Zero-coupon bonds yield curve

Bond market participants take a keen interest in both the cash and the zero-coupon (spot) yield curves. In markets where an active zero-coupon bond market exists, the spreads between implied and actual zero-coupon yields also receive much artention. 

Chapter 3 discussed how a coupon-bond yield curve could be used to derive spot (zero-coupon) and implied forward rates. A forward rate is the interest rate for a term beginning at a future date and maturing one period later. Forward rates form the basis of binomial interest rate trees. 

Of course, interest rates are not constant but Equation (3.1) is valuable as it is used later in constructing a model. By using Equation (3.1), we are able to produce a yield curve, given a set of zero-coupon bond prices. For modelling purposes, we require a definition of the short rate, or the current interest rate for borrowing a sum of money that is paid back a very short period later (in fact, almost instantaneously). This is the rate payable at time t for repayment at time t+M where Af is an incremental passage of time. This is given by... 

Like the bond price function, the yield on a zero-coupon bond is a function of the short-rate r and follows a normal distribution the yield curve is a function of the short-rate r, the time t and the time to maturity T. The Imig-run average future interest over the time to maturity (t, T) is normally distributed and given by ... 

In this chapter, we have considered both equilibrium and arbitrage-free interest-rate models. These are one-factor Gaussian models of the term structure of interest rates. We saw that in order to specify a term structure model, the respective authors described the dynamics of the price process, and that this was then used to price a zero-coupon bond. The short-rate that is modelled is assumed to be a risk-free interest rate, and once this is modelled, we can derive the forward rate and the yield of a zero-coupon bond, as well as its price. So, it is possible to model the entire forward rate curve as a function of the current short-rate only, in the Vasicek and Cox-Ingersoll-Ross models, among others. Both the Vasicek and Merton models assume constant parameters, and because of equal probabilities of forward rates and the assumption of a normal distribution, they can, xmder certain conditions relating to the level of the standard deviation, produce negative forward rates. 

In seeking to develop a model for the entire term structure, the requirement is to model the behaviour of the entire forward yield curve, that is, the behaviour of the forward short-rate/(f, T) for all forward dates T. Therefore, we require the random process f(T) for all forward dates T. Given this, it can be shown that the yield R on a T-maturity zero-coupon bond at time t is the average of the forward rates at that time on all the forward dates s between t and T, given by Equation (4.1) ... 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

Z-spread The Z-spread or zero volatility spread calculates the yield spread of a corporate bond by taking a zero-coupon bond curve as benchmark. Conversely to other yield spreads, the Z-spread is constant. In fact, it is found as an iterative procedure, which is the yield spread required to get the equivalence between market price and the present value of all its cash flows. The Z-spread is given by Equation (8.2) ... 

Chapter 4 provides a comprehensive discussion of duration. Duration is the change in the price of a bond as a result of a very small shift in its yield. In other words, duration measures the sensitivity of the price of a bond to changes in its yield. An increasingly common measure of duration is effective duration, which is the measure of price sensitivity due to a small parallel shift in the spot curve. One would immediately realise that these two definitions of duration would give identical results for zero-coupon bonds and different results for most other instruments. 

Some texts refer to the graph of coupon-bond yields plotted against maturities as the term structure of interest rates. It is generally accepted, however, that this phrase should be used for zero-coupon rates only and that the graph of coupon-bond yields should be referred to instead as the yield curve. Of course, given the law of one price—which holds that two bonds having the same cash flows should have the same values—the zero-coupon term structure is related to the yield to maturity curve and can be derived from it. 

Equilibrium interest rate models also exist. These make the same assumptions about the dynamics of the short rate as arbitrage models do, but they are not designed to match the current term structure. The prices of zero-coupon bonds derived using such models, therefore, do not match prices seen in the market. This means that the prices of bonds and interest rate derivatives are not given purely by the short-rate process. In brief, arbitrage models take as a given the current yield curve described by the... 

