ADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason 2002 Vault Inc.

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1 ADVA FINAN QUAN 00 Vault Inc. VAULT GUIDE TO ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS

2 Copyrght 00 by Vault Inc. All rghts reserved. All nformaton n ths book s subject to change wthout notce. Vault makes no clams as to the accuracy and relablty of the nformaton contaned wthn and dsclams all warrantes. No part of ths book may be reproduced or transmtted n any form or by any means, electronc or mechancal, for any purpose, wthout the express wrtten permsson of Vault Inc. Vault.com, and the Vault logo are trademarks of Vault Inc. For nformaton about permsson to reproduce selectons from ths book, contact Vault Inc., 50 West nd Street, New York, New York 00-77, () Lbrary of Congress CIP Data s avalable. ISBN Prnted n the Unted States of Amerca

3 Vault Gude to Advanced Fnance and Quanttatve Intervews Table of Contents INTRODUCTION Your Frst Step Problem Solvng Strateges Sample Questons and Answers BOND FUNDAMENTALS 5 Bond Bascs Tme Value of Money Bond Prces and Relatonshps to Yelds Taylor Seres Expanson Bond Prce Dervatves Sample Questons and Answers Summary of Formulas STATISTICS 35 Random Varables Key Statstcal Fgures Permutatons and Combnatons Functons of Random Varables Dstrbutons Regresson Analyss Sample Questons and Answers Summary of Formulas Vst the Vault Fnance Channel, the complete onlne resource for fnance careers, featurng frm profles, message boards, the Vault Fnance Job Board, and more. C A R E E R L I B R A R Y

4 Vault Gude to Advanced Fnance and Quanttatve Intervews Table of Contents DERIVATIVES 8 Introducton to Dervatves Forward Contracts Futures Contracts Swaps Optons: An Overvew+RC Combnng Optons Opton Valuaton I: Introducng Black-Scholes Opton Valuaton II: Other Soluton Technques Opton Senstvtes: the Greeks Exchange-traded optons Exotc Optons Sample Questons and Answers Summary of Formulas FIXED INCOME 87 Bond and Fxed Income Market Issuers Types of Fxed Income Securtes Quotng Bond and Fxed Income Prces Analyzng and Valung Bonds and Fxed Income Securtes...96 Fxed-ncome Specfc Dervatves Mortage-backed Securtes Sample Questons and Answers Summary of Formulas Equty Markets Equty Valuaton Overvew Dvdend Dscount Model Multples Analyss Return on an Asset CAPM Equty Indexes Hybrds Equty-specfc dervatves Sample Questons and Answers C A R E E R L I B R A R Y 00 Vault.com Inc.

5 Vault Gude to Advanced Fnance and Quanttatve Intervews Table of Contents CURRENCY AND COMMODITY MARKETS 83 Currency Swaps Comodty Swaps Sample Questons and Answers RISK MANAGEMENT 97 Measurng Market Rsk: Value at Rsk Types of Rsk Currency Rsk Fxed-ncome Rsk Equty Rsk Currency and Commodty Rsk n the News Hedgng Rsk Portfolo Rsk and Correlaton Arbtrage Sample Questons and A+R[-47]Cnswers APPENDIX Advanced Fnance Glossary About the Author Vst the Vault Fnance Channel, the complete onlne resource for fnance careers, featurng frm profles, message boards, the Vault Fnance Job Board, and more. C A R E E R L I B R A R Y

6 Competton on the Street and beyond s heatng up. Wth the fnance job market tghtenng, you need to be your best. We know the fnance ndustry. And we ve got experts that know the fnance envronment standng by to revew your resume and gve you the boost you need to snare the fnancal poston you deserve. Fnance Resume Wrtng and Resume Revews Have your resume revewed by a practcng fnance professonal. For resume wrtng, start wth an e-maled hstory and - to -hour phone dscusson. Our experts wll wrte a frst draft, and delver a fnal draft after feedback and dscusson. For resume revews, get an n-depth, detaled crtque and rewrte wthn TWO BUSINESS DAYS. Thank you, thank you, thank you! I would have never come up wth these changes on my own! W.B., Assocate, Investment Bankng, NY Havng an experenced par of eyes lookng at the resume made more of a dfference than I thought. R.T., Managng Drector, SF I found the coachng so helpful I made three appontments! S.B., Fnancal Planner, NY Fnance Career Coachng Have a pressng fnance career stuaton you need Vault s expert advce wth? We ve got experts who can help. Tryng to get nto nvestment bankng from busness school or other careers? Swtchng from one fnance sector to another for example, from commercal bankng to nvestment bankng? Tryng to fgure out the cultural ft of the fnance frm you should work for? For more nformaton go to

