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Originally published as: Fixed income analysis for the chartered financial analyst program. . readable, treatment of the key topics in fixed income analysis. Advanced Fixed Income Analysis This page intentionally left blank Advanced Fixed Income Analysis Moorad ChoudhryA. Prepare cash flow and profit & loss forecasts. • Get backers to invest. Plan. 25YEARS. THE LEADING. BUSINESS PLAN. BO.

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pdf. (CFA Institute Investment Series) Barbara Petitt, Jerald Pinto, Wendy L. Pirie- Fixed FIXED INCOME ANALYSIS CFA Institute is the premier association for. A fixed income security is a financial obligation of an entity (the issuer) who spread measure for valuation and relative value analysis. fixed income securities, e.g. options and futures on bonds or interest rates, caps The key concept in the analysis of fixed income securities and interest rate.

Personal information is secured with SSL technology. Free Shipping No minimum order. Description Each new chapter of the Second Edition covers an aspect of the fixed income market that has become relevant to investors but is not covered at an advanced level in existing textbooks. This is material that is pertinent to the investment decisions but is not freely available to those not originating the products. While the level of mathematical sophistication is both high and specialized, he includes a brief introduction to the key mathematical concepts. This is a book on the financial markets, not mathematics, and he provides few derivations and fewer proofs. He draws on both his personal experience as well as his own research to bring together subjects of practical importance to bond market investors and analysts.

Commercial Mortgage-Backed Securities 6. Credit Risk 6. Non-Mortgage Asset-Backed Securities 7. Auto Loan Receivable-Backed Securities 7. Credit Card Receivable-Backed Securities 8. Collateralized Debt Obligations 8. Structure of a CDO Transaction 8. Illustration of a CDO Transaction 9. The Meaning of Arbitrage-Free Valuation 2. The Law of One Price 2. Arbitrage Opportunity 2. The Binomial Interest Rate Tree 3. Determining the Value of a Bond at a Node 3.

Constructing the Binomial Interest Rate Tree 3. Valuing an Option-Free Bond with the Tree 3. Pathwise Valuation 4. Monte Carlo Method 5. Bonds with Embedded Options Learning Outcomes 1.

Overview of Embedded Options 2. Simple Embedded Options 2. Complex Embedded Options 3. Valuation and Analysis of Callable and Putable Bonds 3. A Refresher 3. Valuation of Risky Callable and Putable Bonds 4. Duration 4. Effective Convexity 5. Valuation of a Capped Floater 5. Valuation of a Floored Floater 6. Valuation and Analysis of Convertible Bonds 6. Defining Features of a Convertible Bond 6. Analysis of a Convertible Bond 6.

Valuation of a Convertible Bond 6. Bond Analytics 8. Spot Rates and Forward Rates 2. The Forward Rate Model 2. Yield Curve Movement and the Forward Curve 2. Active Bond Portfolio Management 3. The Swap Rate Curve 3. The Swap Spread 3. Spreads as a Price Quotation Convention 4. Local Expectations Theory 4. Liquidity Preference Theory 4.

Segmented Markets Theory 4. Preferred Habitat Theory 5. Modern Term Structure Models 5. Equilibrium Term Structure Models 5. Arbitrage-Free Models: The Ho—Lee Model 6.

Yield Curve Factor Models 6. Factors Affecting the Shape of the Yield Curve 6. Managing Yield Curve Risks 7. Managing Funds against a Bond Market Index 3. Classification of Strategies 3. Indexing Pure and Enhanced 3.

Active Strategies 3. Managing Funds against Liabilities 4. Dedication Strategies 4. Cash Flow Matching Strategies 5. Other Fixed-Income Strategies 5. Combination Strategies 5. Leverage 5. Derivatives-Enabled Strategies 6. International Bond Investing 6. Active versus Passive Management 6. Currency Risk 6.

Breakeven Spread Analysis 6. Emerging Market Debt 7. Selecting a Fixed-Income Manager 7. Historical Performance as a Predictor of Future Performance 7. Developing Criteria for the Selection 7. Comparison with Selection of Equity Managers 8. Credit Relative-Value Analysis A. Relative Value B. Classic Relative-Value Analysis C. Relative-Value Methodologies 3. Total Return Analysis 4. Primary Market Analysis A. 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.

