Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned in you into finance couse to some of the concepts that you have leaned in this couse. heefoe, this note should be seen only as a complement to the othe couse mateials. 1. onds In ode to undestand how floating ate bonds ae piced, you need to ecall that when a bond pays coupons equivalent to the maket ate (i.e. the discount ate applicable to that bond), the bond sells at pa. Let s stat with a vey simple example that poves this point. Example 1: onside a bond that pays coupons semiannually and that has 1 yea to matuity. Futhe, assume that the maket ate,, is 10% (compounded semiannually), and assume that the coupon ate on a bond,, is also 10% (compounded semiannually). What is the value of the bond today? Fist, we find the dolla value of the coupon: Now we can find the value of the bond. 0.1 ( ) ( 100) 5 5 105 0 / ( / ) 1.05 ( 1.05) 100 whee is the coupon payment 0 is the value of the bond at time 0 is the pa value of the bond (in this case 100). 1 Fo simplicity, we ignoe day-count issues in this note. his should be a eview of what you coveed in you into finance couse (i.e. us 31). Updated: June 1, 010.
Joge uz Lopez us 316: Deivative Secuities In geneal, any time that ( ) ( ) 0 ( / ) / ( ( / ) / ) ( / ) / (1 ( / ) / ) ( / ) / / ( / ) / (1 / ) / Keep in mind that is the maket ate that is appopiate fo discounting the cash flows of this bond. In othe wods, takes into account the isk implied by the bond cash flows. Riskie bonds (i.e. bonds with iskie cash flows) equie a highe, othe things being equal. We can extend the pevious esults by using the fomula to calculate the pesent value of an annuity. In geneal, fo any matuity, if we have ( ) ( ) ( 1 ) 0 ( / ) / 1 1 ( / ) ( / ) ( / ) ( / ) 1 1 ( / ) ( / ) heefoe, we have the following esult: Result 1: when the coupon ate is equal to the maket ate, the bond sells at pa. his is the key pinciple behind floating-ate bond pices. Updated: June 1, 010.
Joge uz Lopez us 316: Deivative Secuities. oupons on a Floating-Rate ond he coupons of a floating ate bond ae set at the beginning of each coupon peiod. Specifically, we set at the beginning of each peiod. A coupon peiod stats ight afte a coupon payment is made, and it ends ight befoe the next coupon payment. Example : onside a bond that pays LIOR 1%, with coupons occuing evey 6 months. he bond just made a coupon payment (i.e. we ae at the beginning of the coupon peiod) and the 6 month LIOR is.5% compounded semiannually. What is the value of the next coupon payment? he next coupon payment will be set at.5% 1% 3.5% of. In dolla tems, on a bond with a face value of $100. 3.5%* ½ *100 $17.5 3. Valuing a Floating Rate ond he value of the floating ate bond, fl, ight at the beginning of the coupon peiod is. his statement is always tue because a floating-ate bond sets at the beginning of the coupon peiod. his point is illustated in Examples 3 and 4, below. Example 3: oday is June 1, 009. Find the value of a floating-ate bond that has six months left to matuity. Assume that the bond just made a coupon payment and that the bond pays LIOR 1%. LIOR today is 5% compounded semiannually. he following figue shows the timeline of the elevant cash flows. June 1, 09 (ODAY) Dec 1, 09 ime Fist, we find the appopiate maket ate: LIOR 1%, 6% Updated: June 1, 010.
Joge uz Lopez us 316: Deivative Secuities ecause today is the beginning of the coupon peiod, we set, June 1 09,, whee, June 1 09 and, ae the maket ate and the coupon ates on June 1, 009, espectively. So fom equation (1) we know that the next coupon payment will be Dec 1 09,, ( ) ( ) and fom Result 1 above, we know that because fl, Dec 1 09 (, June1 09 / ) (, June1 09 ) / (, June1 09 ) / fl, whee Dec 1 09 is the dolla coupon payment expected on Decembe 1, 009 and fl, is the value of the floating-ate bond on June 1, 009 (today). Example 4: onside the bond pesented in Example 3, howeve, now assume that the bond has 1 yea left to matuity; that is, today is Decembe 1, 008. What is the value of the bond today? Since today is the beginning of a coupon peiod, we set, Dec 1 08, Dec 1 08, just as we did befoe. Fom (1) we know that the next coupon payment is June 1 09, Dec 1 08, Dec 1 08 ( ) ( ) whee, Dec 1 08 and, Dec 1 08 ae the maket and the coupon ates on Decembe 1, 008, espectively. June 1 08 is the dolla coupon payment expected on June 1, 009. Updated: June 1, 010.