7 ADVA FINAN INTRODUCTION QUAN

8 Introducton Your Frst Step Congratulatons on takng your frst step to succeedng n your advanced fnance ntervews. Ths book was wrtten to gve you the techncal background needed to master that ntervew compled nto one convenent volume. Quanttatve and Wall Street ntervews are notorously tough, and wth good reason. These types of jobs pay very well and a lot of people want them. Ths book wll gve you the edge you need to succeed. Ths s the book the wrters and edtors wsh they had when they were ntervewng. It s the dstllaton of years of experence n the fnance feld, n teachng fnance and n numerous ntervews. Vault edtors have even taken ntervews just to fnd out what knds of questons ntervewers are currently askng, n order to brng you the latest n ths book. In quanttatve ntervews, mastery of the subject matter s assumed t s your startng pont. You wll also have to convnce your ntervewer(s) you are the rght ft for the frm and have the experence and background that they are lookng for. Of course, no book can gve you that though the Vault Gude to Fnance Intervews gves you helpful ponters n that drecton. What ths book can do s help you revew and master the requred subject matter, wthout whch no amount of charm wll get you by. (Although charm s always good.) Also unque to ths book are strateges to help you succeed on those tough ntervew questons that you may not be prepared for. Some questons you may get are delberately desgned to be mpossble to answer. The ntervewer just wants to see how you thnk and how you approach problems. Remember, all of the easy problems have already been solved. The problems you wll see on the job wll lkely be thngs that no one has qute seen before. Stll, you wll fnd some ntervewers who wll ask questons straght out of textbooks (one nsder reports recevng the followng queston n a recent ntervew wth Bloomberg: What s an equvalence statement n FORTRAN and why would t be used? ) It s smply the style of certan companes and ntervewers to ask questons from textbooks, so you should be prepared for ths f you want to land a job. For nsde nformaton on ntervewer style, you may want to check out the Vault message boards. For everythng else, let ths book be your gude. Wherever possble, we ve used questons from actual ntervew experence, ncludng the ntervewer s comments on what they were lookng for (when we could get t). It s our hope that you wll fnd the problem solvng strateges and the materal n ths book ndspensable to you even after you land your job. Good luck!

9 Introducton Problem Solvng Strateges What do you do when confronted wth an ntervew queston you have absolutely no dea how to solve? We recommend the followng strateges t should help you handle most anythng thrown at you. Strategy # Strategy # Strategy #3 Strategy #4 Strategy #5 Strategy #6 Strategy #7 Strategy #8 Strategy #9 Cte from memory Draw a fgure Work backwards Formulate an equvalent problem Enumerate all cases Search for a pattern Bracket the answer solve the extreme cases Relate to somethng you know Take advantage of symmetry Remember to RELAX. Try to see these ntervews smply as conversatons. It s a chance for ntervewers to evaluate you, but remember, you are also decdng f you want to work there as well. The more relaxed and calm you are, the easer t wll be for you to thnk creatvely, whch s often what s requred n fnance ntervews. Also, try to thnk of the tough ntervew questons as amusng lttle problems (the ntervewer probably does). One recent ntervewee reports havng an ntervewer grll her relentlessly on currency forwards, nterest rate party and so on. When the job seeker fnally reported beng unsure of the approach to one queston, The ntervewer laughed and sad, Don t worry. If you had known the answer to ths problem, I would have found somethng else that you don t know. That s my job. Remember, sometmes you wll be able to use one strategy by tself to answer a queston, but magne what a powerful approach t s when you can combne two or more. Those tough ntervew questons won t have a chance. You wll see the above strateges used throughout ths book, and dentfed to help you remember them. Often, problems can be approached from more than one angle, so don t feel that you must use the approach we show.

10 Introducton Sample Questons. You have a sheet of paper and an nfnte supply of tokens. I also have an nfnte supply of tokens. We take turns placng tokens on the paper, one token at a tme. We cannot place tokens on top of other tokens (no overlappng), and the tokens cannot extend over the edges of the paper. The last player to place a token on the paper wns. What s your wnnng strategy? (Ths s called a strategy game queston, and s an actual queston recently asked on a hedge fund ntervew.) Soluton: Don t freak out f you see somethng lke ths. The ntervewer s just tryng to get a sense of how you attack a new problem. Let s go through our lst of tactcs. Tactc # wll not work here. Tactc # has promse: Try breakng t down nto smaller sub-problems. What f the paper were so small that only a sngle con could ft on t? In ths case your strategy would be to go frst. After you place your con, your opponent has no place to place hs, and you wn. Next, what f the paper were bg enough for two cons? Here, you place your con n the dead center of the sheet so your opponent can t place hs con. Agan, your strategy would be to go frst. Prohbted Ths tactc can be repeated untl you have derved the correct answer: You always move frst, and f you play the game properly, you wll always wn.. What do you thnk s the major factor mpactng the proftablty of an arlne? (Ths was an actual queston asked n a Goldman Sachs equty quanttatve research ntervew.) Soluton: Ths s another queston that the ntervewer doesn t expect you to have memorzed, but expects you to go through a reasonng process enumeratng possble factors affectng arlne proftablty to come up wth the most mportant one. You could say, passenger meals, labor costs, weather delays, leasng costs, marketng, mantenance, prce wars, but the major cost drver s probably fuel. 3. Would the volatlty of an enterprse be hgher or lower than the volatlty of ts equty? (Actually asked by a Goldman Sachs ntervewer who kept comng back to ths n one form or another durng the ntervew.) 3

11 Introducton Soluton: Ths s a straghtforward Statstcs or Corp Fnance 0 queston. Even f you have never seen ths exact queston before, t can be reasoned out. In the followng response we employ a combnaton of tactcs # and #5. Corporatons usually have both debt and equty (we reason.) So, suppose you have a portfolo consstng of w percent equty and (-w) percent debt. We calculate volatlty as the square of the standard devaton of p w E w D w wcovd, E E D stock returns. Then Assume that the covarance s zero. Then Now, consder three cases. p w E w D Case one: There s no debt. Then the varance of the enterprse ( p above) s equal to the varance of the equty. Case two: There s no equty. Then the varance of the enterprse ( p above) s equal to the varance of the debt. Case three: There s a combnaton of the two. We have bracketed the answer already n cases one and two: the result must le n between these. Now a judgment must be made. Assume that the volatlty of the debt s lower than the volatlty of the equty. Then the volatlty of the enterprse wth both debt and equty wll be lower than the volatlty of the equty alone, snce the volatlty of the enterprse s somewhat of a weghted average of both debt and equty. Also, note that we multply the volatlty of the equty by w, a fracton assumed to be less than one. The only way that we could have volatlty of the enterprse hgher than that of equty alone would be f D > E and f w were negatve (mpossble). To see ths, rearrange the equaton to p w E D D. If the vol of debt equals vol of equty, vol of the enterprse stll equals vol of the equty. 4

12 ADVA FINAN QUAN BOND FUNDAMENTALS

13 Increase your T/NJ Rato (Tme to New Job) Use the Internet s most targeted job search tools for fnance professonals. Vault Fnance Job Board The most comprehensve and convenent job board for fnance professonals. Target your search by area of fnance, functon, and experence level, and fnd the job openngs that you want. No surfng requred. VaultMatch Resume Database Vault takes match-makng to the next level: post your resume and customze your search by area of fnance, experence and more. We ll match job lstngs wth your nterests and crtera and e-mal them drectly to your n-box.

