DEBT INSTRUMENTS AND MARKETS Zeroes and Coupon Bonds Zeroes and Coupon Bonds
Ouline and Suggesed Reading Ouline Zero-coupon bonds Coupon bonds Bond replicaion No-arbirage price relaionships Zero raes Buzzwords Zeroes, STRIPS, dedicaion, implied zeroes, semi-annual compounding Suggesed reading Tuckman, Chapers and 2 Defaul-Free Fixed Income Securiies The mos basic deb insrumen is a "zero-coupon" or "pure discoun" bond -- a securiy wih a single cash flow equal o face value a mauriy. Concepually, hese "zeroes" are he building blocks of all securiies wih fixed cash flows. Combining zeroes in a porfolio creaes an asse wih muliple fixed cash flows. We can srucure porfolios of zeroes o replicae exising securiies, such as coupon bonds. I is also possible o consruc a porfolio of coupon bonds ha replicaes a zero. The possibiliy of replicaing one asse from a porfolio of ohers means ha, in he absence of arbirage, heir prices mus be relaed. Zeroes and Coupon Bonds 2
Zero Prices or Discoun Facors Le d denoe he -year discoun facor, he price oday of an asse which pays off $ in years, or he price of a -year zero as a fracion of par value. Because of he "ime value of money," a dollar oday is worh more han a dollar o be received in he fuure, so he price of a zero is always less han is face value: d < The Discoun Funcion gives he discoun facor as a funcion of mauriy. Because of he ime value of money, longer zeroes have lower prices. he discoun funcion is always downward-sloping. Discoun Funcion From U.S. Treasury STRIPS Zeroes and Coupon Bonds 3
Coupon Bonds A Treasury bond pays semi-annual coupons every six monhs a an annualized fixed coupon rae c and par value a mauriy T. Coupon bond cash flows as a percen of par value: c/2 c/2 c/2 + c/2 0.5 years year.5 years T years A Coupon Bond as a Porfolio of Zeroes Example: $0,000 par of a one and a half year, 8.5% Treasury bond makes he following paymens: 6 monhs year /2 years $425 $425 $0425 This is he same as a porfolio of hree differen zeroes: $425 par of a 6-monh zero $425 par of a -year zero $0425 par of a /2-year zero Zeroes and Coupon Bonds 4
The Principle of No Arbirage or The Law of One Price Two asses which offer exacly he same cash flows mus sell for he same price. Why? If no, hen one could buy he cheaper asse and sell he more expensive, making a profi oday wih no cos in he fuure. This arbirage opporuniy canno persis in equilibrium. Valuing a Coupon Bond Using Zero Prices In he absence of arbirage, he coupon bond mus have he same price as he corresponding package of zeroes. Equivalenly, in erms of he discoun funcion, V = $ 425 d d 0.5 + $425 d + $ 0425.5 Zeroes and Coupon Bonds 5
Valuing a Coupon Bond Using Zero Prices Valuing $0,000 par of a.5-year 8.5% coupon bond. The discoun facors come from STRIPS prices from he WSJ. Mauriy Price of $00 Par of Zero Discoun Facor Bond Cash Flow Value 0.5 $97.30 0.9730 $425 $44.0 $94.76 0.9476 $425 $403.5 $92.22 0.9222 $0425 $964 Toal $0430 On he same day, he WSJ priced a.5-year 8.5%-coupon bond a 04 0/32 (=04.325). An Arbirage Opporuniy Wha if he.5-year 8.5% coupon bond were worh only 04% of par value? You could buy, say, $ million par of he bond for $,040,000 and sell he cash flows off individually as zeroes for oal proceeds of $,043,000, making $3000 of riskless profi. Similarly, if he bond were worh 05% of par, you could buy he porfolio of zeroes, reconsiue hem, and sell he bond for riskless profi. Zeroes and Coupon Bonds 6
Summary: Zeroes and Asses wih Fixed Cash Flows Securiies wih fixed cash flows can be viewed as packages or porfolios of zeroes. If an asse pays cash flows K, K 2,, K n, a imes, 2,, n, hen i is he same as K -year zeroes plus K 2 2 - year zeroes plus plus K n n -year zeroes. Therefore no arbirage requires ha he asse s value V is V = K d + K V = 2 n j= d or K j 2 +... + K d j n d n Consrucing Zeroes from Coupon Bonds No only can we consruc a given bond from zeroes, we can also go he oher way. Example: Consrucing a -year zero from 6-monh and - year coupon bonds. Coupon Bonds: Bond # Mauriy Coupon Price in 32nds Price in Decimal 0.5 4.250% 99-3 99.40625 2.0 4.375% 98-3 98.96875 Zeroes and Coupon Bonds 7
