1. Chapter 2 Pricing of Bonds

2. 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). MARKET VALUE You must be comfortable with TVM: PV and FV of lump sums and annuities. Your text has a good review of the TVM concepts needed for this course.

3. Two Important PV Formulas PV of a lump sum: PV of an annuity (Formula 2.5, where CF = A):

4. Pricing A Bond We begin with a simple bullet bond: Non-callable (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.

5. 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 (Formulas 2.6, 2.7, 2.8): Note: All inputs to the bond pricing formula are fixed except for r. As r changes so does P.

6. 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 r = 0.055

7. Pricing Zero-Coupon Bonds Zero-coupons bonds (zeros) are so called because they pay no coupons (i.e., C = 0): They have only maturity value:

8. 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% Formula inputs: M = 1,000 n = 30 r = 0.047 An investor would pay $252.12 today and receive $1,000 in 15 years.

9. Price-Yield Relationship A fundamental property of bond pricing is the inverse relationship between bond yield and bond price. Yield Price

10. 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 r < 7%, the bond sells at a premium For r > 7% the bond sells at a discount For r = 7%, the bond sells at par value YieldPrice 5.01,307.45 5.51,218.01 6.01,137.65 6.51,065.29 7.01,000.00 7.5940.95 8.0887.42 8.5838.80 9.0794.53

11. Price-Yield Relationship The price-yield relationship can be summarized: yield 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. (Time Passes) 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. (Credit Spreads)

12. Complications to Bond Pricing We have assumed the following so far: 1.Next coupon is due in six months. 2.Cash flows are known with certainty 3.We can determining the appropriate required yield. 4.One discount rate applies to all cash flows. These assumptions may not be true and therefore complicate bond pricing.

13. 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:

14. Complications to Bond Pricing: CFs May Not Be Known For a non-callable bond cash flows are known with certainty (assuming issuer does not default) However, most 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).

15. Complications to Bond Pricing: Determining Required Yield The required yield for a bond is: R = r f + RP r f is obtained from an appropriate maturity Treasury security. RP (Risk Premium) should be obtained from RPs of bonds of similar risk. This process requires some judgement.

16. 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.

17. 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 RateSpread Floaters can have restrictions on the coupon rate: Cap: A maximum coupon rate. Floor: A minimum coupon rate.

18. 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.

19. Inverse Floater Example (pg. 30 text) 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

20. 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.

21. Price Quotes on Bonds 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.”

22. Clean vs. Dirty Price pg 31 Suppose a bond just sold for 87.01 (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: $87.01 – (120/180)($4) = $84.34 Dirty price: $87.01

23. Clean vs. Dirty Bonds example Bond Pricing Example TermValue Par value1,000.00 Full market value$985.50 Dirty price98.55 Accrued interest$2.00 Flat market value$983.50 Clean price98.35 A US bond has a coupon rate of 7.2% and pays 4 times a year, on the 15th of January, April, July, and October. It uses the 30/360 US day count convention.day count convention A trade for 1,000 par value of the bond settles on January 25th. The prior coupon date was January 15th. The accrued interest reflects ten days' interest, or $2.00 (7.2% of 1,000 * (10 days/360 days)).par valueaccrued interest The full (Dirty) value of these bonds is set by the market at $985.50