14 Bond Fundamentals Ths s the chapter that you wll need to know f an ntervewer or headhunter asks, Do you know bond math? and you want to answer, Yes. Bond Bascs A bond s a contract to provde payments accordng to a specfc schedule. Bonds are long-term securtes wth maturtes exceedng one year, n contrast to blls, or other short-term debt such as commercal paper, whch have maturtes of less than one year. The bond unverse s huge. There are treasury bonds, agency bonds, junk bonds, corporate bonds, zero coupon bonds, muncpal bonds, soveregn bonds, tax-free bonds and so forth. In addton to all of these, there are optons on bonds, optons on optons on bonds, and so on. (These wll be covered n detal n later sectons.) Fnally, most bonds are hghly senstve to nterest rates, so we wll have to study the yeld curve n some detal. For now, we wll cover the fundamental fnancal concepts requred n valuaton of bonds. Bonds are dfferent from equty In the contractual agreement of a bond, there s a stated maturty and a stated par value. Ths s unlke an equty, whch has no maturty and no guaranteed prce at maturty. To express ths defnte prce at maturty, we say that bonds converge to par value at maturty. Ths defned par value makes the volatlty of a bond generally lower than a share of stock (equty), especally as maturty draws close. However, don t get the dea that bonds are wthout rsk or unnterestng. Qute the contrary. Accordng to a February 8, 000, BusnessWeek Onlne artcle ( Is the Bond Market Ready for Day Traders? ), Bonds are no longer the stodgy nvestments they once were What most people don't know s that the 30-year Treasury bond has the same volatlty as an Internet stock. Constructng models for bond valuaton s one of the tougher challenges out there. Bonds have many nherent rsks, ncludng default rsk, bass rsk, credt rsk, nterest rate rsk and yeld curve rsk, all of whch may not apply to equty or equty-lke securtes. Bond ratngs Bonds are generally consdered to be less rsky than equty (except for junk bonds), so they can generally be expected to have lower rates of return. In the world of bonds, we are concerned wth the credt ratng of the company or muncpalty that ssued the bond. Credt ratngs are provded by major ratngs agences, ncludng Moody s, Standard & Poor s, and Ftch. You may wsh to famlarze yourself wth these ratngs ( Recently, a Goldman Sachs ntervewer quzzed one of ths book s edtors on ratngs of corporate bonds and subsdary lablty n case of default. Of course, you may not have to worry about ths f you do not have ths type of experence lsted on your resume, but remember, anythng on your resume, no matter how long ago or obscure, s far game.) Bond ratngs affect the ease and cost of obtanng credt for the corporaton ssung the bonds. The hgher the ratng AAA s the hghest S&P ratng, for example the lower the cost of credt. As the corporaton s credt ratng declnes, t gets more and more expensve for the corporaton to rase new funds. Most corporate treasures are concerned wth possble ratngs downgrades and check frequently wth ratngs agences before undertakng somethng that could potentally result n a downgrade. Downgrades can also affect nvestors, as many fxed ncome managers n asset management frms have mandates to hold only corporate-grade bonds and better. If a corporaton s bonds fall to the junk category (see, for example, Xerox, May 00), the nsttutonal nvestors n the company may have to sell ther holdngs to comply 7

15 Bond Fundamentals wth clent requrements. Ths dumpng whch could be large holdngs of the bonds makes the prce of the bonds drop. Before pushng forward wth valuaton of bonds under scenaros such as the above, we have to revew the tme value of money, whch s possbly the most mportant concept n fnance. You wll see ths over and over agan n varous forms, so there s no tme lke the present. 8

16 Bond Fundamentals Tme Value of Money Dscountng, present and future value If you depost $,000 n a bank or money market account today, you expect to earn some nterest on your nvestment so that, as tme passes, the value of your nvestment grows. (If t dd not, you would probably not put your money n the bank.) If you are earnng a rate of 0% a year, at the end of the year you wll have your orgnal $,000 plus the nterest earned, 0% of the prncpal nvested ($,000) or $00. The total value of your nvestment s then $,000 + $00 = $,00 = $,000( + r), where r s the nterest rate n percent for the perod. In general, the future value of an ntal nvestment P 0 over n compoundng perods s gven by the formula P N P0 r n nt where: n = number of compoundng perods n tme nterval t t = tme nterval r = the nterest rate earned per compoundng perod (assumed constant) P 0 = the ntal value (prncpal) P N = the fnal value The above formula then gves the future value of money. Ths type of formula s called smple compoundng. Many mportant results n fnance are based on ths very smple prncpal of the tme value of money. NOTE: The value P N s often called the future value and P 0 the present value. So we could also wrte FV PV r n nt Example (Annual Compoundng) What s the value of $,000 after one year f nterest s only compounded once per year? Here n =, t = year, r = 0%/year, P 0 = $, P $,000 =$,00 Example (Sem-Annual Compoundng): What s the value of $,000 after one year f nterest s compounded twce per year? Here n = and: 0.0 P $,000 =$,0.5 Example (Quarterly Compoundng) What s the value of $,000 after one year f nterest s compounded four tmes per year? Here n = 4 and: P $,000 =$,

