Théorie Financière 2. Valeur actuelle Professeur André Farber

Present Value: general formula Cash flows: C1, C2, C3, … ,Ct, … CT Discount factors: v1, v2, … ,vt, … , vT Present value: PV = C1 × v1 + C2 × v2 + … + CT × vT An example: Year 0 1 2 3 Cash flow -100 40 60 30 Discount factor 1.000 0.9803 0.9465 0.9044 Present value -100 39.21 56.79 27.13 NPV = - 100 + 123.13 = 23.13 We now have a general formula to compute present values. The formula is very simple. You simply multiply each cash flow by the corresponding discount factor and you take the sum. Keep in mind that the discount factors are market prices for zero-coupon. An analogy is helpful to understand the logic. Suppose that you are offered the following deal. You pay 100 EUR and you receive 30 USD (US dollar), 40 GBP (British pound) and 60 CHF (Swiss franc). Is this a good deal? Before accepting, you will convert all cash flows into one unit: the euro using current spot exchange rates. If the spot rates are: 1 USD = 1 EUR 1 GBP = 1.50 EUR 1 CHF = 0.70 EUR the net present value is NPV = - 100 + 30 * 1 + 40 * 1.50 + 40 * 0.70 = - 100 + 132 = +32 This is a good deal. Present value calculation is similar. Instead of cash flows being denominated in different currencies, they take place at different points in time. The NPV is obtained by valuing each cash flow at the appropriate “exchange rate”. Tfin 02 Present Value

Using prices of U.S. Treasury STRIPS Separate Trading of Registered Interest and Principal of Securities Prices of zero-coupons Example: Suppose you observe the following prices Maturity Price for $100 face value 1 98.03 2 94.65 3 90.44 4 86.48 5 80.00 The market price of $1 in 5 years is DF5 = 0.80 NPV = - 100 + 150 * 0.80 = - 100 + 120 = +20 A zero-coupon is a pure discount bond. You pay a price today and you receive one single payment (the face value of the zero-coupon) at one future date (the maturity of the zero-coupon). In the early 1980s, several U.S. investment banks started “stripping” U.S. Treasury issues. They would buy a bond (consisting of a number of coupons and the principal) and sell each element separately as a zero-coupon using feline names as CATS, TIGRS, COUGARS and LIONS1. The idea was taken over by the U.S. Treasury in 1985 who started offering its own zero-coupon instruments under the denomination STRIPS (nothing to do with strip tease, at least as far as I know..). Prices of U.S. Treasury strips are regularly published in the WSJ. Note that prices are quoted with colons that represent 32nds. For instance, 93:16 means 93 + 16/32 = 93.50. Strips were later on introduced in various European markets. In the UK, Gilt strips exist (a gilt is a long term British government bond) and quotations can be found in the FT. France introduced stripped OATs (OAT means Obligation Assimilée du Trésor). In Belgium, strips on Belgian government bonds are known as “obligations démambrées”. 1 CATS: Certificates of Accrual on Treasury Securities (Salomon Brothers) TIGRS: Treasury Investment Growth Receipts (Merrill Lynch) COUGARS: Certificates on Government Receipts (A. G. Beckers Paribas) LIONS: Lehman Investment Opportunity Notes (Lehman Brothers) Tfin 02 Present Value

Present value and discounting How much would an investor pay today to receive €Ct in t years given market interest rate rt? We know that 1 €0 => (1+rt)t €t Hence PV  (1+rt)t = Ct => PV = Ct/(1+rt)t = Ct  vt The process of calculating the present value of future cash flows is called discounting. The present value of a future cash flow is obtained by multiplying this cash flow by a discount factor (or present value factor) vt The general formula for the t-year discount factor is: Tfin 02 Present Value

Spot interest rates Back to STRIPS. Suppose that the price of a 5-year zero-coupon with face value equal to 100 is 75. What is the underlying interest rate? The yield-to-maturity on a zero-coupon is the discount rate such that the market value is equal to the present value of future cash flows. We know that 75 = 100 * v5 and v5 = 1/(1+r5)5 The YTM r5 is the solution of: The solution is: This is the 5-year spot interest rate Tfin 02 Present Value

Term structure of interest rate Relationship between spot interest rate and maturity. Example: Maturity Price for €100 face value YTM (Spot rate) 1 98.03 r1 = 2.00% 2 94.65 r2 = 2.79% 3 90.44 r3 = 3.41% 4 86.48 r4 = 3.70% 5 80.00 r5 = 4.56% Term structure is: Upward sloping if rt > rt-1 for all t Flat if rt = rt-1 for all t Downward sloping (or inverted) if rt < rt-1 for all t Starting from the prices of zero-coupons with different maturities, we calculate the yield-to-maturity of each of them to obtain the various spot interest rates. In general, spot rates vary with maturity. The term structure of interest rates is the relationship between spot interest rates (yields of zero-coupons) and maturity. The term structure of interest rates for the European Monetary Union is available of the web site of the European Commission: http://europa.eu.int/comm/eurostat. Click on Euro yield curve. Tfin 02 Present Value

