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The Time Value of Money Introduction to Time Value of Money, TVM Future Value, FV Lump-sum amount Annuity Uneven cash flow Present Value, PV Lump-sum amount Annuity Uneven cash flow FV and PV Comparison Solving for r and n Intra-year Interest Compounding Amortization 1 Time Value of Money Why is it important to understand and apply time value of money concepts? What is the difference between a present value amount and a future value amount? What is an annuity? What is the difference between the Annual Percentage Rate and the Effective Annual Rate? What is an amortized loan? How is the return on an investment determined? 2 The Time Value of Money Time value of money is considered the most important concept in finance Mathematics of finance “Nuts & Bolts” of financial analysis— apply of TVM concepts to determine value Interest = Rate of return = r = i = k = Y 3 The Time Value of Money “$1 received today is more valuable than $1 received in one year.” Why? Because if you have the opportunity to earn a positive return, investing the $1 today will cause it to grow to greater than $1 in one year. For example, $1 invested at 5 percent will grow to $1.05 in one year because 5¢ of interest will be earned. 4 Future Value and Present Value Future Value (FV)—determine to what amount an investment will grow over a particular time period re-invested interest (earned in previous periods) earns interest compounding—interest compounds or grows the investment Present value (PV)—determine the current value of an amount that will be paid, or received, at some time in the future PV is the future amount restated in current dollars; future interest has not been earned, thus it is not included in the PV discounting—deflate, or discount, the future amount by future interest that can be earned (“deinterest” the FV) 5 Lump-Sum Amounts, Annuities, and Uneven Cash Flow Streams Lump-sum amount—a single amount invested (received) today or in the future; growth in value is the result of interest only Annuity—equal payments made (received) at equal intervals; growth in value is the result of additional payments as well as interest ordinary annuity—end of period payments annuity due—beginning of period payments Uneven Cash Flows—payments that are not all equal that are generally made (received) at equal intervals; growth in value is the result of additional payments as well as interest 6 Cash Flow Time Lines Helps you to visualize the timing of the cash flows associated with a particular situation Constructing a cash flow time line is easy: Time 0 r = 10% Cash Flows -500 1 2 3 4 FVn = ? 7 Approaches to TVM Solutions Time line solution Solve using a cash flow time line Equation (numerical) solution Use equations to solve the problem Financial calculator solution Financial calculators are programmed to solve time value of money problems using the numerical solution Spreadsheet solution Spreadsheets contain functions that can be used to solve time value of money problems using the numerical solution Interest tables Obsolete 8 Future Value Determine to what amount an investment will grow over a particular time period if it is invested at a positive rate of return. Compounding Lump-sum amount Annuity Uneven cash flow stream 9 Future Value, FV, of a Lump-Sum Amount Example: If you invest $500 today at 10%, what will the investment be worth in four years if interest is paid annually? Time 0 r = 10% Cash Flows -500 1 2 3 4 FVn = ? 