Corporate Finance Bonds Valuation Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES

A.Farber Vietnam 2004 |2|2 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

A.Farber Vietnam 2004 |3|3 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

A.Farber Vietnam 2004 |4|4 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

A.Farber Vietnam 2004 |5|5 Interest rates Source: FT.com

A.Farber Vietnam 2004 |6|6 Government bonds Source: FT.com

A.Farber Vietnam 2004 |7|7

|8|8 Government Bonds Spreads

A.Farber Vietnam 2004 |9|9 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

A.Farber Vietnam 2004 | 10 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

A.Farber Vietnam 2004 | 11 Valuing a level coupon bond Example: If r = 5% Note: If P 0 >: 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 = Expected return: [ ( – )]/ = 5%

A.Farber Vietnam 2004 | 12 A level coupon bond as a portfolio of zero- coupons « Cut » level coupon bond into 5 zero-coupon Face value Maturity Value Zero Zero Zero Zero Zero Total

A.Farber Vietnam 2004 | 13 Bond prices and interest rates Bond prices fall with a rise in interest rates and rise with a fall in interest rates

A.Farber Vietnam 2004 | 14 Sensitivity of zero-coupons to interest rate

A.Farber Vietnam 2004 | 15 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

A.Farber Vietnam 2004 | 16 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

A.Farber Vietnam 2004 | 17 Duration - example Back to our 5-year 6.50% coupon bond. Face value Value w t Zero % Zero % Zero % Zero % Zero % Total Duration =.0581× × × × ×5 = 4.44 For coupon bonds, duration < maturity

A.Farber Vietnam 2004 | 18 Price change calculation based on duration General formula: In example: Duration = 4.44 (when r=5%) If Δr =+1% : Δ ×4.44 × 1% = % Check: If r = 6%, P = ΔP/P = ( – )/ = % Difference due to convexity

A.Farber Vietnam 2004 | 19 Duration -mathematics If the interest rate changes: Divide both terms by P to calculate a percentage change: As: we get:

A.Farber Vietnam 2004 | 20 Yield to maturity Suppose that the bond price is known. Yield to maturity = implicit discount rate Solution of following equation:

A.Farber Vietnam 2004 | 21 Yield to maturity vs IRR The yield to maturity is the internal rate of return (IRR) for an investment in a bond.

A.Farber Vietnam 2004 | 22 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

A.Farber Vietnam 2004 | 23 Examples SAVING BANK LIFE INSURANCE COMPANY

A.Farber Vietnam 2004 | 24 Immunization: D Equity = 0 As: D Asset * A = D Equity * E + D Liabilities * L D Equity = 0 → D Asset * A = D Liabilities * L

A.Farber Vietnam 2004 | 25 Spot rates Spot rate = yield to maturity of zero coupon Consider the following prices for zero-coupons (Face value = 100): Maturity Price 1-year year 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%

A.Farber Vietnam 2004 | 26 Measuring spot rate Data: = 106 * d = 9 * d * d = 6.5 * d * d * d = 8 * d * d * d * d 4 To recover spot prices, solve: Solution:

A.Farber Vietnam 2004 | 27 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: × (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:

A.Farber Vietnam 2004 | 28 Forward rates :example Maturity Discount factor Spot rates Forward rates Details of calculation: 3-year spot rate : 1-year forward rate from 3 to 4

A.Farber Vietnam 2004 | 29 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

A.Farber Vietnam 2004 | 30 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 : = 100 × 1.05 × 1.06 = 100 × (1+r 1 ) × (1+E(r 1 )) STRATEGY 2 : Buy year zero coupon, face value = 100

A.Farber Vietnam 2004 | 31 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