ࡱ> 7 |UbjbjUU )d7|7|dQ l.^.^.^8f^<^bbbbb#e#e#eg[``$ ?#ee#e#e#e?hbboThhh#edbbh#ehhnTu^Vwbb p3_d*[.^ev,Vwj0*w,hVwhChapter 3 Time value of Money This chapter discusses how to calculate the present value, future value, internal rate of return, and modified internal rate of return of a cash flow stream. Understanding how (and when) to use these formulas is essential to your success as a financial manager! Formulas and examples are included with these notes. Numbers are rounded to 4 decimal places in tables and formula. However, the actual (non-rounded) numbers are used in the calculations. Time Value of Money Concepts Time-Line Conventions $1 received today (cash inflow) $1 paid in five years (cash outflow) $1 received at the end of the third year $1 received at the beginning of the third year Four-year annuity of $1 per year, first cash flow received at t = 1 (ordinary annuity) Four-year annuity of $1 per year, first cash flow received at t = 0 (annuity due) Four-year annuity of $1 per year, first cash flow received at t = 2 (deferred annuity) 012345A.$1B.-$1C.$1D.$1E.$1$1$1$1F.$1$1$1$1G.$1$1$1$1 Notation C0 = cash flow at time 0 C = cash flow (used when all cash flows are the same) r = discount rate or interest rate t = time period (e.g., t = 4), or number of years (e.g., t years in the future) m = number of compounding periods per year (e.g., with monthly compounding, m = 12) g = growth rate in cash flow Annual Compounding, Single Payments Future value of $1 as of time 1. Interest rate = 5%. 012345$1$0$0$0$0$0 Formula: C0 (1 + r)t = $1(1.05)1 = $1.0500 Financial Calculator: N = 1, I/Y = 5, PV = -1, PMT = 0, FV = Answer Note on financial calculators The calculator inputs described above are for a Texas Instruments BAII Plus calculator. (Many other financial calculators require similar inputs.) Notice that you enter a -1 as the PV and the solution is +1.05. Here is the intuition: deposit $1 in the bank (negative cash flow), withdraw $1.05 in one year (positive cash flow). If you had entered +1 as the PV, the solution would be 1.05. Future value of $1, as of time 5. Interest rate = 5%. 012345$1$0$0$0$0$0 Formula: C0 (1 + r)t = $1(1.05)5 = $1.2763 Financial Calculator: N = 5, I/Y = 5, PV = -1, PMT = 0, FV = Answer Present value of $1, received at time 1. Discount rate = 5%. 012345$0$1$0$0$0$0 Formula: C1 / (1 + r)t = $1/(1.05)1 = $0.9524 Financial Calculator: N = 1, I/Y = 5, PV = Answer, PMT = 0, FV = -1 Present value of $1, received at time 5. Discount rate = 5%. 012345$0$0$0$0$0$1 Formula: C5 / (1 + r)t = $1/(1.05)5 = $0.7835 Financial Calculator: N = 5, I/Y = 5, PV = Answer, PMT = 0, FV = -1 Compounding periods less than one year Future value of $1, as of time 5. Interest rate = 5%, compounded m times per year. 012345$1$0$0$0$0$0 General (non-continuous) formula: C0 (1 + r/m)tm Continuous compounding formula: C0 ert Note: e = 2.718281828 Semi-annual compounding: $1(1 + (0.05/2))(5)(2) = $1.280085 Monthly compounding: $1(1 + (0.05/12))(5)(12) = $1.283359 Daily compounding: $1(1 + (0.05/365))(5)(365) = $1.284003 Continuous compounding: $1 e(5)(0.05) = $1.284025 Note: Some may use 360 days as the length of one year, other may take into account leap years (366 days every four years). The effects of these changes (from a 365-day year) are extremely small. Financial Calculator (for semi-annual): N = 10, I/Y = 5/2, PV = -1, PMT = 0, FV = Answer Financial Calculator (for monthly): N = 60, I/Y = 5/12, PV = -1, PMT = 0, FV = Answer Financial Calculator (for daily): N = 1825, I/Y = 5/365, PV = -1, PMT = 0, FV = Answer Present value of $1, received at time 5. Discount rate = 5%, compounded m times per year. 012345$0$0$0$0$0$1 General (non-continuous) formula: C5 / (1 + r/m)tm Continuous compounding formula: C5 / ert Semi-annual compounding: $1 / [1 + (0.05/2)](5)(2) = $0.781198 Monthly compounding: $1 / [1 + (0.05/12)](5)(12) = $0.779205 Daily compounding: $1 / [1 + (0.05/365)](5)(365) = $0.778814 Continuous compounding: $1 / e(5)(0.05) = $0.778801 Financial Calculator (for semi-annual): N = 10, I/Y = 5/2, PV = Answer, PMT = 0, FV = -1 Financial Calculator (for monthly): N = 60, I/Y = 5/12, PV = Answer, PMT = 0, FV = -1 Financial Calculator (for daily): N = 1825, I/Y = 5/365, PV = Answer, PMT = 0, FV = -1 Constant Finite Annuities Four-year annuity of $1 per year, first cash flow received at t = 1. Interest and discount rate = 5%. 