10 Future Value Graphically, these computations are: 0 10% -500× 1.10 End of year amount 2 1 × 1.10 = 550.00 3 × 1.10 = 605.00 4 × 1.10 = 665.50 = 732.05 FV4 = 500(1.10 x 1.10 x 1.10 x 1.10) = 500(1.10)4 = 732.05 11 Future Value The future value of an amount invested today for n years, FVn, can be found using the following equation: FVn = PV(1 + r)n = PV(interest multiple) FVn PV r n = = = = future value in period n present, or current, value interest rate per period number of periods interest is earned 12 Equation (Numerical) Solution Determined by applying the appropriate equation: FV4n = PV 500 x (1 + r)4nn = 732.05 (1.10) In our example: PV = $500, r = 10.0%, n = 4 13 Financial Calculator Solution In our example: PV = $500, r = 10.0%, n = 4 4 10 -500 0 N I/Y PV PMT ? FV 732.05 14 Future Value of an Annuity Annuity—a series of equal payments that are made at equal intervals Ordinary annuity—end of the period Annuity due—beginning of the period The future value of an annuity, FVA, can be computed by solving for the future value of a lump-sum amount 15 Future Value of an Annuity, FVA Time Cash Flows 0 7% 1 2 3 -100 -100 -100 x x (1.07)2 x (1.07)0 (1.07)1 100.00 107.00 114.49 FVA = 321.49 FVA = 100(1.07)2 + 100(1.07)1 + 100(1.07)0 = 321.49 = 100[(1.07)2 + (1.07)1 + (1.07)2] = 100(3.2149) = 321.49 16 FVA— Equation (Numerical) Solution n (1 r) -1 t FVAn PMT (1 r) PMT r 0 n1 In our example: PMT = $100, r = 7%, n = 3 (1.07)3 - 1 FVA n 100 0.07 100(3.2149) 321.49 17 FVA—Annuity Due Annuity due is an annuity with cash flows that occur at the beginning of the period. When compared to an ordinary annuity, which has end-of-period cash flows, the cash flows of an annuity due earn one additional period of interest. 18 FVA—Annuity Due Time Cash Flows 0 -100 7% 1 2 3 -100 -100 -100 -100 x (1.07) x (1.07) x (1.07)1 x (1.07) x (1.07)2 x 107.00 (1.07)0 100.00 107.00 114.49 114.49 122.50 FVA = = 321.49 343.99 FVA(DUE) 19 FVA(DUE)—Equation (Numerical) Solution (1 r)n - 1 (1 r) FVA FVA(DUE) = PMT n r In our example: PMT = $100, r = 7%, n = 3 (1.07)3 - 1 FVA(DUE)n 100 1.07 0.07 100(3.4399) 343.99 20 FVA—Financial Calculator Solution In our example: n = 3, r = 7%, PMT = $100 3 7.0 0 -100 N I/Y PV PMT ? FV FVA = 321.49 3 7.0 0 N I/Y PV BEGIN -100 PMT ? FV FVA(DUE) = 343.99 21 Solutions: Future value computations FV of $25,000 lump-sum amount: N = 5, I/Y = 5, PV = -25,000, PMT = 0, FV = ? 31,907 FV of $5,499.40 annuity: N = 5, I/Y = 5, PV = 0, PMT = -5,499.40, FV = ? 31,907 Uneven Cash Flow Streams Uneven cash flow stream—cash flows that are not all the same (equal) Simplifying techniques (that is, using a single equation) used to compute FVA cannot be used 23 FV—Uneven Cash Flow Streams 0 4% 1 2 3 -600 -400 -200 (1.04)0 1 (1.04) (1.04)2 FV 600(1.04) 2 400(1.04) 1 200(1.04) 0 200.00 416.00 648.96 _______ 1,264.96 24 FV of Uneven Cash Flow Streams— Equation (Numerical) Solution 1 2 n FVn CF1 (1 r) CF2 (1 r) CFn (1 r) n CFt (1 r) t t 1 2 1 0 FVn 600(1.04) 400(1.04) 200(1.04) 600(1.0816) 400(1.0400) 200(1.0000) 1,264.96 25 FV of Uneven CF Streams—Calculator Solution Input the cash flows, find the present value, PV, and then compute the future value, FV, of PV. Discussed in the next section. 26 Present Value Determine the current value of an amount that will be paid, or received, at a particular time in the future. Finding the present value (PV), or discounting, an amount to be received (paid) in the future is the reverse of compounding, or determining the future value of an amount invested today. We find the PV by “de-interesting” the FV. 27 Present Value—Lump-Sum Amount What is the PV of $800 to be received in four years if your opportunity cost is 8 percent? Stated differently: How much would you be willing to pay today for an investment that will pay $800 in four years if you have the opportunity to invest at 8 percent per year? 28 Present Value—Lump-Sum Amount Time Cash Flows 0 8% PV = ? 