Time Value Money
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chapter_9.docxChapter 9: Time Value Analysis
Learning Objectives
After studying this chapter, readers will be able to:
· Explain why time value analysis is so important to healthcare financial management.
· Find the present and future values for lump sums, annuities, and uneven cash flow streams.
· Solve for interest rate and number of periods.
· Explain and apply the opportunity cost principle.
· Measure the financial return on an investment in both dollar and percentage terms.
· Create an amortization table.
· Describe and apply stated, periodic, and effective annual interest rates.
Introduction
The monetary (economic) value of any asset, whether a financial asset, such as a stock or a bond, or a real asset, such as a piece of diagnostic equipment or an ambulatory surgery center, is based on future cash flows. However, a dollar to be received in the future is worth less than a current dollar because a dollar in hand today can be invested in an interest-bearing account and hence can be worth more than one dollar in the future.[1] Because current dollars are worth more than future dollars, financial management decisions must account for cash flow timing differences.
The process of assigning appropriate values to cash flows that occur at different points in time is called time value analysis or discounted cash flow analysis. It is an important part of healthcare financial management because most financial analyses involve the valuation of future cash flows. In fact, of all the financial analysis techniques that are discussed in this book, none is more important than time value analysis. The concepts presented here are the cornerstones of most financial analyses, so a thorough understanding of time value concepts is essential to good financial decision making.
[1]Even if no investment opportunities existed, a dollar in hand would still be worth more than a dollar to be received in the future because a dollar today can be used for immediate consumption, whereas a future dollar cannot.
Time Lines
The creation of a time line is the first step in time value analysis, especially when first learning time value concepts. Time lines make it easier to visualize when the cash flows in a particular analysis occur. To illustrate the time line concept, consider the following five-period time line:
Time 0 is any starting point (typically the time of the first cash flow in an analysis); Time 1 is one period from the starting point, or the end of Period 1; Time 2 is two periods from the starting point, or the end of Period 2; and so on. Thus, the numbers on top of the tick marks represent end-of-period values. Often, the periods are years, but other time intervals such as quarters, months, or days are also used when needed to fit the timing of the cash flows being evaluated. If the time periods are years, the interval from 0 to 1 would be Year 1, and the tick mark labeled 1 would represent both the end of Year 1 and the beginning of Year 2.
Cash flows are shown on a time line directly below the tick marks that indicate the point in time that they are expected to occur. The interest rate that is relevant to the analysis is sometimes shown directly above the time line in the first period. Additionally, unknown cash flows—the ones to be determined in the analysis—are sometimes indicated by question marks. To illustrate, consider the following time line:
In this situation, the interest rate for each of the three periods is 5 percent, an investment of $100 is made at Time 0, and the Time 3 value is the unknown. The $100 is an outflow because it is shown as a negative cash flow. (Outflows are sometimes designated by parentheses rather than by minus signs.) In more complicated analyses, it is essential to use the proper signs to get the correct answer. Furthermore, many financial calculators and some spreadsheet functions require that signs be attached to cash flows in time value analyses, even simple ones, before the calculation can be completed. Thus, to ensure that readers are familiar with the sign convention used in time value analyses, we will use them on most illustrations.
Time lines are essential when learning time value concepts, but even experienced analysts use time lines when dealing with complex problems. The time line may be an actual line, as illustrated above, or it may be a series of columns (or rows) on a spreadsheet. Time lines will be used extensively in the remainder of this book, so get into the habit of creating time lines when conducting analyses that involve future cash flows.
Future Value of A Lump Sum (Compounding)
The process of going from today's values, or present values, to future values is called compounding. Although compounding is not used extensively in healthcare finance, it is the best starting point for learning time value analysis. To illustrate lump sum compounding, which deals with a single starting amount, suppose that the manager of Meridian Clinics deposits $100 in a bank account that pays 5 percent annual interest (interest is credited to the account at the end of each year). How much would be in the account at the end of one year? To begin, here are the terms that are used in this time value analysis:
· PV = $100 = present value, or beginning amount, of the account.
· I = 5% = interest rate the bank pays on the account per year. The interest amount, which is paid at the end of the year, is based on the balance at the beginning of each year. Note that in "by hand" time value calculations, I must be expressed as a decimal, so I = 0.05.
· INT = dollars of interest earned during each year, which equals the beginning amount multiplied by the interest rate. Thus, INT = PV × I.
· FVn = future value, or ending amount, of the account at the end of N years. Whereas PV is the value now, or present value, FVn is the value N years into the future after the interest earned has been added to the account.
