FINANCE 3. Present Value Professor André Farber Solvay Business School Université Libre de Bruxelles Fall 2007 MBA 2007 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 DF 5 = 0. 80 • NPV = - 100 + 150 * 0. 80 = - 100 + 120 = +20 MBA 2007 Present value 2
Present Value: general formula • Cash flows: • Discount factors: C 1, C 2, C 3, … , Ct, … CT DF 1, DF 2, … , DFt, … , DFT • Present value: PV = C 1 × DF 1 + C 2 × DF 2 + … + CT × DFT • An example: • Year • Cash flow • Discount factor • Present value 0 -100 1. 000 -100 1 40 0. 9803 39. 21 2 60 0. 9465 56. 79 3 30 0. 9044 27. 13 • NPV = - 100 + 123. 13 = 23. 13 MBA 2007 Present value 3
Several periods: future value and compounding • Invests for € 1, 000 two years (r = 8%) with annual compounding • After one year FV 1 = C 0 × (1+r) = 1, 080 • After two years FV 2 = FV 1 × (1+r) = C 0 × (1+r) • = C 0 × (1+r)² = 1, 166. 40 • • Decomposition of FV 2 C 0 × 2 × r C 0 × r² Principal amount Simple interest Interest on interest 1, 000 160 6. 40 • Investing for t years FVt = C 0 (1+r)t • Example: Invest € 1, 000 for 10 years with annual compounding Principal amount 1, 000 • FV 10 = 1, 000 (1. 08)10 = 2, 158. 82 Simple interest 800 Interest on interest 358. 82 MBA 2007 Present value 4
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 5
Discount factors Interest rate per year # y e ar s 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 0. 9901 0. 9804 0. 9709 0. 9615 0. 9524 0. 9434 0. 9346 0. 9259 0. 9174 0. 9091 2 0. 9803 0. 9612 0. 9426 0. 9246 0. 9070 0. 8900 0. 8734 0. 8573 0. 8417 0. 8264 3 0. 9706 0. 9423 0. 9151 0. 8890 0. 8638 0. 8396 0. 8163 0. 7938 0. 7722 0. 7513 4 0. 9610 0. 9238 0. 8885 0. 8548 0. 8227 0. 7921 0. 7629 0. 7350 0. 7084 0. 6830 5 0. 9515 0. 9057 0. 8626 0. 8219 0. 7835 0. 7473 0. 7130 0. 6806 0. 6499 0. 6209 6 0. 9420 0. 8880 0. 8375 0. 7903 0. 7462 0. 7050 0. 6663 0. 6302 0. 5963 0. 5645 7 0. 9327 0. 8706 0. 8131 0. 7599 0. 7107 0. 6651 0. 6227 0. 5835 0. 5470 0. 5132 8 0. 9235 0. 8535 0. 7894 0. 7307 0. 6768 0. 6274 0. 5820 0. 5403 0. 5019 0. 4665 9 0. 9143 0. 8368 0. 7664 0. 7026 0. 6446 0. 5919 0. 5439 0. 5002 0. 4604 0. 4241 10 0. 9053 0. 8203 0. 7441 0. 6756 0. 6139 0. 5584 0. 5083 MBA 2007 Present value 0. 4632 0. 4224 0. 3855 6
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 * DF 5 and DF 5 = 1/(1+r 5)5 • The YTM r 5 is the solution of: • The solution is: • This is the 5 -year spot interest rate MBA 2007 Present value 7
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 r 1 = 2. 00% • 2 94. 65 r 2 = 2. 79% • 3 90. 44 r 3 = 3. 41% • 4 86. 48 r 4 = 3. 70% • 5 80. 00 r 5 = 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 8
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 9
Constant perpetuity • Ct =C for t =1, 2, 3, . . . Proof: PV = C d + C d² + C d 3 + … PV(1+r) = C + C d² + … PV(1+r)– PV = C/r • Examples: Preferred stock (Stock paying a fixed dividend) • Suppose r =10% Yearly dividend = 50 • Market value P 0? • Note: expected price next year = • Expected return = MBA 2007 Present value 10
