Fixed Income Finance: A Quantitative Approach - Hardcover

Wise, Mark; Bhansali, Vineer

Hardcover

ISBN 10: 0071621202 ISBN 13: 9780071621205

Publisher: McGraw Hill, 2010

Synopsis

A complete guide forprofessionals with advancedmathematical skills but little orno financial knowledge . . .

You’re smart. Logical. Mathematicallyadept. One of those people who canmake quick work of long, difficult equations.But when it comes to managing a financialportfolio and managing risk, you wonder ifyou’re missing out.

Fixed Income Finance is the book for you.It’s the perfect introduction to the concepts,formulas, applications, and methodology,all derived from first principles, that youneed to succeed in the world of quantitativefinance―with a special emphasis on fixedincomes. Written by two of the sharpest analyticalminds in their fields, this instructiveguide takes you through the basics of fixedincome finance, including many new andoriginal results, to help you understand:

• Treasury Bonds and the Yield Curve
• The Macroeconomics behind TermStructure Models
• Structural Models for CorporateBonds and Portfolio Diversification
• Options
• Fixed Income Derivatives
• Numerical Techniques

Filled with step-by-step equations, clearand concise concepts, and ready-to-useformulas, this essential workbook bridges thegap between basic beginners’ primers andmore advanced surveys to provide hands-on tools you can begin to use immediately. It’s allyou need to put your math skills to work―and make the money work for you.

Brilliantly researched, impeccably detailed,and thoroughly comprehensive, Fixed IncomeFinance is applied mathematics at its best andmost useful.

About this title

About the Author

Mark Wise is the John A. McConeProfessor of High Energy Physics at theCalifornia Institute of Technology. He is thewinner of the 2001 J.J. Sakurai Prize of theAmerican Physical Society and a member ofthe American Academy of Arts and Sciencesand National Academy of Sciences. He is alsothe coauthor of Heavy Quark Physics.
Vineer Bhansali is an executive vicepresident, portfolio manager, firm-wide headof analytics for portfolio management, and asenior member of PIMCO’s portfolio managementteam. He is the author of Pricingand Managing Exotic and Hybrid Options andcurrently serves as an associate editor forthe International Journal of Theoretical andApplied Finance.

From the Back Cover

If you have a talent for math, you've got a head start on building wealth . . .

A hands-on user’s guide to the world of quantitative finance, this much-neededbook shows you how to apply your advanced mathematical skills to a vast arrayof financial opportunities available to those investing in fixed income. Written by twoanalytical experts, Fixed Income Finance shows you how to:

• Obtain a deep understanding of the risks and rewards of bonds
• Use tools of modern fixed income finance to creatively solve new problems
• Value bonds and their derivatives using rigorous foundations
• Properly manage the risk-reward tradeoffs in bond portfolios
• Build a toolkit that you can apply to other practical problems

Filled with step-by-step derivations, many of them original and detailed calculations,and other applied mathematics not easily found in existing literature, Fixed IncomeFinance helps you manage the ups and downs, ins and outs, of quantitative finance—quickly, easily, and profitably.

Excerpt. © Reprinted by permission. All rights reserved.

FIXED INCOME FINANCE

A QUANTITATIVE APPROACH

By MARK B. WISE, VINEER BHANSALI

The McGraw-Hill Companies, Inc.

Copyright © 2010 The McGraw-Hill Companies, Inc.
All rights reserved.

ISBN: 978-0-07-162120-5

Contents

Preface
1 Bond Basics
2 The Mathematics of Financial Modeling
3 Structural Models
4 Bond Portfolio Management
5 Term Structure Models
6 Derivatives of Bonds
7 Trees
Index

