ad covexity Defie Macaulay duratio D Mod = r 1 = ( CF i i k (1 + r k) i ) (1.) (1 + r k) C = ( r ) = 1 ( CF i i(i + 1) (1 + r k) i+ k ) ( ( i k ) CF i

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1 Fixed Icome Basics Cotets Duratio ad Covexity Bod Duratios ar Rate, Spot Rate, ad Forward Rate Flat Forward Iterpolatio Forward rice/yield, Carry, Roll-Dow Example Duratio ad Covexity For a series of cash flows {CF i }, the et preset value (NV) is a fuctio of aualized iterest rate r. Assume k-th periodic compoudig, the price or NV is give by Apply Taylor expasio (r) = V i (r) = CF i (1 + r k) i (r) = (r 0 ) + r r + 1 r ( r) By defiig = (r) (r 0 ), it ca be re-writte as = ( r 1 ) r + ( r ) ( r) letiaquat.wordpress. = D Mod r + 1 C( r) (1.1) where equatio (1.1) is referred to as Secod order approximatio. The modified duratio is defied as

2 ad covexity Defie Macaulay duratio D Mod = r 1 = ( CF i i k (1 + r k) i ) (1.) (1 + r k) C = ( r ) = 1 ( CF i i(i + 1) (1 + r k) i+ k ) ( ( i k ) CF i (1 + r k) D Mac = i ) V i = ( i V i k ) Notice that ( i k ) is the maturity of ith cash flow CF i. Therefore Macaulay duratio is a weighted average of maturities. From the defiitio of Macaulay duratio (1.), it is easy to see its relatioship with modified duratio, Dollar Duratio ad DV01 (V01) are defied as D Mac D Mod = 1 + r k D $ = r = D Mod DV01 = r ( r = 1bps) = D Mod ( r = 1bps) letiaquat.wordpress. This gives the liear (first order) approximatio of the price movemet due to 1bps jump of iterest rate. So far the yield curve is implicitly assumed to be flat. I reality, it is rarely the case. The parallel shift duratio (Fisher-Weil duratio) measures the effect of a small parallel shift o the spot yield curve. Cosider the market zero curve with term structure {r i }, where r i is the aualized zero rate of correspodig teor i. The I the formula, s is the parallel shift / spread. (s) = V i (s) = CF i (1 + r i + s k )i

3 D FS = 1 ( s ) s=0 = 1 CF i (1 + r i + s k )i i k (1 + r i + s k ) s=0 = 1 ( CF i i k (1 + r i k) i ) (1 + r i k) C FS = 1 ( s ) = 1 ( CF i i(i + 1) (1 + r s=0 i k) i+ k ) which equals the modified duratio oly if the zero curve is flat. I practice, they are differet but fairly close. I additio to the parallel shift, key rates duratio is itroduced to tackle o-parallel shifts. T.Ho (199) suggests usig key rates that correspod to the teors of o-the-ru treasuries (3M, 1Y, Y, 3Y, 5Y, 7Y, 10Y, 15Y, 0Y, 5Y, ad 30Y). Oe shifts oe key rate at a time while keeps the others uchaged. Key rate duratio ca be defied either o the spot rate curve or o the par rate curve (see below). Key rate duratio is computed as + Key Rate Duratio = 1bps 0 where ad + are bod price after a ±1bps chages o that duratio, respectively. Bod Duratios Let s cosider specific cash flows cotaied i a bod. Zero-Coupo Bod A zero-coupo bod, such as a T-bill, has oly oe paymet o the maturity date. The paymet equals its face value F. Cosider a T-bill of maturity T ad semi-aual compoudig, (=, k = ). The D Mac = V = ( V k = ) = T D Mod = T 1 + r C = T(T + 1 ) (1 + r ) Therefore the Macaulay duratio of a zero-coupo bod is its time to maturity. Fixed Coupo Bod letiaquat.wordpress.

