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Chapter 3 The Time Value of Money Essentials of Managerial Finance by S. Besley & E. Brigham Slide 1 of 48 The Role of Time Value in Finance • Most financial decisions involve costs & benefits that are spread out over time. • Time value of money allows comparison of cash flows from different periods. Which investment would you choose? (a)An investment of €100,000 that would return €200,000 after one year (b)An investment of €100,000 that would return €220,000 after two years Essentials of Managerial Finance by S. Besley & E. Brigham Slide 2 of 48 The Role of Time Value in Finance • In general, all else being equal, the sooner a € is received, the more quickly is re-invested • However a short-term investment is not necessarily more valuable because this depends on the re- investment interest rate Answer The investment (a) is more valuable if it can re- invested at an annual interest rate >10%, otherwise the investment (b) is more valuable Essentials of Managerial Finance by S. Besley & E. Brigham Slide 3 of 48 Cash Flow Time Lines • The cash flow time lines help to visualize the timing of the cash flows associated with a particular situation •The construction of a cash flow time line is fairly easy: Time 0 1 2 3 4 k = 10% Cash Flows -500 FVn = ? Essentials of Managerial Finance by S. Besley & E. Brigham Slide 4 of 48 Difference between simple interest and compounded interest With simple interest, an investor doesn’t earn interest on interest • Year 1: 5% of €100 = €5 + €100 = €105 • Year 2: 5% of €100 = €5 + €105 = €110 • Year 3: 5% of €100 = €5 + €110 = €115 With compounded interest, an investor earns interest on interest • Year 1: 5% of €100.00= €5.00 + €100.00 = €105.00 • Year 2: 5% of €105.00= €5.25 + €105.00 = €110.25 • Year 3: 5% of €110.25= €5.51+ €110.25= €115.76 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 5 of 48 Future Value Definition and Formula • 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 • FVn = PV0(1+k)n = PV(FVIFk,n) Essentials of Managerial Finance by S. Besley & E. Brigham Slide 6 of 48 Effects of compounding Essentials of Managerial Finance by S. Besley & E. Brigham Slide 7 of 48 Future Value - Example • Suppose an investment of €100 for one year at 5% per year. What is the future value in one year? – The compounding rate is given as 5%. Hence the value of current Euros in terms of future Euros is 1.05 future Euros per current Euro. Hence future value is 100(1.05) = €105. •Suppose that money is left in for another year. What is the future value in two years from now? – Assume that the money in one year as present value and the money in two years as future value. Therefore the price of one-year-from-now money in terms of two-years-from-now money is 1.05. Therefore 105 of one-year-from-now Euros in terms of two years-from-now Euros is 105(1.05) = 100 (1.05)(1.05) = 100(1.05)2 = 110.25 – By making the same assumptions, the FV in 3 years would be 115.76 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 8 of 48 Future Value– Example using Excel An investment of €100 is made today at 5% interest. How much money will this investment yield in 3 years? Excel Function PV 100 (assumes compounded interest k 5,00% as capital is being re-invested) n 3 =FV (interest, periods, pmt, PV) FV? 115,76 =FV (.05, 3, , 100) Essentials of Managerial Finance by S. Besley & E. Brigham Slide 9 of 48 Financial Calculator Solution In the previous example: PV = €100, k = 5.0%, n = 3 3 5 -100 0 ? N I PV PMT FV 115,76 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 10 of 48 A Graphic view of Future Value Relationship among Future Value, Growth or Interest Rates and Time Essentials of Managerial Finance by S. Besley & E. Brigham Slide 11 of 48 Present Value Definition and Formula • 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 • PV0 = FVn[1/(1+k)n] = FV(PVIFk,n) Essentials of Managerial Finance by S. Besley & E. Brigham Slide 12 of 48 Present Value - Example • Suppose an investor needs €10,000 in two years for the down payment on a new car. If the investor can earn 6% annually, how much does he need to invest today? – PV = 10,000 / (1.06)2 = 8899.96 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 13 of 48 Present Value– Example using Excel How much is needed for an investment today in order to have €10,000 in 2 years if the investor can earn 6% interest on his investment? FV 10.000 Excel Function k 6,00% =PV (interest, periods, pmt, FV) n 2 PV? 8.899,96 =PV (.06, 2, , 10,000) Essentials of Managerial Finance by S. Besley & E. Brigham Slide 14 of 48 Financial Calculator Solution In the previous example: FV = €10000, k = 6.0%, n = 2 2 6 ? 0 10000 N I PV PMT FV -8899,96 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 15 of 48 A Graphic view of Present Value Relationship among Present Value, Growth or Interest Rates and Time Essentials of Managerial Finance by S. Besley & E. Brigham Slide 16 of 48 Solving for interest rates - Example • If a mutual fund investment that was bought six years ago at a price of €1000 is now worth €5525, what rate of return (k) has the investor already earned today? – FV = PV (1+k)n – 5525 = 1000 (1+k)6 – and hence k = 33%. Essentials of Managerial Finance by S. Besley & E. Brigham Slide 17 of 48 Solving for interest rates with Excel What rate of return has an investor earned from a €1000 investment bought 6 years ago that is worth today €5525? 