Excerpt

<h2>CHAPTER 1</h2><p><b>Bond Basics</p><br><p>1.1 Treasury Bonds and the Yield Curve</b></p><p>A zero coupon bond pays its holder an amount <i>P</i> at some time <i>T</i>years in the future. <i>P</i> is called the principal of the bond, and<i>T</i> is the time of maturity. The bonds can be issued, for example, bycorporations or by the U.S. Federal Reserve. Suppose an investor purchases azero coupon bond today; how much should the investor pay? This depends on anumber of factors. For example, how sure is the investor that the institutionthat issued the bond will be able to make the principal payment at the time ofmaturity <i>T</i>? If the institution is a corporation, it might go bankruptbefore that date and not be able to make the full principal payment <i>P</i>.This possibility is a source of risk for the investor; it is called <i>creditrisk</i>. If the institution is the U.S. Treasury, the credit risk isnonexistent. Yet even for a Treasury bond, we would not pay $100 today to get$100 at some time in the future because the present value is degraded byinflation and the amount of future inflation is uncertain. The present value,i.e., the amount an investor would pay today for a zero coupon Treasury bondthat matures in <i>T</i> years and has principal <i>P</i>, can be written as</p><p>Price = 1/(1 + <i>Y</i><sub>1</sub>)<i><sup>T</sup> P</i> (1.1.1)</p><p>We can think of <i>Y</i><sub>1</sub> in <b>Equation 1.1.1</b> as a yearlyinterest rate by which we are discounting the value of the payment <i>P</i>that occurs at time <i>T</i>. After all, if you had invested that amount todayand received yearly interest at a rate <i>Y</i><sub>1</sub> that was compoundedannually, then the value of your investment at maturity <i>T</i> would equalthe principal <i>P. Y</i><sub>1</sub> is called the <i>yield to maturityT</i>.</p><p>The yield is often quoted in units of percent or basis points (bp). One hundredbasis points equals 1 percent. If <i>T</i> = 10 years and the (annual) yield tothis maturity is equal to 5 percent or 500 bp, the price of a zero couponTreasury bond with principal $100 is $61.39.</p><p><b>Equation 1.1.1</b> is written in a way that suggests that the principal isdiscounted annually. However, there is nothing special about discountingannually. Suppose we discount every 1/<i>n</i> years by an amount<i>Y<sub>n</sub>/n</i>, where <i>n</i> is a natural number greater than 1.Then <b>Equation 1.1.1</b> becomes</p><p>Price = 1/(1 + <i>Y<sub>n</sub>)<sup>nT</sup> P</i> (1.1.2)</p><p>Equating the prices in <b>Equations 1.1.1</b> and <b>1.1.2</b> gives</p><p><i>Y<sub>n</sub> = n</i> [(1 + <i>Y</i><sub>1</sub>])<sup>1/<i>n</i></sup>] (1.1.3)</p><p>Taking the limit of <b>Equation 1.1.2</b>, keeping <i>Y<sub>n</sub></i> fixedat <i>Y</i>, as <i>n</i> -> ∞ gives the formula</p><p>Price = <i>e<sup>-YT</sup> P</i> (1.1.4)</p><p>which corresponds to discounting continuously in time by the fixed yield<i>Y</i>. In this limit, the discount factor for each infinitesimal timeinterval [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where we haveneglected quadratic terms in the infinitesimal time interval <i>dt</i>.Repeating this infinitesimal discounting for each successive time interval oflength <i>dt</i> gives the price in <b>Equation 1.1.4</b> written as theexponential of an integral,</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.5)</p><p>as <i>n</i> -> ∞. The yield for discounting continuously in time<i>Y</i> is related to the yield for yearly discounting <i>Y</i><sub>1</sub>by</p><p><i>Y</i> = log(1 + <i>Y</i><sub>1</sub>) (1.1.6)</p><p>In other words, annual discounting and continuous discounting are equivalent.They are just different ways of writing the same price.</p><p>There is no reason that all maturities should be discounted by the same factor.Suppose that the discounting rate for the time interval [<i>t, t + dt</i>](i.e., the "short rate" at time <i>t</i&gtt;) is <i>y(t)</i>; the price thenbecomes</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.7)</p><p>and the yield to maturity <i>T</i> can be written as</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.8)</p><p>Multiplying the above by <i>T</i> and differentiating with respect to thematurity, the "short rate" at any time <i>T</i> is given by</p><p><i>y(T) = d[TY(T)]/dT</i> (1.1.9)</p><p><i>YY(T)</i> is also called the <i>spot rate</i>. It is a function of thematurity <i>T</i>, and this function is known as the <i>yield curve</i>. Undertypical circumstances, we can expect that the spot rate will be an increasingfunction of <i>T</i>. We may be confident that inflation will be contained overthe near term, but as the period of time increases, that confidence diminishes,and the investor who purchases a zero coupon bond should demand compensation forthat source of risk. Of course, if investors feel that the economy is about togo into recessiiiion, then we might expect that inflationary pressures and interestrates will fall in the future, and in that case <i>Y(T)</i> could, for somerange of <i>T</i>, be a decreasing function of <i>T</i>. This is called an<i>inverted yield curve</i>.