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Chapter 2 Pricing of Bonds Time Value of Money (TVM) The price of any security equals the PV of the security’s expected cash flows. So, to price a bond we need to know: • The size and timing of the bond’s expected cash flows. • The required return (commensurate with the riskiness of the cash flows). You must be comfortable with TVM: • PV and FV of lump sums and annuities. Two Important PV Formulas PV of a lump sum: PV C (1 y ) n PV of an ordinary annuity: 1 1 (1 y ) n PV C y Time Value Future Value P n P o (1 r ) n where: n = number of periods Pn = future value n periods from now (in dollars) Po = original principal (in dollars) r = interest rate per period (in decimal form) Future Value of on Ordinary Annuity (1 r ) 1 Pn A r n Bond Pricing price = PV of all future cash flows to find price, you need expected CFs • coupon payments • par value yield P n t 1 C (1 r ) t M (1 r ) n Pricing A Bond We begin with a simple bullet bond: • • • • Noncallable (maturity is known with certainty) Coupons are paid every six months. The next coupon is received exactly six months from now. The interest rate at which the coupons can be invested is fixed for the life of the bond. • Principal is paid at maturity (no amortizing). • Coupon fixed for the life of the bond. Bond Pricing Formula Notation: • • • • P = price of the bond (in $) n = number of periods (maturity in years 2) C = semiannual coupon (in $) M = maturity value The bond price is: P C (1 y ) n P t 1 C (1 y ) 2 C (1 y ) 3 C (1 y ) 1 1 n (1 y ) C M C n n y (1 y ) (1 y ) n M (1 y ) M n (1 y ) n Note: All inputs to the bond pricing formula are fixed except for y. As y changes so does P. Example Price a 20-year 10% coupon bond with a face value of $1,000 if the required yield on the bond is 11%. Formula inputs: • • • • The coupon is: 0.10 1,000 = $100. The semiannual coupon, C, is: $50. n = 40 y = 0.055 1 1 n (1 y ) P C y 1 1 40 M (1.055) 50 n (1 y ) 0.055 1, 000 8 0 2 .3 1 1 1 7 .4 6 9 1 9 .7 7 40 (1.055) A few good points… Projecting cash flows for fixed income securities is relatively straightforward – but sometimes it may be harder, for example: if the issuer or the investor has the option to change the contractual due date for the payment of the principal (callable bonds, putable bonds) if the coupon payment is reset periodically by a formula based on some value or values of reference rates (floating rate securities) if the investor has the choice to convert or exchange the security into common stock (convertible bond) Pricing Zero-Coupon Bonds Zero-coupon bonds (zeros) are so called because they pay no coupons (i.e., C = 0): They have only maturity value: P P 0 (1 y ) M (1 y ) n 0 (1 y ) 2 0 (1 y ) 3 0 (1 y ) n M (1 y ) n Example Price a zero that expires 15 years from today if it’s maturity value is $1,000 and the required yield is 9.4% An investor would pay Formula inputs: $252.12 today and receive $1,000 in 15 years. • M = 1,000 • n = 30 • y = 0.047 P M (1 y ) n 1, 000 (1.047 ) 30 2 5 2 .1 2 Price-Yield Relationship A fundamental property of bond pricing is the inverse Price relationship between bond yield and bond price. Yield Price-Yield Relationship For a plain vanilla bond all bond pricing inputs are fixed except yield. Therefore, when yields change the bond price must change for the bond to reflect the new required yields. Example: Examine the price-yield relationship on a 7% coupon bond. • For y < 7%, the bond sells at a premium • For y > 7% the bond sells at a discount • For y = 7%, the bond sells at par value Yield Price 5.0 1,307.45 5.5 1,218.01 6.0 1,137.65 6.5 1,065.29 7.0 1,000.00 7.5 940.95 8.0 887.42 8.5 838.80 9.0 794.53 Price-Yield Relationship The price-yield relationship can be summarized: • yield < coupon rate ↔ bond price > par (premium bond) • yield > coupon rate ↔ bond price < par (discount bond) • yield = coupon rate ↔ bond price = par (par bond) Bond prices change for the following reasons: • Discount or premium bond prices move toward par value as the bond approaches maturity. • Market factors – change in yields required by the market. • Issue specific factors – a change in yield due to changes in the credit quality of the issuer. Example Suppose that you are reviewing a price sheet for bonds and see the following prices (per $100 par value) reported. You observe what seem to be several errors. Without calculating the price of each bond, indicate which bonds seem to be reported incorrectly and explain why. Bond Price Coupon % Yield % U V W X Y Z 90 96 110 105 107 100 6 9 8 0 7 6 9 8 6 5 9 6 Complications to Bond Pricing We have assumed the following so far: 1. 2. 3. 