1999 1.000 2000 1.127 PV 1.000 Excel Function =Rate(periods, pmt, PV, 2001 1.158 FV 5.525 FV) 2002 2.345 2003 3.985 n 6 =Rate(6, ,1000, 5525) 2004 4.677 2005 5.525 k? 33,0% Essentials of Managerial Finance by S. Besley & E. Brigham Slide 18 of 48 Financial Calculator Solution In the previous example: PV = €1000, FV = €5525, n = 6 6 ? -1000 0 5525 N I PV PMT FV 33 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 19 of 48 Solving for period - Example • If a security worth €712 is invested at 6 percent, how long will it take to grow to €848? – FV = PV (1+k)n – 848 = 712 (1+0.06)n – and hence n = 3. Essentials of Managerial Finance by S. Besley & E. Brigham Slide 20 of 48 Financial Calculator Solution In the previous example: PV = €712, FV = €848, k = 6% ? 6 -712 0 848 N I PV PMT FV 3 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 21 of 48 Relationship between interest rates and present value • For a given interest rate – the longer the time period, the lower the present value • For a given time period – the higher the interest rate, the smaller the present value Essentials of Managerial Finance by S. Besley & E. Brigham Slide 22 of 48 Future Value of an annuity Definition and Formula • Annuity—a series of equal payments that are made at equal intervals – Ordinary annuity—has cash flows that occur at the end of each period – Annuity due—has cash flows that occur at the 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 • FVAn = A (1+k)n - 1 = A(FVIFAk,n) k Essentials of Managerial Finance by S. Besley & E. Brigham Slide 23 of 48 FV of Ordinary Annuity - Example • Suppose an equal cash flow of deposits of €100 at the end of each year for five years at 3% per year. How much will these deposits grow? – The growth rate is given as 3%. Therefore FVA = 100(1.03)0 + 100(1.03)1+ 100(1.03)2 + 100(1.03)3 + 100(1.03)4 = 530,91 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 24 of 48 Future Value of an Ordinary Annuity – using Excel How much will the deposits grow if the initial deposit is €100 at the end of each year at 3% interest for five years. PMT 100 Excel Function k 3,0% =FV (interest, periods, pmt, PV) n 5 FV? 530,91 =FV (.03, 5,100, ) Essentials of Managerial Finance by S. Besley & E. Brigham Slide 25 of 48 Financial Calculator Solution In the previous example: PMT = €100, k = 3%, n=5 5 3 0 -100 ? N I PV PMT FV 530,91 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 26 of 48 FV of an Annuity Due - Example • Suppose an equal cash flow of deposits of €100 at the beginning of each year for five years at 3% per year. How much will these deposits grow? – The growth rate is given as 3%. Therefore FVA = 100(1.03)1 + 100(1.03)2+ 100(1.03)3 + 100(1.03)4 + 100(1.03)5 = 546,84 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 27 of 48 Future Value of an Annuity Due – using Excel How much will the deposits grow if the initial deposit is €100 at the beginning of each year at 3% interest for five years. PMT 100 Excel Function k 3,00% =FV (interest, periods, pmt, PV) n 5 =FV (.03, 5,100, ) FV 530,91 FVA? 546,84 =530.91*(1.03) Essentials of Managerial Finance by S. Besley & E. Brigham Slide 28 of 48 Financial Calculator Solution In the previous example: PMT = €100, k = 3%, n=5 (switch calculator to BEGIN) 5 3 0 -100 ? N I PV PMT FV 546,84 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 29 of 48 Present Value of an annuity Definition and Formula • The present value of an annuity, FVA, can be computed by solving for the future value of a lump-sum amount •Annuity due is an annuity with cash flows that occur at the beginning of the period. • PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n) k Essentials of Managerial Finance by S. Besley & E. Brigham Slide 30 of 48 PV of an Ordinary Annuity - Example • Suppose an equal cash flow of payments of €1000 at the end of each year. How much could an investor borrow if he could afford annual payments of $1,000 (which includes both principal and interest) at the end of each year for five years at 10% interest? – The present value of the annuity is calculated as follows : PVA = 1000/(1.1)1 + 1000/(1.1)2+ 1000/(1.1)3 + 1000/(1.1)4 + 1000/(1.1)5 = 3790,79 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 31 of 48 Present Value of an Ordinary Annuity – using Excel •How much could an investor borrow if he could afford annual payments of €1,000 (which includes both principal and interest) at the end of each year for five years at 10% interest? PMT 1000 Excel Function I 10,0% =PV (interest, periods, pmt, FV) n 5 PV? 3.790,79 =PV (.10, 5, 1000, ) Essentials of Managerial Finance by S. Besley & E. Brigham Slide 32 of 48 Financial Calculator Solution In the previous example: PMT = €1000, k = 10%, n=5 5 10 ? -1000 0 N I PV PMT FV 3790,79 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 33 of 48 PV of an Annuity Due- Example • Suppose an equal cash flow of payments of €1000 at the beginning of each year. How much could an investor borrow if he could afford annual payments of $1,000 (which includes both principal