012345$0$1$1$1$1$0 Standard formula for the future value of a finite annuity = C [(1 + r)t 1] / r This standard formula for the future value of a finite annuity gives a value as of the last period of the annuity (time 4 in this example). The 1.05 is raised to the fourth power because there are 4 payments in the annuity. Value as of time 4 = $1 [(1.054 1) / 0.05] =$4.3101  You can calculate the value of the cash flows at other points in time by multiplying or dividing by 1+r, where r (the interest and discount rate) is 5% in this example. For instance, assume you want to know the value of the above cash flow stream at t = 6. Time 6 is two years after time 4. To calculate, use the standard formula to determine the value at t = 4, then multiply by 1.052 to determine the value at t = 6. (Use the second power because you are calculating the value two years after time 4.) The solution is: Value as of time 6 = $1 [(1.054 1) / 0.05] 1.052 =$4.7519  As a second example, assume that you want to know the value of the above cash flow stream at t = 1. Time 1 is three years before time 4. To calculate, use the standard formula to determine the value at t = 4, then divide by 1.053 to determine the value at t = 1. (Use the third power because you are calculating the value three years before time 4.) The solution is: Value as of time 1 = $1 [(1.054 1) / 0.05] / 1.053 =$3.7232  Therefore, multiply when you want to determine the value at a later date, divide when you want to determine the value at an earlier date. Some more examples: Value as of time 0 = $1 [(1.054 1) / 0.05] / 1.054 =$3.5460 Value as of time 1 = $1 [(1.054 1) / 0.05] / 1.053 =$3.7232 Value as of time 2 = $1 [(1.054 1) / 0.05] / 1.052 =$3.9094 Value as of time 3 = $1 [(1.054 1) / 0.05] / 1.051 =$4.1049 Value as of time 4 = $1 [(1.054 1) / 0.05] =$4.3101 Value as of time 5 = $1 [(1.054 1) / 0.05] 1.051 =$4.5256 Value as of time 6 = $1 [(1.054 1) / 0.05] 1.052 =$4.7519 Value as of time 7 = $1 [(1.054 1) / 0.05] 1.053 =$4.9895  Financial Calculator (time 0): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Divide answer by 1.054. Financial Calculator (time 1): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Divide answer by 1.053. Financial Calculator (time 2): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Divide answer by 1.052. Financial Calculator (time 3): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Divide answer by 1.051. Financial Calculator (time 4): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Financial Calculator (time 5): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Multiply answer by 1.051. Financial Calculator (time 6): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Multiply answer by 1.052 Financial Calculator (time 7): N = 4, I/Y = 5, PV = 0, PMT = -1, FV = Answer. Multiply answer by 1.053 You can also use the formula for the present value of a finite annuity to calculate the value of a cash flow stream at different points in time. Standard formula for the present value of a finite annuity = C { [1 (1 / (1 + r))t] / r} The standard formula gives a value one period before the first payment of the annuity (time 0 in this example). The 1.05 is raised to the fourth power because there are 4 payments in the annuity. Value as of time 0 = $1 { [1 (1/1.05)4] / (0.05) } = $3.5460  As before, you can calculate the value at other points in time by multiplying or dividing by 1+r, (1.05 in this example). Two examples: Value as of time 1 = $1 { [1 (1/1.05)4] / (0.05) } 1.051 = $3.7232  Value as of time 6 = $1 { [1 (1/1.05)4] / (0.05) } 1.056 = $4.7519  Financial Calculator (time 0): N = 4, I/Y = 5, PV = Answer, PMT = -1, FV = 0 Financial Calculator (time 1): N = 4, I/Y = 5, PV = Answer, PMT = -1, FV = 0. Multiply answer by 1.051. Financial Calculator (time 6): N = 4, I/Y = 5, PV = Answer, PMT = -1, FV = 0. Multiply answer by 1.056. Three-year annuity of $1 per year, first cash flow received at t = 0. Interest and discount rate = 5%. 