1 Interest rates and bonds

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1 1 Interest rates and bonds 1.1 Copounding There are different ways of easuring interest rates Exaple 1 The interest rate on a one-year deposit is 10% per annu. This stateent eans different things depending on the copounding frequency. With annual copounding it eans that $100 grows to $100 With seiannual copounding it eans that $100 grows to $ ) ) = $ With copounding ties per year it eans that $100 grows to $ ) With continuous copounding let ) it eans that $100 grows to $100 e For ost practical purposes continuous copounding can be thought of as being equivalent to daily copounding, which here eans that $100 grows to $ ) Let R c be a rate of interest with continuous copounding and R the equivalent rate with copounding with copounding ties per year. Then e Rc 1 = ), so and R c = ln 1+ R ) ) R = e Rc/ 1. 1

2 1.2 Zero coupon bonds and zero rates As stated before the ost basic interest rate derivatives are zero coupon bonds. Definition 1 A zero coupon bond with $1 principal and aturity T is a T-clai paying $1 at tie T. The price at tie t of the bond is denoted pt,t) and we will assue that pt,t) = 1. Spot rates are defined fro zero coupon bond prices. They are also known as zero coupon bond rates or zero rates. Definition 2 The continuously copounded) spot rate rt, T) at tie t for the tie interval [t,t] is defined by pt,t) = e rt,t)t t) or rt,t) = lnpt,t). T t The price at tie t of the bond is denoted pt,t) and we will assue that pt,t) = 1. For each aturity T there exists a separate spot rate. The collection of all tie t spot rates rt,t) : T = t,..., is referred to as the ter structure or zero coupon yield curve, or zero curve or yield curve) at tie t. Exaple 2 Given prices of zero coupon bonds T p0,t) 0.3 $ $ $ the ter structure is T r0,t) 0.3 5% 0.6 8% % 1.3 Money arket account Before the oney arket account or risk less asset has been given by B T = e rt if B 0 = 1) where r has been constant, but now we will have in discrete tie) B t+ t = B t e rt,t+ t) t so if we divide the tie interval [t,t] into n intervals of size t = T t)/n n 1 B T = B t e i=0 rt+i t,t+i+1) t) t Here rt,t+ t) is the over night rate, and it is deterined at tie t. 2

3 Exaple 3 Consider a three onth investent of $10 in a oney arket account with onthly capitalization eans that t= 1 onth = 1/12 year). Assue that the one onth rate is 5% per annu the first onth, 5.5% the second, and 6% the third. Then 3 1 B 3/12 = B 0 e i=0 r0+i t,0+i+1) t) t { = 10exp r0, 1 12 ) r 1 12, 2 12 ) r 2 12, 3 } 12 ) 1 12 = 10e ) 1 12 = Note that r1/12,2/12) and r2/12,3/12) are not really known at tie t = 0! 1.4 Fixed coupon bonds Most bonds pay coupons to the holder periodically. The siplest coupon bond is the fixed coupon bond. The foral description is as follows. Fix ties T 0,T 1,...,T n. Here T 0 is thought of as the eission date of the bond and T 1,...,T n as coupon dates. At tie T i, i = 1,...,n the owner of the bond receives the deterinistic coupon c i. At tie T n the owner of the bond receives the principal face value) K. The coupon bond is siply a portfolio of zero coupon bonds. We have that the price of the bond for t < T 1 is p fixed t) = c i pt,t i )+Kpt,T n ). 1) Often the coupons are deterined in ters of return rather than onetary ters. The return forthei:thcouponistypically quotedasasiplerateactingontheprincipalk over[t i 1,T i ]. Here coes a sall digression on siple rates. We have seen that the continuously copounded spot rate rt,t) for the interval [t,t] solves or pt,t) = e rt,t)t t) e rt,t)t t) = 1 pt,t). This is because an investent of $1 at tie t can create a payoff of 1/pt,T) at tie T. Just buy 1/pt,T) T-bonds at tie t, this will cost you exactly $1, and at tie T you will get the payoff 1/pt,T). The siple spot rate for [t,t], henceforth referred to as LIBOR spot rate solves and is defined as 1+Lt,T)T t) = 1 pt,t), Lt,T) = pt,t) 1 T t)pt,t). The zero rates and LIBOR spot rates are thus used to express the sae return, they are just quoted in different ways. 3

