= 1.8% =.018 = 3.1% =.031 = 3.6% =.036 = 3.9% =.039 = 4.3% =.043

Transcription

1 1 Yield Curve (Preliminary Verion) Floyd Ve, Nov. 014 Yield curve are ofen een in financial lieraure and applied o invemen ranacion. For example, ee Wikipedia.org, Inveopedia.com, or yieldcurve.com. They ofen have a horizonal axi for he number of year for variou bond, and a verical axi of yield. Typically he longer he erm of bond, he higher he yield. Differen ype of yield are ploed uch a YTM, po rae, and forward rae. YTM and erm. Conider Table 1 for bond of lengh 1,, 3, 4, 5 year. The YTM in Table 1 could be ploed making a yield curve. Table 1 i for annual pay par bond a $100 wih 5% ($5) coupon a he end of each year, and value a mauriy of $100. A he end of each year hey pay he coupon of $5 and a mauriy hey pay $100 + $5 = $105. Table 1 Price Number of year YTM $ I year y 1 = 1.8% =.018 $ year y = 3.1% =.031 $ year y 3 = 3.6% =.036 $ year y 4 = 3.9% =.039 $ year y 5 = 4.3% =.043 Price of bond. We will calculae he price of each bond in Table 1, given he erm and YTM. For example conider he wo year bond. The YTM = y = 3.1% =.031 dicoun he 1 paymen of $5 and $105 o he Price. Thu Price = 5(1+ y ) + 105(1+ y ). A convenien way o calculae Price i o ue on he TI84 he funcion npv(inere rae, CF0, CFLi), page 14-8 of he manual. To ue npv(, we fir pu in {5,105} L1. Then nd Finance 7 give npv(. Wrie in 3.1, 0, L1 ) Pre Alpha Solve and read Price = which we round o $ for he price of he wo year bond. For eriou calculaion, five or more digi o he righ of he decimal poin are ofen ued, bu for convenience we will ue differen place depending on he iuaion. (See he Exercie.) Spo rae. From Table 1, we will calculae po rae, r 1, r, r 3, r 4, r 5 for he bond. For example o deermine he po rae r for he wo year bond, we will divide i ino wo zero coupon bond, one bond pay $5 a he end of year 1, and he oher pay $105 a he end of year. We will dicoun he $5 paymen a 1.8% giving $(5/1.018) and ubrac hi value from he $ giving /1.08 and olve for he po rae r which dicoun he $105 for wo year o he difference, giving /1.08 = 105/(1+ r ) and olve for po rae r.

2 A an illuraion, for he year bond, we have old he fir half of he year bond for $( 5/1.08) and purchaed he econd half wih he difference requiring a po rae r = 3.13%. We aume ha hi ranacion i wihou any co of ale or purchae, and oher inerference. Thi calculaion can be done wih he TI84 Solver (page -8). Code and commenary: Clear he home creen. On he home creen, Pre Mah 0. You ee eqn:0=. Wrie / /(1+x ) where x = r. Pre Ener. Wrie 0 nex o x and pre Alpha Solve. You read x = We round o 3.13% for r. To clear he Solver, o he op and Clear. To remove he Solver, urn off, on, Clear For calculaing r 3 we ue Price = 3 (1 r ) (1 r ) (1 r ) 1 3 which give = 3 (1.018) (1.0313) (1 r ) 3 o ge r 3 = = %. See Table for po rae for he bond in Table 1. See he Exercie for more calculaion. Table : Spo rae for bond in Table 1. 1 year 1.8% = r 1 year 3.13% = r 3 year % = r 3 4 year % = r 4 5 year 4.39% = r 5 The po rae r i i he implied zero-coupon rae for he erm beginning a ime zero o he end of period i. (See page 4 of Moorad in he Reference.) Forward rae can be calculae from po rae. They will be denoed f[0,1], f[1,], f[,3], f[3,4], and f[4.5]. For example for f[1,] (1 r ) (1.0313) 1 f 1 1 (1 r1 ) (1.018) 1 [1, ]. Solving give f[1,] = = 4.48%. In general (1 r ) (1) 1 f [, ] for > 0. f[0,1] = r 1. I i probably eaier o calculae hi wih (1 r ) a cienific calculaor pad. See Table 3 and Exercie. For an argumen for Formula 1, from he above dicuion if > 0, he po rae r i he annual effecive rae for an invemen over he period [0,]. So $K grow from ime 0 o ime

