1. FINANCE 4. Bond Valuation Professeur André Farber Solvay Business School Université Libre de Bruxelles Fall 2007

2. MBA 2007 Bonds |2|2 Review: present value calculations Cash flows: C 1, C 2, C 3, …,C t, … C T Discount factors: DF 1, DF 2, …,DF t, …, DF T Present value:PV = C 1 × DF 1 + C 2 × DF 2 + … + C T × DF T If r 1 = r 2 =...=r

3. MBA 2007 Bonds |3|3 Review: Shortcut formulas Constant perpetuity: C t = C for all t Growing perpetuity: C t = C t-1 (1+g) r>g t = 1 to ∞ Constant annuity: C t =C t=1 to T Growing annuity: C t = C t-1 (1+g) t = 1 to T

4. MBA 2007 Bonds |4|4 Bond Valuation Objectives for this session : –1.Introduce the main categories of bonds –2.Understand bond valuation –3.Analyse the link between interest rates and bond prices –4.Introduce the term structure of interest rates –5.Examine why interest rates might vary according to maturity

5. MBA 2007 Bonds |5|5 Zero-coupon bond Pure discount bond - Bullet bond The bondholder has a right to receive: one future payment (the face value) F at a future date (the maturity) T Example : a 10-year zero-coupon bond with face value $1,000 Value of a zero-coupon bond: Example : If the 1-year interest rate is 5% and is assumed to remain constant the zero of the previous example would sell for

6. MBA 2007 Bonds |6|6 Level-coupon bond Periodic interest payments (coupons) Europe : most often once a year US : every 6 months Coupon usually expressed as % of principal At maturity, repayment of principal Example : Government bond issued on March 31,2000 Coupon 6.50% Face value 100 Final maturity 2005 2000 2001 2002 2003 2004 2005 6.50 6.50 6.50 6.50 106.50

7. MBA 2007 Bonds |7|7 Valuing a level coupon bond Example: If r = 5% Note: If P 0 > F: the bond is sold at a premium If P 0 <F: the bond is sold at a discount Expected price one year later P 1 = 105.32 Expected return: [6.50 + (105.32 – 106.49)]/106.49 = 5%

8. MBA 2007 Bonds |8|8 When does a bond sell at a premium? Notations: C = coupon, F = face value, P = price Suppose C / F > r 1-year to maturity: 2-years to maturity: As: P 1 > F with

9. MBA 2007 Bonds |9|9 A level coupon bond as a portfolio of zero- coupons « Cut » level coupon bond into 5 zero-coupon Face value Maturity Value Zero 1 6.50 1 6.19 Zero 2 6.50 2 5.89 Zero 3 6.50 3 5.61 Zero 4 6.50 4 5.35 Zero 5 106.50 5 83.44 Total 106.49

10. MBA 2007 Bonds | 10 Bond prices and interest rates Bond prices fall with a rise in interest rates and rise with a fall in interest rates

11. MBA 2007 Bonds | 11 Sensitivity of zero-coupons to interest rate

12. MBA 2007 Bonds | 12 Duration for Zero-coupons Consider a zero-coupon with t years to maturity: What happens if r changes? For given P, the change is proportional to the maturity. As a first approximation (for small change of r): Duration = Maturity

13. MBA 2007 Bonds | 13 Duration for coupon bonds Consider now a bond with cash flows: C 1,...,C T View as a portfolio of T zero-coupons. The value of the bond is: P = PV(C 1 ) + PV(C 2 ) +...+ PV(C T ) Fraction invested in zero-coupon t: w t = PV(C t ) / P Duration : weighted average maturity of zero-coupons D= w 1 × 1 + w 2 × 2 + w 3 × 3+…+w t × t +…+ w T ×T

14. MBA 2007 Bonds | 14 Duration - example Back to our 5-year 6.50% coupon bond. Face value Value w t Zero 1 6.50 6.19 5.81% Zero 2 6.50 5.89 5.53% Zero 3 6.50 5.61 5.27% Zero 4 6.50 5.35 5.02% Zero 5 106.50 83.44 78.35% Total 106.49 Duration D =.0581×1 + 0.0553×2 +.0527 ×3 +.0502 ×4 +.7835 ×5 = 4.44 For coupon bonds, duration < maturity

