YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

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1 . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure of ineres raes. A. There are hree reasons ha bonds wih he same ime o mauriy have differen YTM s. i. Differen probabiliy of defaul ii. Liquidiy iii. Taxes B. Bonds wih defaul risk will always have a posiive risk premium. i. Define he risk premium here as he difference (spread) beween he ineres raes on bonds wih defaul risk and he ineres rae on defaul free bonds. a) Treasuries are essenially defaul risk free. b) Bu corporae bonds and municipals are cerainly no defaul risk free. () How do we know how much defaul risk a corporae bond has? (2) There are wo major firms ha rank he deb of corporaions based on defaul risk. Moody s and S&P. (a) The bes bonds (leas likely o defaul) are hose raed AAA, hen here are AA, A, BBB, and anyhing raed below BBB is considered junk. C. Liquidiy should obviously have an affec on ineres raes. The harder i is o sell he bond he higher he ineres rae mus be o compensae invesors. i. Noe ha Treasuries are he mos liquid of bonds. They have he mos acive and larges marke. I is easy and fas o sell a T-bond. ii. The same canno be said for corporae bonds or muni s. These markes are much hinner and i akes much more ime o conver a bond in hese markes ino cash. D. Taxes if a bond ges special ax reamen hen i may have a lower ineres rae han bonds of similar risk, liquidiy, and mauriy. i. Muni s are exemp from federal axes ii. Flower bonds reasuries ha can be cashed in a par value o pay esae axes if he owner dies. E. Hence, YTM is posiively relaed o defaul risk. YTM is posiively relaed o liquidiy risk. YTM is negaively relaed o special ax reamen. 3. So, why do bonds wih differen imes o mauriy, bu similar risk characerisics have differen YTM s?

2 A. Anoher way o pu he same quesion is: Why does he yield curve (erm srucure) have a given shape? 4. Wha is he yield curve? A. The yield curve is also called he erm srucure of ineres raes B. The yield curve is simply a plo of yields o mauriy versus imes o mauriy. i. The ime o mauriy is also called he erm hence he name erm srucure of ineres raes. ii. Curve is usually drawn for bonds of equal qualiy a) Like Treasuries b) AAA or Junk (bonds wih raings below BBB) for corporae bonds. iii. Here is he erm srucure for Treasury securiies from yeserdays WSJ. iv. How do you read his? a) For Treasuries wih 3 monhs lef o mauriy he yield o mauriy is jus below 2%. For Treasuries wih 2 monhs lef o mauriy he yield o mauriy is abou.75%. For Treasuries wih 0 years lef o mauriy he yield o mauriy is jus below 5%. C. How does one draw a yield curve? 2

3 i. The way he WSJ has done i is o ake he YTM of he Treasury closes o he mauriy. Plo he poins and connec he dos. ii. The way he book does i is a bi more sophisicaed. a) Here hey calculae all he YTM s and imes o mauriies for all available Treasuries. b) Then fi a non-linear spline hrough he daa. This is probably more accurae bu is cerainly no he only way o esimae a curve hrough he daa. D. Typically, he yield curve is upward sloping ha is long raes are ypically higher hen shor raes. Why? 5. Three heories o explain he erm srucure A. Expecaions hypohesis B. Marke segmenaion heory C. Liquidiy preference or Preferred habia heory. D. Expecaions hypohesis: Ineres raes on long erm bonds will equal an average of shor-erm ineres raes ha people expec o occur. i. This heory assumes ha bonds of differen mauriy are perfec subsiues. ii. I.E. invesors are indifferen beween: a) Buying a one-year bond and holding i o mauriy, hen buying anoher one-year bond. b) Buying a wo-year bond. iii. If hese porfolios are perfec subsiues hen he ineres rae on he wo-year bond should be he average ineres rae of he wo one-year bonds. i + i 2 e i 2 = + iv. And we could exend his ou for longer-erm bonds oo. v. Noice wha his implies hough. a) If he yield curve slopes up, hen shor-raes are going o rise b) The yield curve ypically slopes up. c) Tha means if his hypohesis is correc hen we would ypically see shor raes increasing. Since shor raes ofen go up as well as go down, hen we have a problem wih his heory. 6. Marke segmenaion heory (he oher exreme) A. Marke for differen mauriy bonds are compleely separae. B. Bonds of differen mauriies are no subsiues a all. i. Here, people are assumed o only wan o purchase bonds ha mach heir invesmen horizon. - Maybe because hey are immunizing heir porfolio we will ge o his shorly. ii. A each mauriy he supply and demand for bonds deermines he ineres rae for ha mauriy. 3