Although the term zero-coupon rate refers to the interest rate on a discount instrument that pays no coupon and has one cash flow at maturity, constructing a zero-coupon yield curve does not require a functioning zero-coupon bond market. Most financial pricing models use a combination of the following instruments to construct zero-coupon yield curves ... 

This section describes the relationships among spot interest rates and the actual market yields on zero-coupon and coupon bonds. It explains how an implied spot-rate curve can be derived from the redemption yields and prices observed on coupon bonds, and discusses how this curve may be used to compare bond yields. Note that, in contrast with the common practice, spot rates here refer only to rates derived from coupon-bond prices and are distinguished from zero-coupon rates, which denote rates actually observed on zero-coupon bonds trading in the market. 

Spot yields cannot be directly observed in the market. They can, however, be computed from the observed prices of zero-coupon bonds, or strips, if a liquid market exists in these securities. An implied spot yield curve can also, as the previous section showed, be derived from coupon bonds prices and redemption yields. This section explores how the implied and actual strip yields relate to each other. 

Two observations are worth making. First, the zero-coupon bonds are shown to be trading cheap relative to the spot curve throughout the term structure. This indicates that investors at the time were not prepared to hold strips unless they could earn a spread above their theoretical yields. This probably reflected the inverted yield curve during the period, which meant that strips would be expensive relative to coupon bonds of the same maturity. It should be noted, however, that strips were expensive relative to the spot curve from the 11- to the 15-year point on the curve. Second, principal strips trade at lower yields than coupon strips of the same maturity, reflecting the fact that investors prefer holding the former. 

The first method equates a strips value with its spread to a bond having the same maturity. The main drawback of this rough-and-ready approach is that it compares two instruments with different risk profiles. This is particularly true for longer maturities. The second method, which aligns strip and coupon-bond yields on the basis of modified duration, is more accurate. The most common approach, however, is the third. This requires constructing a theoretical zero-coupon curve in the manner described above in connection with the relationship between coupon and zero-coupon yields. 

When the bond yield curve is flat, the spot curve is too. When the yield curve is inverted, the theoretical zero-coupon curve must lie below it. This is because the rates discounting coupon bonds earlier cash flows are higher than the rate discounting their final payments at redemption. In addition, the spread between zero-coupon and bond yields should decrease with maturity. 

The potential profit from stripping a Treasury coupon depends on current market Treasury yields and the implied spot yield curve. Consider a hypothetical 5-year, 8 percent Treasury trading at par—and therefore offering a yield to maturity of 8 percent—in the yield curve environment shown in figure 16.2. A market maker buys the Treasury and strips it with the intention of selling the resulting zero-coupon bonds at the yields indicated in figure 16.2. 

The term structure of interest rates is the spot rate yield curve spot rates are viewed as identical to zero-coupon bond interest rates where there is a market of liquid zero-coupon bonds along regular maturity points. As such a market does not exist anywhere the spot rate yield curve is considered a theoretical construct, which is most closely equated by the zero-coupon term structure derived from the prices of default-free liquid government bonds. 

Using equation 14.16, we can build a forward inflation curve provided we have the values of the index at present, as well as a set of zero-coupon bond prices of required credit quality. Following standard yield curve analysis, we may build the term structure from forward rates and therefore imply the real yield curve, or alternatively we may construct the real curve and project the forward rates. However, if we are using inflation swaps for the market price inputs, the former method is preferred because IL swaps are usually quoted in terms of a forward index value. 

Z-spread is an alternative spread measure to the ASW spread. This type of spread uses the zero-coupon yield curve to calculate the spread, in which in this case is assimilated to the interest-rate swap curve. Z-spread represents the spread needful in order to obtain the equivalence between the present value of the bond s cash flows and its current market price. However, conversely to the ASW spread, the Z-spread is a constant measme. 

