pdf. (CFA Institute Investment Series) Barbara Petitt, Jerald Pinto, Wendy L. Pirie- Fixed Income FIXED INCOME ANALYSIS CFA Institute is the premier association for .. The third edition updates key concepts from the second edition. Fixed Income Analysis is a new edition of Frank Fabozzi's Fixed Income Analysis, Second Edition that provides authoritative and up-to-date coverage of how. A fixed income security is a financial obligation of an entity (the issuer) who . Given the information in the first and third columns, complete the table in the second Eurobond Market,'' Chapter 29 in Frank J. Fabozzi (ed.).

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Wiley, p. Fixed Income Analysis is a new edition of Frank Fabozzi s Fixed Income Analysis, Second Edition that provides authoritative and up-to-date . The essential guide to fixed income portfolio management, from the experts at CFA Fixed Income Analysis is a new edition of Frank Fabozzis Fixed Income. FIXED INCOME. ANALYSIS. Third Edition. Barbara S. Petitt, CFA. Jerald E. Pinto, CFA. Wendy L. Pirie, CFA with. Robin Grieves, CFA. Gregory M. Noronha, CFA.

Drawn from information supplied by The Surrey Hills Partnership. Reproduced with permission. Moorad Choudhry lives in Surrey, England. We also look at the procedure involved in estimating and fitting the yield curve. This topic is one of the most heavily researched subjects in financial market economics, and indeed research is ongoing. There are a number of ways to estimate and fit the yield curve, and there is no one right or wrong method. It is important, both when discussing the subject matter or writing about it, to remember to place the relevant ideas in context, otherwise there is the danger of becoming too theoretical. The aim is to confine the discussion within the boundary of user application; there is a great deal of published material that is, quite simply, rather too theoretical, not to mention highly technical. We must always try to keep in touch with the markets themselves. The chapters are written from the point of view of both the market practitioner and the research student. In the following chapters we look at bond relative value trading as well as interest-rate modelling.

Our interactive player makes it easy to find solutions to Fixed Income Analysis 3rd Edition problems you're working on - just go to the chapter for your book. Hit a particularly tricky question? Bookmark it to easily review again before an exam. The best part? As a Chegg Study subscriber, you can view available interactive solutions manuals for each of your classes for one low monthly price.

Why download extra books when you can get all the homework help you need in one place? Can I get help with questions outside of textbook solution manuals? You bet! Just post a question you need help with, and one of our experts will provide a custom solution.

You can also find solutions immediately by searching the millions of fully answered study questions in our archive. We review the main one-factor models that were initially developed to model the term structure. In most cases the model result is given and explained, rather than the full derivation. The objective here is to keep the content accessible, and pertinent to practitioners and most postgraduate students. A subsequent book is planned that will delve deeper into the models themselves, and the latest developments in research.

Finally, we review some techniques used to estimate and fit the zero-coupon curve using the prices of bonds observed in the market, with an illustration from the United Kingdom gilt market. The last part of the book considers some advanced analytical techniques for indexlinked bonds. Chapter 9 is a look at some of the peculiar properties of very long-dated bond yields, including the convexity bias inherent in such yields, and their relative volatility.

In Chapter 10 we review some concepts that apply to the analysis of the credit default risk of corporate bonds, and how this might be priced. The dynamics of the yield curve In Chapter 2 of the companion volume to this book in the Fixed Income Markets 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 given today's spot 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 is 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.

If using a spot rate curve for valuation purposes, banks use what are known as arbitragefree yield curve models, where the derived curve has been matched to the current spot yield curve.

So, if one was valuing a two-year bond that was put-able by the holder at par in one year's time, it could be analysed as a one-year bond that entitled the holder to reinvest it for another year.

The rule of no-arbitrage pricing states that an identical price will be obtained whichever way one chooses to analyse the bond. When matching derived yield curves therefore, correctly matched curves will generate the same price when valuing a bond, whether a derived spot curve is used or the current term structure of spot rates.

The dynamics of interest rates and the term structure is the subject of some debate, and the main difference between the main interest-rate models is in the way that they choose to capture the change in rates over a time period. However, although volatility of the yield curve is indeed the main area of difference, certain models are easier to implement than others, and this is a key factor a bank considers when deciding which model to use. The process of calibrating the model, that is setting it up to estimate the spot and forward term structure using current interest rates that are input to the model, is almost as important as deriving the model itself.

So the availability of Preface xv data for a range of products, including cash money markets, cash bonds, futures and swaps, is vital to the successful implementation of the model. As one might expect the yields on bonds are correlated, in most cases very closely positively correlated. This enables us to analyse interest-rate risk in a portfolio for example, but also to model the term structure in a systematic way.

Much of the traditional approach to bond portfolio management assumed a parallel shift in the yield curve, so that if the 5-year bond yield moved upwards by 10 basis points, then the year bond yield would also move up by 10 basis points.

This underpins traditional duration and modified duration analysis, and the concept of immunisation. To analyse bonds in this way, we assume therefore that bond yield volatilities are identical and correlations are perfectly positive. Although both types of analysis are still common, it is clear that bond yields do not move in this fashion, and so we must enhance our approach in order to perform more accurate analysis.

Factors influencing the yield curve From the discussion in Chapter 2 of the companion volume to this book in the Fixed Income Markets Library, Corporate Bonds and Structured Financial Products we are aware that there are a range of factors that impact on the shape and level of the yield curve. A combination of economic and non-economic factors are involved.

A key factor is investor expectations, with respect to the level of inflation, and the level of real interest rates in the future. In the real world the market does not assume that either of these two factors is constant, however given that there is a high level of uncertainty over anything longer than the short-term, generally there is an assumption about both inflation and interest rates to move towards some form of equilibrium in the long-term. It is possible to infer market expectations about the level of real interest rates going forward by observing yields in government index-linked bonds, which trade in a number of countries including the US and UK.