and interest) at the end of each year for five years at 10% interest? – The present value of the annuity is calculated as follows : PVA = 1000/(1.1)0 + 1000/(1.1)1+ 1000/(1.1)2 + 1000/(1.1)3 + 1000/(1.1)4 = 4169,87 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 34 of 48 Present Value of an Annuity Due– using Excel •How much could an investor borrow if he could afford annual payments of €1,000 (which includes both principal and interest) at the beginning of each year for five years at 10% interest? PMT 1000 Excel Function k 10,00% =PV (interest, periods, pmt, FV) n 5 PV 3790,79 =PV (.10, 5, 1000, ) PVA? 4169,87 = 3790,79*(1,1) Essentials of Managerial Finance by S. Besley & E. Brigham Slide 35 of 48 Financial Calculator Solution In the previous example: PMT = €1000, k = 10%, n=5 (switch calculator to begin) 5 10 ? -1000 0 N I PV PMT FV 4169,87 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 36 of 48 Solving for interest rates with annuities- Example • If an investor pays €846,80 for an investment that promises to pay €250 per year for the next four years, what rate of return (k) will the investor earn on the investment? – Assuming that payments are made at the end of each year, this is an ordinary annuity. The solution from the annuity equation provides k=7%. Beware that trial and error process should be used. Essentials of Managerial Finance by S. Besley & E. Brigham Slide 37 of 48 Financial Calculator Solution In the previous example: PV = €846,80, PMT = €250, n = 4 4 ? -846,80 250 0 N I PV PMT FV 7 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 38 of 48 Solving for interest rates - Example • If an investor pays €1685 for an investment that promises to pay him back €400 per year, how many payments must he receive to earn a 6% return? – Assuming that payments are made at the end of each year, this is an ordinary annuity. The solution from the annuity equation provides n=5%. Beware that trial and error process should be used. Essentials of Managerial Finance by S. Besley & E. Brigham Slide 39 of 48 Financial Calculator Solution In the previous example: PV = €1685, PMT = €400, k = 6 ? 6 -1685 400 0 N I PV PMT FV 5 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 40 of 48 Present Value of a Perpetuity • A perpetuity is a special kind of annuity. • With a perpetuity, the periodic annuity or cash flow stream continues forever. PV = Annuity/k • For example, how much would an investor have to deposit today in order to withdraw €1,000 each year forever if the investor can earn 8% return? PV = €1,000/.08 = $12,500 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 41 of 48 Uneven Cash Flow Streams • In an uneven cash flow stream, the cash flows are not the same (equal). • Simplifying techniques, i.e. the use of a single equation to compute PV cannot be used Essentials of Managerial Finance by S. Besley & E. Brigham Slide 42 of 48 Present Value of an uneven Cashflow Stream - Example • Calculate the present value of the following uneven cashflow stream assuming a required return of 9%. Year Cash Flow PVIF9% ,N PV 1 400 0,917 366,80 2 800 0,842 673,60 3 500 0,772 386,00 4 400 0,708 283,20 5 300 0,650 195,00 PV 1.904,60 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 43 of 48 Present Value of an uneven Cashflow Stream - Example using Excel • Find the present value of the following uneven cashflow stream assuming a required return of 9%. Ye ar Cas h Flow 1 400 Excel Function 2 800 3 500 =NPV (interest, cells containing CFs) 4 400 5 300 =NPV (.09,B3:B7) NPV 1.904,76 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 44 of 48 Compounding More Frequently than Annually • Interest is compounded more than once per year— quarterly, monthly, or daily. The more frequent the compounding, the more the investor earns because he is earning on interest more frequently • Therefore, the effective interest rate (the rate of return per year considering interest compounding) is greater than the nominal (annual) interest rate. Essentials of Managerial Finance by S. Besley & E. Brigham Slide 45 of 48 Annual and Semi-annual compounding • For example, what would be the difference in future value if the depositors puts €100 for 5 years in the bank and earns 3% annual interest compounded (a) annually, (b) semiannually? Annually: 100 x (1 + .03)5 = €115.92 Semiannually: 100 x (1 + .015)10= €116.05 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 46 of 48 Nominal & Effective Rates • The nominal interest rate is the stated or contractual rate of interest charged by a lender or promised by a borrower. • The effective interest rate is the rate actually paid or earned. • In general, the effective rate > nominal rate whenever compounding occurs more than once per year EAR = (1 + k/m) m -1 Essentials of Managerial Finance by S. Besley & E. Brigham Slide 47 of 48 Nominal & Effective Rates • Example: Paul bought a vacation package for winter holidays and charged it to his credit card. What is the effective rate of interest on Paul’s credit card if the nominal rate is 18% per year, compounded monthly? EAR = (1 + .18/12) 12 -1 EAR = 19.56% Essentials of Managerial Finance by S. Besley & E. Brigham Slide 48 of 48

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