4. Next coupon is due in six months. Cash flows are known with certainty We can determine the appropriate required yield. One discount rate applies to all cash flows. These assumptions may not be true and therefore complicate bond pricing. Complications to Bond Pricing: Next Coupon Due < 6 Months What if the next coupon payment is less than six months away? Then the accepted method for pricing bonds is: n P (1 y ) t 1 v C v (1 y ) t 1 M (1 y ) (1 y ) v n 1 # days betw een settlem ent and next coupo n # days in a six-m onth period Complications to Bond Pricing: CFs May Not Be Known For a noncallable bond cash flows are known with certainty (assuming issuer does not default) However, lots of bonds are callable. Interest rates then determine the cash flow: • If interest rates drop low enough below the coupon rate, the issuer will call the bond. Also, CFs on floaters and inverse floaters change over time and are not known (more on this later). Complications to Bond Pricing: Determining Required Yield The required yield for a bond is: R = rf + RP • rf is obtained from an appropriate maturity Treasury security. • RP should be obtained from RPs of bonds of similar risk. • This process requires some judgement. Complications to Bond Pricing: Cash Flow Discount Rates We have assumed that all bond cash flows should be discounted using one discount rate. However, usually we are facing an upward sloping yield curve: • So each cash flow should be discounted at a rate consistent with the timing of its occurrence. In other words, we can view a bond as a package of zero-coupon bonds: • Each cash coupon (and principal payment) is a separate zero-coupon bond and should be discounted at a rate appropriate for the “maturity” of that cash flow. Pricing Floaters Coupons for floaters depend on a floating reference interest rate: • coupon rate = floating reference rate + fixed spread (in bps) • Since the reference rate is unpredictable so is the coupon. Example: • Coupon rate = rate on 3-month T-bill + 50bps Reference Rate Spread Floaters can have restrictions on the coupon rate: • Cap: A maximum coupon rate. • Floor: A minimum coupon rate. Pricing Inverse Floaters An inverse floater is a bond whose coupon goes up when interest rates go down and vice versa. Inverse floaters can be created using a fixed-rate security (called the collateral): • From the collateral two bonds are created: (1) a floater, and (2) an inverse floater. These bonds are created so that: • Floater coupon + Inverse floater coupon ≤ Collateral coupon • Floater par value + Inverse floater par value ≤ Collateral par value Equivalently, the bonds are structured so that the cash flows from the collateral bond is sufficient to cover the cash flows for the floater and inverse floater. Inverse Floater Example Consider a 10-yr 15% coupon bond (7.5% every 6 months). Suppose $100 million of bond is used to create two bonds: • $50 million par value floater and $50 million par value inverse floater. Assume a 6-mo coupon reset based on the formula: • Floater coupon rate = reference rate + 1% • Inverse coupon rate = 14% - reference rate Notice: Floater coupon rate + Inverse coupon rate = 15% • Problem: if reference rate > 14%, then inverse floater coupon rate < 0. • Solution: put a floor on the inverse floater coupon of 0%. • However, this means we must put a cap in the floater coupon of 15%. The price of floaters and inverse floaters: • Collateral price = Floater price + Inverse floater price Price Quotes on Bonds We have assumed that the face value of a bond is $1,000 and that is often true, but not always: • So, when quoting bond prices, traders quote the price as a percentage of par value. • Example: A quote of 100 means 100% of par value. Prices Converted into a Dollar Price (1) Price Quote (2) Converted to a Decimal [ = (1)/100] 97 0.9700000 (3) Par Value $ 10,000 (4) Dollar Price [ = (2) x (3)] $ 9,700.00 85 1/2 0.8550000 100,000 85,500.00 90 1/4 0.9025000 5,000 4,512.50 80 1/8 0.8012500 10,000 8,012.50 76 5/32 0.7615625 1,000,000 761,562.50 86 11/64 0.8657588 100,000 86,172.88 100 1.0000000 50,000 50,000.00 109 1.0900000 1,000 1,090.00 103 3/4 1.0375000 100,000 103,750.00 105 3/8 1.0537500 25,000 26,343.75 103 19/32 1.0358375 1,000,000 1,035,937.50 Clean vs. Dirty Price Most bond trades occur between coupon payment dates. • Thus at settlement, the buyer must compensate the seller for coupon interest earned since the last coupon payment. • This amount is called accrued interest. • The buyer pays the seller: Bond price + Accrued Interest (often called the dirty price). • The bond price without accrued interest is often called the “clean price.” Clean vs. Dirty Price Suppose a bond just sold for 84.34 (based on par value of $100) and pays a coupon of $4 every six months. The bond paid the last coupon 120 days ago. What is the clean price? What is the dirty price? Clean price: • $84.34 Dirty price: • $84.34 + 120/180($4) = $87.01