Tfin 02 Present Value

Zero coupon yield curve Euro 5-aug-2005 Source:http://epp. eurostat Tfin 02 Present Value

Using one single discount rate When analyzing risk-free cash flows, it is important to capture the current term structure of interest rates: discount rates should vary with maturity. When dealing with risky cash flows, the term structure is often ignored. Present value are calculated using a single discount rate r, the same for all maturities. Remember: this discount rate represents the expected return. = Risk-free interest rate + Risk premium This simplifying assumption leads to a few useful formulas for: Perpetuities (constant or growing at a constant rate) Annuities (constant or growing at a constant rate) Tfin 02 Present Value

Constant perpetuity Ct =C for t =1, 2, 3, ..... Proof: PV = C d + C d² + C d3 + … PV(1+r) = C + C d + C d² + … PV(1+r)– PV = C PV = C/r Ct =C for t =1, 2, 3, ..... Examples: Preferred stock (Stock paying a fixed dividend) Suppose r =10% Yearly dividend = 50 Market value P0? Note: expected price next year = Expected return = Tfin 02 Present Value

Growing perpetuity Ct =C1 (1+g)t-1 for t=1, 2, 3, ..... r>g Example: Stock valuation based on: Next dividend div1, long term growth of dividend g If r = 10%, div1 = 50, g = 5% Note: expected price next year = Expected return = Tfin 02 Present Value

Constant annuity A level stream of cash flows for a fixed numbers of periods C1 = C2 = … = CT = C Examples: Equal-payment house mortgage Installment credit agreements PV = C * v1 + C * v2 + … + C * vT + = C * [v1 + v2 + … + vT] = C * Annuity Factor Annuity Factor = present value of €1 paid at the end of each T periods. Tfin 02 Present Value

Constant Annuity Ct = C for t = 1, 2, …,T Difference between two annuities: Starting at t = 1 PV=C/r Starting at t = T+1 PV = C/r ×[1/(1+r)T] Example: 20-year mortgage Annual payment = €25,000 Borrowing rate = 10% PV =( 25,000/0.10)[1-1/(1.10)20] = 25,000 * 10 *(1 – 0.1486) = 25,000 * 8.5136 = € 212,839 Tfin 02 Present Value

Growing annuity Ct = C1 (1+g)t-1 for t = 1, 2, …, T r ≠ g This is again the difference between two growing annuities: Starting at t = 1, first cash flow = C1 Starting at t = T+1 with first cash flow = C1 (1+g)T Example: What is the NPV of the following project if r = 10%? Initial investment = 100, C1 = 20, g = 8%, T = 10 NPV= – 100 + [20/(10% - 8%)]*[1 – (1.08/1.10)10] = – 100 + 167.64 = + 67.64 Tfin 02 Present Value

Useful formulas: summary Constant perpetuity: Ct = C for all t Growing perpetuity: Ct = Ct-1(1+g) r>g t = 1 to ∞ Constant annuity: Ct=C t=1 to T Growing annuity: Ct = Ct-1(1+g) t = 1 to T Tfin 02 Present Value

Compounding interval Up to now, interest paid annually If n payments per year, compounded value after 1 year : Example: Monthly payment : r = 12%, n = 12 Compounded value after 1 year : (1 + 0.12/12)12= 1.1268 Effective Annual Interest Rate: 12.68% Continuous compounding: [1+(r/n)]n→er (e= 2.7183) Example : r = 12% e12 = 1.1275 Effective Annual Interest Rate : 12.75% Tfin 02 Present Value

Juggling with compounding intervals The effective annual interest rate is 10% Consider a perpetuity with annual cash flow C = 12 If this cash flow is paid once a year: PV = 12 / 0.10 = 120 Suppose know that the cash flow is paid once a month (the monthly cash flow is 12/12 = 1 each month). What is the present value? Solution 1: Calculate the monthly interest rate (keeping EAR constant) (1+rmonthly)12 = 1.10 → rmonthly = 0.7974% Use perpetuity formula: PV = 1 / 0.007974 = 125.40 Solution 2: Calculate stated annual interest rate = 0.7974% * 12 = 9.568% Use perpetuity formula: PV = 12 / 0.09568 = 125.40 Tfin 02 Present Value

