the
“visualization” of the yield relationship to term may be neither com- pletely useful nor adequate.
Term Structure Modeling 97
More generally, term structure models are called on to describe the evolution of a set of interest rates over time. This motivates the follow- ing distinction in classifying term structure models:
■ Static modelsof the term structure offer a mechanism to establish the
“present value of a future dollar” in a deterministic economy. That is, no allowance for uncertainty or interest-rate volatility is explicitly incorporated into the model.
■ Dynamic models of the term structure, in contrast to static models, explicitly allow for uncertainty in the future course of interest rates.
Ideally, a dynamic model of the term structure should have useful static models embedded within. That is, with no contingency on the receipt of a future cash payment or when there is an assumption of neg- ligible volatility, a dynamic model should correspond to a consistent static model.
The essence of term structure modeling is the process of converting the market description of a sector’s reference set (the data) into a math- ematical set of relationships that characterizes all issues in a sector. This is by no means trivial to do correctly. For example, the same model that correctly values a note in the Treasury market should also correctly value an option on that note, the futures contract into which that note may be deliverable, and an option on that futures contract. It should also reveal if the traded basis on that note is rich or cheap relative to the cash, futures, and options markets. It should also be able to describe any stripping or reconstitution opportunities between coupon and prin- cipal strips and the cash market. These analyses should not be the result of several models, but of a single term structure model.
A key element of the modeling process is to eliminate distinguishing characteristics associated with each constituent of the reference set. For example, in the on-the-run set of Treasury issues, there are bills as well as notes and bonds. The bills have different conventions for day count- ing, pricing, and yield expression from those of the coupon paying issues of the sector. These characteristics need to be removed prior to developing the mathematical relation of the term structure model (as do the distinguishing characteristics for notes and bonds). In this simple example, a model of the Treasury term structure might be the spot curve or the discount function, as opposed to a “connect-the-dots” model to which no yield adjustments have been made.
The mathematical relationship of a term structure model can be used to characterize all issues in a sector. As is the case for the Treasury sector, every instrument can be considered a collection of zero-coupon bonds (the maturities of which correspond to the coupon/principal payment dates, the
5-Audley/Chin-TermStructModel Page 97 Thursday, August 29, 2002 10:00 AM
denominations of which correspond to the amount of coupon/principal paid). Accordingly, the discount function or equivalently, its correspond- ing spot-yield curve, furnishes a pricing technique for each zero-coupon bond and, therefore, for each of the instruments. With this insight, the utility of an equivalence between the spot-yield curve and discount func- tion, which are derived from the original reference set, is readily apparent.
It will be seen later that a technical discussion of term structure mod- els is really equivalent to a discussion of the (zero-coupon) spot-yield curve. The theory of the term structure of interest rates focuses on a term structure model that models the movement of the spot (zero-coupon) yield over time. Once such a term structure model is developed, any coupon paying bond may be viewed in terms of its constituent zero-coupon bonds and analyzed in the context of this term structure model.
Dynamic Term Structure Models
Modern financial markets are predicated on the notions of contingency and uncertainty. Many recent financial innovations are directed at cop- ing with the uncertainty of markets and the contingency of obligations.
As part of this evolutionary process, dynamic models of securities and their behavior in the markets are at the forefront of financial economic research and application. In the fixed-income markets, this condition dominates and drives the need for dynamic term structure models.
The dynamic term structure model of a market sector, as defined by a reference set of securities, is a mathematical set of relationships that can be used to characterize any security in that market sector in which mar- ket uncertainty dominates the expected timing and receipt of cash flows.
There are several qualitative essentials that need to be accommodated by a useful modeling approach. The ability to value fixed-income securities at any point in time (present or future) for conventional or forward set- tlement is a necessary first step. This is especially true in the valuation of compound or derivative instruments. Indeed, before the value of a bond option may be determined, the ability to calculate the (probabilistic) expected value of the bond on the future exercise date (conditioned on current market condition) is needed. Complementing this, reasonable variations from this expectation also need to be determined and weighed relative to the expected outcome. It is essentially this same idea that allows for the analysis of a futures contract, an interest-rate cap, or an option on a swap. In addition, to determine the performance risk that results from market moves, a rationale for incorporating market changes needs to be embedded into the modeling process.
With these premises in mind, the following assertions regarding dynamic models for the term structure of interest rates are postulated:
Term Structure Modeling 99
■ The model must have the capability to extrapolate into the future an equilibrium evolution of the term structure of interest rates, given its form on a specified day, and must preclude riskless arbitrage.
■ The model must allow a probabilistic description of how the term structure may deviate from its expected extrapolation while maintain- ing the model’s equilibrium assumption.
■ The model must embody a rationale to incorporate perturbationsfrom the equilibrium that correspond to the economic fundamentals that drive the financial markets.
This treatise is focused on a dynamic term structure model that responds to the imperatives outlined.