Joge uz Lopez us 316: Deivative Secuities Fom Example 3 we know that at the beginning of the next coupon peiod (i.e. in six months) the bond will ALWAYS be woth no matte what; that is, fl,. hus, we only need to conside the next coupon payment,, and value of the bond in six months, fl,, to calculate the bond pice. Notice that we do not conside the coupon and pincipal payment on Decembe 1, 009 because these cash flows ae aleady accounted fo by fl,. he following figue shows the timeline of the elevant cash flows. fl, June 1-09 June 1-09 June 1-09 Dec 1, 008 (ODAY) June 1, 009 Dec 1, 009 ime he value of the bond today is fl, (, Dec1 08 / ) ( ) / ( ) / (, Dec1 08 ) / fl, Dec 1 08, Dec1 08, Dec1 08 It can be shown that this patten holds fo any matuity, because we know that ight at the beginning of the next coupon peiod the bond will be woth fl. heefoe, we have the following impotant esult. Result : fl ight afte a coupon payment (i.e. ight at the beginning of a coupon peiod) because the next coupon is set exactly equal to the obseved floating ate on that date (i.e. the maket ate). Now, what if we want to value a floating-ate bond ight befoe a coupon payment is made? Well, because we know that exactly afte a coupon payment is made the bond is woth, it must be tue that Result 3: fl ight befoe a coupon payment (i.e. ight at the end of a coupon peiod). Updated: June 1, 010.
Joge uz Lopez us 316: Deivative Secuities hus, the value of a floating-ate bond at any time t should be the discounted value of ; that is, plus the coupon payment that we expect to eceive at the end of the cuent bond peiod. athematically this becomes whee fl ( ) (1 t*, t / ), is the maket ate obseved at time t, and t t* is the time to the next coupon peiod measued in the same fequency as the maket ate (e.g. in six month intevals fo six month ates). Example 5: oday is Apil 1, 009. What is the value of the bond descibed in the pevious two examples if the cuent LIOR ate is % compounded semiannually and the LIOR ate on Decembe 1, 008 was 4% compounded semiannually? he timeline fo this poblem is shown below. fl, June 1-09 June 1-09 June 1-09 Dec 1, 008 Apil 1, 009 (ODAY) June 1, 009 Dec 1, 009 ime Fist, emembe that the coupons of a floating ate bond ae set at the beginning of each coupon peiod (see Section of this note). heefoe, the coupon ate fo this bond was set on Decembe 1, 008 at LIOR 1% 4% 1% 5%, which tanslates to the following dolla value, Dec 1 08 June he elevant maket ate on Apil 1, 009 is hus, the value of the bond today is 0.5 1 09 (100).5, Api 1 09 LIOR 1% % 1% 3% [1 ( ) / ] 10.5 (1 0.03/ ) fl, Apil 1 09 t* / 6, Api 1 09 101.996 Updated: June 1, 010.
Joge uz Lopez us 316: Deivative Secuities 4. Hull s (008) Notation he pinciples illustated in this note apply to the discussion of floating-ate bonds in the textbook. In this note, howeve, discete ates wee used so that you can undestand floatingate bonds given the tools that you leaned in you into finance couse. whee Hull (008) uses the following fomula to find the value of a floating-ate bond, ( L ( *)( t*) k*) e ( 3 ) L is the value of the bond. t* is the time to the next coupon payment measued in yeas. k* is the floating payment that will be made at t* (i.e. the coupon payment that was set at the beginning of the cuent coupon peiod). We denoted this by in the pevious sections of this note. * is the LIOR/swap zeo ate fo a matuity t*. So you can see that eveything is the same as in (). he only diffeence is that in (3) we use continuous ates to discount the payoff expected at the end of the next coupon peiod. Updated: June 1, 010.