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1 Deb Isrumes ad Markes Professor Carpeer Duraio Oulie ad Readig Oulie Ieres Rae Sesiiviy Dollar Duraio Duraio Buzzwords Parallel shif Basis pois Modified duraio Macaulay duraio Readig Tuckma, Chapers 5 ad 6 Duraio

2 Deb Isrumes ad Markes Professor Carpeer Duraio Defiiio: The duraio of a bod is a liear approximaio of he perce chage i is price give a 00 basis poi chage i ieres raes. (00 basis pois % 0.0) For example, a bod wih a duraio of 7 will gai abou 7% i value if ieres raes fall 00 bp. For zeroes, his measure is easy o defie ad compue wih a formula. For securiies or porfolios wih muliple fixed cash flows, we mus make assumpios abou how raes shif ogeher. We shall assume all zero raes move by he same amou. To compue duraio for oher isrumes requires furher assumpios ad umerical esimaio. Oher Duraio Coceps Cocep : Perce chage i he bod's price give 00 bp chage i raes Cocep : Average mauriy of he bod's cash flows, weighed by prese value. Cocep 3: Holdig period over which reur from ivesig i he bod is riskless, or immuized from immediae parallel shifs i ieres raes. Mahemaical fac: For a securiy wih fixed cash flows, hese ur ou o be he same. For securiies wih radom cash flows, such as callable bods, cocep does' really make sese. We'll focus o cocep. Duraio

3 Deb Isrumes ad Markes Professor Carpeer Dollar Duraio Sar wih he oio of dollar duraio. Cocep: dollar duraio - Applicaio : chage i dollar value chage i ieres raes chage i value - dollar duraio x chage i raes Example: Suppose a bod has a dollar duraio of 50,000. How much will is value chage if raes fall bp? Approx. chage i value -50,000 x ( -0.00) $55 Dollar Duraio ad DV0 DV0 DVBP Dollar Value of a Basis Poi How much will a bod value chage if raes chage bp? Approx. chage i value -$dur x chage i raes DV0 $dur x Example: Bod wih $dur 50,000 has DV0 5. bp rae chage causes *DV0$55 price chage. Duraio 3

4 Deb Isrumes ad Markes Professor Carpeer Compuig Dollar Duraio for a Zero-Coupo Bod For zero-coupo bods, here is a simple formula relaig he zero price o he zero rae. We use his price-rae formula o ge a formula for dollar duraio. The Price-Rae Fucio for a Zero A a rae of 5%, he price is 0.73 If raes fall o 4%, he price is The acual chage is Duraio 4

5 Deb Isrumes ad Markes Professor Carpeer The Price-Rae Fucio for a Zero d 30 ( + r / ) Usig a liear approximaio, he chage is abou A a rae of 5%, he price is 0.73 If raes fall o 4%, he price is The acual chage is 00 bp Compuig Dollar Duraio for a Zero... Recall dollar duraio - chage i dollar value chage i ieres raes By his defiiio, he dollar duraio of he zero is direcly relaed o he slope of he price-rae fucio. Example: The dollar duraio of $ par of a 30-year zero a a ieres rae of 5% is 6.65, as illusraed i he las slide: /(-0.0)0.0665/ We ca use calculus o compue he slope of he price-rae fucio ad ge a explici formula for he dollar duraio of ay zero. Duraio 5

6 Deb Isrumes ad Markes Professor Carpeer Formula for he Dollar Duraio of $ Par of a Zero- Coupo Bod d ( r ) ( + ' d ( r ) ( + r / ) r / ) + ( / ) 30 ( / ) To avoid workig wih egaive umbers, chage he sig. The dollar duraio of $ par of a -year zero is 60 6 ( r ) ( + r / ) 30 ( / ) $ ' dur d Example Wha's he dollar duraio of $ par of a.5-year zero if he.5-year discou rae is 5.47%? ( + r / ).5 ( / ) If he rae falls o 5.40%, how much will he price rise? Usig he dollar duraio approximaio, he price will rise by x d (5.47%) d (5.40%) The acual price rise is Duraio 6

