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# Microsoft PowerPoint - Ch04-ppt

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### Microsoft PowerPoint - Ch04-ppt

1. 1. Chapter 4 The Time Value of Money © 2007 Thomson/South-Western 1 Essentials of Chapter 4 Why is it important to understand and apply time value to 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 Time Value of Money The most important concept in finance A dollar today is worth more than a dollar tomorrow. Used in nearly every financial decision Business decisions Personal finance decisions 3 1
2. 2. Compound Interest Simple interest (interest is earned only on the original principal) Compound interest (interest is earned on principal and on interest received) Year Beginning Balance Interest Earned Ending Amount 1 \$100 \$10 \$110 2 \$110 \$11 \$121 3 \$121 \$12 \$133 4 \$133 \$13 \$146 5 \$146 \$15 \$161 Total Interest \$61 4 Effects of Compounding 5 Cash Flow Time Lines Graphical representations used to show timing of cash flows Time: 0 1 2 3 6% FV = ? Cash Flows: PV=100 Time 0 is today Time 1 is the end of Period 1, etc. 6 2
3. 3. Future Value The amount to which a cash flow or series of cash flows will grow over a period of time when compounded at a given interest rate. 7 Terms Used in Calculating Future Value PV = Present value, or the beginning amount that is invested r = Interest rate paid on the account each period FVn = Future value of the account at the end of n periods n = number of period interest is earned 8 Future Value In general, FVn = PV (1 + r)n 9 3
4. 4. Four Ways to Solve Time Value of Money Problems Use Cash Flow Time Line Use Equations Use Financial Calculator Use Electronic Spreadsheet 10 Time Line Solution The Future Value of \$100 invested at 10% per year for 3 years Time(t): 0 1 2 3 10% x (1.10) x (1.10) x (1.10) Account balance: 100.00 110.00 121.00 133.10 11 Numerical (Equation) Solution The Future Value of \$100 invested at 10% per year for 3 years FVn = PV(1+ r) n FV3 =\$100(1.10)3 =\$100(1.331) =133.10 12 4
5. 5. Financial Calculator Solution The Future Value of \$100 invested at 10% per year for 3 years INPUTS 3 10 -100 0 ? N I/YR PV PMT FV OUTPUT 133.10 13 Future Value Relationships Please note that the graph should start at \$1 14 Present Value Present value is the value today of a future cash flow or series of cash flows. Discounting is the process of finding the present value of a future cash flow or series of future cash flows; it is the reverse of compounding. 15 5
6. 6. What is the PV of \$100 due in 3 years if r = 10%? Time(t): 0 1 2 3 10% x (1.10) x (1.10) x (1.10) Account balance: 75.13 82.64 90.90 100.00 16 What is the PV of \$100 due in 3 years if r = 10%? ⎡ 1 ⎤ PV = FVn ⎢ n⎥ ⎣ (1+ r) ⎦ ⎡ 1 ⎤ PV = \$100 ⎢ 3⎥ ⎣(1.10) ⎦ = \$100(0.7513)= \$75.13 17 What interest rate would cause \$100 to grow to \$125.97 in 3 years? 0 1 2 3 r=? 100 (1 + r )3 = 125.97 PV = 100 ] 100 = 125.97/ (1 + r )3 [ 125.97 = FV 18 6
7. 7. What interest rate would cause \$100 to grow to \$125.97 in 3 years? \$100 (1 + r )3 = \$125.97 INPUTS 3 ? -100 0 125.97 N I/YR PV PMT FV OUTPUT 8% 19 How many years will it take for \$68.30 to grow to \$100 at an interest rate of 10%? 0 1 2 n-1 n=? 10% ... 68.30 (1.10 )n = 100.00 PV = 68.30 ] 68.30 = \$100.00 /(1.10 )n [100.00 = FV 20 How many years will it take for \$68.30 to grow to \$100 if interest of 10% is paid each year? FVn = PV (1 + r ) n \$100.00 = \$68.30(1.10) n \$100 (1.10) n = = 1.46413 \$68.30 ln[(1.10) n ] = n[ln(1.10)] = ln(1.46413) ln(1.46413) n= = 4 .0 ln(1.10) 21 7