Pdf fixed income analysis

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. Continuous distributions are commonly encountered in finance theory.

The normal or Gaussian distribution is perhaps the most important. P0 Expected values A probability distribution function describes the distribution of a random variable X. A squared measure has little application so commonly the square root of the variance, the standard deviation is used.

Regression analysis A linear relationship between two variables, one of which is dependent, can be estimated using the least squares method. This can be transformed into a form that is linear and then fitted using least squares. Stochastic processes This is perhaps the most difficult area of financial mathematics. Most references are also very technical and therefore difficult to access for the non-mathematician. We begin with some definitions. A random process is usually referred to as a stochastic process.

This is a collection of random variables X t and the process may be either discrete or continuous. A Markov process is one where the path is dependent on the present state of the process only, so that all historical data, including the path taken to arrive at the present state, is irrelevant.

So in a Markov process, all data up to the present is contained in the present state. The dynamics of asset prices are frequently assumed to follow a Markov process, and in fact it represents a semi-strong form efficient market.

Selected bibliography Baxter, A. Gujarati, D. Hogg, R. Kreyszig, E. Ross, S. Rather, we will discuss certain approaches that have worked in the past and should, given the right circumstances, work again at some point in the future. It is the intention however, to focus on real-world application whilst maintaining analytical rigour. The term trading covers a wide range of activity.

Market makers who are quoting two-way prices to market participants may be tasked with providing a customer service, building up retail and institutional volume, or they may be tasked with purely running the book at a profit and trying to maximise return on capital.

The nature of the market that is traded will also impact on their approach. In a highly transparent and liquid market such as the US Treasury or the UK gilt market the price spreads are fairly narrow,1 although increased demand has reduced this somewhat in both markets.

However this means that opportunities for profitable trading as a result of mispricing of individual securities, whilst not completely extinct, are rare. It is much more common for traders in such markets to take a view on relative value 1 In fact, in the late s spreads in the gilt market were beginning to widen as excess demand over supply, particularly at the long end of the yield curve, drove down yields and reduced liquidity.

In the Treasury market at the start of the yield curve had inverted, with the yield on the long bond at 6. The volatility level in the market was at a two-year high at that time. These developments in the two markets have led to wider price quotes and lower liquidity.

A sustained public sector deficit has many implications for the debt markets, if governments start to repay national debt and cease issuing securities. This is an important topic which is currently the subject of some debate. A significant reduction in government debt levels, while advantageous in many respects, will pose new problems. This is because government bonds play an important part in the financial systems of many countries. In the first instance, government bonds are used as the benchmark against which many other instruments are priced, such as corporate bonds.

An illiquid market in government debt could have serious implications for the corporate bond markets, with investors possibly becoming reluctant to invest in corporate paper unless yield levels rise. Derivatives may also suffer from pricing problems, particularly bond futures contracts. In addition, while long-horizon institutional investors such as pension funds may find themselves short of investment products, many central banks and sovereign governments are big holders of securities such as US Treasuries, gilts and bunds.

A shortage of supply in these instruments, particularly Treasuries, might have implications for all these investors unless an alternative instrument is made available. The continuing inverted yield curve in the UK, which dated from July , and the inversion of the US curve in February , is put down partly to shortage of longdated government stock. The OECD, as reported in The Economist 12 February has suggested a policy whereby governments maintain a minimum level of gross public debt, with this minimum being an amount sufficient to maintain bond market liquidity.

This may not be a practical solution for large economies however, especially that of the United States, but is certainly viable for other developed economies. The issue of alternative benchmarks is currently being researched by the author.

This is also called spread trading. A large volume of trading on derivatives exchanges is done for hedging purposes, but speculative trading is also prominent. Very often bond and interest-rate traders will punt using futures or options contracts, based on their view of market direction. Ironically, market makers who have a low level of customer business, perhaps because they are newcomers to the market, for historical reasons or because they do not have the appetite for risk that is required to service the high quality customers, tend to speculate on the futures exchanges to relieve tedium, often with unfortunate results.