A landmark development in interest-rate modelling has been the specification of the dynamics of the complete term stracture. In this case, the volatility of the term structure is given by a specified functiOTi, which may be a function of time, term to maturity or zero-coupon rates. A simple approach is described in the Ho-Lee model, in which the volatility of the term structure is a parallel shift in the yield curve, the extent of which is independent of the current time and the level of current interest rates. The Ho-Lee model is not widely used, although it was the basis for the HJM model, which is widely used. The HJM model describes a process whereby the whole yield curve evolves simultaneously, in accordance with a set of volatility term structures. The model is usually described as being one that describes the evolution of the forward rate however, it can also be expressed in terms of the spot rate or of bond prices (see, e.g., James and Webber (1997), Chapter 8). For a more detailed description of the HJM framework refer to Baxter and Rennie (1996), Hull (1997), Rebonato (1998), Bjork (1996) and James and Webber (1997). Baxter and Reimie is very accessible, while Neftci (1996) is an excellent introduction to the mathematical background. 

As above, assuming a constant average inflation rate, which is then used to calculate the value of the bond s coupon and redemption payments. The duration of the cash flow is then calculated by observing the effect of a parallel shift in the zero-coupon yield curve. By assuming a constant inflation rate and constant increase in the cash flow stream, a further assumption is made that the parallel shift in the yield curve is as a result of changes in real yields, not because of changes in inflation expectations. Therefore, this duration measure becomes in effect a real yield duration ... 

The change in the short rate will result in a 50-basis point decline in all the expected future interest rates. However, this will not result in a uniform fall in all bond yields. The impact on the zero-coupon curve and the forward rate curve is shown in Figure 7.4. 

In chapter 2 of the companion volume to this book in the boxed-set library, Corporate Bonds and Structured Financial Products, we introduced the concept of the yield curve, and reviewed some preliminary issues concerning both the shape of the curve and to what extent the curve could be used to infer the shape and level of the yield curve in the future. We do not know what interest rates will be in the future, but given a set of zero-coupon (spot) rates today we can estimate the future level of forward rates using a yield curve model. In many cases however we do not have a zero-coupon curve to begin with, so it then becomes necessary to derive the spot yield curve from the yields of coupon bonds, which one can observe readily in the market. If a market only trades short-dated debt instruments, then it will be possible to construct a short-dated spot curve. 

It is important for a zero-coupon yield curve to be constructed as accurately as possible. This because the curve is used in the valuation of a wide range of instruments, not only conventional cash market coupon bonds, which we can value using the appropriate spot rate for each cash flow, but other interest-rate products such as swaps. 

Exhibit 16.5 shows very clearly the price responses for three 20-year bonds offering 10%, 6% annual coupon, and zero-coupon payments. One feature worth noting is that the curvilinear, price/yield relationship is not constant. It varies at different points on the curve for each type of... 

We derive daily zero-coupon yield curves from five countries of the Eurozone (France, Germany, Italy, Spain, and the Netherlands) during the period from 2 January 2001 to 21 August 2002, using zero-coupon rates with 26 different maturities ranging from one month to 30 years. The yield curves are extracted from daily Treasury bond market prices by using a standard cubic B-splines method. Our input baskets are composed of... 

Part Three presents the author s insights into trading, based on his experiences working as a gilt-edged market maker and sterling-bond proprietary trader. The topics covered include implied spot and market zero-coupon yields, yield-curve spread trading, and butterfly spreads. 

Between the short- and long-term horizon dates is one at which the net effect of the change in reinvestment rate on the bond s future value is close to zero. At this date, the bond behaves like a single-cash-flow or zero-coupon security, and its future value can be predicted with greater certainty, no matter what the yield curve does after its purchase. Defining this date as Sh interest periods after the purchase date and Ph as the value of the bond at that point, it can be shown that the bond s rate of return up to this horizon date is the value for rntn that solves equation (16.6). 

This section discusses the factors that must be assessed in analyzing the relative values of government bonds. Since these securities involve no credit risk (unless they are emerging-market debt), credit spreads are not among the considerations. The zero-coupon yield curve provides the framework for all the analyses explored. 

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