Growing perpetuity • Ct =C 1 (1+g)t-1 for t=1, 2, 3, . . . r>g • Example: Stock valuation based on: • Next dividend div 1, long term growth of dividend g • If r = 10%, div 1 = 50, g = 5% • Note: expected price next year = • Expected return = MBA 2007 Present value 11
Constant annuity • A level stream of cash flows for a fixed numbers of periods • C 1 = C 2 = … = CT = C • Examples: • Equal-payment house mortgage • Installment credit agreements • PV = C * DF 1 + C * DF 2 + … + C * DFT + • = C * [DF 1 + DF 2 + … + DFT] • = C * Annuity Factor • Annuity Factor = present value of € 1 paid at the end of each T periods. MBA 2007 Present value 12
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 13
Annuity Factors Interest rate per year # y e ar s 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 0. 9901 0. 9804 0. 9709 0. 9615 0. 9524 0. 9434 0. 9346 0. 9259 0. 9174 0. 9091 2 1. 9704 1. 9416 1. 9135 1. 8861 1. 8594 1. 8334 1. 8080 1. 7833 1. 7591 1. 7355 3 2. 9410 2. 8839 2. 8286 2. 7751 2. 7232 2. 6730 2. 6243 2. 5771 2. 5313 2. 4869 4 3. 9020 3. 8077 3. 7171 3. 6299 3. 5460 3. 4651 3. 3872 3. 3121 3. 2397 3. 1699 5 4. 8534 4. 7135 4. 5797 4. 4518 4. 3295 4. 2124 4. 1002 3. 9927 3. 8897 3. 7908 6 5. 7955 5. 6014 5. 4172 5. 2421 5. 0757 4. 9173 4. 7665 4. 6229 4. 4859 4. 3553 7 6. 7282 6. 4720 6. 2303 6. 0021 5. 7864 5. 5824 5. 3893 5. 2064 5. 0330 4. 8684 8 7. 6517 7. 3255 7. 0197 6. 7327 6. 4632 6. 2098 5. 9713 5. 7466 5. 5348 5. 3349 9 8. 5660 8. 1622 7. 7861 7. 4353 7. 1078 6. 8017 6. 5152 6. 2469 5. 9952 5. 7590 10 9. 4713 8. 9826 8. 5302 8. 1109 7. 7217 7. 3601 7. 0236 MBA 2007 Present value 6. 7101 6. 4177 6. 1446 14
Growing annuity • Ct = C 1 (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 = C 1 – Starting at t = T+1 with first cash flow = C 1 (1+g)T • Example: What is the NPV of the following project if r = 10%? Initial investment = 100, C 1 = 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 15
Review: general formula • Cash flows: C 1, C 2, C 3, … , Ct, … CT • Discount factors: DF 1, DF 2, … , DFt, … , DFT • Present value: PV = C 1 × DF 1 + C 2 × DF 2 + … + CT × DFT If r 1 = r 2 =. . . =r MBA 2007 Present value 16
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=C t=1 to T • Growing annuity: Ct = Ct-1(1+g) t = 1 to T MBA 2007 Present value 17
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% e 12 = 1. 1275 • Effective Annual Interest Rate : 12. 75% MBA 2007 Present value 18
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: 1. Calculate the monthly interest rate (keeping EAR constant) (1+rmonthly)12 = 1. 10 → rmonthly = 0. 7974% 2. Use perpetuity formula: PV = 1 / 0. 007974 = 125. 40 Solution 2: 1. Calculate stated annual interest rate = 0. 7974% * 12 = 9. 568% 2. Use perpetuity formula: PV = 12 / 0. 09568 = 125. 40 MBA 2007 Present value 19
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) H 1, 000 H 1, 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 20
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