The market's view on the future level of interest rates may also be inferred from the shape and level of the current yield curve. We know that the slope of the yield curve also has an information content.

There is more than one way to interpret any given slope, however, and this debate is still open. The fact that there are a number of factors that influence changes in interest rates and the shape of the yield curve means that it is not straightforward to model the curve itself. In Chapter 6 we consider some of the traditional and more recent approaches that have been developed. Approaches to modelling The area of interest-rate dynamics and yield curve modelling is one of the most heavily researched in financial economics.

There are a number of models available in the market today, and generally it is possible to categorise them as following certain methodologies.

By categorising them in this way, participants in the market can assess them for their suitability, as well as draw their own conclusions about how realistic they might be.

Let us consider the main categories. One-factor, two-factor and multi-factor models The key assumption that is made by an interest-rate model is whether it is one-factor, that is the dynamics of the yield change process are based on one factor, or multi-factor. From xvi Preface observation we know that in reality there are a number of factors that influence the price change process, and that if we are using a model to value an option product, the valuation of that product is dependent on more than one underlying factor.

For example, the payoff on a bond option is related to the underlying bond's cash flows as well as to the reinvestment rate that would be applied to each cash flow, in addition to certain other factors. Valuing an option therefore is a multi-factor issue.

In many cases, however, there is a close degree of correlation between the different factors involved. If we are modelling the term structure, we can calculate the correlation between the different maturity spot rates by using a covariance matrix of changes for each of the spot rates, and thus obtain a common factor that impacts all spot rates in the same direction.

This factor can then be used to model the entire term structure in a one-factor model, and although two-factor and multifactor models have been developed, the one-factor model is still commonly used. In principle it is relatively straightforward to move from a one-factor to a multi-factor model, but implementing and calibrating a multi-factor model is a more involved process. This is because the model requires the estimation of more volatility and correlation parameters, which slows down the process.

Readers will encounter the term Gaussian in reference to certain interest-rate models.

Put simply, a Gaussian process describes one that follows a normal distribution under a probability density function. The distribution of rates in this way for Gaussian models implies that interest rates can attain negative values under positive probability, which makes the models undesirable for some market practitioners.

Nevertheless, such models are popular because they are relatively straightforward to implement and because the probability of the model generating negative rates is low and occurs only under certain extreme circumstances. The short-term rate and the yield curve The application of risk-neutral valuation 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 shortterm rate. They assume that changes in the short-term interest rate is a Markov 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 to the solution of a single partial differential equation. Short-rate models are composed of two components. The first attempts to capture the average rate of change, also called the drift, of the short-term rate at each instant, while the second component measures this drift as a function of the volatility of the short-term rate.

In most models the drift rate term is determined through a numerical technique that matches the initial spot rate yield curve, while in some models an analytical solution is available. So the Vasicek and CIR models are models of the short-term rate, and both incorporate the same form for the drift term, which is a tendency for the short-term rate to rise when it is below the long-term mean interest rate, and to fall when it is above the long-term mean.

This is known as mean reversion.

In the Vasicek model, the rate dependence of the volatility is constant, in the CIR model it is proportional to the square-root of the short rate. In both models, because the dynamics of the short-rate cover all possible moves, it is possible to derive negative interest rates, although under most conditions of initial spot rate and volatility levels, this is quite rare.

Essentially the Vasicek and CIR models express the complete forward rate curve as a function of the current short-term rate, which is why later models are sometimes preferred. The drift rate term is not known analytically in these models. In the BDT model the short-term rate volatility is related to the strength of the mean reversion in a way that reduces the volatility over time. Arbitrage-free and equilibrium modelling In an arbitrage-free model, the initial term structure described by spot rates today is an input to the model.

In fact such models could be described not as models per se, but essentially a description of an arbitrary process that governs changes in the yield curve, and projects a forward curve that results from the mean and volatility of the current short-term rate. An equilibrium term structure model is rather more a true model of the term structure process; in an equilibrium model the current term structure is an output from the model.

An equilibrium model employs a statistical approach, assuming that market prices are observed with some statistical error, so that the term structure must be estimated, rather than taken as given.

Risk-neutral probabilities When valuing an option written on say, an equity the price of the underlying asset is the current price of the equity.

When pricing an interest-rate option the underlying is obtained via a random process that describes the instantaneous risk-free zero-coupon rate, which is generally termed the short rate. Mathematics primer The level of mathematics required for a full understanding of even intermediate concepts in finance is frighteningly high.

To attempt to summarise even the basic concepts in just a few pages would be a futile task and might give the impression that the mathematics was being trivialised. Our intention is quite the opposite. As this is a financial markets book, and not a mathematics textbook, a certain level of knowledge has been assumed, and a formal or rigorous approach has not been adopted.

Hence readers will find few derivations, and fewer proofs. What we provide here is a very brief introduction to some of the concepts; the aim of this is simply to provide a starting point for individual research.

We assist this start by listing recommended texts in the bibliography. Random variables and probability distributions In financial mathematics random variables are used to describe the movement of asset prices, and assuming certain properties about the process followed by asset prices allows us to state what the expected outcome of events are. A random variable may be any value from a specified sample space. The specification of the probability distribution that applies to the sample space will define the frequency of particular values taken by the random variable.

A discrete random variable is one that can assume a finite or countable set of values, usually assumed to be the set of positive integers. The sum of the probabilities is 1.

Discrete probability distributions include the Binomial distribution and the Poisson distribution. Continuous random variables The next step is to move to a continuous framework. The normal or Gaussian distribution is perhaps the most important.