Speculative trading is undertaken on the basis of the views of the trader, desk or head of the department. The former is an assessment of macroeconomic and microeconomic factors affecting not just the specific bond market itself but the economy as a whole. Those running corporate debt desks will also concentrate heavily on individual sectors and corporations and their wider environment, because the credit spread, and what drives the credit spread, of corporate bonds is of course key to the performance of the bonds.

Technical analysis or charting is a discipline in its own right, and has its adherents. It is based on the belief that over time the patterns displayed by a continuous time series of asset prices will repeat themselves. Therefore detecting patterns should give a reasonable expectation of how asset prices should behave in the future. Many traders use a combination of fundamental and technical analysis, although chartists often say that for technical analysis to work effectively, it must be the only method adopted by the trader.

A review of technical analysis is presented in Chapter 63 of the author's book The Bond and Money Markets. In this chapter we introduce some common methods and approaches, and some not so common, that might be employed on a fixed interest desk.

The essential features of futures trading are volatility and leverage. To establish a futures position on an exchange, the level of margin required is very low proportional to the notional value of the contracts traded.

For speculative purposes traders often carry out open, that is uncovered trading, which is a directional bet on the market. So therefore if a trader believed that short-term sterling interest rates were going to fall, they could download a short sterling contract on LIFFE.

This may be held for under a day, in which case if the price rises the trader will gain, or for a longer period, depending on their view. The trade can be carried out with any futures contract; the same idea could be carried out with a cash market product or a FRA, but the liquidity, narrow price spread and the low cost of dealing make such a trade easier on a futures exchange.

It is much more interesting however to carry out a spread trade on the difference between the rates of two different contracts. Consider Figures 1. Approaches to Trading and Hedging 3 and 1. The specification for this contract is summarised in Chapter 35 of Choudhry From Chapter 2 we know that forward rates can be calculated for any term, starting on any date.

In Figure 1. Figure 1. It is possible to trade a strip of contracts to replicate any term, out to the maximum maturity of the contract. This can be done for hedging or speculative purposes. Note from Figure 1. A trader can take positions on cash against futures, but it is easier to transact only on the futures exchange. Short-term money market interest rates often behave independently of the yield curve as a whole. A money markets trader may be aware of cash market trends, for example an increased frequency of borrowing at a certain point of the curve, as well as other market intelligence that suggests that one point of the curve will rise or fall relative to others.

One way to exploit this view is to run a position in a cash instrument such as CD against a futures contract, which is a basis spread trade. However the best way to trade on this view is to carry out a spread trade, shorting one contract against a long position in another trade.

Consider Figure 1. This is not a market directional trade, rather a view on the relative spread between two contracts. The trade must be carried out in equal weights, for example lots of the June against lots of the September.

Chapter 1: Approaches to Trading and Hedging 5 direction that the trader expects, the trade will generate a profit. There are similar possibilities available from an analysis of Figure 1. Spread trading carries a lower margin requirement than open position trading, because there is no directional risk in the trade.

It is also possible to arbitrage between contracts on different exchanges. If the trade is short the near contract and long the far contract, so the opposite of our example, this is known as downloading the spread and the trader believes the spread will widen. The opposite is shorting the spread and is undertaken when the trader believes the spread will narrow. Note that the difference between the two price levels is not limitless, because the theoretical price of a futures contract provides an upper limit to the size of the spread or the basis.

The spread or the basis cannot exceed the cost of carry, that is the net cost of downloading the cash security today and then delivering it into the futures market on the contract expiry. The two associated costs for a short-sterling spread trade are the notional borrowing and lending rates from having bought one and sold another contract. If the trader believes that the cost of carry will decrease they could sell the spread to exercise this view.

The trader may have a longer time horizon and trade the spread between the shortterm interest-rate contract and the long bond future. This is usually carried out only by proprietary traders, because it is unlikely that one person would be trading both threemonth and year or year, depending on the contract specification interest rates. A common example of such a spread trade is a yield curve trade.

Income pdf fixed analysis

If a trader believes that the sterling yield curve will steepen or flatten between the three-month and the year terms, they can download or sell the spread by using the LIFFE short-sterling contract and the long gilt contract.

To be first-order risk neutral however the trade must be durationweighted, as one short-sterling contract is not equivalent to one gilt contract. We use 1.

