Chapter 4: Introduction to Valuation the Time Value of Money

FIN3403: Corporate FinanceChapter 4: Introduction to Valuation: The Time Value of MoneyIn the most general sense, the phrase time value of money refers to the fact that a dollar inhand today is worth more than a dollar promised at some time in the future.On a practical level, one reason for this is that you could earn interest while you waited;so a dollar today would grow to more than a dollar later.The trade-off between money now and money later thus depends on, among other things,the rate you can earn by investing.Future Value and CompoundingFuture value (FV) refers to the amount of money an investment will grow to over someperiod of time at some given interest rate.Put another way, future value is the cash value of an investment at some time in thefutureInvesting for a Single PeriodIn general, if you invest for one period at an interest rate of your will growto(1per dollar invested.Investing for More Than One PeriodGoing back to our $100 investment, what will you have after two years, assuming theinterest rate doesn't change?If you leave the entire $110 in the bank, you will earn $110 + 10 = $11 in Interest duringthe second year, so you will have a total of $110 + 11 = $121.This $121 is the future value of $100 in two years at 10 percent.Another way of looking at it is that one year from now you are effectively investing $110at 10 percent for a year.This is a single-period problem, so you'l end up with $1.10 for every dollar invested, or$110 X 1.1 = $121 total.Compounding the interest means earning interest on interest, so we call the resultcompound interest.Compound Interest: Interest earned on both the initial principle and the interestreinvested from prior periodsWith Simple Interest, the interest is not reinvested, so interest is earned each period onlyon the original principle.As our examples suggest, the future value of $1 invested for / periods at a rate of r perperiod is:PVFuturevalueS1The expression (1 + r)^1 is sometimes called the future value interest factor (or futurevalue factor) for $1 invested at r percent for I periods.To solve future value problems, we need to come up with the relevant future valuefactors. There are several different ways of doing this.In our example, we could have multiplied 1.1 by itself five times. This would work fine,but it would get to be very tedious for, say, a 30-year investment.Fortunately, there are several easier ways to get future value factors. Most calculatorshave a key labeled "y^x" FIN3403: Corporate FinanceChapter 4: Introduction to Valuation: The Time Value of MoneyYou can usually enter 1.1, press this key, enter 5. and press the key to get the answer.This is an easy way to calculate future value factors because it's quick and accurate.Alternatively, you can use a table that contains future value factors for some commoninterest rates and time periods.To use the table, find the column that corresponds to 10 percent Then look down therows until you come to five periods. You should find the factor that we calculated,1.6105.Interest Rates# of Periods5%10%15%20%11.05001.10001.15001.20001.10251.21001.3225.440031.15761.33101.52091.728041.21551.46411.74902.073651.276361062.01142.4883Tables like this one are not as common as they once were because they predateinexpensive calculators and are only available for a relatively small number of rates.Interest rates often are quoted to three or four decimal places, so the tables needed to dealwith these accurately would be quite large.As result, the "real world" has moved away from using them. We will emphasize theuse of a calculator in this chapter.Calculator Hints*Using A Financial CalculatorAlthough there are the various ways of calculating future values we have described sofar, many of you will decide that a financial calculator is the way to go.If you are planning on using one, you should read this extended hint; otherwise, skip it.A financial calculator is an ordinary calculator with a few extra features. In particular, itknown some of the most commonly used financial formulas, so it can directly computethings like futures values.Financial calculators have the advantage that they handle a lot of the computation, butthat is really all. In other words, you still have to understand the problem; the calculatordoes some of the arithmetic.In fact, there is an old joke (somewhat modified) that goes like this: Anyone can make amistake on a time value of money problem, but to really screw one up takes a financialcalculator!We therefore have two goals for this section. First, we'll discuss how to compute futurevalues. After that, we'll show you how to avoid the common mistakes people make whenthey start using financial calculatorsHow to Calculate Future Values with a Financial Calculator*Examining a typical