</p><p>The value of a zero coupon Treasury bond increases with time until maturity,when it is equal to its principal. Let <i>V</i><sub>rf</sub><i>(t)</i> denotethe value of a zero coupon Treasury bond at some time <i>t</i> in the future.The subscript rf emphasizes that the cash flows of Treasury bond investments arerisk-free, i.e., the investor is certain to get the full principal returned atthe time of maturity. The value of a zero coupon bond at some time t in thefuture is related to its value today, <i>V</i><sub>rf</sub>(0), by</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.10)</p><p>This exponential growth is just undoing the effect of the discounting. Theinstantaneous rate of return for investments in Treasury bonds is equal to theshort rate, <i>y(t)</i>, since</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.11)</p><p>So far we have only discussed zero coupon bonds. There are more complicatedbonds that can be made by putting together zero coupon bonds of varyingmaturities. For example, suppose you purchase a coupon-paying bond that pays afixed "coupon" <i>cP</i> every year until it matures at a time <i>T</i> yearsin the future. At maturity, the principal <i>P</i> is also paid along with thefinal coupon payment. Each of the coupon payments can be thought of as a zerocoupon bond with principal <i>cP</i> and a maturity that corresponds to thedate on which that coupon payment is made. Hence if you understand everythingabout zero coupon bonds, you understand coupon-paying bonds as well. Consider aTreasury bond with principal <i>P</i> and maturity <i>T</i> years that pays acoupon <i>cP</i> annually. Assuming that the yield curve is flat, <i>Y(T) =Y</i>, independent of <i>T</i>, it has the price</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.12)</p><p>where we have used the familiar expression for the sum of a geometric series,</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.13)</p><p>The price in <b>Equation 1.1.12</b> is equal to the principal <i>P</i> for acoupon, <i>cP</i>, when</p><p><i>c = e<sup>Y</sup></i> - 1 = <i>Y</i><sub>1</sub> (1.1.14)</p><p>A coupon-paying bond that has a price equal to its principal is said to trade atpar. If <i>c</i> is greater than <i>Y</i><sub>1</sub>, its price is greaterthan its principal and the bond is said to trade at a premium, and if <i>c</i>is less than <i>Y</i><sub>1</sub>, the bond is said to trade at a discount. Asimilar analysis holds for coupon payments made at other intervals. For example,if the bond pays a coupon <i>cP</i> quarterly, then it trades at par if <i>c =Y</i><sub>4</sub>/4. Coupon payments provide a fixed income stream that manyinvestors find attractive. They are the origin of the phrase <i>fixed-incomefinance</i>, which is used to describe the part of finance involving bonds.</p><br><p><b>1.2 Duration and Convexity</b></p><p>As an investor in a zero coupon bond, you might be interested in selling it toanother investor sometime in the future. At that future time, the yield curvemight be different from what it is today. Hence it is interesting to understandthe risk associated with changes in the yield curve. Consider the change in bondprice ΔPrice associated with an overall shift in the yield curve <i>Y(T)-> Y(T) + ΔY</i>. Expanding in a power series in Δ<i>Y</i>,</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.15)</p><p>where we have included only the first two terms in the power series. Theduration <i>D</i> and convexity <i>C</i> depend on the bond's maturity and arerelated to the first and second derivatives of the price with respect to changesin overall yield level:</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.16)</p><p>Using the price formula for a zero coupon bond in <b>Equation 1.1.4</b>, wefind that the duration and convexity of a zero coupon bond are</p><p><i>D = T C = T</i><sup>2</sup> (1.2.17)</p><p>A duration of 10 years means that for a change in the annualized yield of ±1percent or equivalently ±100 bp, the first-order fractional shift in thebond price is [??] 10<sup>-2</sup> x 10 = [??] 0.1 = [??] 10 percent. A convexity of (10 years) means that for a change in theannualized yield of ±1 percent, the second-order fractional shift in the bondprice is 0.5 x 100 x 10<sup>-4</sup> = 0.5 percent. The convexity term increasesthe bond price no matter what sign the change in yield is. Hence bondholdersgain more from a yield decrease than they lose from a yield increase of the samesize.