3 3 o K(1 + r. If > hen K(1 + r ) hould grow o K(1 + r ). From ime o ime, i ) grow a he rae f[,] o ha K(1 r ) 1 f [, ] K(1 r ). Solving give (1 r ) 1 f [, ] a in Formula 1 above. f [, 1] i a rae for a conrac now, and i he 1 r ) forward one year rae deferred year. f[0,1] = 1.8% f[1,] = 4.48% Table 3: Forward rae for bond in Table 1 and. f[,3] = 4.688% f[3,4] = 4.9% f[4,5] = 6.13% Forward rae are a mahemaical calculaion of where rae hould be in he fuure baed on curren po rae and he rule of arbirage. However, hey are no predicaion of he fuure ince he fuure rae are baed on erm of he fuure, which are no likely o be he ame a curren erm. (Moorhead, page 3) Rae of inere for riky bond. Aume a one year $100 governmen bond pay % and a value a mauriy of $10. We aume he governmen bond i compleely afe. Conider a riky $100 bond which pay 4% and $104 a mauriy. The analy who priced he bond had in mind a probabiliy p of failure of he bond o pay, where he inveor receive $0 a mauriy. We ak, wha probabiliy p did he analy have in mind? Thi give Expeced value = $10 = 0(p) + 104(1-p). Solving for p we have 10 = p, = 104(p), p = nohing. 104 =.019 = 1.9% chance of failure where he inveor pay $100 and receive We ak he queion in revere. If he analy ha calculae he probabiliy of complee failure of he $100 bond o be 1.9%, a wha inere rae hould he bond be quoed, how much hould i pay a mauriy = M? Thi give = 0(.019) + M( ) and olve for M. M = = $104. The $100 bond hould be paying 4%, and $104 a mauriy, if he invemen i ucceful. (See he Exercie.) Dicoun rae. U. S. Treaury Bill are old in $10,000 denominaion by he U. S. Treaury a a dicoun rae baed on face value of $10,000 for erm in week up o 5 week. We give a imulaion drawn from Treaury Bill ale in 1988 wih a dicoun rae d = 7.48% = for a 5 week Bill. A 7.48% bid a he U. S. Treaury aucion mean ha he inveor i willing o pay 100% % = 9.5% of he $10,000 face value. The inveor bid 9.5% of $10,000 = $95. The dicoun rae d = 7.34% = We calculae he yield y of

4 4 95(1+y) = and olve for y = = From hi example we have 10,000(1- d)(1 + y) = 10,000 and (1 - d)(1+ y) = 1 a he maer relaionhip beween dicoun rae d and yield y. (See he Exercie for 014 level of he dicoun rae d for Treaury Bill a he U. S. Treaury Aucion for 5 week Bill.) (Search he inerne for Daily Treaury Bill Rae Daa Deparmen of he Treaury.) 1 y We can olve (1 d)(1 + y) = 1 for d giving d 1 and d. The laer can 1 y 1 y be inerpreed a he inere on $1 dicouned a he yield rae y. Alo d i he raio of he inere y on $1 divided by wha $1 will be in one year. d Solving for y give y which i he raio of he dicoun rae d divided by $1 1 d dicouned a he rae d. If all hi i for one year, we have an effecive dicoun rae and an effecive yield rae. Compounded dicoun rae. For impliciy, y and d are in annual rae and in year, he Fuure Value S = P(1 + y ) = d 1 P 1 P P(1 d) 1d 1d d For he Preen Value P S(1 y) S 1 S(1 d) 1 d.. (See he Exercie. ) Simple dicoun. The borrower i o pay back $100 a he end of = year a imple dicoun rae D =.03 he ge 1001 (.03) = $94. If he borrower receive $94, he payback of $100 = 941 (.03) 1. The amoun paid back can be called he loan amoun. If K i he payback, he borrower receive K(1 - D). If he borrower receive $1, he payback i 1(1 - D ) 1. Simple inere rae. If P dollar i inveed a he imple inere rae R for year, he fuure value S = P(1+R). Noice here i a difference beween imple dicoun and imple inere rae. (See he Exercie and Reference.) Simple inere can be more complicaed. Side Bar Noe: Bond yield are expeced o rie. From rend ince Jan. 013, i ha been exrapolaed ha US wo year governmen bond yield will increae from abou.4% o over % by 10/16. (See a graph in T. Rowe Price Repor, Fall 014.) Median wage wa $38,685 for 1983 and wa $41,444 for 013. Wha wa he annual compounded rae of increae? Wha wa he arihmeic rae of increae. Wha wa he rae of inflaion during hi period? (Bloomberg Buineweek, Nov. 014) Real and UnReal rae. In 1981, he 30 year zero-coupon Treaury bond yielded 15.1%. If you owned $100,000, wha wa i worh in 30 year? Currenly 30 year Treaury zeroe yield