Analysis pdf income fixed

Therefore in practice one would use the duration of the cheapest-to-deliver bond. A butterfly spread is a spread trade that involves three contracts, with the two spreads between all three contracts being traded.

This is carried out when the middle contract appears to be mispriced relative to the two contracts either side of it. The trader may believe 6 Advanced Fixed Income Analysis that one or both of the outer contracts will move in relation to the middle contract; if the belief is that only one of these two will shift relative to the middle contract, then a butterfly will be put on if the trader is not sure which of these will adjust.

For example, consider Figure 1. The prices of the front three contracts are The trader feels that the September contract will rise, but will that be because June and December prices fall or because the September price will rise? In markets where an active zero-coupon bond market exists, much analysis is undertaken into the relative spreads between derived and actual zero-coupon yields.

In this section we review some of the yield curve analysis used in the market. Customer business apart, decisions to download or sell securities will be a function of their views on: All three areas are related but will react differently to certain pieces of information.

A report on the projected size of the government's budget deficit for example, will not have much effect on two-year bond yields, whereas if the expectations came as a surprise to the market it could have an adverse effect on long-bond yields. The starting point for analysis is of course the yield curve, both the traditional coupon curve plotted against duration and the zero-coupon curve.

For a first-level analysis, many market practitioners will go no further than Figure 1. An investor who had no particular view on the future shape of the yield curve or the level of interest rates may well adopt a neutral outlook and hold bonds that have a duration that matches their investment horizon. If they believed interest rates were likely to remain stable for a time, they might hold bonds with a longer duration in a positive sloping yield curve environment, and pick up additional yield but with higher interest-rate risk.

Once the decision has been made on which part of the yield curve to invest in or switch into, the investor must decide on the specific securities to hold, which then brings us on to relative Chapter 1: Approaches to Trading and Hedging 7 6. Yield and duration of gilts, 21 October For this the investor will analyse specific sectors of the curve, looking at individual stocks.

An assessment of a local part of the yield curve will include looking at other features of individual stocks in addition to their duration.

This recognises that the yield of a specific bond is not only a function of its duration, and that two bonds with near-identical duration can have different yields. The other determinants of yield are liquidity of the bond and its coupon.

To illustrate the effect of coupon on yield consider Table 1. This shows that, where the duration of a bond is held roughly constant, a change in coupon of a bond can have a significant effect on the bond's yield. In the case of the long bond, an investor could under this scenario both shorten duration and pick up yield, which is not the first thing that an investor might expect.

However an anomaly of the markets is that, liquidity issues aside, the market does not generally like high coupon bonds, so they usually trade cheap to the curve. The other factors affecting yield are supply and demand, and liquidity. A shortage of supply of stock at a particular point in the curve will have the effect of depressing yields at that point.

A reducing public sector deficit is the main reason why such a supply shortage might exist. Duration and yield comparisons for bonds in a hypothetical inverted curve environment. Institutional investors prefer to hold the benchmark bond, which is the current two-year, five-year, ten-year or thirty-year bond and this depresses the yield on the benchmark bond. A bond that is liquid also has a higher demand, thus a lower yield, because it is easier to convert into cash if required.

This can be demonstrated by valuing the cash flows on a six-month bond with the rates obtainable in the Treasury bill market. We could value the six-month cash flows at the six-month bill rate. The lowest obtainable yield in virtually every market4 is the T-bill yield, therefore valuing a six-month bond at the T-bill rate will produce a discrepancy between the observed price of the bond and its theoretical price implied by the T-bill rate; as the observed price will be lower.

The reason for this is simple: We have therefore determined that a bond's coupon and liquidity level, as well as its duration, will affect the yield at which it trades. These factors can be used in conjunction with other areas of analysis, which we look at next, when deciding which bonds carry relative value over others. This is because it does not highlight any characteristics of the yield curve other than its general shape; this does not assist in the making of trading decisions.

To facilitate a more complete picture, we might wish to employ the technique described here. The other curve in Figure 1. Using the two figures together, an investor can see the impact of coupons, the shape of the curve and the effect of yield on different maturity points of the curve.