17 Bond Fundamentals Example (Daly Compoundng) What s the value of $,000 after one year f nterest s compounded each tradng day? Here n = 50 and: P $,000 =$, The value of the nvestment ncreases as we compound more and more frequently snce nterest s beng compounded on nterest. In the lmt as n approaches nfnty, we have contnuous compoundng, whch gves the future value of money as: Pt P0 Example (Contnuous Compoundng) What s the value of $,000 after one year f nterest s compounded contnuously? e rt 0.* P $, 000e =$,05.7. It should not surprse you that ths answer s very close to the result that we obtaned wth daly compoundng. What f you wll be nvestng over a perod of tme n, say a savngs account for retrement? To develop the formula, suppose you nvest $,000 for the next fve years at a constant 0%/year. How much wll you have at the end of the year? Assume you nvest at the begnnng of each year and nterest s pad once per year. Year Zero: Intal $,000 nvested, wll stay n account for fve years. Year One: Another $,000 nvested, wll stay n account for four years. Year Two: Another $,000 nvested, wll stay n account for three years. Year Three: Another $,000 nvested, wll stay n account for two years. Year Four: Another $,000 nvested, wll stay n account for one year. So our account wll contan an amount of $,000(+0.) 5 +$,000(+0.) 4 +$,000(+0.) 3 +$,000(+0.) +$,000(+0.) =$6,75.6 after fve years. In terms of a formula, P t n CF n r CF r snce the nvestments at each year (CF ) are the same each year. Ths s an nterestng applcaton for retrement savng: If a 0-year old could earn 0%/year on average, how much would they have at age 65? Solve the problem by fgurng how much there would be after 0 years, then use the smple compoundng formula to take t forward another 5 years. What f a person wats untl age 40 to start savng for retrement? How much would they have to save per year to end up wth the same amount as the saver who began at age 0? Solvng ths, you ll understand why so many people are plannng to work past age 65 these days. Present value of a future dollar The same formulas can be used to solve for the present value of a future payment. We just have to solve for P 0. For n compoundng ntervals per year, 0

18 Bond Fundamentals P 0 PN r n nt and for contnuous compoundng, P 0 = P t e -rt. NOTE: In ths case, the rate r s called the dscount rate snce P 0 < P t always. Example You are promsed a payout of $,000,000 ten years from now. (Fnancal applcaton: ths s used to value a zero coupon bond.) If the dscount rate s 0%, what s ths payout worth today? 0.*0 Use the contnuously compounded formula. P 0 $,000, 000e =$367,879. You should be ndfferent between a payout today of ths amount and a future payment of $,000,000 n ten years. Now, what f you have a seres of cash payments? (Ether these termnate at some future tme or go on to nfnty -- such types of payments are called perpetutes.) In the frst case, suppose you buy a bond payng 8% per year for the next 0 years. At the end of 0 years, you wll receve your last nterest payment plus return of your prncpal. Assume that the prncpal s $,000. What s ths bond worth today? (Later we wll see that there are three scenaros that can occur dependng on what the nvestment rate r s. For now, assume that you can earn 0% by placng money n a savngs bank.) All we do s take each payment and dscount t back to the present. We are pad 0.08*$,000 each year or $80 ( the coupon payment.) We dscount the frst payment over a one-year perod, the second payment over a two-year perod and so on. At the end of 0 years, we have $80 plus the return of our prncpal for a total of $,080 to be dscounted back ten years. It s really lke workng ten ndependent problems and summng together. PV $80 $80 $80 $80 $80 $,080 r r r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 0 PV = $877. In terms of a formula, we have where the values of CF for = to n- are the coupon payments, and the last cash flow, CF n, s that year s coupon payment plus the return of par. Observe that ths formula reduces to the one we had earler when n =. Also we have assumed that the dscount rate r s constant over the lfe of the nvestment. Techncal Note: To be more general, we should actually dscount each year by the prevalng dscount rate, r, at year. Then we have: $80 n CF PV r $80 $80 $80 n CF PV r The precedng formula s very mportant and wll be used over and over. Annuty: If the payments CF are constant over a perod, ths s called an annuty. Common examples nclude mortgage and car loan payments.

19 Bond Fundamentals Perpetuty: Now, what f we receve a payment of CF forever? We take the lmt as n approaches nfnty and use ths result nstead. To use ths approach, r has to be constant. It turns out that Want to check wthout usng any calculus? Just use a spreadsheet wth any and r you want. Let n n CF CF CF lm n so PV Perpetuty r r r ncrease untl the answer stops changng (n Excel, the functon PV s used as =PV(r,n,pmt) or =PV(0.,0,80) for r = 0., n = 0 and PMT = 80). You wll fnd that you approach the value of 800, or CF/r = $80/0.. Remember the above formula because you wll t agan. Ths formula can be used to value perpetual debt that a corporaton may have. The corporaton may have fxed ncome labltes on ts books that have a fnte expraton perod, but f t can keep rollng over the debt, t can be valued as a perpetuty. Here, the rate r s the average coupon payment on the corporaton s debt. Ths formula can also be used to value a company n the mature stage where t s stable and payng out a constant dvdend. Then, r plays the role of the rsk of the company, or hurdle rate. Ths model can be used f t s assumed that the company s a gong concern,.e. t wll operate nto perpetuty. As an example, what f a company s payng out dvdends of $.35/share at a hurdle rate of 0%? What s the company worth on the bass of ths model? Value = $.35/0.0 =$6.75/share. Gordon growth model Ths s used for valung cash flows such as debt or stock dvdends that are projected to grow at a constant rate g. Then, CF0 g PV r CF0 g r CF0 g 3 r 3 CF n 0 g r n n In the lmt as n approaches nfnty, we get CF0 PV r g For example, n the above, suppose that the company s dvdend polcy s to grow the dvdend 0%/year for perpetuty. The value of the company should be calculated usng the Gordon Growth Model. We need to know the current dvdend, CF 0 = $.65. Then, the value wth no growth, and t s. $.65 PV =$6.5/share. Ths should be hgher than Slck Trck: Now that we know the formulas for annutes and perpetutes, we can come up wth a shortcut for calculatng the value of an annuty that pays from = to n. Ths means we won t have to sum that long seres agan. Frst, look at the cash flow dagram of a perpetuty payng cash flow C at dscount rate r:

20 Bond Fundamentals C C C C C C C =0 = = =n =n+ The present value of ths perpetuty (call t PV ) has already been shown to be C PV r Next, consder a perpetuty payng C that doesn t start untl tme = n+. The value of ths C perpetuty at tme n s, and to get the present value at =0, just dscount back by dvdng by r (+r) n. So the value of the perpetuty at =0 (call t PV ) s PV C n r r Now we can fnd the value of the annuty runnng from = 0 to =n. We just take the nfnte perpetuty runnng from tme =0 to nfnty, PV, and chop off the part we don t want: the value of the perpetuty runnng from = n+ to nfnty. Hence, C PV r C r C r n n r r Ths can be used to value a stream of coupons from a bond. The formula can be easly modfed to handle the full coupon bond by just addng on a term representng the PV of the prncpal repayment, so we would have: PV coupbond C r P r n r n Example: Let s go back to that 0-year, 8% coupon bond at a dscount rate of 0%. Tedous calculatons gave ts value as $877. Usng the above formula gves: =$49.56+$ = $877. PV coupbond ,

21 Bond Fundamentals Bond Prces and Relatonshps to Yelds We have lad all of the necessary foundaton for prcng bonds. In fact, we have already started prcng bonds. It s now tme to talk about three types of bonds: dscount bonds, par bonds and premum bonds. There s only one thng that dfferentates these three types of bonds: the spread between the dscount rate r and the coupon rate. Dscount bond In the prevous example, our bond sold at less than the face value of $,000. Ths s an example of a dscount bond. The reason t sold for less than ts face value ( less than par ) s because the coupon nterest rate, 8%, s lower than the dscount rate r. Thnk about t: The dscount rate r s the expected nterest rate an nvestor could earn by nvestng n a vehcle such as hgh-yeld treasures. If the nvestor could earn 0% elsewhere, but ths bond s only payng 8%, shouldn t the nvestor be compensated for takng a lower nterest rate? The bond s sad to be sellng at a dscount to par. Par bond If the bond s sellng for par, that s, the present value s equal to the face value, here $,000, the bond s sad to be sellng at par or a par bond. We look at the general formula PV C r C r r n r n = f C = r where C s the coupon rate and r s the dscount rate. P Snce we know that the PV of a payment P at tme n s just, the last term above, the only way that n r ths can be s f = r. If ths occurs, the bond s payng nterest at the rate r and we have a par bond. Premum bond On the other hand, f the bond pays a rate that s hgher than the prevalng rate r, t wll be prced hgher than par and s called a premum bond. We also have to now nterject realty nto the story. Unless the yeld curve s flat, nterest rates do change wth tme to maturty ( tenor ). Let s take a look at a yeld curve from May 3, 00. The followng data s from Bloomberg.com (but you can see yeld curves n many places). 4

22 Bond Fundamentals Tenor, months Yeld, % Yeld Curve 03 May 00 Yeld, % Tenor, Months The fact that the yeld curve s not flat means that we have to use dfferent dscount rates to value each cash flow. For example, f we had a cash flow at two years, we would use the rate of 3.% from the table above; a cash flow ten years out would be dscounted at a rate of 5.09% and so on. Mssng data n between the gven ponts (such as three years, four years and so forth) must be calculated. We wll have much more to say about these topcs n the Fxed Income Secton. For now, to make thngs easy, assume that the followng has been calculated: Tenor, Yeld, months % Let s value a fve-year bond payng a coupon of 3.5%. On a bond wth face value of $,000 we wll then receve $35 each year (assumng for smplcty that we have annual compoundng.) Here s a table of our cash flows, dscount rate and present value of each cash flow. Tenor, months Yeld, % Cash Flow 5 PV Flow Cash

23 Bond Fundamentals PV Bond $ Why s the bond prced below par? We have dfferent dscount rates. In some sense, the average rate used must be lower than the coupon rate. We can determne ths average rate by settng the present value n C Par Thus, snce PV, for our example, n y y of the prce of our bond equal to the present value of a bond wth the same cash flows but a unform rate r. $96.97 $ $35 $, y $ y y y We solve by y by tral and error, usng a spreadsheet and the Solver functon, or wth our formula for an annuty. We fnd that the value of y that satsfes the above equalty s 4.34%. Ths specal y s called the yeld of the bond. That s all yeld s: just a mathematcal concept that s used to allow us to compare dfferent bonds on a level playng feld. Otherwse, how would we rank bonds? Is t correct to say that a bond wth a hgher coupon s a better nvestment? You can t just rank by coupon snce dfferent bonds have dfferent maturtes. Note that on Bloomberg s page, they show the current yeld as 4.35%, very close to ours, but we assume that they use slghtly dfferent nterpolaton methods. Now we are n a poston to explore the very crtcal relatonshp of prce to yeld. $35 Prce and yeld are nversely proportonal: as yeld ncreases, prce decreases. As yeld decreases, prce ncreases. (We repeat ths statement because t s so mportant. You can thnk of t as the frst law of bond dynamcs f you want.) You can see from the equaton above that f yeld s hgher than 4.34% requred to mantan the equalty, we wll be dvdng by a larger number so the PV (prce of bond) should decrease. And f we decrease the yeld we are dvdng by a smaller number, so the prce wll ncrease. Practcally, ths makes sense. Bonds must converge to par at maturty $35 $35 $35 $35 $,035 y Yeld s the mathematcal mechansm by whch we get from the present to the future. So, f a prce s low, we have to have a whoppng large yeld to clmb to par. If prce s already hgh, say, close to par, we don t need very much growth to get to par. Ths s such a crucal fact that you may want to buld your own bond model on a spreadsheet and explore the effects of varyng yeld. 6