7 Deb Isrumes ad Markes Professor Carpeer Dollar Duraio of a Porfolio of Fixed Cash Flows Cosider he dollar sesiiviy of a porfolio o a chage i ieres raes. Remember ha he porfolio value is a fucio of all of he differe zero raes associaed wih is cash flows. For simpliciy, we will approximae he chage i he porfolio value assumig all raes chage by he same amou. I oher words, we will measure he sesiiviy of he porfolio value o a parallel shif i ieres raes. How useful will his measure be? Of course, raes do o always chage by exacly he same amou, bu hey do ed o move ogeher. Dollar Duraio for a Porfolio of Fixed Cash Flows Suppose a porfolio (or bod) has cash flows,,... a imes,,... Is value is he sum of he values of he compoes: V d + d + If raes chage, is value will chage by he sum of he chages i value of he compoes: V We ca approximae he chage i each zero price usig is dollar duraio: d $ dur r... d + d +... Duraio 7

8 Deb Isrumes ad Markes Professor Carpeer Dollar Duraio for a Porfolio of Fixed Cash Flows... The approximae chage i he porfolio value is: V ( $ dur r + $ dur r + Suppose all rae chages are he same. Tha is, he yield curve makes a parallel shif: r r r... r The he porfolio value chage is: 3 V $ dur + $ dur +...) r (...) Dollar Duraio for a Porfolio of Fixed Cash Flows... The he porfolio dollar duraio is: porfolio $dur V ( $ dur + V porfolio $ dur r chage i value chage i raes $ dur + ) r $ dur V r + $ dur + I oher words, he dollar duraio of he porfolio is he sum of he dollar duraios of is cash flows: porfolio $ dur $ dur + $ dur + Duraio 8

9 Deb Isrumes ad Markes Professor Carpeer Example Wha is he dollar duraio of a porfolio cosisig of $500 par of he.5-year zero ad $00 par of he 30-year zero? (500 x.35) + (00 x 6.65) 340 This meas he porfolio value will chage abou $3.40 for every 00 basis poi shif i ieres raes. Why? Each 00 bp chage i he.5-year rae chages he value of he.5-year zero abou 500 x.35 x Each 00 bp chage i he 30-year rae chages he value of he 30-year zero abou 00 x 6.65 x The oal porfolio chage is abou ((500 x.35) + (00 x 6.65)) x x 0.0. Duraio Duraio approximaes he perce chage i price for a 00 basis poi chage i raes: duraio perce chage i value per 00 bp chages i raes dollar chage i value per 00bp 00 iiial value dollar duraio iiial value dollar duraio iiial value Duraio 9

10 Deb Isrumes ad Markes Professor Carpeer Duraio for a Zero dollar duraio duraio price The duraio of a -year zero is: ( + r / ) duraio ( + r / ) ( + / ) Noice ha he duraio of a zero is us slighly less ha is mauriy. + r Example:.5-Year Zero A a ieres rae of 5.47%, he duraio of he.5- year zero is duraio dollar duraio price duraio (+ r / ) / If raes rise 00 basis pois o 6.47%, he price falls abou.46% from 0.9 o Duraio 0

11 Deb Isrumes ad Markes Professor Carpeer Example: 30-Year Zero A a ieres rae of 5%, he duraio of he 30-year zero is duraio dollar duraio price duraio (+ r / ) / If raes fall 00 basis pois o 4%, he price rises abou 9.6% from 0.73 o abou If raes fall oly 50 bp o 4.5%, he price rises oly half as much, abou 4.63% o abou Duraio of a Porfolio Jus as wih a zero, he duraio of a porfolio is is dollar duraio divided by is marke value. The duraio gives he perce chage i value for each 00 basis poi chage i all raes. duraio dollar duraio value $ dur d + + $ dur d Duraio