In a government bond market, there is no credit risk consideration unless it is an emerging market government market , and 3 4 5 The requirements of the MFR were removed during and the UK gilt yield curve exhibited a conventional positive-sloping shape shortly afterwards. The author is not aware of any market where there is a yield lower than its shortest-maturity T-bill yield, but that does not mean such a market doesn't exist!

Approaches to Trading and Hedging 9 6. T-bill and par yield curve, October There are a number of factors that can be assessed in an attempt to identify relative value. The objective of much of the analysis that occurs in bond markets is to identify value, and identifying which individual securities should be downloadd and which sold.

At the overview level, this identification is a function of whether one thinks interest rates are going to rise or fall. At the local level though, the analysis is more concerned with a specific sector of the yield curve, whether this will flatten or steepen, whether bonds of similar duration are trading at enough of a spread to warrant switching from one into another.

The difference in these approaches is one of identifying which stocks have absolute value, and which have relative value. A trade decision based on the expected direction of interest rates is based on assessing absolute value, whether interest rates themselves are too low or too high. Yield curve analysis is more a matter of assessing relative value. On very! However this is not really a real-life situation.

Instead, a trader might find himself assessing the relative value of the three-year bond compared to much shorter- or longer-dated instruments. That said, there is considerable difference between comparing a short-dated bond to other short-term securities and comparing say, the two-year bond to 3 2 Spread bps 1 0 —1 0.

Structure of bond yields, October Although it looks like it on paper, the space along the x-axis should not be taken to imply that the smooth link between one-year and five-year bonds is repeated from the five-year out to the thirty-year bonds. It is also common for the very short-dated sector of the yield curve to behave independently of the long end.

One method used to identify relative value is to quantify the coupon effect on the yields of bonds. The relationship between yield and coupon is given by 1. The coefficient c reflects the effect of a high coupon on the yield of a bond. In reality this relationship may not be purely linear; for instance the yield spread may widen at a decreasing rate for higher coupon differences.

Therefore 1. The same analysis can be applied to bonds with coupons lower than the same-duration par bond. The value of a bond may be measured against comparable securities or against the par or zero-coupon yield curve. In certain instances the first measure may be more appropriate when for instance, a low coupon bond is priced expensive to the curve itself but fair compared to other low coupon bonds. In that case the overpricing indicated by the par yield curve may not represent unusual value, rather a valuation phenomenon that was shared by all low coupon bonds.

Having examined the local structure of a yield curve, the analysis can be extended to the comparative valuation of a group of similar bonds. This is an important part of the analysis, because it is particularly informative to know the cheapness or dearness of a single stock compared to the whole yield curve, which might be somewhat abstract.

Instead we would seek to identify two or more bonds, one of which was cheap and the other dear, so that we might carry out an outright switch between the two, or put on a spread trade between them. Using the technique we can identify excess positive or negative yield spread for all the bonds in the term structure. This has been carried out for our five gilts, together with other less liquid issues as at October and the results are summarised in Table 1.

From the table as we might expect the benchmark securities are all expensive to the par curve, and the less liquid bonds are cheap. Note that the 6. This analysis used mid-prices, which would not be available in practice. Yields and excess yield spreads for selected gilts, 22 October We now look at the issues involved in putting on a spread trade.

Generally there is no analytical relationship between changes in a specific yield spread and changes in the general level of interest rates. That is to say, the yield curve can flatten when rates are both falling or rising, and equally may steepen under either scenario as well. The key element of any spread trade is that it is structured so that a profit or any loss is made only as a result of a change in the spread, and not due to any change in overall yield levels.

That is, spread trading eliminates market directional or first-order market risk. If a trader believed that the yield curve was going to flatten, but had no particular strong feeling about whether this flattening would occur in an environment of falling or rising interest rates, and thought that the flattening would be most pronounced in the two-year versus ten-year spread, they could put on a spread consisting of a short position in the two-year and a long position in the ten-year.

This spread must be duration-weighted to eliminate first-order risk. At this stage we must point out, and it is important to be aware of, the fact that basis point values, which are used to weight the trade, are based on modified duration measures. This measure is an approximation, and will be inaccurate for large changes in yield. Therefore the trader must monitor the spread to ensure that the weights are not going out of line, especially in a volatile market environment.