1 2 3 4 800 29 Present Value—Equation (Numerical) Solution Remember that FV is computed as follows: FV = PV x (1.08) (1 + r)4nn 800 800 = PV x 1.36049 In our example, r = 8.0% PV =FV 800/1.36049 588.02 4 = 800, n ==4, 30 Present Value—Equation (Numerical) Solution 1 PV Equation: PV FV n (1r) 1 800 4 (1.08) 800(0.7350 3) 588.02 In our example: FV = $800, r = 8.0%, and N = 4 31 Present Value—Time Line Solution Graphically, this computation is: 0 8% 588.02 1 End of year amount 1.08 2 1 635.07 1 1.08 3 685.87 1 1.08 740.74 4 800.00 1 1.08 32 PV Lump-Sum Amount—Financial Calculator Solution In our example: FV = $800, r = 8.0%, n = 4 4 8.0 ? 0 N I/Y PV PMT 800 FV -588.02 33 Present Value of an Annuity, PVA 0 7% 1 100 1 1 (1.07) 93.46 87.34 81.63 262.43 = PVA 2 3 100 100 1 (1.07) 2 1 (1.07) 3 1 1 1 100 100 100 1 2 3 (1.07) (1.07) (1.07) 1 1 1 100(2.6243 ) 262.43 100 1 2 3 (1.07) (1.07) (1.07) 34 PVA—Equation (Numerical) Solution 1 1 1 (1 r) n PVA PMT PMT t 1 (1 r) r n In our example: PMT = $100, r = 7%, n = 3 1 1 - (1.07) 3 PVA 100 0.7 100(2.6243 ) 262.43 35 PVA—Annuity Due Annuity due is an annuity with cash flows that occur at the beginning of the period. 36 PVA—Annuity Due 0 7% 100 100.00 93.46 1 1 (1.07) 1 1 (1.07) (1.07) (1.07) 1 (1.07) 2 (1.07) 1 3 (1.07) 93.46 87.34 1 2 3 100 100 100 100 87.34 81.63 280.80 262.43 = PVA (DUE) 37 PVA(DUE)—Equation (Numerical) Solution 1 - (1 1r) n PVA PVA(DUE) n = PMT r x (1 + r) In our example: PMT = $100, r = 7%, n = 3 1 1 3 (1.07) PVA(DUE)3 100 1.07 0.7 100(2.8080) 280.80 38 PVA—Financial Calculator Solution In our example: n = 3, r = 7%, PMT = $100 3 7.0 ? 100 0 N I/Y PV PMT FV -262.43 = PVA BEGIN 3 7.0 N I/Y ? 100 0 PV PMT FV -280.80 = PVA(DUE) 39 Calculator Solution: N = 5, I/Y = 5, PMT = -5,499.40, FV = 0, PV = ? 25,000 Numerical Solution: 1 1 (1.05)5 PVA 5,499.40 0.05 25,000 x(1.05) 5,499.40( 4.54595) Uneven Cash Flow Streams Uneven cash flow stream—cash flows that are not all the same (equal) Simplifying techniques (that is, using a single equation) used to compute PVA cannot be used 41 Present Value of an Uneven Cash Flow Stream 0 576.92 369.82 177.80 4% 1 600 1 (1.04) 1 2 3 400 200 1 2 (1.04) 1 3 (1.04) 1 1 1 400 200 1,124.54 600 1 2 3 (1.04) (1.04) (1.04) 42 PV of Uneven Cash Flows— Equation (Numerical) Solution 1 PV CF1 CF2 1 (1 r) 1 (1 r)2 CFn 1 (1 r)n 1 CFt t (1 r) 1 1 1 1 600 400 200 1 2 3 (1.04) (1.04) (1.04) n 1,124.54 43 PV of Uneven Cash Flows—Financial Calculator Solution Use the cash flow (CF) register (see calculator instructions) Input CFs in the order they occur—that is, first input CF1, then input CF2, and so on CF0—most calculators require you to input a value for before entering any other cash flows Enter the value for I NPV = PV of uneven cash flows 44 PV of Uneven Cash Flows—Financial Calculator Solution CF0 = 0 CF1 = 600 CF2 = 400 CF3 = 200 r = 4% Compute NPV = 1,124.54 45 FV of Uneven CF Streams—Calculator Solution Input the cash flows, find the present value, PV, and then compute the future value, FV, of PV In our example, PV = $1,124.54, so the future value is: FV = $1,124.54(1.04)3 = $1,264.95 46 Numerical solution: PV = ($2 million)/(1.06)1 + ($4 million)/(1.06)2 + ($5 million)/(1.06)3 = ($2 million)(0.943396) + ($4 million)(0.889996) +($5 million)(0.839619) = 1.8868 + 3.5600 + 4.1981 = 9.6449 Calculator solution: CF0 = 0 CF1 = 2 million CF2 = 4 million CF3 = 5 million I=6 NPV = 9.6448 