12 Deb Isrumes ad Markes Professor Carpeer Example The duraio of he porfolio cosisig of $500 par of he.5-year zero ad $00 par of he 30- year zero is dollar duraio 340 duraio.8 marke value This meas ha he porfolio value will chage abou.8% for every 00 basis poi chage i ieres raes. Duraio of a Porfolio as Average Duraio We ca hik of he porfolio duraio as he average of he duraios of he idividual cash flows, weighed by heir prese value or "marke" value. Recall ha he dollar duraio of each zero is is duraio imes is price: $ dur d dur So he porfolio duraio is d dur... + d dur + porfolio dur d + d +... porfolio dur where w d w dur + d + w d dur +... is he pv weigh of cash flow +... Duraio

13 Deb Isrumes ad Markes Professor Carpeer Duraio of a Porfolio porfolio dur d ( + r dur / ) d ( + r + + ( + r / ) d d dur +... / ) +... Example Recall he porfolio cosisig of $500 par of he.5-year zero ad $00 par of he 30-year zero. The marke value of he.5-year zero is 500 x 0.94 $46.. Is duraio is.46. The marke value of he 30-year zero is 00 x 0.73 $.73. Is duraio is 9.6. The duraio of he porfolio is: ($46..46) + ($ ) $46. + $.73.8 Duraio 3

14 Deb Isrumes ad Markes Professor Carpeer Example... I erms of he marke value weighs, he duraio of he porfolio is as follows: w %, w % porfolio duraio Summary duraio (mius he) perce chage i price per 00 bp chage i raes chage i value dollar duraio - chage i raes (i decimal) dollar duraio duraio value duraio of a zero dollar duraio of $par of a zero dollar duraio of a porfolio duraio of a porfolio dur + r / $dur + ( + r / ) sum of dollar duraios $dur + $dur +... dollar duraio average duraio weighed by pv value dur ( + r + + r / ) ( r ( / ) / ) ( + r / ) ( + r / ) Duraio 4

15 Deb Isrumes ad Markes Professor Carpeer Modified Duraio I pracice, people compue wha's called he modified duraio of a securiy by usig he securiy's yield isead of he differe zero raes associaed wih each cash flow. ( + y / ) ( + y / ) modified duraio The modified duraio of a porfolio is he average modified duraio of is securiies weighed by heir marke value. ( + y / ) Example Origial daa Par Coupo (%) Mauriy (years) Yield (%) Marke Value Duraio Dollar Duraio Modified duraio of he coupo bod Par Coupo (%) Mauriy (years) Yield (%) Marke Value Duraio Dollar Duraio Duraio 5

16 Deb Isrumes ad Markes Professor Carpeer Modified Duraio... Compuig he bod's modified duraio is like compuig is duraio assumig he yield curve is fla, i.e., ha all raes are equal o he bod's yield. Whe he yield curve is o fla, usig securiy yield isead of idividual zero raes creaes a differece ha is usually sligh. Modified Duraio: Icosisecy vs. Pracicaliy Usig modified duraio ca lead o logical icosisecies if he yield curve is o fla. If he yield curve is o fla, he wo porfolios wih ideical cash flows ca have slighly differe modified duraios. For example, hik of a coupo bod ad he correspodig porfolio of zeroes. I he previous example, he coupo bod had a modified duraio of The replicaig porfolio of zeroes would have a modified duraio of BUT: If zero raes are o readily available, modified duraio is easier o compue. Duraio 6

17 Deb Isrumes ad Markes Professor Carpeer Fially: Macaulay Duraio The firs measure of duraio was developed by Frederick Macaulay i 938: ( + y / ) Macaulay duraio modified duraio (+ y/) ( + y / ) Noe ha he Macaulay duraio of a -year zero is us is ime o mauriy,. The Macaulay duraio of a securiy is he average mauriy of each cash flow weighed by he cash flow's prese value a he yield o he securiy. This gives a iuiive way o guess he ieres rae sesiiviy of a bod. Duraio 7