4 A fixed coupo bod, such as a T-Note/T-Bod, has semi-aual coupo paymets ad the redemptio of the face value at the maturity. Deote coupo rate by c, the the semiaual coupo paymets equal F c. I the case of fixed coupo bod, the iterest rate r is also referred to as yield to maturity (YTM), the sigle flat rate that equates the NV with the quoted market dirty price. Floatig Rate Bod = F c F (1 + r + )i (1 + r ) D Mac = ( 1 F c (1 + r )i) (i ) + ( 1 F (1 + r D Mod = D Mac 1 + r )) (T) C = 1 ( F c i(i + 1) F T( + 1) (1 + r ) i+ ) + 4 (1 + r ) + Cosider a quarterly settled, paid i arrear floatig rate ote (FRN). After the floatig rate beig fixed at LIBOR rate, the holder of the FRN is expected to receive periodic iterest paymet o the ext paymet date whe the bod will is priced at par agai. Therefore its duratio equals to a zero-coupo bod matures o the ext paymet date. ar Rate, Spot Rate, ad Forward Rate letiaquat.wordpress. ar coupo rate of a fixed coupo bod is the coupo rate this bod should pay if it is priced at par. That is, the coupo rate c satisfies F = F c (1 + r + i )i F (1 + r ) The coupo rates of o-the-ru bods are close to the par coupo rates sice they are always issued with price ear par.

5 If the bod is priced at par, the par coupo rate should be equal to the yield to maturity. This rate is referred to as par rate. ar rate ca be calculated directly by solvig the equatio, or it ca be calculated from BS (basis-poit sesitivity) or VB (preset value of 1 bp). VB is defied as VB = c (1bps) = F (1 + r (1bps) = BS (1bps) )i the curret coupo rate should icrease/decrease ( F ) basis poits i order to reach the par coupo VB rate. If a bod is priced at par, par rate equals its yield-to-maturity. Therefore VB approximates V01 from 1bp (parallel) shift of YTM. Spot rate is the rate oe gets by holdig a zero coupo bod ow. Forward rate is the rate oe is expected to get by holdig a zero coupo bod i the future. The yield curves are referred to as spot (rate) curve/zero curve, (-moth) forward (rate) curve, ad par rate curve/par yield curve, respectively. There is aother oe called discout curve. Ay oe curve ca imply the other three (with the help of iterpolatio). Flat Forward Iterpolatio Curves are be built through bootstrap or global fittig (such as cubic splie). But o matter how oe builds curve, a iterpolatio method is ievitable. I this sectio we itroduce a popular iterpolatio method flat forward iterpolatio. Notatio is iherited from Brigo ad Mercurio (006). Specifically, deote zero coupo bod price at time t by (t, T). The the term structure o T at time 0 (that is, (0, T)) is the discout curve at time 0. The (cotiuously compouded) spot rate/zero curve is R(0, T), where R(0, T) is defied as e R(0,T)T (0, T) = 1 R(0, T)T = l(0, T) The istataeous forward rate f(0, T) is defied as letiaquat.wordpress. l(0, T) f(0, T) = T T l(0, T) = f(0, u)du 0

6 The flat forward iterpolatio assumes costat istataeous forward rate betwee ay two kots T 1 ad T. I the followig we show the iterpolatio procedure ad poit out that it is equivalet to the logliear iterpolatio o the discout curve. Give the zero rates o T 1 ad T, the aim is to get the zero rates o ay T [T 1, T ] via iterpolatio. Note that R(0, T 1 )T 1 = l(0, T 1 ) = 0 T 1 f(0, u)du T f(0, u)du = R(0, T )T R(0, T 1 )T 1 T 1 By assumig costat istataeous forward rate we have The for ay T [T 1, T ], R(0, T )T = l(0, T ) = f(0, u) f [T1,T ] where u [T 1, T ] f [T1,T ] = R(0, T )T R(0, T 1 )T 1 T T 1 R(0, T)T = R(0, T 1 )T 1 + f(0, u)du T T 1 = R(0, T 1 )T 1 + f [T1,T ](T T 1 ) = R(0, T 1 )T 1 + R(0, T )T R(0, T 1 )T 1 (T T T T 1 ) 1 = T T R(0, T T T 1 )T 1 + T T 1 R(0, T 1 T T )T 1 0 T f(0, u)du which idicates liear iterpolatio o R(0, T)T. By usig the equatio R(0, T)T = l(0, T), it ca be re-writte as letiaquat.wordpress. l(0, T) = T T T T 1 l(0, T 1 ) + T T 1 T T 1 l(0, T ) I sum, the flat forward iterpolatio o the zero curve is equivalet to the logliear (liear o the log) iterpolatio o the discout curve. This method is very stable, easy to implemet, ad provide meaigful results o both spot rates ad forward rates. Therefore it is widely used i the idustry.