15. MBA 2007 Bonds | 15 Price change calculation based on duration General formula: In example: Duration = 4.44 (when r=5%) If Δr =+1% : Δ ×4.44 × 1% = - 4.23% Check: If r = 6%, P = 102.11 ΔP/P = (102.11 – 106.49)/106.49 = - 4.11% Difference due to convexity

16. MBA 2007 Bonds | 16 Duration -mathematics If the interest rate changes: Divide both terms by P to calculate a percentage change: As: we get:

17. MBA 2007 Bonds | 17 Yield to maturity Suppose that the bond price is known. Yield to maturity = implicit discount rate Solution of following equation:

18. MBA 2007 Bonds | 18 Yield to maturity vs IRR The yield to maturity is the internal rate of return (IRR) for an investment in a bond.

19. MBA 2007 Bonds | 19 Asset Liability Management Balance sheet of financial institution (mkt values): Assets = Equity + Liabilities → ∆A = ∆E + ∆L As: ∆P = -D * P * ∆r (D = modified duration) -D Asset * A * ∆r = -D Equity * E * ∆r - D Liabilities * L * ∆r D Asset * A = D Equity * E + D Liabilities * L

20. MBA 2007 Bonds | 20 Examples SAVING BANK LIFE INSURANCE COMPANY

21. MBA 2007 Bonds | 21 Immunization: D Equity = 0 As: D Asset * A = D Equity * E + D Liabilities * L D Equity = 0 →D Asset * A = D Liabilities * L

22. MBA 2007 Bonds | 22 Spot rates Spot rate = yield to maturity of zero coupon Consider the following prices for zero-coupons (Face value = 100): Maturity Price 1-year 95.24 2-year 89.85 The one-year spot rate is obtained by solving: The two-year spot rate is calculated as follow: Buying a 2-year zero coupon means that you invest for two years at an average rate of 5.5%

23. MBA 2007 Bonds | 23 Measuring spot rate Data: 99.06 = 105 * d 1 103.70 = 9 * d 1 + 109 * d 2 97.54 = 6.5 * d 1 + 6.5 * d 2 + 106.5 * d 3 100.36 = 8 * d 1 + 8 * d 2 + 8 * d 3 + 108 * d 4 To recover spot prices, solve: Solution:

24. MBA 2007 Bonds | 24 Forward rates You know that the 1-year rate is 5%. What rate do you lock in for the second year ? This rate is called the forward rate It is calculated as follow: 89.85 × (1.05) × (1+f 2 ) = 100 → f 2 = 6% In general: (1+r 1 )(1+f 2 ) = (1+r 2 )² Solving for f 2 : The general formula is:

25. MBA 2007 Bonds | 25 Forward rates :example Maturity Discount factor Spot rates Forward rates 1 0.9500 5.26 20.8968 5.60 5.93 30.8444 5.80 6.21 40.7951 5.90 6.20 50.7473 6.00 6.40 Details of calculation: 3-year spot rate : 1-year forward rate from 3 to 4

26. MBA 2007 Bonds | 26 Term structure of interest rates Why do spot rates for different maturities differ ? As r 1 r 1 = r 2 if f 2 = r 1 r 1 > r 2 if f 2 < r 1 The relationship of spot rates with different maturities is known as the term structure of interest rates Time to maturity Spot rate Upward sloping Flat Downward sloping

27. MBA 2007 Bonds | 27 Forward rates and expected future spot rates Assume risk neutrality 1-year spot rate r 1 = 5%, 2-year spot rate r 2 = 5.5% Suppose that the expected 1-year spot rate in 1 year E(r 1 ) = 6% STRATEGY 1 : ROLLOVER Expected future value of rollover strategy: ($100) invested for 2 years : 111.3 = 100 × 1.05 × 1.06 = 100 × (1+r 1 ) × (1+E(r 1 )) STRATEGY 2 : Buy 1.113 2-year zero coupon, face value = 100

28. MBA 2007 Bonds | 28 Equilibrium forward rate Both strategies lead to the same future expected cash flow → their costs should be identical In this simple setting, the foward rate is equal to the expected future spot rate f 2 =E(r 1 ) Forward rates contain information about the evolution of future spot rates