012345$1$1$1$0$0$0 The standard future value annuity formula gives a value as of the last year of the annuity (year 2 in this example). This is a three-year annuity. Therefore, 1.05 is raised to the third power in the formula. Value as of time 2 = $1 [(1.053 1) / 0.05] =$3.1525  The value at other points in time can be calculated by multiplying or dividing by 1.05, raised to the appropriate power. Value as of time 0 = $1 [(1.053 1) / 0.05] / 1.052 =$2.8594 Value as of time 4 = $1 [(1.053 1) / 0.05] 1.052 =$3.4756  The standard present value annuity formula gives a value one period before the first payment of the annuity. Therefore, the formula will give you a value at t = -1. You need to multiply by 1 + r to get the value by t = 0. Value as of time 0 = $1 { [1 (1/1.05)3] / (0.05) } 1.051 = $2.8594  The values at time 2 and 4: Value as of time 2 = $1 { [1 (1/1.05)3] / (0.05) } 1.053 = $3.1525 Value as of time 4 = $1 { [1 (1/1.05)3] / (0.05) } 1.055 = $3.4756  Five-year annuity of $1 per year, first cash flow received at t = 3. Interest and discount rate = 5%. 012345678$0$0$0$1$1$1$1$1$0 The standard future value annuity formula gives the value as of the last year of the annuity (t = 7 in this example). This is a five-year annuity. Therefore, 1.05 is raised to the fifth power. Value as of time 7 = $1 [(1.055 1) / 0.05] =$5.5256  Values at different points in time using the future value annuity formula. A couple of examples Value as of time 0 = $1 [(1.055 1) / 0.05] / 1.057 =$3.9270 Value as of time 4 = $1 [(1.055 1) / 0.05] / 1.053 =$4.7732 Value as of time 8 = $1 [(1.055 1) / 0.05] 1.051 =$5.8019  The standard present value annuity formula gives the value as of the year before the first payment of the annuity (t = 2 in this example). Value as of time 2 = $1 { [1 (1/1.05)5] / (0.05) } = $4.3295  Values at different points in time using the present value annuity formula. A couple of examples: Value as of time 0 = $1 { [1 (1/1.05)5] / (0.05) } / 1.052 = $3.9270 Value as of time 4 = $1 { [1 (1/1.05)5] / (0.05) } 1.052 = $4.7732 Value as of time 8 = $1 { [1 (1/1.05)5] / (0.05) } 1.056 = $5.8019  Growing Finite Annuities Four-year growing annuity, growing at 10% per year. First cash flow (equal to $1) received at t = 1. Interest and discount rate = 5%. 012345$0$1$1.1$1.21$1.331$0 Standard formula for the present value of a finite growing annuity (for when r is not equal to g) = Cfirst [1 [(1 + g) / (1 + r)]t ] / (r g). This formula gives the value one period before the first payment (t = 0 in this example). Cfirst is the first cash flow of the annuity. In this above example, Cfirst = C1 = $1. Value as of time 0 = $1 { [1 (1.10/1.05)4] / (0.05 0.10) }$4.0904  Values at different points in time using the present value growing annuity formula. Multiply or divide by 1+r (raised to the appropriate power) to determine the value at other points in time. Value as of time 2 = $1 { [1 (1.10/1.05)4] / (0.05 0.10) } 1.052$4.5096 Value as of time 4 = $1 { [1 (1.10/1.05)4] / (0.05 0.10) } 1.054$4.9719 Value as of time 5 = $1 { [1 (1.10/1.05)4] / (0.05 0.10) } 1.055$5.2205  Four-year growing annuity, growing at 10% per year. First cash flow (equal to $1) received at t = 3. Interest and discount rate = 5%. 