Default-free bond

A potential model structure requiring multiple-Fields could e.g. come from the need to model the spread dynamics of corporate bonds separately from the underlying default-free bond dynamics. 

The general rule of corporate bonds is that they are priced at a spread to the government yield curve. In absolute terms, the yield spread is the difference between the yield to maturity of a corporate bond and the benchmark, generally a yield to maturity of a govermnent bond with the same maturity. Corporate bonds include a yield spread on a risk-free rate in order to compensate two main factors, liquidity premium and credit spread. The yield of a corporate bond can be assumed as the sum of parts of the elements as shown in Figure 8.1, in which the yield spread relative to a default-free bond is given by the sum of default premium (credit spread) and liquidity premium. 

The credit spread is defined as the difference between the risky rate of a defaul-table bond and the risk-free rate of a default-free bond. In this case, with bonds priced at par, between coupon and risk-free rate, the pricing is performed like a valuation of a straight bond, including the default risk adjustment. The price is given by Equation (8.25) ... 

The research compares the model spread to the one observed in the market. In order to determine the term structure of credit spread. Eons uses historical probabilities by Moody s database, adopting a recovery rate of 48.38%. The empirical evidence is that bonds with high investment grade have an upward credit spread curve. Therefore, the spread between defaultable and default-free bonds increases as maturity increases. Conversely, speculative-grade bonds have a negative or flat credit yield curve (Figure 8.7). 

Consider also that because the OAS is applied over the risk-free yield curve, it includes the credit risk and liquidity risk between defaultable and default-free bonds. Figure 11.3 shows an example of the OAS Bloomberg screen for Mittel s callable bond. 

George Courtadon, The Pricing of Default-Free Bonds, Journal of Financial and Quantitative Analysis 17 (March 1982), pp. 75-100. 

The Monte Carlo method, however, is prone to model risk. If the stochastic process chosen for the underlying variable is unrealistic, so will be the estimate of VaR. This is why the choice of the underlying model is particularly important. The geometric Brownian motion model described above adequately describes the behavior of some financial variables, but certainly not that of short-term fixed-income securities. In the Brownian motion, shocks on prices are never reversed. This does not represent the price process for default-free bonds, which must converge to their face value at expiration. 

Part One describes fixed-income market analysis and the basic concepts relating to bond instruments. The analytic building blocks are generic and thus applicable to any market. The analysis is simplest when applied to plain vanilla default-free bonds as the instruments analyzed become more complex, additional techniques and assumptions are required. 

The discussion in this chapter assumes a liquid market of default-free bonds, where hoth zero-coupon and coupon bonds are freely bought and sold. Prices are determined by the economy-wide supply of and demand for the bonds at any time. The prices are thus macroeconomic, rather than being set by individual bond issuets or traders. 

Carleton and Cooper (1976) describes an approach to estimating tetm structure that assumes default-free bond cash flows, payable on specified discrete dates, to each of which a set of unrelated discount factors are applied. These discount factors are estimated as regression coefficients, with the bond cash flows beir the independent variables and the bond price at each payment date the dependent variable. This type of simple linear regression produces a discrete discount fimction, not a continuous one. The forward-rate curves estimated from this fimction are accordir ly very ja ed. 

The continuously compounded gross redemption yield at time ton a default-free zero-coupon bond that pays 1 at maturity date 7 is x. We assume that the movement in X is described by... 

A default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

No arbitrage opportunities-. They assume the existence of an unique martingale measure Q in which default-free and risky zero-coupon bonds are martingales ... 

The minimum interest rate that an investor should require is the yield available in the marketplace on a default-free cash flow. For bonds whose cash flows are denominated in euros, yields on European government securities serve as benchmarks for default-free interest rates. In some European countries, the swap curve serves as a benchmark for pricing spread product (e.g., corporate bonds). For now, we can think of the minimum interest rate that investors require as the yield on a comparable maturity benchmark security. 

In this chapter we consider only default-free securities. We use interchangeably the notation ( ) = E - Ft). We shall denote by p(t,T) the price at time t of a pure discount bond with maturity T and obviously p(t,t) = p T,T)=l. 

Ramon Rabinovitch, Pricing Stock and Bond Options when the Default-Free Rate is Stochastic. ... 

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. 

Another indicator of credit risk is the credit risk premium the spread between the yields on corporate bonds and those of government bonds in the same currency. This spread is the compensation required by investors for holding bonds that are not default-free. The size of the credit premium changes with the market s perception of the financial health of individual companies and sectors and of the economy in general. The variability of the premium is illustrated in FIGURES 10.2 and 10.3 on the following page, which show the spreads between the U.S.-dollar-swap and Treasury yield curves in, respectively, February 2001 and February 2004. 

This chapter examines a number of issues relevant to participants in the fixed-income markets. The analysis presented is based on government-bond trading and is confined to generic bonds that are default-free, with no consideration given to factors that apply to corporate bonds, asset- and mortgage-backed bonds, convertibles, or other nonvanilla securities, or to issues such as credit risk and prepayment risk. Nevertheless, the principles adduced are pertinent to all relative-value fixed-income analysis. 

All bond instruments are characterized by the promise to pay a stream of future cash flows. The term structure of interest rates and associated discount function is crucial to the valuation of any debt security and underpins any valuation framework. Armed with the term structure, we can value any bond, assuming it is liquid and default-free, by breaking it down into a set of cash flows and valuing each cash flow with the appropriate discount factor. Further characteristics of any bond, such as an element of default risk or embedded option, are valued incrementally over its discounted cash flow valuation. 

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. ... 

Using the spot rate structure at Table 12.1, the price of this bond is calculated to be 98.21. This would be the bonds fair value if it were liquid and default free. Assume, however, that the bond is a corporate bond and carries an element of default risk, and is priced at 97.00. What spread over the risk-free price does this indicate We require the spread over the implied forward rate that would result in a discounted price of 97.00. Using iteration, this is found to be 67.6 basis points. The calculation is... 

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. 

The credit rating of a company is a major determinant of the yield that will be payable by that company s bonds. The yield spread of a corporate bond over the risk-free bond yield is known as the default premium. In practice, the default premium is composed of two elements, the compensation element specific to the company and the element related to market risk. This is because, in an environment where the default of one company was completely unrelated to the default of other companies, the return from a portfolio of corporate bonds would equal that of the risk-free bond. The gains from bonds of companies that did not default compensated for the loss from those that did default. The additional part of the default premium, the risk premium, is the compensation for risk exposure that cannot be diversified away in a portfolio, known as systematic or non-diversifiable risk. Observation of the market tells us that in certain circumstances, the default patterns of companies are related for example, in a recession there are more corporate defaults, and this fact is reflected in the risk premium. 

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