24 Bond Fundamentals Taylor Seres Expanson Suppose you have nformaton about a functon at one pont and want nformaton about that functon at some other pont. For a smple example, suppose Gordon s now located 00 mles east of Chcago. He s travelng due west at 60 mles/hour and s very anxous to make hs class. How would he be from Chcago after one hour of drvng f he decdes to accelerate hs speed at a steady 0 mles per hour over that hour (so that hs speed after one hour s 70 mph)? Or, consder another example: suppose you have the prce of a bond at a certan yeld. What would the prce of that bond be f the yeld changes by one percent? You may not realze t, but to solve these and smlar problems you use the prncples of Taylor Seres expansons. Taylor Seres are even used to derve the Black-Scholes equaton and Ito s Lemma, whch we wll come to later. In fact, f you have taken physcs and are famlar wth the equaton of tme poston of a partcle x(t) = x 0 + v 0 t + / a 0 t, you have are already used Taylor Seres. So, here s the theory that you need to know: A contnuous, dfferentable functon f may be expanded n a Taylor Seres about a pont k as follows: n '' ''' f k n f k f k 3 f ( x) x k f ( k) f '( k) x k x k x k n0 n!! 3! The dstance x-k must be small and the dervatves must exst at k. Note that the Taylor Seres ncludes an nfnte number of terms. In practce, we can only take a fnte number of terms, and there wll be truncaton error due to the contrbuton of the terms that are dropped. So, f(x) s approxmated by a fnte-number-of-terms Taylor Seres, plus a truncaton error. For example, the second-order Taylor Seres expanded about the pont k s gven as: ( n) n f ( k) f ( x) R 0 n!! n x k R3 f ( k) f '( k)( x k) f ' '( k)( x k) 3 where R 3 s the truncaton error, consstng of the sum of terms n = 3 to nfnty. Also note that f we are just approxmatng polynomals of degree n, the Taylor Seres of order n wll gve an exact result. (The Taylor Seres of order n s a polynomal of order n.) Graphcally, what we are tryng to do s ths: 7

25 Bond Fundamentals f(x)=? f(k) f k x The assumptons are that the functon exsts over the range of nterest of expanson (that s, t s contnuous between k, the known pont, and x, the pont you are tryng to forecast), and that the dervatves above exst. Note that the rght hand sde s completely known so you can just add t up to get your forecast for f at the desred pont. So, a Taylor Seres expanson s smply a technque to make an approxmaton of the behavor of the functon f(x) over the nterval (k,x). If you knew the actual functon f(x), you could smply evaluate t at the desred value x. The assumpton s that you do not know what f wll be at x, and need some way to estmate t. Actual practce tp: You don t even really need to know the actual functon f(x) as long as you know, or can estmate, the values of f and ts dervatves at the pont k. You ll see ths n prcng bonds usng duraton and convexty. Let s try an example. (We ll do a math problem frst, for confdence, and then we ll move on to fnance.) Example Let the functon be f(x) = x 3 x. We already know that f() = 3 = 8 3 = 5. To check Taylor Seres, assume that you only have nformaton about f and ts dervatves at the pont x = (ths wll be the k n the Taylor Seres equaton, the known pont), and we seek the value of the functon f at, the unknown pont.) We need to have all of the values of f and dervatves at the known pont to forecast what f wll be at. How many dervatves s enough? The more you use, the better the approxmaton, and there are formulas that tell how far you must go to fall wthn an acceptable error. In ths case, we wll compare approxmatons wth usng the frst dervatve only, the frst and second dervatves, and the frst three dervatves (whch wll gve the exact soluton for ths cubc functon). f(x) = x 3 x f() = 3 = - f (x) = 3 x f () = 3() = f (x) =6 x f (x) =6() = 6 f (x) = 6 f () = 6 Usng frst dervatve only: f() f() + f ()(-) = -+(-) =. Error = 5 = 4 Usng frst two dervatves: f() f() + f ()(-) + f ()(-) / = + 6/= 4. Error = 5 4 =. Usng frst three dervatves: f() f() + f ()(-) + f ()(-) / + f ()(-) 3 /6 = 4+6/6 = 5. Error = 5 5 = 0. 8