· N = number of years involved in the analysis.
In this example, N = 1, so FVn can be calculated as follows:
The future value at the end of one year, FV1, equals the present value multiplied by (1.0 plus the interest rate). This future value relationship can be used to find how much $100 will be worth at the end of one year, if it is invested in an account that pays 5 percent interest:
What would be the value of the $ 100 if Meridian Clinics left the money in the account for five years? Here is a time line that shows the amount at the end of each year:
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Beginning amount —$100 |
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Interest earned |
$ 5 |
$ 5.25 |
$ 5.51 |
$ 5.79 |
$ 6.08 |
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End-of-year amount |
105 |
110.25 |
115.76 |
121.55 |
127.63. |
Note the following points:
· The account is opened with a deposit of $100. This is shown as an outflow at Year 0.
· Meridian earns $100 × 0.05 = $5 of interest during the first year, so the amount in the account at the end of Year 1 is $100 + $5 = $105.
· At the start of the second year, the account balance is $105. Interest of $105 × 0.05 = $5.25 is earned on the now larger amount, so the account balance at the end of the second year is $105 + $5.25 = $110.25. The Year 2 interest, $5.25, is higher than the first year's interest, $5, because $5 × 0.05 = $0.25 in interest was earned on the first year's interest.
· This process continues, and because the beginning balance is higher in each succeeding year, the interest earned increases in each year.
· The total interest earned, $27.63, is reflected in the final balance, $127.63, at the end of Year 5.
To better understand the mathematics of compounding, note that the Year 2 value, $110.25, is equal to:
Furthermore, the balance at the end of Year 3 is:
Continuing the calculation to the end of Year 5 gives:
These calculations show that a pattern clearly exists in future value calculations. In general, the future value of a lump sum at the end of N years can be found by applying this equation:
Future values, as well as most other time value problems, can be solved three ways: regular calculator, financial calculator, or spreadsheet.
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Regular Calculator Solution |
To use a regular calculator, multiply the PV by (1 + I) for N times or use the exponential function to raise (1 + I) to the Nth power and then multiply the result by the PV. Perhaps the easiest way to find the future value of $100 after five years when compounded at 5 percent is to enter $100, then multiply this amount by 1.05 for five times. If the calculator is set to display two decimal places, the answer would be $127.63:
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As denoted by the arrows, compounding involves moving to the right along the time line. In fact, the term compounding is used for finding future values because future values increase, or compound, over time.
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Financial Calculator Solution |
Financial calculators have been programmed to solve many types of time value analyses, including future value of a lump sum. In effect, the future value equation is programmed directly into the calculator. With a financial calculator, the future value is found using three of the following five time value input keys[2]:
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Note that these keys correspond to the five time value variables that are commonly used:
· N = number of periods.
· I = interest rate per period.
· PV = present value.
· PMT = payment. (This key is used only if the cash flows involve an annuity, which is a series of equal payments. Annuities are discussed in a later section.)
· FV = future value.
Also, note that this chapter deals with time value analyses that involve only four of the variables at any one time. Three of the variables will be known, and the calculator will solve for the fourth, unknown variable. In Chapter 11, when bond valuation is discussed, all five variables will be included in the analysis.
To find the future value of $100 after five years at 5 percent interest using a financial calculator, just enter PV = 100, 1 = 5, and N = 5, and then press the FV key. The answer, 127.63 (rounded to two decimal places), will appear. As stated previously, many financial calculators require that cash flows be designated as either inflows or outflows (entered as either positive or negative values). Applying this logic to the illustration, Meridian deposits the initial amount, which is an outflow to the firm, and takes out, or receives, the ending amount, which is an inflow to the firm. If the calculator requires this sign convention, the PV would be entered as —100. (If the PV was entered as 100, a positive value, the calculator would display —127.63 as the answer.) The calculator solution can be shown pictorially as follows:
Also, some calculators require the user to press a Compute key before pressing the FV key. Finally, financial calculators permit specifying the number of decimal places that are displayed, even though 12 (or more) significant digits are actually used in the calculations. Two places are generally used for answers in dollars or percentages, and four places for decimal answers. The final answer, however, should be rounded to reflect the accuracy of the input values; it makes no sense to say that the return on a particular investment is 14.63827 percent when the cash flows are highly uncertain. The nature of the analysis dictates how many decimal places should be displayed.
