Useful Concepts for Valuation of (interest rate) Swaps • (Fixed) Coupon bond: A bond that pays a pre-speciﬁed amounts at ﬁxed interval of times, and principal at maturity. • ”Par rate”: value coupon as fraction of principal so that Fixed coupon bond is worth par when issue. • Floating rate bond: Bond that pays ﬂoating rate (variable) at intervals of time, and pays principal at maturity • ”A ﬂoating rate bond is worth par” (after each payment date). • dates 0 = t0, t1, ..., tj , ..., tn • Current date: t0, maturity tn. • Zero coupon rates between ti and tj : ri,j (current zero rates: r0,i) • Value at t = 0 of one dollar to be received at tj : e−r0,j ti • Value of ANY bond with payments Ci at ti at time t0 : i=1 n X e−r0,i ti Ci • Coupon bond with Ci = C for i = 1, 2, ..., n − 1 and C + P at tn : i=1 n X e−r0,i ti C + e−r0,n tn P • Coupon bond issue at par: coupon C must solve: P = i=1 n X e−r0,i ti C + e−r0,n tn P • Example: What is the value of C/P (par rate) if the term structure is ﬂat and ti are equally spaced? • In symbols: r0,i = r for all i, ti+1 − ti = ti − ti−1 = ∆ P = P 1 − e−rtn ³ ³ ´ ´ i=1 n X n X e−r ti C + e−r tn P e−r ti C =⇒ ³ = = = 1 − e−r∆ n i=1 µ ¶X C n 1 − e−rtn µ ¶ ´ C ³ −r∆ −r∆i = −r∆ i e e e P i=1 P i=0 µ ¶³ ´³ ´ C C e−r∆ 1 − e−r∆ n =⇒ = er∆ P P ´ = i=1 ´ n−1 ³ X n X −r ti C e P - Par rate = constant interest rate per period In general: a ﬁxed coupon bond is issue at par if : P = or par rate n X e−r0,i ti C + e−r0,tn P ³ ´ i=1 C = Pn −r0,i ti P i=1 e 1 − e−r0,n tn Now we consider a Floating Rate Bond. • Floating rate bond: Bond that pays ﬂoating rate (variable) at intervals of time, and pays principal at maturity • We use an arbitrage argument to show that ”A ﬂoating rate bond is worth par” (after each payment date). • Cash ﬂow of ﬂoating bond: t0 −P P ... ... P h h t1 t2 ... i h i r0,1 (t1−t0) − 1 P er1,2 (t2−t1) − 1 ... e ti ... tn i h i ri−1,i (ti−ti−1) − 1 ... P ern−1,n (tn−tn−11) − 1 + P e • Floating rate bond is worth P at t0, t1, ..., tn−1. • Suppose is worth more at t0. Short the bond, and invest in the short rate P at t0 . At t1 receive P er0,1(t1−t0) at t1. At t1 pay the ﬂoating rate from the short position and invest P. Keep doing this at every date, including maturity. • Using the previous concepts (ﬁxed coupon bond, ﬂoating bond) we can value a swap. • Recall that a swap receiving variable rate and paying ﬁxed rate has the same cash ﬂows a long position in a ﬂoating bond and a short position in a coupon bond. • We just showed that a ﬂoating bond is worth par at inception. • The ﬁxed rate of the swap (the ”swap rate”) is determined so that the swap is worth zero at inception • Then the swap rate is the par rate. ³ ´ • Determining the swap rate, i.e. the value of C at which a ﬁxed for P ﬂoating swap is issued. • Recall that ﬁxed coupon bond issue at par: P = or i=1 n X e−r0,i ti C + e−r0,tn P ³ C = Pn P e−r0,i ti i=1 1 − e−r0,n tn ´ • Floating (variable) rate bond issue at par: P. • Term structure of swap rate (paying every ti − ti−1 6 months) up to tn years, see WSJ, Mkt Data center, key interest rate, swap rates. Swap cash ﬂow (pay ﬁxed, receives ﬂoating): t0 : P −P =0 P ti : i ri−1,i (ti−ti−1) − 1 − C P e P h h t1 : i r0,1 (t1−t0) − 1 − C e P ... ... ... h tn : i rn−1,n (tn−tn−1) − 1 − C + P − P ... P e P Clearly it can also be valued as portfolio of FRA.