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CHAPTER 5 The Time Value of Money CHAPTER ORIENTATION In this chapter the concept of a time value of money is introduced, that is, a dollar today is worth more than a dollar received a year from now. Thus if we are to logically compare projects and financial strategies, we must either move all dollar flows back to the present or out to some common future date. CHAPTER OUTLINE I. Compound interest results when the interest paid on the investment during the first period is added to the principal and during the second period the interest is earned on the original principal plus the interest earned during the first period. A. Mathematically, the future value of an investment if compounded annually at a rate of i for n years will be FVn = PV (l + i)n where n = the number of years during which the compounding occurs i = the annual interest (or discount) rate PV = the present value or original amount invested at the beginning of the first period FVn = the future value of the investment at the end of n years 1. The future value of an investment can be increased by either increasing the number of years we let it compound or by compounding it at a higher rate. 2. If the compounded period is less than one year, the future value of an investment can be determined as follows: i mn FVn = PV 1 + m Where m = the number of times compounding occurs during the year 90 II. Determining the present value, that is, the value in today's dollars of a sum of money to be received in the future, involves nothing other than inverse compounding. The differences in these techniques come about merely from the investor's point of view. A. Mathematically, the present value of a sum of money to be received in the future can be determined with the following equation: 1 PV = FVn n (1 + i) where: n = the number of years until payment will be received, i = the opportunity rate or discount rate PV = the present value of the future sum of money FVn = the future value of the investment at the end of n years 1. The present value of a future sum of money is inversely related to both the number of years until the payment will be received and the opportunity rate. III. An annuity is a series of equal dollar payments for a specified number of years. Because annuities occur frequently in finance, for example, bond interest payments, we treat them specially. A. A compound annuity involves depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow. 1. This can be done by using our compounding equation and compounding each one of the individual deposits to the future or by using the following compound annuity equation: n 1 FVn = PMT (1 i) t t 0 where: PMT = the annuity value deposited at the end of each year i = the annual interest (or discount) rate n = the number of years for which the annuity will last FVn = the future value of the annuity at the end of the nth year B. Pension funds, insurance obligation, and interest received from bonds all involve annuities. To compare these financial instruments we would like to know the present value of each of these annuities. 1. This can be done by using our present value equation and discounting each one of the individual cash flows back to the present or by using the following present value of an annuity equation: n 1 PV = PMT t 1 (1 i) t 91 where: PMT = the annuity withdrawn at the end of each year i = the annual interest or discount rate PV = the present value of the future annuity n = the number of years for which the annuity will last C. This procedure of solving for PMT, the annuity value when i, n, and PV are known, is also the procedure used to determine what payments are associated with paying off a loan in equal installments. Loans paid off in this way, in periodic payments, are called amortized loans. Here again we know three of the four values in the annuity equation and are solving for a value of PMT, the annual annuity. IV. Annuities due are really just ordinary annuities where all the annuity payments have been shifted forward by one year, compounding them and determining their present value is actually quite simple. Because an annuity due merely shifts the payments from the end of the year to the beginning of the year, we now compound the cash flows for one additional year. V. A perpetuity is an annuity that continues forever, that is every year from now on this investment pays the same dollar amount. A. An example of a perpetuity is preferred stock which yields a constant dollar dividend infinitely. B. The following equation can be used to determine the present value of a perpetuity: pp PV = i where: PV = the present value of the perpetuity pp = the constant dollar amount provided by the perpetuity i = the annual interest or discount rate VI. (Financial Tables) V. Spreadsheets and the Time Value of Money. A. While there are several competing spreadsheets, the most popular one is Microsoft Excel. Just as with the keystroke calculations on a financial calculator, a spreadsheet can make easy work of most common financial calculations. Listed below are some of the most common functions used with Excel when moving money through time: Calculation: Formula: Present Value = PV(rate, number of periods, payment, future value, type) Future Value = FV(rate, number of periods, payment, present value, type) Payment = PMT(rate, number of periods, present value, future value, type) Number of Periods = NPER(rate, payment, present value, future value, type) Interest Rate = RATE(number of periods, payment, present value, future value, type, guess) 92 where: rate = i, the interest rate or discount rate number of periods = n, the number of years or periods payment = PMT, the annuity payment deposited or received at the end of each period future value = FV, the future value of the investment at the end of n periods or years present value = PV, the present value of the future sum of money type = when the payment is made, (0 if omitted) 0 = at end of period 1 = at beginning of period guess = a starting point when calculating the interest rate, if omitted, the calculations begin with a value of 0.1 or 10% 93