4 Back to the coupon bonds. A siple rate r i acting on K over [T i 1,T i ] results in K[1+r i T i T i 1 )] and by definition if the i:th coupon has a return r i and the principal is K then c i = Kr i T i T i 1 ). Reark 1 Recall that the rate R copounded ties per year akes $1 grow to 1+ R ) Over a period of 1/ years $1 grows to 1+R 1 ) which is the siple rate for the period 1/! Deterining zero rates by bootstrapping Given arket bond prices, how do we deterine the zero coupon curve? Recall that ost traded bonds are coupon bonds, particularly for the longer aturities. Since the price of a fixed coupon bond is ade up of prices of zero coupon bonds, see Equation 1), which are in turn deterined by the zero rates, there is a one-to-one correspondence, between bond prices and zero rates. Starting fro the short end of the ter structure that is short tie to aturity) zero rates are obtained fro zero coupon bonds. When oving to longer aturities bonds are coupon bonds, but now we can use that we have the short zero rates to back out the longer rates. Exaple 4 Assue that in addition to the zero coupon bonds given in Exaple 2 there is also a coupon bond traded. This bond pays a $5 annual coupon, has a principal of $100, and atures in 1.6 years. The price of the coupon bond is $ We can use this bond to find the 1.6-year zero rate. The price is given by p fixed t) = c i pt,t i )+Kpt,T n ) = 5e rt;t+0.6) )e rt;t+1.6)1.6. Now using that we know that p fixed = and that rt;t + 0.6) = 8%, we can solve for rt;t+1.6) and obtain rt;t+1.6) = e ln We can thus extend the table of zero rates to T t rt,t) 0.3 5% 0.6 8% % %

5 1.4.2 Yield and duration Definition 3 The yield to aturity of a fixed coupon bond is the interest rate y that when used to discount all coupons and the principal results in the arket price. The yield thus solves p arket t) = c i e yti t) +Ke ytn t). Reark 2 The yield to aturity of a zero coupon bond is siply the zero rate. This represents the bonds internal rate of interest and the above definition is an attept to extend the concept to fixed coupon bonds. Exaple 5 The yield to aturity of the coupon bond considered in Exaple 4 solves = 5e y )e y 1.6. It should lie between 8% and 11% the zero rates for aturities 0.6 and 1.6, and it should be closer to 11%, since ost weight is given to the large final payent. Trial and error gives y 10.94%. To siplify notation includethe principal K in the last coupon so c new n that t = 0. Also let p = p fixed, then p = c i e yti t). Definition 4 The duration D of a fixed coupon bond is defined as n T i c i e yt i c i e yt i D = = T i. p p = K+c old n and assue The duration is thus a weighted average of the coupon dates of the bond where the weight for a certain coupon date is the present value of the coupon payent divided by the value of the bond which is the present value of all the payents). Duration is a easure of how long on average the holder of the bond has to wait before receiving cash payents or ean tie to coupon payent. Note that for a zero coupon bond with aturity T the duration is T. Duration also acts as a easure of sensitivity of the bond price to changes in the yield. Proposition 1 With notation as above we have dp dy = d { n } c i e yt i t) dy = Dp So duration is essentially for bonds with respect to yield) what delta is for derivatives with respect to the underlying price). Approxiately it holds that p Dp y or p p D y For concrete coputations, see Exaple 6.5 in Hull. 5

6 Reark 3 Note that the change in yield is supposed to be the sae for all aturities. This eans that we are looking at what happens when there is a parallel shift in the yield curve zero curve). Corresponding to gaa for derivatives there is convexity for bonds. It is defined as 1.5 Forward rates C = d2 p dy 2 We have seen that the continuously copounded spot rate rt,t) solves e rt,t)t t) = 1 pt,t). 2) This is because an investent of $1 at tie t can create a risk less payoff of $1/pt,T) at tie T. Now fix t < S < T ans suppose t is today and we want to offer an interest rate, deterined today, over the future interval [S,T]. We will extend the arguent that led to 2). At tie t sell an S-bond. This will earn you pt,s). Invest the oney in T-bonds. This will give you pt,s)/pt,t) T-bonds. The net investent at tie t is zero. At tie S you will have to pay $1 to the owner of the S-bond. At tie T you will receive $pt,s)/pt,t) 1. Thus an investent of $1 at tie S results in a payoff of pt,s)/pt,t) at tie T. We would therefore offer the rate e ft;s,t)t S) = pt,s) pt,t). 3) Definition 5 The continuously copounded forward rate for [S, T] contracted at t is defined as ft;s,t) = lnpt,t) lnpt,s) 4) T S Reark 4 Note that as S t the forward rate tends to the spot rate lift;s,t) = rt,t). S t There exists a ter structure of forward rates as well; it is in two diensions since for each future tie S there exists one forward rate for each T S. Forally we have that the ter structure of forward rates is given by {ft;s,t) : S [t,t], T [S, )}. If we use that pt,t) = e rt,t)t t) in 4) we get the following relationship between forward rates and spot rates rt,t)t t) = rt,s)s t)+ft;s,t)t S). The ter structure of forward rates is therefore deterined by the ter structure of spot rates. For a concrete exaple of coputations using the relation, see Table 4.5 in Hull. 6

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