26 Bond Fundamentals Note how the error between the true (known) value and our approxmaton decreases as we ncrease the number of dervatves used. What we are dong s provdng more and more nformaton n order to more closely approxmate the unknown value. Also, note that t really sn t necessary to know the defnton of the functon. In the real world, we mght just know the dervatve values. We could have just as easly solved ths just knowng the values f(), f (), f () and so on, but the values would have to be known to us n some way. Note: If the functon f depends on two varables, such as x and y, we just dfferentate wth respect to each. We have df f f f f! x x dx, y dy) f ( x, y dx dy dx dxdy dy x y xy Ths s called the Multvarate Taylor Seres. It can be used to estmate to the value of f at a pont (dx, dy) away from a pont (x,y) at whch f s known. It requres knowledge of the above dervatves at (x,y). Example Let f(x,y) = x y. Suppose we only know the value of f for x = and y = 3, so that f(,3) = 3. Let s say f represents the prce of an opton, x the prce of a stock and represents the tme parameter. We are nterested n estmatng the value of f f x and y change a small amount, say by each. (Of course, we could just plug n the x and y values to get f(,4) = 6, but we want to try out our Taylor Seres approxmaton here. Later we wll use just such a method to derve the Black-Scholes equaton for prcng optons.) To estmate usng Taylor Seres, we need the dervatves. f f y f f f f f f f f xy, x, y,, x x y x x x xdy x y y x y Then, df = xydx x dy ydx xdxdy 0dy = ()(3)() () () (3)() ()()() 0() f 0 y y =6++3+ =. Ths s the change n f caused by movng from (,3) to (,4), so the new value of f s the sum of the old value of f plus the change, f(,4) = f(,3) + df = 3 + = 5. Note that snce we truncated the seres after the second order terms, we stll have truncaton error to account for. But t wll do for a frst approxmaton. Example The prce change of a bond caused by the change n yeld can be estmated by expandng P as a functon of y n a Taylor Seres. Usng just the frst two terms, we have: dp dp dy d P y y dy... What s the prce change f y = %, the frst dervatve of P wth respect to y s equal to 6,7 and the second dervatve of P wth respect to y s 60,600? Just plug nto the formula to get dp = -$64.8. In followng sectons, we wll see more Taylor Seres, ncludng fndng out where the above dervatves came from and what they mean, and n dervng the Black-Scholes equaton and numercal approxmatons for ts soluton. Why do you need to know the Taylor Seres? It s often the case that we have nformaton about somethng at a certan pont, say, n tme, for example, and want to know what t mght be another other pont n tme (ths s called forecastng.) If you know the rate at whch the functon s changng, and have reasonable 9

27 Bond Fundamentals expectatons that ths rate wll reman constant over the tme nterval of nterest, you can use Taylor Seres to project the future value. Also, notce how addng more terms mproves our estmate of the unknown value. (Of course, the more nformaton we have, the better). Ths s the theory underlyng the convexty of a bond dea, comng up n the next sectons. 0

28 Vault Gude to Advanced and Quanttatve Fnance Intervews Bond Fundamentals Bond Prce Dervatves Let s delve deeper to see more precsely how prce changes wth yeld. To do so, we have to take a dervatve. To make t easy on ourselves we wll just use the constant cash flow C as from a coupon-payng bond. Then, If we factor out y then the denomnator wll look lke t dd for P: Now dvde both sdes by P. We then have (The negatve sgn s used snce we defne postve duraton as occurrng when an ncrease n yeld causes a decrease n prce, the normal result. We wll see later that certan specal fxed ncome nstruments can have negatve duraton.) Dollar duraton Duraton s used to make an estmate of how our bond s prce wll change n response to a change n yeld. Duraton measures the bond s frst-order senstvty to a change n yeld. It can most easly be thought of as the change n prce for a 00bp yeld. The unts of duraton are tme and t wll have the same unts as the coupon payment nterval (one year, one-half year, one-quarter year etc.) Before movng to an example, a defnton: bass pont, or bp for short, s just an alas for /00 of a percent. Bass ponts are a frequent unt of measure n fxed ncome. There are 00bp per %, so 00bp s a way of sayng % n n y npar y nc y C y C y C y P n n n n y npar y C y y npar y nc y C y C y C y y P 3 3 MAC n n D y y npar y C P y y P P y P P Where n n MAC y npar y C P D s defned as Macauley Duraton. The Modfed Duraton, D MOD, s defned as MAC MOD D y D, so MOD D y P P n n n n y Par y C y C y C y C y y Par y C y y P 3

29 Bond Fundamentals Example: If the Macauley duraton of a bond s known to be 7.5 years when the prce s $,000, yeld s 8% and the yeld changes by 00 bp, what wll be the change n the bond s prce? Just use the formula and solve for P: Py P y D MAC $,000(0.0) So, P 7. 5 =-$67.3. Let s see how good a job ths dd. We already used ths bond 0.08 before, ths s the ten-year, 8% par bond. Snce t s a par bond ts prce s known: $,000. If the yeld changes by 00 bp so that t s now 8% + % = 9%, the prce of the bond wll be 0 80 $, P = =$ So, the actual change n prce s $,000 $935.8 = $64.8. (Notce how handy our lttle shortcut s.) Here s how to do ths n Excel: We make a column of coupon payment tmes ( =,, 0, for ths example); a column for cash flows (coupon rate/number of payments/year tmes par value); a column for present value of each coupon payment (PVCF); and a column wth tme-weghted values of the cash flows, *PVCF. Then duraton = the sum of *PVCF over the sum of PV of cash flows. Duraton of a Bond Coupon Rate 8% Par Value of Bond $,000 Term (years) 0 Intal Yeld 8% Number of coupons/year Coup Tme Cash Flow PV of CF t * PVCF $ $ $ $ $ $ $ $ $ $, Sum $, Macauley Duraton Modfed Duraton 6.70

30 Bond Fundamentals Duraton of zero coupon bond For a zero coupon bond, the duraton wll be the same as the tenor of the bond, because we only receve one cash flow and t s at the end of the perod. How senstve are zeros to prce changes? Snce there are no coupons, we can go back to bascs and fnd that snce P zero Par y n P y npar n Par n y y y P n P, y P y n n D y MAC n For a zero at par and yeld of 8%, a 00 bp change n yeld would cause a prce change of Py $,000(0.0) P n 0 y 0.08 =$9.59. Because ths s the same formula we had for the coupon-payng bond (except D MAC s replaced by n, whch s larger than D MAC for a coupon bond), the prces of zero coupon bonds are extremely senstve to changes n yeld. These duratons are called dollar duratons because they are expressed n terms of currency. Dollar convexty Now we need to talk about why there s an error between the change n prce calculated usng duraton and the actual change n prce that would occur. If we plot bond prce as a functon of yeld (agan usng our 0-year 8% bond) we get a graph lke the followng. $,000 Bond Prce as a Functon of Yeld $,500 Prce $,000 $500 $0 0% % 4% 6% 8% 0% % 4% 6% 8% 0% Yeld, % 3