Comparison of PVA, FVA, and Lump-Sum Amount PMT = $100; r = 7%; n = 3 FVA = $321.49 PVA = $262.43 C 0 1 7% A PVA = 262.43 100 2 100 FV = 262.43 x (1.07)3 = 321.49 3 100 B FVA = 321.49 PV = 321.49/(1.07)3 = 262.43 48 PVA, FVA, and Lump-Sum Amount PMT = $100; r = 7%; n = 3; PVA = $262.43 Year 1 2 3 Beginning Interest Balance @ 7% $262.43 180.80 93.46 $18.37 12.66 6.54 Ending Balance Payment/ Withdrawal $280.80 193.46 100.00 $100.00 100.00 100.00 FVA = 321.49 49 Solving for Time (n) and Interest Rates (r)—Lump Sums The computations for lump-sum amounts included four variables: n, r, PV, and FV. If three of the four variables are known, then we can solve for the unknown variable—e.g., if n, PV, and FV are known, we can solve for r. 50 Solving for Interest Rates, r—Lump-Sum Amount If $200 that was invested three years ago is now worth $245, what rate of return (r) did the investment earn? Time Cash Flows 0 -200 r=? 1 2 3 245 51 Solving for r for a Lump-Sum—Equation (Numerical) Solution FV = PV (1+r)n 245 = 200 (1+r)3 245 3 (1 r) 200 1/3 245 (1 r) 200 0.333 r 2.45 - 1.0 = 7.0% 52 Solving for r for a Lump-Sum—Financial Calculator Solution In our example: PV = $200, FV = $245, n = 3 3 ? -200 0 245 N I/Y PV PMT FV 7.00 53 Solving for Number of Years, n—LumpSum Amount If $712 is invested at 6 percent, how long will it take to grow to $848? Time Cash Flows 0 r = 6% 1 -712 2 … n=? 848 54 Solving for n for a Lump-Sum—Equation (Numerical) Solution FV PV(1 r)n 848 712(1.06)n 848 n (1.06) 712 55 Solving for n for a Lump-Sum—Financial Calculator Solution In our example: PV = $712, FV = $848, r = 6% ? 6.0 -712 0 848 N I/Y PV PMT FV 3.00 56 Solving for Time (n) and Interest Rates (r)—Annuities The computations for annuities included four variables: n, r, PMT, and PVA or FVA. If three of the four variables are known, then we can solve for the unknown variable—e.g., if n, PVA (or FVA), and PMT are known, we can solve for r. 57 Solving for Interest Rates, r, for Annuities The current value of an investment that will pay $300 each year for three years is $817. What rate of return (r) will the investment earn? 0 PVA = -817 r=? 1 2 3 300 300 300 58 Solving for r for Annuities—Equation (Numerical) Solution 1 - (1 1r)n PVA PMT r 1 - (1 1r) 3 817 300 r To solve, use a trial-and-error process Solution = 5.0% 59 Solving for r for Annuities—Financial Calculator Solution In our example: PMT = $300, PVA = $817, n = 3 3 ? N I/Y -817 PV 300 0 PMT FV 5.00 60 Solving for Number of Years, n, for Annuities If $480 is invested each year at 8 percent, how long will take to grow to $2,816? Time Cash Flows 0 r = 8% 1 -480 2 … -480 n=? -480 FVA = 2,816 61 Solving for n for Annuities—Equation (Numerical) Solution (1 r)n - 1 FVA PMT r (1.08)n - 1 2,816 480 0.08 To solve, use a trial-and-error process Solution = 5 years 62 Solving for n for Annuities—Financial Calculator Solution In our example: PMT = $480, FVA = $2,816, r = 8% ? 8.0 N I/Y 0 PV -480 2,816 PMT FV 5.00 63 Solving for r for Uneven Cash Flows Internal Rate of Return (IRR)—average rate of return an investment earns Capital budgeting decisions—decisions concerning what investments a firm should purchase 64 Intra-Year Interest Compounding Interest is compounded more than once per year—quarterly, monthly, or daily Adjustments to computations: Use the interest rate per compounding period and the number of interest compounding periods during the life of the investment, or Use the effective annual rate, EAR, and the number of years to maturity 65 Intra-Year Interest Compounding— Example How much will an amount invested today grow to in two years