8. 8. How many years will it take for \$68.30 to grow to \$100 if interest of 10% is paid each year? INPUTS ? 10 -68.30 0 100.00 N I/YR PV PMT FV OUTPUT 4.0 22 Present Value Relationships 23 Future Value of an Annuity Annuity: A series of payments of equal amounts at fixed intervals for a specified number of periods. Ordinary (deferred) Annuity: An annuity whose payments occur at the end of each period. Annuity Due: An annuity whose payments occur at the beginning of each period. 24 8
9. 9. Ordinary Annuity Versus Annuity Due Ordinary Annuity 0 1 2 3 r% PMT PMT PMT Annuity Due 0 1 2 3 r% PMT PMT PMT 25 What’s the FV of a 3-year Ordinary Annuity of \$100 at 5%? 0 1 2 3 5% 100 100 100 105 110.25 FV = 315.25 26 Numerical Solution: ⎡n−1 ⎤ ⎡ (1 + r) n −1⎤ FVA n = PMT⎢∑ (1 + r) n ⎥ = PMT⎢ ⎥ ⎣t = 0 ⎦ ⎣ r ⎦ ⎡ (1.05)3 −1⎤ FVA3 = \$100⎢ ⎥ ⎣ 0.05 ⎦ = \$100(3.15250) = \$315.25 27 9
10. 10. Financial Calculator Solution INPUTS 3 5 0 -100 ? N I/YR PV PMT FV OUTPUT 315.25 28 Find the FV of an Annuity Due 0 1 2 3 5% 100 100 100 x[(1.05)0](1.05) 105.00 x[(1.05)1](1.05) 110.25 x[(1.05)2](1.05) 115.76 FVA(DUE)3 = 331.01 29 Numerical Solution ⎡n ⎤ ⎡⎧ (1+ r)n −1⎫ ⎤ FVA(DUE)n = PMT⎢∑ (1+ r) t ⎥ = PMT⎢⎨ ⎬ x(1+ r)⎥ ⎣t=1 ⎦ ⎣⎩ r ⎭ ⎦ ⎡⎧ (1.05)3 −1⎫ ⎤ FVA(DUE)3 = \$100⎢⎨ ⎬ x(1.05)⎥ ⎣⎩ 0.05 ⎭ ⎦ = \$100[3.15250x1.05] = \$100[3.310125] = 331.01 30 10
11. 11. Financial Calculator Solution Switch from “End” to “Begin”. INPUTS 3 5 0 -100 ? N I/YR PV PMT FV OUTPUT 331.01 31 Present Value of an Annuity PVAn = the present value of an annuity with n payments. Each payment is discounted, and the sum of the discounted payments is the present value of the annuity. 32 What is the PV of this Ordinary Annuity? 0 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.69 = PV 33 11
12. 12. Numerical Solution ⎡n 1 ⎤ ⎡1 - 1 n ⎤ PVA n = PMT⎢∑ t ⎥ = PMT⎢ (1 +r) ⎥ ⎣t=1 (1 + r) ⎦ ⎢ r ⎥ ⎣ ⎦ ⎡1 - 1 3 ⎤ (1.10) PVA 3 = \$100 ⎢ ⎥ ⎢ 0.10 ⎥ ⎣ ⎦ = \$100(2.48685) = \$248.69 34 Financial Calculator Solution INPUTS 3 10 ? -100 0 N I/YR PV PMT FV OUTPUT -248.69 35 Find the PV of an Annuity Due 0 1 2 3 5% 100 100 100 (1.05)x[1/(1.05)1]x 100.00 (1.05)x[1/(1.05)2]x 95.24 (1.05)x[1/(1.05)3]x 90.70 PVA(DUE)3= 285.94 36 12
13. 13. Financial Calculator Solution INPUTS 3 5 ? -100 0 N I/YR PV PMT FV OUTPUT 285.94 37 Solving for Interest Rates with Annuities You pay \$864.80 for an investment that promises to pay you \$250 per year for the next four years, with payments made at the end of each year. What interest rate will you earn on this investment? 0 1 2 3 4 r=? - 864.80 250 250 250 250 38 Numerical Solution Use trial-and-error by substituting different values of r into the following equation until the right side equals \$864.80. ⎡1 - 1 4 ⎤ \$864.80 = \$250⎢ (1+r) ⎥ ⎢ r ⎣ ⎥ ⎦ 39 13
14. 14. Financial Calculator Solution INPUTS 4 ? -846.80 250 0 N I/YR PV PMT FV OUTPUT 7.0 40 Perpetuities Annuities that go on indefinitely Payment PMT PVP = = Interest rate r 41 Uneven Cash Flow Streams A series of cash flows in which the amount varies from one period to the next: Payment (PMT) designates constant cash flows— that is, an annuity stream. Cash flow (CF) designates cash flows in general, both constant cash flows and uneven cash flows. 42 14
15. 15. What is the PV of this Uneven Cash Flow Stream? 0 1 2 3 4 10% 100 300 300 -50 90.91 247.93 225.39 -34.15 530.08 = PV 43 Numerical Solution ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ 1 ⎤ PV = CF1 ⎢ ⎥ + CF2 ⎢ ⎥ + ... + CFn ⎢ n⎥ ⎣ (1 + r)1 ⎦ ⎣ (1 + r)2 ⎦ ⎣(1 + r) ⎦ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡ 1 ⎤ PV = 100 ⎢ 1⎥ + 300 ⎢ 2⎥ + 300 ⎢ 3⎥ + (−50) ⎢ 4⎥ ⎣ (1.10) ⎦ ⎣ (1.10) ⎦ ⎣ (1.10) ⎦ ⎣ (1.10) ⎦ = \$100(0.90909) + \$300(0.82645) + \$300(0.75131) + (−\$50)(0.68301) = \$530.09 44 Financial Calculator Solution Input in “CF” register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV) 45 15