5 5 around 3%. In 30 year, wha i hi worh? An UnReal example: Suppoe ha in 1981, he zero coupon Treaury rae on he remaining 30 year uddenly drop o a marke rae of 3%, wha i he immediae benefi of owning he 15.1% bond? Forbe ay Since 1981 he zero-coupon Treaury bond have rien ix ime more han he S&P 500 oal reurn, whaever hi mean. (Forbe, Nov. 4, 014, page 76). Wha i he income ax on zero-coupon bond? How do he rae of reurn on zero-coupon STRIP compare o he comparable emiannual pay bond, and he conrac ha pay only coupon? The world large new ock offering. Jack Ma brough hi gian Alibaba o he public ock marke in New York in Sep Look up Alibaba (BABA) a finance.yahoo.com. Look up he ock price hiory and marke-cap. Read he Profile. Repor. Anoher heroic inveor. Shelby Davi bea he marke by over 10% a year over a period of 57 year beween he 1940 and He grew $100,000 o a forune of $900 million, ranked number 4 inveor in U. S. hiory. (The Morningar Approach o Inveing, Andrew Leckey, referenced in wikipedia.org, Search Shelby Davi). A repeced porfolio manager, he managed he Davi New York Venure Fund beginning in Hi reurn were conien, no running ho and cold, inveing in a combinaion of value and growh ock. He wa able o po major rend, inveing in energy relaed ock in he 1970, inere rae eniive ock and brand name ock in he 1980, and financial ervice in he According o AAII Journal, Nov. 014, he wa ouranked long erm by only Warren Buffe. Hi firm Davi Selec Advior managed $100 billion in everal fund. He began hi work a The Bank of New York, i a graduae of Princeon Univeriy, and i acive in philanhropy. Facor which bea he indexe. Of 466 managed equiy fund creened from 1999 o 013, wih 10 year rolling period, here were 509 fund wih expene in he lowe quarile of which 44 percen bea heir index. For he 447 fund wih he highe manager ownerhip, 71 per cen bea heir index. For he 85 fund wih boh characeriic, 100 percen bea heir index. Of all he fund, only 7% bea heir index. Thi udy wa conduced by Capial Reearch which manage American Fund. In a longer erm udy, American Fund equiy fund bea he marke 73 percen of he ime in 10 year rolling period. Did he reurn include ale commiion? (Sco Burn, Denon Record Chronicle, Nov. 16, 014) A 8% reurn, how long would $100,000 la wih $10,000 wihdrawn each and a he beginning of he year? Wih an 8% commiion wihdrawn, how long doe he money la? Exercie: Show your work. Label anwer, variable, and number. Conduc dicuion in complee enence. Name your compuing device and give ome code. #1. Calculae for he price for he hree year bond in Table 1, by wo or more mehod. #. Calculae for he four year bond in Table 1 he po rae r 4. #3. Calculae for he hree year bond in Table 1 he forward rae f [,3]. #4. On a grapher uch a he TI84, pu he Term and YTM in Li and graph hem. Ue he device o fi mooh curve hrough he poin. Give he equaion of your curve and an indicaion of he accuracy of he fi. Dicu he meaning of he curve and comparion wih

