FIN3403: Corporate Finance Chapter 4: Introduction to Valuation: The Time Value of Money In the most general sense, the phrase time value of money refers to the fact that a dollar in hand 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 Compounding Future value (FV) refers to the amount of money an investment will grow to over some period of time at some given interest rate. Put another way, future value is the cash value of an investment at some time in the future Investing for a Single Period In general, if you invest for one period at an interest rate of your will grow to (1 per dollar invested. Investing for More Than One Period Going back to our $100 investment, what will you have after two years, assuming the interest rate doesn't change? If you leave the entire $110 in the bank, you will earn $110 + 10 = $11 in Interest during the 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 $110 at 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 result compound interest. Compound Interest: Interest earned on both the initial principle and the interest reinvested from prior periods With Simple Interest, the interest is not reinvested, so interest is earned each period only on the original principle. As our examples suggest, the future value of $1 invested for / periods at a rate of r per period is: PV Future value S1 The expression (1 + r)^1 is sometimes called the future value interest factor (or future value factor) for $1 invested at r percent for I periods. To solve future value problems, we need to come up with the relevant future value factors. 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 calculators have a key labeled "y^x"
FIN3403: Corporate Finance Chapter 4: Introduction to Valuation: The Time Value of Money You 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 common interest rates and time periods. To use the table, find the column that corresponds to 10 percent Then look down the rows until you come to five periods. You should find the factor that we calculated, 1.6105. Interest Rates # of Periods 5% 10% 15% 20% 1 1.0500 1.1000 1.1500 1.2000 1.1025 1.2100 1.3225 .4400 3 1.1576 1.3310 1.5209 1.7280 4 1.2155 1.4641 1.7490 2.0736 5 1.2763 6106 2.0114 2.4883 Tables like this one are not as common as they once were because they predate inexpensive 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 deal with these accurately would be quite large. As result, the "real world" has moved away from using them. We will emphasize the use of a calculator in this chapter. Calculator Hints *Using A Financial Calculator Although there are the various ways of calculating future values we have described so far, 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, it known some of the most commonly used financial formulas, so it can directly compute things like futures values. Financial calculators have the advantage that they handle a lot of the computation, but that is really all. In other words, you still have to understand the problem; the calculator does some of the arithmetic. In fact, there is an old joke (somewhat modified) that goes like this: Anyone can make a mistake on a time value of money problem, but to really screw one up takes a financial calculator! We therefore have two goals for this section. First, we'll discuss how to compute future values. After that, we'll show you how to avoid the common mistakes people make when they start using financial calculators How to Calculate Future Values with a Financial Calculator *Examining a typical financial calculator, you will find five keys of particular interest. They usually look like this: N I/Y PMT PV FV
FIN3403: Corporate Finance Chapter 4: Introduction to Valuation: The Time Value of Money For now, we need to focus on four of these. The keys labeled PV and FV are what you would guess: present value and future value. The key labeled N refers to the number of periods, which is what we have been calling 1. Finally, I/Y stands for the interest rate, which we have called r. The Single-Period Case In 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 money forward into the future. we discount it back to the present. Saving Up As you have probably recognized by now, calculating present values is quite similar to calculating 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 to discount a future cash flow it is often called a discount factor. With this name, it is not surprising 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 is called the present value interest factor (or just present value factor) for $1 at r percent for I periods and is sometimes abbreviated as PVIF (r, 1). Finally, calculating the present value of future cash flow to determine its worth today commonly called discounted cash flow (DCF) valuation. Present versus Future Value What we called the present value factor is just the reciprocal of (i.e., 1 divided by) the future value factor: Future value factor = (1 Present value factor=1/(1+r) Stand for the future value after I periods, then the relationship between future value and present 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 the text. Determining the Discount Rate It will turn out that we frequently need to determine what discount rate is Implicit in an Investment 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 can always find the fourth.
FIN3403: Corporate Finance Chapter 4: Introduction to Valuation: The Time Value of Money Finding the Number of Periods To come up with the exact answer, we again can manipulate the basic present value equation. The present value is $25,000, and the future value is $50,000. With a 12 percent discount 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 future value 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 to explicitly solve for t. Using A Spreadsheet For Time value of Money Calculations More and more, businesspeople from many different areas (and not just finance and accounting)rely on spreadsheets to do all the different types of calculations that come up in the real world. As a result, in this section, we show you how to use a spreadsheet to handle the various time 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. With a spreadsheet, there is a separate formula for each. In Excel, these are as follows: To Find Enter This Formula Future 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, the spreadsheet requires that the rate be entered as a decimal. Second, as with most financial calculators, you have to put negative sign on either the present value or the future value to solve for the rate or the number of periods. For the same reason, if you solve for a present value, the answer will have a negative sign unless you input a negative future value. The same is true when you compute a future value.