31 Bond Fundamentals Note that the graph s not lnear, but has a slght curve to t. Ths curve s known as convexty. Ths means that as yelds ncrease, the curve flattens: the bond prce becomes less senstve to changes n yeld. When yelds are low, the prce of the bond s extremely senstve to changes n yeld. Just usng duraton alone assumes that the bond s equally senstve to yeld changes at any yeld. So, we see that usng duraton alone to estmate prce senstvty s not such a problem at hgh yelds, but can lead to large errors when yelds are low. Usng Duraton alone to estmate prce changes s reasonable only for small changes n yeld, where the prce-yeld curve can be assumed to be approxmately lnear. Computng the prce approxmaton How do we nclude the effects of convexty n our prce calculatons? Recall the Taylor Seres expanson. We can expand prce n terms of y n order to solve for P. The expanson of P n terms of y s: P P y y y Py y y P y. Solvng for P and substtutng our defnton of duraton gves: P P P P MOD y y y y y Py P y y PD y y Truncaton Error The second term s the adjustment that needs to be added to our prce to account for the effects of P convexty. Defnng dollar convexty as C then P PD y C y Truncaton Error MOD y Because here the convexty s postve, convexty has value: t ncreases the prce of the bond. Unts of convexty are n the unt of tme, squared. A formula for use n a spreadsheet can be determned by takng the second dervatve of the P(y) equaton wth respect to y; the result s: P y n y C n n n y Par For example, re-computng the prce change of our 0-year bond ncludng the convexty of 60,53 gves the total prce change due to a change of 00bp n the yeld as change due to duraton + change due to convexty: P $, = =-$ Ths s much closer to the actual prce change of P(0.09) P(0.08) = $,000 = -$64.8. Prce value of a bass pont (PVBP) A common measure of duraton s the prce value of a bass pont. You may sometmes see ths referred to as the dollar value of a bass pont. Ths s a measure of bond prce volatlty. As the name mples, 4

32 Bond Fundamentals ths has to do wth the prce change resultng from a one-bass pont, or 0.0%, change n yeld. (We calculated the prce mpact on our 0-year 8% bond resultng from a 00 bp change. Ths s the same calculaton but wth a change of bp.) Prce of Bond at 8% Prce of Bond at 8.0% Dfference (PVBP) $,000 $ Note that t does not matter f we ncrease or decrease the yeld by bp; t s such a small amount that t makes no dfference. You should get the same answer ether way. Sometmes ths may be quoted per $00 of par value so be aware of the conventons beng used. Estmatng effectve duraton and effectve convexty If we dvde the formula for P by P, we get an expresson for the percentage prce change of the bond. Then P DE y C P E y where D E and C E are known as the effectve duraton and effectve convexty of the bond, respectvely. (Ths duraton s the same duraton we have been usng, except the convexty s now dvded by P.) If we have prces, we can estmate duraton and convexty usng fnte-dfference approxmatons of the dervatves P P,. From Taylor Seres Approxmatons these are derved as: y y P P( y y) P( y y), so y y D MOD P( y y) P( y y) Py P P( y y) P( y) P( y y) C, y y C E P P( y y) P( y) P( y y) P y Py Example: A summary of three bond prces s summarzed below. Yeld, % Prce 7, , The estmate of duraton for a 00bp change n yeld for our $,000 bond at a yeld of 8% s D MOD P(8% %) P(8% %) (000)(0.0) (000)(0.0) 5

33 Bond Fundamentals whch s a pretty good approxmaton. Now, the convexty s calculated as: P( y y) P( y) P( y y) P(9%) P(8%) P(7%) (,000) C 60,600 y Ths s also reasonably close to the 60,53 we calculated n Excel. The effectve convexty s C/P = The percentage prce change s then P DE y C E y 6.7(0.0) 0.5(60.6)(0.0) P so P = ($,000) = -$64.7 as before Portfolo duraton and convexty Suppose a fxed ncome manager s tryng to decde between two portfolos. Portfolo A s a bullet portfolo (one made up of bonds wth maturtes clustered at a sngle pont on the yeld curve) consstng of our 0-year, 8% coupon bond. Portfolo B s a barbell portfolo (one made up of bonds wth maturtes concentrated at both the short and long ends of the yeld curve) consstng of a -year, 5% bond and a 0- year, % bond. The manager s concerned wth the effect of yeld curve shfts on the performance of the portfolos and wants to choose the best one. Snce the duraton of a portfolo of bonds s just the sum of weghted duratons of each bond (where the weghts are the percentage held of each bond), t s easy to choose the weghts of portfolo B so that the duraton matches the duraton of Portfolo A. Ths s called a duraton-matched portfolo. We have the followng duratons: Bond Mod Duraton, years Year.86 0 Year Year 7.47 For the barbell portfolo B we have D B = w D + w 0 D 0 = w D + (-w )D 0 = w.86 + (-w )7.47. Ths should be set equal to the duraton of portfolo A, or 6.7 years. Solvng for w, we fnd w = 0.355, so 3.55% s nvested n the -year bond and 86.45% s nvested n the 0-year bond. Now the fxed ncome manager constructs scenaros of expectatons of future yeld curve shfts and wshes to know the prce change of the portfolos under each scenaro. Yeld Curve Shft Scenaros Yeld Curve Scenaro Scenaro Pont shft, % shft, % Year Year Year