FI 4000 Bond Valuation Formulas Application of formula 2 5 page 18 in F to price of a bond As we have used them in class and as they are used on page 20 in F let P price of the bond r per period required yield on the bond M maturity value or par or principal value of the bond n number of coupon payment periods to maturity of the bond C per period coupon payment The promised cash flows in a typical bond have two components One is the annuity of coupon interest of C per period while the other is the maturity value of the bond M The definition of the rate r is that it is the rate that when you use it to present value the bond s promised cash flow the answer you get is the price of the bond P so P C PVAF r n M PVF r n The first term on the right is the present value of the promised interest annuity for n periods It is written as the coupon payment C times the present value annuity factor for n periods at rate r Part of what we are going to do here is find a formula for PVAF r n The quantity M is also called the par or principal value of the bond because if you present value the promised cash flows of the bond at the coupon rate on the bond the answer you get is M This is easy to do for a bond with n 1 by substituting C cM i e you should convince your self that M C PVAF c n M 1 c n 1 where PVAF c n denotes the present value of an annuity of 1 for n periods at the rate c By 2 6 and 2 7 on page 20 of F we have that 1 1 1 r n P C r M n 1 r Substituting C cM we have that 1 1 1 r n P cM r M n 1 r 1 n 1 1 r M c r 1 1 r n c 1 1 M 1 n n r 1 r 1 r c M 1 q q r M s where q 1 1 r n 2 c and s 1 q q r This expression 2 or the price of the bond is very convenient It allows us to verify the price par relationship expressed on page 24 of F rigorously since P M is just the number s The number s is just a convex combination of the ratio of the coupon rate to the require yield c and the number 1 for we can write s as r c c s 1 q q 1 q 1 q 1 r r When c r then c c 1 and any convex combination of and 1 must lie between the r r two i e we have c s 1 In this case P M s 1 Thus as we know intuitively r when the coupon rate exceeds the required yield the bond sells at a premium to par Similarly when c r then c c 1 and any convex combination of and 1 must lie r r between the two i e we have that c s 1 In this case P M s 1 Thus as we r know intuitively when the coupon rate is less than the required yield the bond sells at a 1 For r 0 we have that 0 q 1 discount to par Finally when c r then c 1 and s 1 1 q 1 q 1 In this r case as we know very well from simple bond arithmetic P M The expression 2 s also convenient from a computational point of view If you forget how to key into your calculator the details for bond calculations it is easy to compute q and s Since M is typically 1000 or 10000 it is easy to compute Ms For suppose we want the price of a bond such as the one in the example following 2 7 on page 20 in F There n 40 r 055 and c 05 Thus q 1 1 r n 1 1 055 40 117463 and c 05 s 1 q q 1 117463 117463 r 055 9090909 882537 117463 91977 Thus P Ms is 919 77 The price par relationship is intuitive and 2 just provides rigorous justification for it We can however use 2 to determine something that is not at all obvious at first namely how a bond s price changes with changes in its term to maturity What do you think Does the price of the bond go up when its maturity goes up Does the price go down with an increase in the bonds term to maturity Well the answer is it depends on the coupon rate required rate relationship From the above discussion it is clear that the price does not change at all with an increase in term to maturity if the coupon rate and the required yield are the same The price of the bond is its par value regardless of its maturity Some intuition for what we will discover here is given in the answer to Problem 4 page 331 in BKM Looking at the right hand side of 2 it is clear that the only term that depends on the term to maturity n of the bond is the quantity q 1 1 r n Rearranging 2 to isolate this term we get that c P M 1 q q r c c M q 1 r r c c M M 1 q r r 3 With this decomposition the change in P that results from a change in n must come from the change in the second term on the right in 3 since the first term on the right does not depend on n Since the second term on the right in 3 only depends on n through q the change in this second term is easy to determine If we denote the change in P with a change in n by P then by 3 c P M 1 q r 4 where q denotes the change in the quantity q with a change in n We know that q 0 in the sense that as n increases q 1 1 r n decreases since for r 0 1 1 Since 1 r M 0 M q 0 From this fact and 4 the sign of P depends on the sign of the c term 1 If this term is positive then P is negative while if this term is negative r then P is positive We thus have the following If c r then an increase in the term to maturity of the bond will decrease the price of the bond If c r then an increase in the term to maturity of the bond will increase the price of the bond In this case the extra coupon payment is worth more than the loss in value of …