6 6 curve in he lieraure. Prin, or race by hand he graph and label axe. Publihed yield curve are ofen arrived a by imilar mehod and inerpolaion. #5 On a grapher, do he ame for po rae a in problem 4. For he po rae curve, give he equaion of your curve and an indicaion of he fi. Exrapolae o r 6. Apply he dy/dx funcion a yearly poin on he po rae curve and build a able. Dicu. How would you define a angen line o he curve a a poin. How would you define a angen line o a circle a a poin. Wha abou dy/dx for he circle? #6. On a grapher, do he ame for forward rae a in problem 4. Dicu in imple erm how curve fiing i done. #7. Dicu he relaionhip beween he hree curve in problem 4, 5, and 6. Pu hem on he ame axe if you can. #8. Arbirage. To read abou arbirage, ee Wikipedia.org, Inveopedia.org, ec. Repor. #9. Dicu everal reaon why, for longer erm bond, he marke may require greaer rae of reurn. #10. In Table 1, he Price = $ for he 5 year bond a y 5 = 4.3% =.043 look ou of paern. Calculae hi price o check i. #11. Go o bankrae.com and ge he op APY rae for CD for 1 year, year, 3 year, 4 year, and 5 year. Conruc a able and a CD yield curve for op APY. Do i for average APY. For CD, wha happen o he dividend? Wha doe he APY repreen? Wha doe Rae mean? Wha kind of compounding i offered? #1. A one year bond priced a $100 pay $.50 in ix monh, and $ a mauriy. Calculae he YTM wih a general Solver and wih a TVM Solver. Compare he anwer. You could ue he TI84. When you check he anwer, which one look he be? #13. Hiorically, wha are he differen hape of yield curve, and wha are he caue? (Find reference.) #14. Applicaion: Conider a firm ha need o borrow $100,000 for wo year. They have he opion of borrowing a he wo year rae in Table 1. Alo an opion i o borrow for he fir year, a he year 1 po rae in Table. For he econd year hey calculae o borrow a he forward rae f[1,] in Table 3. Which opion i he be? f[1,] i he forward rae deferred one year and applied o he following one year. Which opion i be? #15. Give an argumen ha f[1, ] r. #16. Find he oal preen value of paymen of $1000 a he end of each year for five year uing he po rae in Table. Wha level yield produce he equivalen value? #17. Aume a $100 one year governmen bond pay 3% and $103 a mauriy. A riky $100 one year bond pay $106 a mauriy and 6%. Wha probabiliy of failure of he riky bond ha he analy calculaed?

7 7 #18. Aume a $1000 one year governmen bond pay 3% and $1030 a mauriy. An analy ha calculaed ha a cerain riky one year $1000 bond ha he probabiliy of complee failure of 10%. A wha inere rae hould he $1000 riky bond be priced? Wha hould i pay a mauriy? #19. On he inerne, from: Daily Treaury Bill Rae Deparmen of he Treaury (Nov. 014), we ge for 5 week bill, a dicoun rae d =.10% =.0010 for a face value of $10,000. How much i he bill old for? Wha i he yield y? Show ha (1 - d)(1 + y) =1. Solve for d and olve for y. #0. If you borrow he amoun K a he compounded dicoun rae d for n year, wha i he amoun of dicoun for he loan? #1. Bill wan o inve money a % dicoun per year o accumulae $3000 in five year. How much hould he inve? Explain he anwer in erm of a equence of one year invemen. #. For he imple inere rae R and ime in year, wha i he preen value P of S? #3. If he payback for a imple inere loan i $100 a R =.04 and = year, how much doe he borrower ge? Compare hi o imple dicoun. Acually, imple inere i more complicaed. See he Reference. Reference: Daniel, Jame W. and Lelie Jane Federer Vaaler, Mahemaical Inere Theory, Pearon/PreniceHall, 007. Thi book i auhored by faculy a he Univeriy of Texa Mahemaic Deparmen and augh in ha Deparmen. Moorad Choudhry, The yield curve, and po and forward inere rae. Search he inerne. Shane, Harold D., Mahemaic for Buine Applicaion, Charle E. Merrill, In he following free download coure, here are a lea a half-dozen aricle on bond. For a free coure in financial mahemaic, wih emphai on peronal finance, for upper high chool and undergraduae college, ee COMAP.com. Regier and hey will you a paword. Simply click on an aricle in he annoaed bibliography, download i, and each i. Uni 1: The Baic of Mahemaic of Finance, Uni : Managing Your Money, Uni 3: Long- Term Financial Planning, Uni 4: Inveing in Bond and Sock, Uni 5: Inveing in Real Eae, Uni 6: Solving Financial Formula for Inere Rae, For abou hireen more advance or echnical aricle, ee he UMAP Journal a COMAP. The la ecion i Addiional Aricle on Financial Mahemaic or Relaed o Peronal Finance. In all, here are abou eighy aricle.

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