</p><p>Next consider a bond of principal <i>P</i> that pays an annual coupon<i>cP</i> and matures in <i>T</i> years. We can define an average payment timeby weighting the time at which the payment is made with the present value of thepayment. This average payment time is the same as the duration <i>D</i>,</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.18)</p><p>The duration depends on the definition of the yield used. For example, if weused the annual discounting yield <i>Y</i><sub>1</sub> to define duration,then, since</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.19)</p><p>the duration, defined as the bond price sensitivity with respect to changes of<i>Y</i><sub>1</sub>, is, for a flat yield curve, <i>D<sub>1</sub> = D/(1 +Y<sub>1</sub>)</i>. The duration <i>D</i>, defined with respect to thecontinuous discounting yield, is usually referred to as the Macaulay duration.</p><p>Duration, which characterizes the impact of small overall changes in the yieldcurve on bond prices, is the most important risk variable for most Treasury bondinvestors. As an example, consider a 10-year bond with principal $100 that paysa coupon <i>cP</i> that is 7 percent of the principal. Suppose the yield curveis flat (i.e., independent of maturity <i>T</i>) at a level of 5 percent. Thisbond trades at a premium and, using <b>Equation 1.1.12</b>, its price is$114.37. Differentiating the price with respect to the yield (and multiplying byminus one divided by the price) gives a Macaulay duration of 7.6935 years, andtaking the second derivative, the convexity is 69.02 years<sup>2</sup>. For anincrease in the (continuously compounding) yield of 1 percent, the durationresults in a change in the bond price of -7.69 x 0.01 x $114.37 = -$8.80. Theconvexity term increases the bond price by 0.5 x 69.02 x 10<sup>-4</sup> x$114.37 = $0.39. The total change in the bond price is -$8.41. On the otherhand, if the yield decreased by 1 percent, the total change in the bond price is$8.80 + $0.39 = $9.19.</p><br><p><b>1.3 Corporate Bonds and Credit Risk</b></p><p>For Treasury bond investors, the instantaneous (i.e., short) risk-free rate ofreturn is <i>y(t)</i>. Since the corporate bond investor is exposed to defaultrisk, the short corporate bond rate of return <i>y<sub>c</sub>(t)</i> isgreater than the Treasury short rate. We write <i>y<sub>c</sub>(t) = y(t) +λ(t)</i> for an infinitesimal time <i>dt</i> in which the company doesnot default. Note that <i>y<sub>c</sub>(t)</i> is the correct rate of returnonly in time intervals in which the company does not default. It is sometimescalled the <i>promised corporate rate of return</i>. The difference between thepromised corporate short rate of return and the risk-free short rate is theexcess promised short rate of return λ<i>(t)</i>. It is needed in orderto compensate the corporate bond investor for the credit risk. To give a formulafor how, on average, the value of an investment in a zero coupon corporate bondincreases with time, we assume that if the company defaults, the investorimmediately gets a recovery fraction <i>R</i> times the promised value of thebond at the time of default returned to him and reinvests it at the risk-freerate. More explicitly, if the time of default is <i>t<sub>d</sub></i>, thevalue amount returned to the investor at that time is</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.20)</p><p>where <i>V<sub>c</sub></i>(0) is the initial value of the zero coupon corporatebond, i.e., the price paid for it. Before default, investors do not know whatthe recovery fraction will be. We will treat <i>R</i> as a fixed quantity thatthe investor estimates using historical recovery fractions from a wide range ofcorporations and from information specific to the company that issued the bond.</p><p>It is useful to introduce the function <i>P<sub>S</sub>(t)</i>, the probabilityof the company's surviving to time t without defaulting. Since<i>-(dP<sub>S</sub>(t)/dt)dt</i> is the probability of the company's defaultingin the time interval between <i>t</i> and <i>t + dt</i>, the formula for how,on average, the value of an investment in a zero coupon corporate bond increaseswith time is</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.21)</p><p>The term in <b>Equation 1.3.21</b> without the factor of <i>R</i> takes intoaccount the case in which the corporation does not default between time zero and<i>t</i>, and this happens with probability <i>P<sub>S</sub>(t)</i>. The termproportional to <i>R</i> takes into account the case in which the corporationdoes default. It is more complicated than the first term because the default canoccur at any time <i>s</i> before <i>t</i> and the compounding factor changesto the risk-free Treasury rate after the default, since we have assumed that theamount recovered is reinvested in a Treasury bond.