4 iii. More demand for shor bonds implies higher prices of bonds and ha implies lower ineres raes. iv. Here we have a heory ha is consisen wih an upward sloping yield curve ha does no sugges ha shor-erm raes will always be rising. C. BUT in his heory here is no reason for bonds of differen mauriies o be relaed a all. So, why do long raes ypically move up when shor raes move up? i. This heory canno accoun for he fac ha ineres raes a differen mauriies are relaed. 7. Liquidiy preference heory (preferred habia) A. This heory is a combinaion of he wo heories above. B. Ineres raes on long-erm bonds are an average of expeced shor-erm raes PLUS a erm or liquidiy premium ha depends on he supply and demand for bonds a ha mauriy. i + i 2 e + 2 = + k2 i 8. Here, if an invesor prefers he one period habia hey will normally buy one period bonds, bu if he premium for he wo-period bond is big enough hen hey can be induced o subsiue a wo-year bond for a one-year bond. 9. Now we have a heory ha explains why raes move ogeher and we can explain why he erm srucure ypically slopes up wihou he need o say ha shor raes mus be expeced o increase. A. The problem wih his heory is wha exacly deermines he erm premium. ) Wha is duraion? D = P = T ( + R) = P = R V c T = D = P ( + R ) c T ( + R ) ( + ) R = P R c 2) Well ha's nice bu wha does i mean? a) D = presen-value weighed average of he mauriy of each cash flow. b) D measures he effec of unexpeced changes in ineres raes on an asse's rae of reurn. c) Think of i like his. i) Toal risk = marke risk plus unique (idiosyncraic) risk ii) The risk premium depends on marke risk, no unique risk 4

5 iii) Sandard deviaion measures oal risk iv) Bea measures marke risk which is nohing more hen changes in he rae of reurn of an asse due o unexpeced changes in general economic condiions such as naional income or ineres raes. v) Duraion is he par of bea ha depends on unexpeced changes in ineres raes. 3) Duraion is simply a measure of he ineres rae sensiiviy of an asse (or liabiliy). a) Large D means ha he price of ha asse is highly sensiive o changes in ineres raes. b) In fac, i is common o see he following approximaion. dp P dr D + R which says ha he percenage change in price is equal o he percenage change in he ineres rae imes negaive duraion. 4) A reasonable quesion o ask is jus how accurae is ha approximaion? a) The answer is i depends on he size of he ineres rae change or shock as i is someimes called. b) Duraion accuraely measures he price sensiiviy of fixed-income securiies for ineres rae changes on he order of one basis poin. Is ha good enough for mos ineres rae changes? i) The approximaion is definiely good enough for he ypical movemens in ineres raes. ii) Bu wha if Federal Reserve Chairman Greenspan decides o up he Fed Funds rae? These changes are almos always 25 or 50 basis poin changes. iii) For large ineres rae increases, duraion will over predic he fall in bond prices. iv) For large ineres rae decreases, duraion will under predic he increase in bond prices. 5) Pu differenly, for rae increase he capial loss effec ends o be smaller han he capial gain effec from rae decreases. i) This phenomenon is known as convexiy. ii) Le's look a an example. Suppose we have an 8%, $000 bond selling a par wih six years ill mauriy and paying annual coupons. This informaion is in he firs row of Table. 6) I used he Excel duraion funcion o calculae D = a) Wha happens o he price as he ineres rae changes? 7) This can be seen in he firs wo columns of Table. a) If he ineres rae suddenly goes o zero hen he price will become $480. 5

6 b) If he ineres rae jumps o 4% hen he price will become $ These prices are based on he sandard bond pricing equaion: he presen value of level sream of cash flows and he presen value of he face. c) The las column of he able shows he percenage change in he price given he changes in ineres raes. d) Remember ha we are looking a he change in price when he ineres rae changes from 8%. Table : Duraion Approximaion N Coupon Rae Price Duraion 6 $ % $, New Rae New Price -D*dR/(+R) New Duraion based price dr/(+r) Duraion dp/p True dp/p 0.0% $, $, % $, $, % $, $, % $, $, % $, $, % $, $, % $, $, % $, $, % $ $ % $ $ % $ $ % $ $ % $ $ % $ $ % $ $

7 Duraion Approximaion vs. True Price 0.33 Dp/p DR/(+R) e) We can also esimae ha percenage change in price using he duraion formula above. i) Column 3 has he percenage change in price given he change in he ineres rae in column. ii) Again I used he Excel funcion for duraion. f) The prices based on he duraion approximaion are in column 4. 8) Already you can see ha he prices implied by he duraion approximaion are differen from he rue price changes. a) The las hree columns of he able have he percenage change in he ineres rae, he percenage change in price according o duraion, and he rue percenage change in price respecively. b) The figure plos he duraion esimaed percenage change in price and he acual percenage change in price agains he percenage change in ineres raes. c) In he graph i is easy o see ha he larger he ineres rae change he worse he esimae of price change prediced by D. 9) Remember he curved or convex line is he ruh here. 0) The amoun of curvaure in he line wha we mean by he erm convexiy. a) Aribues of convexiy i) Convexiy is a good hing. () The greaer he convexiy of a securiy or a porfolio of securiies, he more ineres rae proecion agains ineres rae increases and he greaer he gains for ineres rae decreases. (2) The higher he level of convexiy, he bigger he error made by using he duraion approximaion. 7

8 b) All fixed income securiies exhibi convexiy. c) Convexiy increases wih mauriy d) Longer erm bonds have more convexiy i) This is a posiive aribue of long erm bonds e) Convexiy varies wih coupon i) For wo bonds ha are exacly alike excep one has a lower coupon, he lower coupon bond will have more convexiy and higher duraion ii) For wo bonds wih equal duraion and yields, he lower coupon bond will have less convexiy. 8

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