Math 177 UCLA Winter 2023 Introduction 1 Lecture notes Bond valuation De nition 1 A bond once purchased entitles the investor to regular coupon payments and one larger nal payment The nal payment is referred to as the redemption amount Some bonds have no coupons only the nal payment These types of bonds are called zero coupon 2 Example 2 Consider a 5 year bond that pays yearly coupons starting one year from now with a redemption value 100 at the end of the 5th The coupon rate is 10 per year and the yield rate on the bond is 12 year per year What is the present value of the bond 3 r m e ective coupon rater per coupon period The face value and e ective coupon rate determine the coupon amount Notation for bonds m number of coupons per year F face value nominal yearly coupon rate n number of coupons j e ective yield rate per coupon period P price C redemption amount When F C we say that the bond is redeemable at par or par valued Unless stated otherwise we will always assume our bond are par valued The basic price formula 4 Example 3 A twelve year 2000 8 par valued bond with quarterly coupons is purchased for 2200 Find the nominal yield rate convertible quarterly 5 6 Modifying the basic price formula Throughout we assume par valued bonds i e F C In this case we the basic price formula is P F vn j F ran j Using the easily veri ed identity vn j 1 jan j we have Alternatively using the identity an j 1 vn the redemption amount be denoted by j j and letting the present value of K F vn j we get the following price equation This is known as Makeham s formula Bonds bought or redeemed at a premium or discount 7 Assuming again that F C par valued examine again the price formula P F F r j an j We can use this equation to characterize when we have P F P F and P F 8 De nition 4 Recall that P denotes the price of a bond and C its redemption amount If P C the bond is said to be bought at a premium If P C the bond is said to be bought at par If P C the bond is said to be bought at a discount Assume again F C when P C so the bound is bought at a premium the amount of premium in the purchase price can be regarded as a loan from the buyer to the seller The case where F cid 54 C The modi ed coupon rate is g F r C 9 Using the modi ed coupon rate we can express our price equations using C instead of F which is useful in the case that F is not necessarily equal to C becomes For example the basic price equation P Cvn j F ran j Similarly we have the equations P C C g j an j P K C K g j De nition 5 When C F the bond is said to be redeemed at par 10 When C F the bond is said to be redeemed at a premium When C F the bond is said to be redeemed at a discount Callable bonds 11 The issuer of a bond sometimes designs the bond contract so that they have more options regarding the maturity date This added uncertainty is a disadvantage for the investor who therefore is usually compensated with a higher yield rate From the investor s point of view P C means coupons overpay P C means coupons underpay After the callable bond is purchased the investor has not say in the call date it is completely the issuer s decision However the investor s decision to buy the bond and the price they are willing to pay are likely to be a ected by the call options in the bond contract 12 Example 6 A 10 bond with semi annual coupons and with face and redemption value 1 000 000 is issued with the condition that the re demption can take place on any coupon date between 12 and 15 years from the issues date of i 2 12 At the time the bond is purchased the investor does not know the actual redemption Find the price paid by an investor wishing a minimum yield 13 Transferring bonds Consider an investor who buys a bond from an issuer 14 Suppose the investor holds the bond for several coupon periods but then wishes to sell the bond to a third party before the maturity date How do we determine how much the third party should pay for the bond The exact timing of the sale matters a lot 1 Purchase right after a coupon payment OR 2 Purchase in between coupons We examine both scenarios 1 Purchase right after a coupon payment 15 Suppose the new investor purchases the bond right after the kth coupon pay ment redemption amount The price should be the present value of the remaining coupons plus the 2 Purchase in between coupons Extending our notation above we want to know how to de ne Pk s where k is an integer and 0 s 1 16 As we saw above the dirty price function f t Pt has some unappealing properties 17 Can we x this so that we have clean price 18 Example 7 A 10 bond with semiannual coupons and face amount of 100 000 000 is issued on June 18 2010 1st coupon Dec 18 2010 Maturity June 18 2030 Assume i 2 05 What s the original price What are the clear an dirty prices on August 1 2020 19 20 Amortization of a bond 21 For taxation and accounting purposes it is sometimes necessary to determine the amount of interest received and principal returned in a bound coupon or redemption payment We can simply think of a bond as an amortized loan At the time of each coupon payment the outstanding balance is just the price of the bond at that time This is also called the bond s book value at that time 22 Example 8 A 10 bond with face amount 10 000 matures 4 years after issue Coupons are paid semiannually Construct the amortization schedule for the bond over its term for a nominal annual yield rate of 10 23