financial calculator, you will find five keys of particular interest. Theyusually look like this:NI/YPMTPVFV FIN3403: Corporate FinanceChapter 4: Introduction to Valuation: The Time Value of MoneyFor now, we need to focus on four of these. The keys labeled PV and FV are what youwould guess: present value and future value. The key labeled N refers to the number ofperiods, which is what we have been calling 1. Finally, I/Y stands for the interest rate,which we have called r.The Single-Period CaseIn other words, we know the future value here is $1, but what is the present value (PV)?The answer isn't too hard to figure out.Present value is thus the reverse of future value Instead of compounding the moneyforward into the future. we discount it back to the present.Saving UpAs you have probably recognized by now, calculating present values is quite similar tocalculating future values, and the general result looks much the same.The present value of $1 to be received / periods into the future at a discount rate of r is:PV = $1 X [1/(1 + =The quantity in brackets, 1/(1 + r)^t, goes by several different names. Because it's used todiscount a future cash flow it is often called a discount factor. With this name, it isnotsurprising that the rate used in the calculation is often called the discount rate.We tend to call it this in talking about present values. The quantity in brackets also iscalled the present value interest factor (or just present value factor) for $1 at r percent forI periods and is sometimes abbreviated as PVIF (r, 1).Finally, calculating the present value of future cash flow to determine its worth todaycommonly called discounted cash flow (DCF) valuation.Present versus Future ValueWhat we called the present value factor is just the reciprocal of (i.e., 1 divided by) thefuture value factor:Future value factor = (1Present value factor=1/(1+r)Stand for the future value after I periods, then the relationship between future value andpresent value can be written as one of the following:We will call this last result the basic present value equation, and we use it throughout thetext.Determining the Discount RateIt will turn out that we frequently need to determine what discount rate is Implicit in anInvestment We can do this by looking at the basie present value equation:There are only four parts to this equation: the present (PV), the future value (FV),the discount rate (r), and the life of the investment (t). Given any three of these, we canalways find the fourth. FIN3403: Corporate FinanceChapter 4: Introduction to Valuation: The Time Value of MoneyFinding the Number of PeriodsTo come up with the exact answer, we again can manipulate the basic present valueequation. The present value is $25,000, and the future value is $50,000. With a 12 percentdiscount rate, the basic equation takes one of the following forms:$25,000 = $50,000/1.12^t$50,000/$25,000 = 1.12^t=We thus have a future value factor of 2 for a 12 percent rate. We now need to solve for 1.If you look down the column, it corresponds to 12 percent, you will see that a futurevalue factor of 1.9738 occurs at six periods.It will thus take about six years, as we calculated. To get the exact answer, we have toexplicitly solve for t.Using A Spreadsheet For Time value of Money CalculationsMore and more, businesspeople from many different areas (and not just finance andaccounting)rely on spreadsheets to do all the different types of calculations that come upin the real world.As a result, in this section, we show you how to use a spreadsheet to handle the varioustime value of money problems we presented in this chapter.We will use Microsoft Excel, but the commands are similar for other types of software.We assume you are already familiar with basic spreadsheet operations.As we have seen, you can solve for any one of the following four potential unknowns:future value, present value, the discount rate, or the number of periods. Withaspreadsheet, there is a separate formula for each. In Excel, these are as follows:To FindEnter This FormulaFuture Value= FV (rate, nper, pmt, pv)Present value= PV (rate, nper, pmt, pv)Discount rate- RATE (nper, pmt, pv, fv)Number of periods= NPER (rate, pmt, pv, fv)In these formulas, pv and fv are present and future value; nper is the number of periods;and rate is the discount, or interest, rate.There are two things that are a little tricky here. First, unlike a financial calculator, thespreadsheet requires that the rate be entered as a decimal. Second, as with most financialcalculators, you have to put negative sign on either the present value or the future valueto solve for the rate or the number of periods. For the same reason, if you solve for apresent value, the answer will have a negative sign unless you input a negative futurevalue. The same is true when you compute a future value.