01234567$0$0$0$1$1.1$1.21$1.331$0 The standard formula for the present value of a growing annuity gives you a value at time 2 (one period before the first payment). This is a 4-year annuity. Therefore, 1.05 is raised to the 4th power. Value as of time 2 = $1 { [1 (1.10/1.05)4] / (0.05 0.10) }$4.0904  Values at different points in time using the present value growing annuity formula. A few examples: Value as of time 0 = $1 { [1 (1.10/1.05)4] / (0.05 0.10) } /1.052$3.7101 Value as of time 5 = $1 { [1 (1.10/1.05)4] / (0.05 0.10) } 1.053$4.7351  Perpetual constant annuities Perpetual constant annuity of $1 (cash flows start at time 1, interest and discount rate = 5%) 012345($0$1$1$1$1$1$1 The standard formula for the present value of a perpetual constant annuity is C / r. The formula gives you the value one period before the first payment. Value as of time 0 = $1 / 0.05$20.0000  Values at different points in time using the present value perpetual constant annuity formula Value as of time 1 = ($1 / 0.05) 1.051$21.0000 Value as of time 4 = ($1 / 0.05) 1.054$24.3101  Perpetual constant annuity of $1 (cash flows start at time 0, interest and discount rate = 5%) 012345($1$1$1$1$1$1$1 The standard formula for the present value of a perpetual constant annuity gives you the value one period before the first payment (t = -1 in this example). Therefore, you need to multiply by 1.05 to get the value at t = 0. Value as of time 0 = ($1 / 0.05) 1.05$21.0000  Values at different points in time using the present value perpetual annuity formula. Two examples: Value as of time 1 = ($1 / 0.05) 1.052$22.0500 Value as of time 4 = ($1 / 0.05) 1.055$25.5256  Perpetual constant annuity of $1 (cash flows start at time 5, interest and discount rate = 5%) 012345($0$0$0$0$0$1$1 The standard formula for the present value of a perpetual constant annuity gives you the value one period before the first payment (t = 4 in this example). Value as of time 4 = ($1 / 0.05)$20.0000  Values at different points in time using the present value perpetual annuity formula. Some examples: Value as of time 0 = ($1 / 0.05) / 1.054$16.4540 Value as of time 5 = ($1 / 0.05) 1.051$21.0000  Perpetual growing annuity, growing at 3% per year (first cash flow, received at time 1, equals $1, interest and discount rate = 5%) 0123($0$1$1.03$1.06093% more The standard formula for the present value of a perpetual growing annuity is Cfirst / (r g). The formula gives you the value one period before the first payment. Value as of time 0 = $1 / (0.05 0.03)$50.0000  Values at different points in time using the present value perpetual growing annuity formula. Some examples: Value as of time 1 = [$1 / (0.05 0.03)] 1.051$52.5000 Value as of time 4 = [$1 / (0.05 0.03)] 1.054$60.7753  Perpetual growing annuity, growing at 3% per year (first cash flow, received at time 11, equals $1, interest discount rate = 5%) 10111213($0$1$1.03$1.06093% more The standard formula for the present value of a perpetual growing annuity gives you the value one period before the first payment (t = 10 in this example). Value as of time 10 = [$1 / (0.05 0.03)]$50.0000  Values at different points in time using the present value perpetual growing annuity formula. A few examples: Value as of time 0 = [$1 / (0.05 0.03)] / 1.0510$30.6957 Value as of time 9 = [$1 / (0.05 0.03)] / 1.051$47.6190 Value as of time 11 = [$1 / (0.05 0.03)] 1.051$52.5000  Two growth-rate example: $1 at time 1, 10% growth rate until time 4, 3% growth rate after time 4 (in perpetuity). Use a 5% discount rate 012345($0$1$1.1$1.21$1.331$1.37093% more Value the first four payments using the finite growing annuity formula. Value the payments starting at time 5 using the perpetual growing annuity formula The first four payments Value as of time 0 = $1 { [1 (1.10/1.05)4] / (0.05 0.10) }$4.0904  The payments starting at time 5 Value as of time 0 = {[($1) (1.13) (1.03)] / [(0.05 0.03)]} / 1.054$56.3934 Solution = $4.0904 + $56.3934 = $60.4837 Note: [($1) (1.13) (1.03)] = $1.3709 = the payment at t = 5 Two growth-rate example: $1 at time 0, 10% growth rate until time 12, 3% growth rate after time 12 (in perpetuity). Use a 5% discount rate 012. . .111213($1$1.1$1.21($2.8531$3.1384$3.23263% more Value the first thirteen payments (t = 0 to t = 12) using the finite growing