Present Value and Discounting in Finance

FINANCE3 . Present Value Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2007

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 MBA 2007 Present value

Present Value: general formula • Cash flows: C1, C2, C3, … ,Ct, … CT • Discount factors: DF1, DF2, … ,DFt, … , DFT • Present value: PV = C1×DF1 + C2×DF2 + … + CT×DFT • 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 MBA 2007 Present value

Several periods: future value and compounding • Invests for €1,000 two years (r = 8%) with annual compounding • After one year FV1 = C0× (1+r) = 1,080 • After two years FV2 = FV1 × (1+r) = C0× (1+r) × (1+r) • = C0× (1+r)² = 1,166.40 • Decomposition of FV2 • C0 Principal amount 1,000 • C0 × 2 × r Simple interest 160 • C0 × r² Interest on interest 6.40 • Investing for t years FVt = C0 (1+r)t • Example: Invest €1,000 for 10 years with annual compounding • FV10 = 1,000 (1.08)10 = 2,158.82 Principal amount 1,000Simple interest 800Interest on interest 358.82 MBA 2007 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  DFt • 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) DFt • The general formula for the t-year discount factor is: MBA 2007 Present value

Discount factors MBA 2007 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 * DF5 and DF5 = 1/(1+r5)5 • The YTM r5 is the solution of: • The solution is: • This is the 5-year spot interest rate MBA 2007 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 MBA 2007 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) MBA 2007 Present value

Constant perpetuity 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 = MBA 2007 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 = MBA 2007 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 * DF1 + C * DF2 + … + C * DFT+ • = C * [DF1 + DF2 + … + DFT] • = C * Annuity Factor • Annuity Factor = present value of €1 paid at the end of each T periods. MBA 2007 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 MBA 2007 Present value

Annuity Factors MBA 2007 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 MBA 2007 Present value

Review: general formula • Cash flows: C1, C2, C3, … ,Ct, … CT • Discount factors: DF1, DF2, … ,DFt, … , DFT • Present value: PV = C1×DF1 + C2×DF2 + … + CT×DFT If r1 = r2 = ...=r MBA 2007 Present value

Review: Shortcut formulas • Constant perpetuity: Ct = C for all t • Growing perpetuity: Ct = Ct-1(1+g) r>g t = 1 to ∞ • Constant annuity: Ct=Ct=1 to T • Growing annuity: Ct = Ct-1(1+g) t = 1 to T MBA 2007 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% MBA 2007 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 MBA 2007 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 MBA 2007 Present value

Present Value and Discounting in Finance