Consrucing he One-Year Zero Find porfolio of bonds and 2 ha replicaes -year zero. Le N be he number (par value) of bond and N2 be he number of bond 2 in he porfolio. A ime 0.5, he porfolio will have a cash flow of N x (+0.0425/2) + N 2 x 0.04375/2 A ime, he porfolio will have a cash flow of N x 0 + N 2 x (+0.04375/2) We need N and N 2 o solve N x (+0.0425/2) + N 2 x 0.04375/2 = 0 N x 0 + N 2 x (+0.04375/2) =00 => N 2 = 97.86 and N = -2.0 Implied Zero Price The price of he replicaing porfolio, long 97.86 par value of he -year bond shor 2.0 par value of he 0.5-year bond, is (97.86 x $0.9896875) - (2.0 x $0.9940625), or $94.7665. In he absence of arbirage, wih no marke fricions, his mus be he price of he -year zero Noe ha he price of he -year STRIP is $94.76 Zeroes and Coupon Bonds 8
Inferring zero prices from bond prices: Shor cu The las example showed how o consruc a porfolio of bonds ha synhesized (had he same cash flows as) a zero. We concluded ha he zero price had o be he same as he price of he replicaing porfolio (no arbirage). If we don' need o compue he replicaing porfolio, we can solve for he implied zero prices more direcly: 3 Price of bond= 99 = (00 + 4.25/ 2) d0.5 32 3 Price of bond 2 = 98 = (4.375/ 2) d0.5 + (00 + 4.375 / 2) d 32 d = 0.973, d = 0.948 0.5 Implied Zero Prices Ofen people would raher work wih Treasury coupon bonds han wih STRIPS, because he marke is more acive. They can imply a discoun funcion from Treasury bond prices insead of STRIPs and use his implied discoun funcion o value more complex securiies. Zeroes and Coupon Bonds 9
Replicaion Possibiliies Since we can consruc zeroes from coupon bonds, we can consruc any sream of cash flows from coupon bonds. Uses Bond porfolio dedicaion--creaing a bond porfolio ha has a desired sream of cash flows funding a liabiliy defeasing an exising bond issue Taking advanage of arbirage opporuniies Marke Fricions In pracice, prices of Treasury STRIPS and Treasury bonds don' fi he pricing relaionship exacly ransacion coss and search coss in sripping and reconsiuing bid/ask spreads Noe: The erms "bid" and "ask" are from he viewpoin of he dealer The dealer buys a he bid and sells a he ask, so he bid price is always less han he ask. The cusomer sells a he bid and buys a he ask. Zeroes and Coupon Bonds 0
Ineres Raes People ry o summarize informaion abou bond prices and cash flows by quoing ineres raes. Buying a zero is lending money--you pay money now and ge money laer Selling a zero is borrowing money--you ge money now and pay laer A bond ransacion can be described as buying or selling a a given price, or lending or borrowing a a given rae. The convenion in U.S. bond markes is o use semi-annually compounded ineres raes. Annual vs. Semi-Annual Compounding A 0% per year, annually compounded, $00 grows o $0 afer year, and $2 afer 2 years: 00.0 = 0 00 (.0) 2 = 2 0% per year semi-annually compounded really means 5% every 6 monhs. A 0% per year, semi-annually compounded, $00 grows o $0.25 afer year, and $2.55 afer 2 years: 00 (.05) 00 (.05) 2 4 = 0.25 = 2.55 Zeroes and Coupon Bonds
Annual vs. Semi-Annual Compounding... Afer T years, a annually compounded rae r A, P grows o F ) Presen value of F o be received in T years wih annually compounded rae r A is T = P( + r A T ( + r A ) In erms of he semi-annually compounded rae r, he formulas become F The key: ( P = 2T = P( + r/ 2) P = 2T 2 + r/ 2) = + r A F F ( + r/ 2) A semi-annually compounded rae of r per year really means r/2 every six monhs. Zero Raes If you buy a -year zero and hold i o mauriy, you lend a rae r where r is defined by d ( + or r = 2 r / 2) 2 (( ) d =, or 2 ) d = ( + r / 2) 2, Call r he -year zero rae or -year discoun rae. Zeroes and Coupon Bonds 2
Example According o convenion, zero prices are quoed using raes. STRIPS raes from he WSJ: 6 monhs: 5.54% year: 5.45% 5 years 5.73% STRIPS prices: 6 monhs: year: 5 years $00 (+ 0.0554 / 2) $00 (+ 0.0545 / 2) $00 (+ 0.0573 / 2) 2 0 0 = 97 32 0 = 94 32 3 = 75 32 Example The -year zero price implied from coupon bond prices was 94.7665. The "implied zero rae" is 2 r = 2 (( ) ) = 5.448% 0.947665 Zeroes and Coupon Bonds 3
Value of a Sream of Cash Flows in Terms of Zero Raes Recall ha any asse wih fixed cash flows can be viewed as a porfolio of zeroes. n The asse price is he sum V = K j d j j= of is cash flows muliplied by he relevan zero prices. n K j Equivalenly, he price is V = 2 j j= he sum of he presen values (+ r j / 2) of he cash flows, discouned a he relevan zero raes. Example $0,000 par of a one and a half year, 8.5% Treasury bond makes he following paymens: 6 monhs year /2 years $425 $425 $0425 $425 V = (+ 0.0554/ 2) $425 + (+ 0.0545/ 2) 2 $0425 + (+ 0.0547/ 2) 3 = $0430 Zeroes and Coupon Bonds 4