</p><p>It is important to know the additional promised short rate of returnλ*<i>(t)</i> that, on average, makes an investment in a corporate bondincrease in value by the same amount in the time <i>t</i> as an investment in arisk-free Treasury bond. It can be expressed in terms of the recovery fractionand survival probabilities by solving the equation</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.22)</p><p>Using <b>Equations 1.3.21</b> and <b>1.1.10</b>, this implies that</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.23)</p><p>Differentiating <b>Equation 1.3.23</b> with respect to the time <i>t</i>, wefind that</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.24)</p><p>which after rearranging becomes</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.25)</p><p>Note that if <i>P<sub>S</sub>(t)</i> = 1, which corresponds to the probabilityof default being zero, then λ* = 0, since there is no credit risk in thiscase. Similarly, if <i>R</i> = 1, there is also no credit risk, since even ifthe company defaults, the investor gets the full value of her investmentreturned, and so in that case λ* = 0 as well. Corporate bond investorsusually demand an average rate of return that is greater than that of a risk-free Treasury bond. So we write</p><p>λ(<i>t</i>) = λ * (<i>t</i>) + µ(<i>t</i>) (1.3.26)</p><p>where µ<i>(t)</i> is called the <i>risk premium</i>. A positive riskpremium makes the average corporate bond return greater than that of a risk-freezero coupon Treasury bond.</p><p>Suppose that if the company survives to time <i>t</i>, there is a probability<i>h(t)dt</i> of default occurring between <i>t</i> and <i>t +dt</i>.<sup>2</sup> The survival probability decreases in the time interval<i>dt</i> because the company may default in that time interval:</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.27)</p><p>Integrating this differential equation using the initial condition<i>P<sub>S</sub></i>(0) = 1 gives</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.28)</p><p>The function <i>h(t)</i> is called the <i>hazard rate</i>, and the excesspromised short rate of return that makes an investment in a corporate bondreturn the same amount as in a Treasury bond is given in terms of it by</p><p>λ * (<i>t</i>) = (1 - <i>R</i>)<i>h(t)</i> (1.3.29)</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>For a constant hazard rate <i>h</i>, the excess promised short rate of returnλ* is also constant.</p><p>On average, the value of an investment made at time <i>t</i> = 0 in a zerocoupon corporate bond with principal <i>P</i> is, at maturity <i>T</i>,</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.30)</p><p>Dividing this by <i>V<sub>c</sub></i>(0) and then using <b>Equation1.3.21</b>, we find that</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.31)</p><p>It is convenient to introduce the analog of the spot Treasury yield forcorporate bonds. We call this quantity <i>Y<sub>c</sub>(T)</i>, and it isdefined by</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.32)</p><p>In terms of the corporate spot yield, the formula for the price of a zero couponcorporate bond of principal <i>P</i> and maturity <i>T</i> is</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.33)</p><p>When <i>R</i> = 0, it is straightforward to see that <b>Equation 1.3.33</b>arises from discounting the expected cash flow (i.e., the principal payment atmaturity times the probability of the company's surviving to maturity withoutdefault) by the risk-free Treasury rate plus the risk premium.</p><p>To review, when the risk premium is zero, on average, a zero coupon corporatebond has the same instantaneous rate of return as an investment in a zero couponrisk-free Treasury bond. In that case, the price of a zero coupon corporate bondis given by <b>Equation 1.3.33</b> with the corporate short rate<i>y<sub>c</sub> = y + λ*</i>, where λ* is given by <b>Equation1.3.25</b>. Note that this value of <i>y<sub>c</sub></i> takes into account theprobability of default, so that even though the average return is the same asthat on a Treasury bond, the short corporate rate <i>y<sub>c</sub></i> isgreater than <i>y</i>.</p><p>The hazard rate can implicitly depend on the yield. The partial derivative</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3.34)</p><p>is usually less than 1. Recall that <i>Y<sub>c</sub></i> is equal to <i>Y</i>plus an additional term that takes into account the default probability and therisk premium. It is this additional term that causesβ<sub><i>c,T</i></sub> to differ from unity. If the economy deteriorates,Y usually decreases; however, default risk increases, resulting in a value forβ<sub><i>c,T</i></sub> that is less than unity.</p><p><i>(Continues...)</i>

(Continues...)

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Edition details

• Publisher: McGraw Hill
• Publication date: 2010
• Language: English
• ISBN 10: 0071621202
• ISBN 13: 9780071621205
• Binding: Hardcover
• Edition number: 1
• Number of pages: 256