annuity formula. Value the payments starting at time 13 using the perpetual growing annuity formula The first thirteen payments Value as of time 0 = $1 { [1 (1.10/1.05)13] / (0.05 0.10) } 1.051$17.4471 The payments starting at time 13 Value as of time 0 = {[($1) (1.112) (1.03)] / [(0.05 0.03)]} / 1.0512$90.0011 Solution = $17.4471 + $90.0011 = $107.4482 Note: [($1) (1.112) (1.03)] = $3.2326 = the payment at t = 13 Two growth-rate example: $1 at time 3, 10% growth rate until time 20, 3% growth rate after time 20 (in perpetuity). Use a 5% discount rate 01234. . .2021($0$0$0$1$1.1($5.0545$5.20613% more Value the first eighteen payments (t = 3 to t = 20) using the finite growing annuity formula. Value the payments starting at time 21 using the perpetual growing annuity formula The first eighteen payments Value as of time 0 = $1 { [1 (1.10/1.05)18] / (0.05 0.10) } / 1.052$23.7689 The payments starting at time 21 Value as of time 0 = {[($1) (1.117) (1.03)] / [(0.05 0.03)]} / 1.0520$98.1063 Solution = $23.7689 + $98.1063 = $121.8752 Note: [($1) (1.117) (1.03)] = $5.2061 = the payment at t = 21 Internal rate of return (IRR) Definition: the IRR = the discount rate that causes the sum of the present values of all cash flows to equal zero. IRR calculation examples Two cash flows 012345-$1$0$0$0$0$2 -$1 + $2 / (1 + r)5 = $0 r = ($2 / $1)(1/5) 1 = 14.8698% = IRR Financial Calculator: N = 5, I/Y = Answer, PV = -1, PMT = 0, FV = 2 012345$1$0$0$0$0-$2 $1 + -$2 / (1 + r)5 = $0 r = ($2 / $1)(1/5) 1 = 14.8698% = IRR Financial Calculator: N = 5, I/Y = Answer, PV = 1, PMT = 0, FV = -2 Perpetual constant annuities 01234(-$10$1$1$1$1$1 -$10 + $1 / r = $0 r = $1 / $10 = 10% = IRR Perpetual growing annuities 01234(-$10$1$1.03$1.0609$1.09273% more -$10 + $1 / (r 3%) = $0 r = ($1 / $10) + 3% = 13% = IRR Other cash flow patterns solve by your calculator or computer. Example finite annuity 012345-$3$1$1$1$1$1 -$3 + $1 { [1 (1/(1+r))5] / r } = $0 r = 19.8577% = IRR Financial Calculator: N = 5, I/Y = Answer, PV = -3, PMT = 1, FV = 0 Modified Internal Rate of Return (MIRR) Step one: Using the discount rate, take a PV (to time zero) of the negative cash flows Step two: Using the interest rate, take a FV (to time t, where t is the time of the last cash flow) of the positive cash flows Step three: Calculate the IRR of the two cash flows calculated in the first two steps Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream? 012345-$3$1$1$1$1$1 Step 1: PV of negative cash flows (at time 0) = -$3 Step 2: FV of positive cash flows (at time 5) = $5.5256 Step 3: IRR = ($5.5256 / $3)(1/5) 1 = 12.9932% = MIRR Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream? 012345+$3-$1-$1-$1-$1-$1 Step 1: PV of negative cash flows (at time 0) = -$4.3295 Step 2: FV of positive cash flows (at time 5) = $3.8288 Step 3: IRR = ($3.8288 / $4.3295)(1/5) 1 = -2.4277% = MIRR Using an interest and discount rate = 5%, what is the MIRR of the following cash flow stream? 012345+$3$0$0-$1-$1$1 Step 1: PV of negative cash flows (at time 0) = -$1.6865 Step 2: FV of positive cash flows (at time 5) = $4.8288 Step 3: IRR = ($4.8288 / $1.6865)(1/5) 1 = 23.4154% = MIRR Application loan amortization schedules A 30-year home loan has an annual interest rate of 8%. Interest is compounded monthly. What is the monthly payment on a fully amortizing, level payment loan for $100,000? $733.7646 Use this payment to filling in the following loan amortization table for the home loan described above. MonthBeginning BalanceTotal PaymentInterest PaymentPrincipal PaymentEnding Balance0$100,000.0001$100,000.000 $733.7646  $666.6667  $67.0979 $99,932.9021 2$99,932.9021  $733.7646  $666.2193  $67.5452 $99,865.3569 3$99,865.3569  $733.7646  $665.7690  $67.9955 $99,797.3613 4$99,797.3613  $733.7646  $665.3157  $68.4488 $99,728.9125 5$99,728.9125  $733.7646  $664.8594  $68.9052 $99,660.0073  The loan balance will be $0 after the 360th payment. 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