7 Forward rice/yield, Carry, Roll-Dow This sectio is adapted from Sadr (009). Similar to the o-arbitrage argumet used for future price determiatio, the fair forward price of a bod is F dirty = Dirty + Fiacig cost FV(Coupo Icome) where the fiacig cost is called carry. Usually bods are fiaced i repo markets, ad the future value of ay coupo icome is supposed to be re-ivested at the repo rate r p. The, F Dirty (T Fwd ) = Dirty (1 + r p T Fwd t 360 ) C (1 + r p where T i are coupo paymet days, t is curret date, ad T Fwd is forward date. T Fwd T i ) 360 The price carry is the differece betwee the curret (spot) clea price ad the forward clea price: rice Carry = Clea F Clea (T Fwd ) The yield carry is the differece betwee the forward ad spot yields (to maturity): Yield Carry = Forward Yield Spot Yield Whe yield term structure is upward sloppig, short term fiacig (repo) rate is usually lower tha the yield of the bod. The borrowig-short-ivest-log strategy offers positive carry gais. Aother gai positively-sloped yield curve offers is the roll-dow retur, which is the capital gai caused by a fallig yield whe a bod is approachig maturity. Assumig that yield curve remais uchaged as time passes, letiaquat.wordpress. Moth Roll Dow = y(t) y(t Moths) Carry reflects the profit from lower fiacig cost. Roll-dow reflects time value of a bod (theta). Example The first part of accompayig C++ code shows relatioship betwee spot rate (curve), forward rate (curve), par rate (curve), ad discout rate (curve). The secod part of this curve shows carry ad rolldow retur. I use o-the-ru securities o

8 O 011-April-11, CB1 is quoted as BID ASK YTM CB1 is quoted at a discout from face value, i percetage, with ACT/360 day cout. Let Y d be the discout rate ad is the price, the 100 Y d = 360 t = F (1 Y d t 360 ) The bill is T+1 settled, or settled o It matures o Hece there are 3 days to maturity, or t = 3 days. Therefore the quoted bid/ask prices are BID = 100 ( % ) = ASK = 100 ( % ) = All the yields are based o the ask price. The bod equivalet yield (BEY) is the yield if oe buys the bill at the ask price ad holds it to maturity. It assumes simple compoudig BEY = 100 ASK 365 = 100 ASK 365 = = 0.01% ASK t ASK 3 The effective aual rate (EAR) aualizes simply-compouded BEY ad trasforms it ito compouded rate. The yield-to-maturity (YTM) equals EAR. O 011-April-11, CT is quoted as BID ASK YTM letiaquat.wordpress. Note ad bod prices are quoted i dollars plus 1/3 factios of a dollar. + stads for half a fractio or 1/64 of oe dollar. Therefore BID = = ASK = = Related Bloomberg commads: CB <Govt>, CT <Govt>, X1, CG I5, ad ALLQ.

9 Referece [1] Brigo, D. ad Mercurio, F (006). Iterest rate models: theory ad practice: with smile, iflatio, ad credit. Spriger Verlag. [] Fabozzi, F.J. (005). The hadbook of fixed icome securities, 7 th. McGraw-Hill. [3] Hull, J. (009). Optios, futures ad other derivatives. earso retice Hall. [4] Sadr, A. (009). Iterest rate swaps ad their derivatives: a practitioer's guide. Joh Wiley & Sos Ic. letiaquat.wordpress.

Chapter Six. Bond Prices 1/15/2018. Chapter 4, Part 2 Bonds, Bond Prices, Interest Rates and Holding Period Return.

Chapter Six. Bond Prices 1/15/2018. Chapter 4, Part 2 Bonds, Bond Prices, Interest Rates and Holding Period Return. Chapter Six Chapter 4, Part Bods, Bod Prices, Iterest Rates ad Holdig Period Retur Bod Prices 1. Zero-coupo or discout bod Promise a sigle paymet o a future date Example: Treasury bill. Coupo bod periodic