Reference yield curve

Making comparison between bonds could be difficult and several aspects must be considered. One of these is the bond s maturity. For instance, we know that the yield for a bond that matures in 10 years is not the same compared to the one that matures in 30 years. Therefore, it is important to have a reference yield curve and smooth that for comparison purposes. However, there are other features that affect the bond s comparison such as coupon size and structure, liquidity, embedded options and others. These other features increase the curve fitting and the bond s comparison analysis. In this case, the swap curve represents an objective tool to understand the richness and cheapness in bond market. According to O Kane and Sen (2005), the asset-swap spread is calculated as the difference between the bond s value on the par swap curve and the bond s market value, divided by the sensitivity of 1 bp over the par swap. 

Depending on the reference yield curve selected and its currency denomination, the ASW spread changes. Eigure 1.3 shows the ASW spread for different reference yield curves for TKAAV. 

Duration measures the sensitivity of a bond s price to a given change in yield. The traditional formulation is derived under the assumption that the reference yield curve is flat and moves in parallel shifts. Simply put, all bond yields are the same regardless of when the cash flows are delivered across time and changes in yields are perfectly correlated. Several recent attempts have been made to address this inadequacy and develop interest rate risk measures that allow for more realistic changes in the yield curve s shape. ... 

Pressure = 0.3 mm Hg nickel cathode. The yield curve for NO2+ ions, not shown here, corresponds closely to that of NO+ ion % yield refers to % total ions observed... 

For 200 C as example, the procedure is shown in Figure 132. Curve A, representing the yield of the hypothetical process with resinification as the only loss, is obtained from equation (7) of the preceding chapter, and curve B is an experimental yield curve given in the literature [126] for 200 °C and for an initial xylose concentration of 0.666 mole/liter (100 g/liter). Experimental yield curves for other temperatures and other initial xylose concentrations are amply available in the same reference. The hatched area between the two curves A and B represents the condensation loss. To round the overall picture, the theoretical yield for the temperature considered is shown by the dashed curve C. 

Rgure 2.1. Mass-yield curves for some fission products, including and Pu. The curve labels refer to the compound nuclei, as in + n = (From Vertes et al. 2003,... 

The intersection of the crack arrest curve with the yield curve (Curve B) is called the fracture transition elastic (FTE) point. The temperature corresponding to this point is normally about 60°F above the NDT temperature. This temperature is also known as the Reference Temperature - Nil-ductility Transition (RTj dt) is determined in accordance with ASME Section III (1974 edition), NB 2300. The FTE is the temperature above which plastic deformation accompanies all fractures or the highest temperature at which fracture propagation can occur under purely elastic loads. The intersection of the crack arrest curve (Curve D) and the tensile strength or ultimate strength, curve (Curve A) is called the fracture transition plastic (FTP) point. The temperature corresponding with this point is normally about 120°F above the NDT temperature. Above this temperature, only ductile fractures occur. 

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. 

Further research has produced a category of models that attempt to describe the jump feature of asset prices and interest rates. Observation of the markets confirms that many asset price patterns and interest rate changes do not move continuously from one price (rate) to another, but sometimes follow a series of jumps. A good example of a jump movement is when a central bank changes the base interest rate when this happens, the entire yield curve shifts to incorporate the effect of the new base rate. There is a considerable body of literature on the subject, and we only refer to a small number of texts here. 

Using this relationship, we build the yield curve by obtaining the relevant reference rate for regular intervals on the term structure, and then obtaining the discount factors for each point along the term structure. This set of discount factors is then used to extract the yield curve. 

An inverse floating-rate note pays coupons that increase if the reference rate decreases. Therefore, this bond gives a benefit at investors with a negative yield curve. The coupon structure of inverse floaters usually is determined as a fixed interest rate less a variable interest rate linked to a reference index. Moreover, they can include floor provisimis. 

The option-adjusted spread (OAS) is the most important measure of risk for bonds with embedded options. It is the average spread required over the yield curve in order to take into account the embedded option element. This is, therefore, the difference between the yield of a bond with embedded option and a government benchmark bond. The spread incorporates the future views of interest rates and it can be determined with an iterative procedure in which the market price obtained by the pricing model is equal to expected cash flow payments (coupons and principal). Also a Monte Carlo simulation may be implemented in order to generate an interest rate path. Note that the option-adjusted spread is influenced by the parameters implemented into the valuation model as the yield curve, but above all by the volatility level assumed. This is referred to volatility dependent. The higher the volatility, the lower the option-adjusted spread for a callable bond and the higher for a putable bond. 

