Zero-coupon bonds spot price

The continuously compounded constant spot rate is r as before. An investor has a choice of purchasing the zero-coupon bond at price P(t, T), which will return the sum of 1 at time T, or of investing this same amount of cash in the money market account, and this sum would have grown to 1 at time T. We know that the value of the money market accoxmt is given by Me If M must have a... 

In this section, we describe the relationship between the price of a zero-coupon bond and spot and forward rates. We assume a risk-free zero-coupon bond of nominal value 1, priced at time t and maturing at time T. We also assume a money market bank account of initial value P t, T) invested at time t. The money market account is denoted M. The price of the bond at time t is denoted P t, T) and if today is time 0 (so that t > 0), then the bmid price today is unknown and a random factor (similar to a future interest rate). The bond price can be related to the spot rate or forward rate that is in force at time t. 

The zero-coupon bond price may also be given in terms of the spot rate r(t, T), as shown in Equation (3.18). From our earlier analysis, we know that... 

Equation (3.22) describes the bond price as a function of the spot rate only, as opposed to the multiple processes that apply for aU the forward rates from t to T. As the bond has a nominal value of 1, the value given by Equation (3.22) is the discount factor for that term the range of zero-coupon bond prices would give us the discount function. 

The expression for the value of the money market account can be used to determine the expression for the zero-coupon bond price, which we denote as P(t, T). The money market account earns interest at the spot rate rit), while the bond price is the present value of 1 discounted at this rate. Therefore, the inverse of Equation (4.8) is required, which is... 

Let us look now at the T-period forward rate again as a function of the range of spot rates from the time f today to point T in more detail than in Section 7.1. If P t, T) is the price today of a zero-coupon bond that has a redemption value of 1 at time T, then this price is given in terms of the instantaneous stmcture of forward rates by Equation (7.9) ... 

However, the price of the zero-coupon bond is also given in terms of the spot rate as the expression in (7.1), where E, is the expectation under the risk-free probability function. Therefore, forward rates are related to the expected level of the instantaneous spot rates, and if we differentiate the expression in (7.10), we obtain a result that states that the forward rate is a weighted average of the range of spot rates in the period t to T. This is given in Equation (7.11), which we encountered earUer as Equation (7.2) ... 

P 0, is the spot price of a zero-coupon bond paying 1 at time... 

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. 

A zero-coupon bond is the simplest fixed-income security. It makes no coupon payments during its lifetime. Instead, it is a discount instrument, issued at a price that is below the face, or principal, amount. The rate earned on a zero-coupon bond is also referred to as the spot interest rate. The notation P t, T) denotes the price at time r of a discount bond that matures at time T, where T >t - The bond s term to maturity, T - t, is... 

Expression (3.14) is the formula for pricing zero-coupon bonds when the spot rate is the nonconstant instantaneous risk-free rate r(s) described above. This is the rate used in formulas (3.12), for valuing a money market account, and (3.15), for pricing a risk-free zero-coupon... 

The approach described in Heath-Jarrow-Morton (1992) represents a radical departure from earlier interest rate models. The previous models take the short rate as the single or (in two- and multifactor models) key state variable in describing interest rate dynamics. The specification of the state variables is the fundamental issue in applying multifactor models. In the HJM model, the entire term structure and not just the short rate is taken to be the state variable. Chapter 3 explained that the term structure can be defined in terms of default-free zero-coupon bond prices or yields, spot rates, or forward rates. The HJM approach uses forward rates. 

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. 

As noted in chapter 2, a Treasury bond can be seen as a bundle of individual zero-coupon securities, each maturing on one of the bond s cash flow payment dates. In this view, the Treasury s price is the sum of the present values of all the constituent zero-coupon bond yields. Assume that the spot rates for the relevant maturities—ri,r2,rg,.rj f—can be observed. If a bond pays a semiannual coupon computed at an annual rate of C from period 1 to period N, its present value can be derived using equation (16.7). 

The first bond matures in precisely six months and thus has no intermediate cash flow before redemption. It can therefore be treated as a zero-coupon bond, and its yield of 6 percent taken as the 6-month spot rate. Using this, the 1-year spot rate can be derived from the price of a 1-year coupon Treasury. The principle of no-arbitrage pricing requires that the price of a 1-year Treasury strip equal the sum of the present value of the coupon Treasury s two cash flows ... 

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. 

We may use the spot rate term structure to value a default-free zero-coupon bond, so for example, a two-period bond would be priced at 89. Using the forward rate, we obtain the same valuation, which is exactly what we expect. ... 

A rise in volatility generates a range of possible future paths around the expected path. The actual expected path that corresponds to a zero-coupon bond price incorporating zero OAS is a function of the dispersion of the rai e of alternative paths around it. This dispersion is the result of the dynamics of the interest-rate process, so this process must be specified for the current term structure. We can illustrate this with a simple binomial model example. Consider again the spot rate structure in Table 12.1. Assume that there are only two possible future interest rate scenarios, outcome 1 and outcome 2, both of equal probability. The dynamics of the short-term interest rate are described by a constant drift rate a, together with a volatility rate a. These two parameters describe the evolution of the short-term interest rate. If outcome 1 occurs, the one-period interest rate one period from now will be... 

In FIGURE 12.1, we present the possible interest-rate paths under conditions of 0 percent and 25 percent volatility levels, and maintaining our assumption that the current spot rate stmcture price for a risk-free zero-coupon bond is identical to the price generated using the structure to obtain a zero... 

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. 

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. 

In an important application of the HIM model, Jarrow and TumbuU (1996) express the price of a zero-coupon risk-free bond as a function of the spot rate r(f), given by... 

Put simply, the Z-spread is the basis point spread that would need to be added to the implied spot yield curve such that the discounted cash flows of the bond are equal to its present value (its current market price). Each bond cash flow is discounted by the relevant spot rate for its maturity term. How does this differ from the conventional asset-swap spread Essentially, in its use of zero-coupon rates when assigning a value to a bond. Each cash flow is discounted using its own particular zero-coupon rate. The bond s price at any time can be taken to be the market s value... 

Part One, Introduction to Bonds, covers bond mathematics, including pricing and yield analytics. This includes modified duration and convexity. Chapters also cover the concept of spot (zero-coupon) and forward rates, and the rates implied by market bond prices and yields yield-curve fitting techniques an account of spline fitting using regression techniques and an introductory discussion of term structure models. 

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