Bond basis point value, 22 October To weight the spread, we use the ratios of the BPVs of each bond to decide on how much to trade. In that case he must sell 0: It is also possible to weight a trade using the bonds' duration values, but this is rare. It is common practice to use the BPV. The payoff from the trade will depend on what happens to the two-year versus ten-year spread.

If the yields on both bonds move by the same amount, there will be no profit generated, although there will be a funding consideration. If the spread does indeed narrow, the trade will generate profit. Note that disciplined trading calls for both an expected target spread as well as a fixed time horizon. So for example, the current spread is If at the end of three weeks the spread has not reached the target, the trader should unwind the position anyway, because that was their original target.

Again, disciplined trading suggests the profit should be taken. If contrary to expectations the spread starts to widen, if it reaches The financing of the trade in the repo markets is an important aspect of the trade, and will set the trade's break-even level.

If the bond being shorted in our example, the two-year bond is special, this will have an adverse impact on the financing of the trade. The repo considerations are reviewed in Choudhry Most bond spread trades are yield curve trades where a view is taken on whether a particular spread will widen or narrow.

Therefore it is important to be able to identify which sectors of the curve to sell. Assuming that a trader is able to transact business along any part of the yield curve, there are a number of factors to consider. In the first instance, the historic spread between the two sectors of the curve.

Approaches to Trading and Hedging 13 7.

Advanced Fixed Income Analysis - 2nd Edition

Other factors to consider are demand and liquidity for individual stocks relative to others, and any market intelligence that the trader gleans. If there has been considerable customer interest on certain stocks relative to others, because investors themselves are switching out of certain stocks and into others, this may indicate a possible yield curve play.

It is a matter of individual judgement. An historical analysis requires that the trader identify some part of the yield curve within which he expects to observe a flattening or steepening. It is of course entirely possible that one segment of the curve will flatten while another segment is steepening, in fact this phenomenon is quite common. This reflects the fact that different segments respond to news and other occurrences in different ways.

A more exotic type of yield curve spread is a curvature trade. Consider for example a trader who believes that three-year bonds will outperform on a relative basis, both two-year and five-year bonds. In our example the trader will download the three-year bond, against short sales of both the two-year and the five-year bonds. All positions are duration-weighted. The principle is exactly the same as the butterfly trade we described in the previous section on futures trading.

Therefore the calculation of a hedge is 14 Advanced Fixed Income Analysis critical. A hedge is a position in a cash or off-balance sheet instrument that removes the market risk exposure of another position. For example a long position in year bonds can be hedged with a short position in year bonds, or with futures contracts.

That is the straightforward part; the calculation of the exact amount of the hedge is where complexities can arise. In this section we review the basic concepts of hedging, and a case study at the end illustrates some of the factors that must be considered. From the sample of gilts in Table 1.

This approach is very common in the market; however it suffers from two basic flaws that hinder its effectiveness. First, the approach assumes implicitly comparable volatility of yields on the two bonds, and secondly it also assumes that yield changes on the two bonds are highly correlated.

Where one or both of these factors do not apply, the effectiveness of the hedge will be compromised. The assumption of comparable volatility becomes increasingly unrealistic the more the bonds differ in terms of market risk and market behaviour.

Consider a long position in twoyear bonds hedged with a short-position in five-year bonds. Using the bonds from Table 1.

Even if we imagine that yields between the two bonds are perfectly correlated, it may well be that the amount of yield change is different because the bonds have different volatilities. For example if the yield on the five-year bond changes only by half the amount that the twoyear does, if there was a 5 basis point rise in the two-year, the five-year would have risen only by 2. This would indicate that the yield volatility of the two-year bond was twice that of the five-year bond.

This suggests that a hedge calculation that matched nominal amounts, due to BPV, on the basis of an equal change in yield for both bonds would be incorrect. In our illustration, the short position in the five-year bond would be effectively hedging only half of the risk exposure of the two-year position.

The implicit assumption of perfectly correlated yield changes can also lead to inaccuracy. Across the whole term structure, it is not always the case that bond yields are even positively correlated all the time although most of the time there will be a close positive correlation. Therefore, using our illustration again, imagine that the two-year and the fiveyear bonds possess identical yield volatilities, but that changes in their yields are uncorrelated.