The application of risk-neutral valuation, which we discussed in chapter 45, requires that we know the sequence of short-term rates for each scenario, which is provided in some interest-rate models. For this reason, many yield curve models are essentially models of the stochastic evolution of the short-term rate. They assume that changes in the short-term interest-rate is aMarkov process. (It is outside the scope of this book to review the mathematics of such processes, but references are provided in subsequent chapters.) This describes an evolution of short-term rates in which the evolution of the rate is a function only of its current level, and not the path by which it arrived there. The practical significance of this is that the valuation of interest-rate products can be reduced by the solution of a single partial differential equation. 

The zero-volatility spread, also referred to as the Z-spread or static spread, is a measure of the spread that the investor would realize over the entire benchmark spot rate curve if the bond were held to maturity. Unlike the nominal spread, it is not a spread at one point on the yield curve. The Z-spread is the spread that will make the present value of the cash flows from the nongovernment bond, when discounted at the benchmark rate plus the spread, equal to the nongovernment bond s market price plus accrued interest. A trial-and-error procedure is used to compute the Z-spread. 

The first two chapters of this section discuss bond pricing and yields, moving on to an explanation of such traditional interest rate risk measures as modified duration and convexity. Chapter 3 looks at spot and forward rates, the derivation of such rates from market yields, and the yield curve. Yield-curve analysis and the modeling of the term structure of interest rates are among the most heavily researched areas of financial economics. The treatment here has been kept as concise as possible. The References section at the end of the book directs interested readers to accessible and readable resources that provide more detail. 

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. 

Chapter 3 introduced the basic concepts of bond pricing and analysis. This chapter builds on those concepts and reviews the work conducted in those fields. Term-structure modeling is possibly the most heavily covered subject in the financial economics literature. A comprehensive summary is outside the scope of this book. This chapter, however, attempts to give a solid background that should allow interested readers to deepen their understanding by referring to the accessible texts listed in the References section. This chapter reviews the best-known interest rate models. The following one discusses some of the techniques used to fit a smooth yield curve to market-observed bond yields. 

The well-known model described in Hull and White (1993) uses Vasicek s model to obtain a theoretical yield curve and fit it to the observed market curve. It is therefore sometimes referred to as the extended Vasicek... 

This section briefly introduces a number of two-factor interest rate models. (The References section indicates sources for further research.) As their name suggests, these models specify the yield curve in terms of two factors, one of which is usually the short rate. A number of factors can be modeled when describing the dynamics of interest rates. Among them are... 

This chapter presents an overview of some of the methods used to fit the yield curve. A selection of useful sources for further study is given, as usual, in the References section. 

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 ... 

Some of the newer models refer to parameters that are difficult to observe or measure direcdy. In practice, this limits their application much as B-S is limited. Usually the problem has to do with calibratii the model properly, which is crucial to implementing it. Galibration entails inputtii actual market data to create the parameters for calculating prices. A model for calculating the prices of options in the U.S. market, for example, would use U.S. dollar money market, futures, and swap rates to build the zero-coupon yield curve. Multifactor models in the mold of Heath-Jarrow-Morton employ the correlation coefficients between forward rates and the term structure to calculate the volatility inputs for their price calculations. 

Fig. 5.16. Spectral yield curves for three members of the InGaAs family, including GaAs, illustrating the drop in above-threshold yield as bandgap is lowered to extend IR response. The S-1 AgCsO curve is for reference [5.15]...

Fission fragments (pre-neutron emission). The yield curves discussed above refer to fission product yields after emission of prompt neutrons. As discussed above, the physical methods based on momentum conservation at scission (double energy, double velocity, or energy and velocity measurements) allow the measurement of the yield distribution of fission fragments (prior to prompt neutron emission). In these cases, simultaneous information is obtained on the kinetic energy of the fragments detected. 

By splitting the yield curve into individual node/tenor pairs, we may work with individual lines within each tenor. A cubic polynomial can then be added to each line to provide the cubic spline. For ease of illustration, we take this one step further and imagine an alternative horizontal axis. This is referred to as Xas shown in FIGURE 5.8. Assume that between each node pair that this horizontal axis Aruns from 0 (at to - x y (at x y,. ). 

The viscosity of each blend is determined, and the four blends, along with fraction 19, constitute five points on the viscosity yield curve also refer to Table 4-10. 

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