if interest is paid quarterly? PV = $200 and r = 8% Quarterly interest = 8%/4 = 2% = r/m Number of interest payments = 2 years x 4 = 8 = n Year 0 Quarter 0 r = 2% 200.00 x 1.02 1 3 2 x 1.02 204.00 x 1.02 208.08 x 1.02 212.24 1 2 4 8 … 216.49 234.33 FVn = PV(1 + r)n = 200(1.02)8 = 200(1.17166) = 234.33 66 Intra-Year Interest Compounding Financial calculator solution: 8 2.0 -200 0 N I/Y PV PMT ? FV 234.33 67 Intra-Year Interest Compounding—Effective Annual Rate Interest = 8%, compounded quarterly r = 8% per year is the simple, or non-compounded rate r/m = 2% per quarter is the effective rate per compounding period (each quarter; m=4) Effective Annual Rate (EAR), rEAR rEAR = (1 + r/m)m – 1.0 m = number of interest payment periods per year rEAR = (1.02)4 – 1.0 = 0.08243 = 8.243% 68 Intra-Year Interest Compounding—Using Effective Annual Rate In our example: PV = $200 n = 2 years rEAR = 8.243% FVn = = = = PV(1 + rEAR)n 200(1.08243)2 234.33 200(1.02)8 69 Annual Percentage Rate, APR, Versus Effective Annual Rate, EAR EAR—the rate of return per year considering interest compounding rEAR = (1 + r/m)m – 1.0 APR—simple rate of return; does not consider compounding APR = r/m x m = r = simple interest rEAR = APR only if interest is paid once per year— that is, annual compounding rEAR > APR if interest is paid more than once per year 70 FVA—Financial Calculator Solution In our example: n = 365, PV = -$1, PMT = 0, FV = $1.06168 365 ? N I -1 PV 0 PMT 0.016399 APR = 0.016399 x 365 = 5.986 Advertise: 5.986% compounded daily 1.06168 FV Amortized Loans Loan agreement requires equal periodic payments A portion of the payment represents interest on the debt and the remainder is applied to the repayment of the debt Amortization schedule—used to determine what portion of the total payment is interest and what portion is repayment of principal Mortgage payment—only the interest portion is considered an expense for tax purposes 72 Amortization Schedule—Example Example: $6,655 home-equity loan; r = 6%; n = 2 years; payments are made quarterly The constant payments per quarter represent an annuity and the amount of the loan ($6,655) represents the present value of the loan payments PVA = $6,655; (n x m) = 2 x 4 = 8 payments; r/m = 6%/4 = 1.5% Financial calculator solution: 8 1.5 6,655 N I/Y PV ? PMT 0 FV -889 73 Amortization Schedule—Example Payment = $889 Begin. Payment Year Pmt # Balance (Pmt) 1 1 $6,655.00 $889 2 5,865.83 889 3 5,064.82 889 4 4,251.79 889 2 5 3,426.57 889 6 2,588.97 889 7 1,738.80 889 8 875.88* 889 Interest (I) Loan Repay. [1.5% x Beg Bal] = Pmt – I $99.83 $789.17 87.99 801.01 75.97 813.03 63.78 825.22 51.40 837.60 38.83 850.17 26.08 862.92 13.14 875.86* * Rounding difference 74 Answers to TVM Questions Why is it important to understand and apply time value of money concepts? To be able to compare various investments. What is the difference between a present value amount and a future value amount? Future value adds interest (compounds); present value subtracts interest (“de-interests”). What is an annuity? A series of equal payments that occur at equal time intervals. 75 Answers to TVM Questions What is the difference between the Annual Percentage Rate and the Effective Annual Rate? APR is a simple interest rate quoted on loans/investments, whereas EAR is the actual interest or rate of return. What is an amortized loan? A loan paid off in equal payments over a specified period, which include payments of interest and principal How is the return on an investment determined? Compute the annual rate based on the amount to which an investment will grow in the future. 76