This means that knowing that the yield on the two-year bond rose or fell by one basis point does not tell us anything about the change in the yield on the five-year bond. If yield changes between the two bonds are indeed uncorrelated, this means that the five-year bonds cannot be used to hedge two-year bonds, at least not with accuracy. Considering volatilities and correlations, Table 1. The standard deviation of weekly yield changes was in fact highest for the short-date paper, and actually Chapter 1: Approaches to Trading and Hedging 15 Segment 2-year 3-year 5-year year year year Volatility bp Yield volatility and correlations, selected gilts October From the table we also note that changes in yield were imperfectly correlated.

We expect correlations to be highest for bonds in the same segments of the yield curve, and to decline between bonds that are in different segments. This is not surprising, and indeed two-year bond yields are more positively correlated with five-year bonds and less so with year bonds. We can use the standard relationship for correlations and the effect of correlation to adjust a hedge.

Therefore for a two-bond position we set the standard deviation of the change in the position as 1. We can rearrange 1. The derivation of 1. A lower correlation leads to a smaller hedge position, because where yield changes are not closely related, this implies greater independence between yield changes of the two bonds. In a scenario where the standard deviation of two bonds is identical, and the correlation between yield changes is 1, 1.

All the possible hedge bonds under consideration are given below. The trader might elect to do the following: He can attempt to match both duration and convexity by constructing two portfolios with a duration of 4. Inspection of the table given above suggests the following bonds would be suitable components of these two portfolios: Approaches to Trading and Hedging 17 Portfolio A: As above, but with the following bonds: Therefore, however the portfolios are combined, the duration will remain at 4.

We now need to determine what amounts of each portfolio are required, such that the combined portfolio convexity is The trader can use simple proportions to determine the amount of each portfolio which would be necessary to form a new portfolio of convexity The exact amounts are: Summary of derivation of optimum hedge equation From equation 1.

Brennan, M. Campbell, J. Choudhry, M. Fisher, L. Neftci, S. Such trades involve simultaneous positions in bonds of different maturity.

Other relative value trades may position high-coupon bonds against low-coupon bonds of the same maturity, as a taxrelated transaction. These trades are concerned with the change in yield spread between two or more bonds rather than a change in absolute interest rate level.

The key factor is that changes in spread are not conditional upon directional change in interestrate levels; that is, yield spreads may narrow or widen whether interest rates themselves are rising or falling. Typically, spread trades will be constructed as a long position in one bond against a short position in another bond. If it is set up correctly, the trade will only incur a profit or loss if there is a change in the shape of the yield curve.

This is regarded as being first-order risk neutral, which means that there is no interest-rate risk in the event of change in the general level of market interest rates, provided the yield curve experiences essentially a parallel shift. In this chapter we examine some common yield spread trades. This market determination is a function of three factors: Government securities such as gilts are default-free and so this factor drops out of the analysis.

Intuitively we associate higher risk with longer-dated instruments, for which investors must be compensated in the form of higher yield. This higher risk reflects greater uncertainty with longer-dated bonds, both in terms of default and future inflation and interest rate levels.

However, for a number of reasons the yield curve assumes an 1 This chapter was presented by the author as an internal paper in July when he was working at Hambros Bank Limited. It previously appeared in Fabozzi The prices quoted are tick prices, fractions of 32nd, identical to US Treasury pricing.

Gilts are now quoted as decimal prices. We summarise some interest-rate models, in a practical way that excludes most of the mathematics.

The aim is firmly to discuss application of the models, and not to derive them or prove the maths. In this way readers should be able to assess the different methodologies for themselves and decide the efficacies of each for their own purposes. As always, selected recommended texts, plus chapter references are listed at the back, and would be ideal as a starting point for further research.

This serves to highlight that this book is very much a summary of the latest developments, rather than a fully comprehensive review of the subject. The topics would be suitable for a separate book in their own right, and such a book would make an ideal companion to this book.

However there is sufficient detail and exposition here to leave the reader with, hopefully, a good understanding of the subject. To begin with we look at relative value trading, and some aspects of this for the bond trader. We then move on to the second part of this book, and introduce the dynamics of asset pricing, which is fundamental to an understanding of yield curve analysis.

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.

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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.

Fixed Income Analysis, 3rd Edition

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.

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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.