Interest rates and inflation: real interest rate Nominal interest rate = 10% Date 0 Date 1 Individual invests $ 1,000 Individual receives $ 1,100 Hamburger sells for $1 $1.06 Inflation rate = 6% Purchasing power (# hamburgers) H1,000 H1,038 Real interest rate = 3.8% (1+Nominal interest rate) = (1+Real interest rate)×(1+Inflation rate) Approximation: Real interest rate ≈ Nominal interest rate - Inflation rate Tfin 02 Present Value

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 Tfin 02 Present Value

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 Tfin 02 Present Value

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 Tfin 02 Present Value

Valuing a level coupon bond Example: If r = 5% Note: If P0 > F: the bond is sold at a premium If P0 <F: the bond is sold at a discount Expected price one year later P1 = 105.32 Expected return: [6.50 + (105.32 – 106.49)]/106.49 = 5% Tfin 02 Present Value

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: P1 > F with Tfin 02 Present Value

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 Tfin 02 Present Value

Law of one price Suppose that you observe the following data: What are the underlying discount factors? Bootstrap method 100.97 = v1 104 105.72 = v1 7 + v2 107 101.56 = v1 5.5 + v2 5.5 + v3 105.5 Tfin 02 Present Value

Bond prices and interest rates Bond prices fall with a rise in interest rates and rise with a fall in interest rates Tfin 02 Present Value

Sensitivity of zero-coupons to interest rate Tfin 02 Present Value

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 Tfin 02 Present Value

Duration for coupon bonds Consider now a bond with cash flows: C1, ...,CT View as a portfolio of T zero-coupons. The value of the bond is: P = PV(C1) + PV(C2) + ...+ PV(CT) Fraction invested in zero-coupon t: wt = PV(Ct) / P • Duration : weighted average maturity of zero-coupons D= w1 × 1 + w2 × 2 + w3 × 3+…+wt × t +…+ wT ×T Tfin 02 Present Value

Duration - example Back to our 5-year 6.50% coupon bond. Face value Value wt 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 Tfin 02 Present Value

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 Tfin 02 Present Value

Duration -mathematics If the interest rate changes: Divide both terms by P to calculate a percentage change: As: we get: Tfin 02 Present Value

Yield to maturity Suppose that the bond price is known. Yield to maturity = implicit discount rate Solution of following equation: Tfin 02 Present Value

Yield to maturity vs IRR The yield to maturity is the internal rate of return (IRR) for an investment in a bond. Tfin 02 Present Value

Asset Liability Management Balance sheet of financial institution (mkt values): Assets = Equity + Liabilities → ∆A = ∆E + ∆L As: ∆P = -D * P * ∆r (D = modified duration) -DAsset * A * ∆r = -DEquity * E * ∆r - DLiabilities * L * ∆r DAsset * A = DEquity * E + DLiabilities * L Tfin 02 Present Value

Examples SAVING BANK LIFE INSURANCE COMPANY Tfin 02 Present Value

Immunization: DEquity = 0 As: DAsset * A = DEquity * E + DLiabilities * L DEquity = 0 → DAsset * A = DLiabilities * L Tfin 02 Present Value

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% Tfin 02 Present Value

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+f2) = 100 → f2 = 6% In general: (1+r1)(1+f2) = (1+r2)² Solving for f2: The general formula is: Tfin 02 Present Value

Forward rates :example Maturity Discount factor Spot rates Forward rates 1 0.9500 5.26 2 0.8968 5.60 5.93 3 0.8444 5.80 6.21 4 0.7951 5.90 6.20 5 0.7473 6.00 6.40 Details of calculation: 3-year spot rate : 1-year forward rate from 3 to 4 Tfin 02 Present Value

Term structure of interest rates Why do spot rates for different maturities differ ? As r1 < r2 if f2 > r1 r1 = r2 if f2 = r1 r1 > r2 if f2 < r1 The relationship of spot rates with different maturities is known as the term structure of interest rates Upward sloping Spot rate Flat Downward sloping Time to maturity Tfin 02 Present Value

Forward rates and expected future spot rates Assume risk neutrality 1-year spot rate r1 = 5%, 2-year spot rate r2 = 5.5% Suppose that the expected 1-year spot rate in 1 year E(r1) = 6% STRATEGY 1 : ROLLOVER Expected future value of rollover strategy: ($100) invested for 2 years : 111.3 = 100 × 1.05 × 1.06 = 100 × (1+r1) × (1+E(r1)) STRATEGY 2 : Buy 1.113 2-year zero coupon, face value = 100 Tfin 02 Present Value

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 f2 =E(r1) Forward rates contain information about the evolution of future spot rates Tfin 02 Present Value