2. Principles of Valuation First Value of financial securities = PV of expected future cash flows To Principles: value financial securities we need to: Estimate future cash flows: Size (how much) and Timing (when) Discount future cash flows at an appropriate rate: The rate should be appropriate to the risk presented by the financial security.
3. Definition of a Bond A bond is a legally binding agreement between a borrower and a lender: Specifies the principal amount of the loan (face value or par value) Specifies the size and timing of the cash flows (coupon payments): In dollar terms (fixed-rate borrowing) As a formula (adjustable-rate or floating-rate borrowing)
4. Example of a Bond Consider a U.S. government bond listed as 6 3/8 of December 2011. The Par Value of the bond is $1,000. Coupon payments are made semi-annually (June 30 and December 31 for this particular bond). Since the coupon rate is 6 3/8 the payment is $31.875. Principal repayment is on December 31, 2011. On January 1, 2006 the size and timing of future cash flows are: $31.875 $31.875 $31.875 $1,031.875 30 / 6 / 11 31 / 12 / 11 1 / 1 / 06 30 / 6 / 06 31 / 12 / 06
5. How to Value Bonds Identify the size and timing of cash flows. Discount at the correct discount rate. If you know the price of a bond and the size and timing of cash flows, then you can calculate the discount rate This discount rate that discounts future cash flows to the current price is the Yield to Maturity or YTM of the bond
6. Pure Discount Bonds Also called zero coupon bonds Information needed for valuing pure discount bonds: Time to maturity (T) = Maturity date - todays date Face value (F) Discount rate (r) $0 $0 $0 $F T 1 T 0 1 2 Present value of a pure discount bond at time 0: F PV = T (1 + r )
7. Pure Discount Bonds: Example Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%. $0 $0 $0 $1,000 0$ 0,1$ 0 0 1 30 229 0 1 2 29 F $1,000 PV = = = $174.11 T 30 (1 + r ) (1.06) 30
8. Level-Coupon Bonds Information needed to value level-coupon bonds: Coupon payment dates and time to maturity (T) Coupon payment (C) per period and Face value (F) Discount rate (r) $C $C $C $C + $ F 0 1 2 T 1 T Value of a Level-coupon bond = PV of coupon payment annuity + PV of face value C 1 F PV = 1 + T r (1 + r ) (1 + r )T
9. Level-Coupon Bonds: Example Find the present value (as of January 1, 2006), of a 6-3/8 coupon T-bond with semi-annual payments, and a maturity date of December 2011 if the YTM is 5-percent. On January 1, 2006 the size and timing of cash flows are: $31.875 $31.875 $31.875 $1,031.875 30 / 6 / 11 31 / 12 / 11 1 / 1 / 06 30 / 6 / 06 31 / 12 / 06 $1,000 $31.875 1 PV = 1 (1.025)12 + (1.025)12 = $1,070.52 .05 2
10. Duration Duration of a Zero Coupon Bond 0 Maturity date Duration
11. Duration (contd.) Duration of a Coupon Paying Bond 0 Duration Maturity date
12. Duration (contd.) Measure of bond price sensitivity to interest rate risk A bond with longer duration has higher relative price change than one with shorter duration when interest rate (YTM) changes
13. Bond Concepts 1. Bond prices and market interest rates move in opposite directions. 2. When coupon rate = YTM, price = par value. When coupon rate > YTM, price > par value (premium bond) When coupon rate < YTM, price < par value (discount bond) 3. A bond with longer maturity has higher relative price change than one with shorter maturity when interest rate (YTM) changes, all other features being identical. 4. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes, all other features being identical.
14. YTM and Bond Value When YTM < coupon, the bond trades at a premium. Bond Value $1400 1300 1200 When YTM = coupon, the bond trades at par. 1100 1000 800 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 6 3/8 0.08 0.09 0.1 YTM When YTM > coupon, the bond trades at a discount.
15. Bond Value Maturity and Bond Price Volatility Consider two otherwise identical bonds. The long-maturity bond will have much more volatility with respect to changes in the discount rate Par Short Maturity Bond C YTM Long Maturity Bond
16. Bond Value Coupon Rate and Bond Price Volatility Consider two otherwise identical bonds. The low-coupon bond will have much more volatility with respect to changes in the discount rate (YTM) High Coupon Bond YTM Low Coupon Bond