16. 16. Semiannual and Other Compounding Periods Annual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added once a year. Semiannual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added twice a year. 46 Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated r constant? Why? LARGER! If compounding is more frequent than once a year—for example, semi-annually, quarterly, or daily—interest is earned on interest—that is, compounded—more often. 47 Compounding Annually vs. Semi-Annually 0 1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10 0 1 2 3 4 5 6 5% 100 134.01 Semi-annually: FV6/2 = 100(1.05)6 = 134.01 48 16
17. 17. Distinguishing Between Different Interest Rates rSIMPLE = Simple (Quoted) Rate used to compute the interest paid per period rEAR = Effective Annual Rate the annual rate of interest actually being earned APR = Annual Percentage Rate = rSIMPLE periodic rate X the number of periods per year 49 Simple (Quoted) Rate rSIMPLE is stated in contracts Periods per year (m) must also be given Examples: 8%, compounded quarterly 8%, compounded daily (365 days) 50 Periodic Rate Periodic rate = rPER = rSIMPLE/m Where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Examples: 8% quarterly: rPER = 8/4 = 2% 8% daily (365): rPER = 8/365 = 0.021918% 51 17
18. 18. Effective Annual Rate The annual rate that causes PV to grow to the same FV as under multi-period compounding. Example: 10%, compounded semiannually: rEAR = (1 + rSIMPLE/m)m - 1.0 = (1.05)2 - 1.0 = 0.1025 = 10.25% 52 How do we find rEAR for a simple rate of 10%, compounded semi-annually? m ⎛ r ⎞ rEAR = ⎜ 1 + SIMPLE ⎟ -1 ⎝ m ⎠ 2 ⎛ 0.10 ⎞ = ⎜1 + ⎟ - 1.0 ⎝ 2 ⎠ = (1.05 ) - 1.0 = 0.1025 = 10.25% 2 53 FV of \$100 after 3 years if interest is 10% compounded semi-annual? Quarterly? ⎛ r ⎞m×n FVn = PV⎜1 + SIMPLE ⎟ ⎝ m ⎠ 2×3 ⎛ 0.10 ⎞ FV3×2 = \$100⎜1 + ⎟ = \$100(1.340 10) = \$134.01 ⎝ 2 ⎠ 4×3 ⎛ 0.10 ⎞ FV3×4 = \$100⎜1 + ⎟ = \$100(1.344 89) = \$134.49 ⎝ 4 ⎠ 54 18
19. 19. Amortized Loans Amortized Loan: A loan that is repaid in equal payments over its life Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, and so forth to determine how much of each payment represents principal repayment and how much represents interest 55 Step 1: Determine the required payments 0 1 2 3 10% -1000 PMT PMT PMT INPUTS 3 10 -1000 ? 0 N I/YR PV PMT FV OUTPUT 402.11 56 Step 2: Find simple interest charge for Year 1 INTt = Beginning balancet (r) INT1 = 1,000(0.10) = \$100.00 57 19
20. 20. Loan Amortization Table 10 Percent Interest Rate YR Beg Bal PMT INT Prin PMT End Bal 1 \$1000.00 \$402.11 \$100.00 \$302.11 \$697.89 2 697.89 402.11 69.79 332.32 365.57 3 365.57 402.11 36.56 365.55 0.02 Total 1,206.33 206.35 999.98 * * Rounding difference 58 Chapter 4 Essentials Why is it important to understand and apply time value to 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 - present value subtracts interest What is an annuity? A series of equal payments that occur at equal time intervals 59 Chapter 4 Essentials What is the difference between the Annual Percentage Rate and the Effective Annual Rate? APR is a simple interest rate quoted on loans. EAR is the actual interest rate or rate of return. What is an amortized loan? A loan paid off in equal payments over a specified period How is the return on an investment determined? The amount to which the investment will grow in the future minus the cost of the investment 60 20