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Spreadsheet Solution |
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Spreadsheet programs, such as Excel, are ideally suited for time value analyses. For simple time value calculations, it is easy to enter the appropriate formula directly into the spreadsheet. For example, you could enter the spreadsheet version of the future value equation into Cell A6: =100*(1.05)^5. Here, = tells the spreadsheet that a formula is being entered into the cell; * is the spreadsheet multiplication sign; and ^ is the spreadsheet exponential, or power, sign. When this formula is entered into Cell A6, the value $127.63 appears in the cell (when formatted with a dollar sign to two decimal places). Note that different spreadsheet programs use slightly different syntax in their time value analyses. The examples presented in this text use Excel syntax.
In most situations, it is more useful to enter a formula that can accommodate changing input values than to embed these values directly in the formula, so it would be better to solve this future value problem with this formula: =A3*(1+A4)^A2, as done in Cell A8. Here, the present value ($100) is contained in Cell A3, the interest rate (0.05, which is displayed as 5%) in Cell A4, and the number of periods (5) in Cell A2. With this formula, future values can be easily calculated with different starting amounts, interest rates, or number of years by changing the values in the input cells.
In addition to entering the appropriate time value formulas, most time value solutions are preprogrammed in the spreadsheet software. The preprogrammed time value formulas are called functions. Like any formula, a time value function consists of a number of arithmetic calculations combined into one statement. By using functions, spreadsheet users can save the time and tedium of building formulas from scratch.
Each function begins with a unique name that identifies the calculation to be performed, along with one or more arguments (the input values for the calculation) enclosed in parentheses. The best way to access the time value functions is to use the spreadsheet's function wizard (also called the -paste function). For this future value problem, first move the cursor to Cell A10 (the cell where you want the answer to appear). Then click on the function wizard, select Financial for the function category and FV (future value) for the function name, and enter A4 for Rate, A2 for Nper (number of periods), and —A3 for Pv. (Note that the Pmt and Type entries are left blank for this problem. Also, note that the cell address entered for Pv has a minus sign. This is necessary for the answer to be displayed as a positive number.) Finally, press OK and the result, $127.63, appears in Cell A10.
Note that most of the spreadsheet solutions shown in this book follow a similar format. The input values and the output are contained in Column A. If a spreadsheet function is used in the solution, the input value (argument) names are shown in Column B to the right of the input values. In addition, the formula or function used to calculate the output is shown in Column B to the right of the output value. Finally, Column C contains the descriptive input names.
The most efficient way to solve most problems that involve time value is to use a financial calculator or spreadsheet.[3] However, the basic mathematics behind the calculations must be understood to set up complex problems before solving them. In addition, the underlying logic must be understood to comprehend stock and bond valuation, lease analysis, capital budgeting analysis, and other important healthcare financial management topics.
The Power of Compounding
The "power of compounding" is a phrase that emphasizes the fact that a relatively small starting value can grow to a large amount, even when the rate of growth (interest rate) is modest, when invested over a long period. For example, assume that a new parent places $1,000 in a stock mutual fund to help pay the child's college expenses, which are expected to begin in 18 years. The investment is assumed to earn a return of 10 percent per year, which is a reasonable estimate by historical standards. After 18 years, the value of the mutual fund account would be $5,560, which is not an inconsequential sum.
Now, assume that the money was meant to help fund the child's retirement, which is assumed to occur 65 years into the future. The value of the mutual fund account at that time would be $490,371, or nearly a half-million dollars. Imagine that: $1,000 grows to nearly half a million all because of the power of compounding. The moral of this story is clear: When saving for retirement, or for any other purpose, start early.
Present Value of A Lump Sum (Discounting)
Suppose that GroupWest Health Plans, which has premium income reserves to invest, has been offered the chance to purchase a low-risk security from a local broker that will pay $127.63 at the end of five years. A local bank is currently offering 5 percent interest on a five-year certificate of deposit (CD), and Group West's managers regard the security offered by the broker as having the same risk as the bank CD. The 5 percent interest rate available on the bank CD is Group West's opportunity cost rate. (Opportunity costs are discussed in detail in the next section.) How much would GroupWest be willing to pay for the security that promises to pay $127.63 in five years?
The future value example presented in the previous section showed that an initial amount of $100 invested at 5 percent per year would be worth $127.63 at the end of five years. Thus, GroupWest should be indifferent to the choice between $100 today and $127.63 at the end of five years. Today's $100 is defined as the present value, or PV, of $127.63 due in five years when the opportunity cost rate is 5 percent. If the price of the security being offered is exactly $100, GroupWest could buy it or turn it down because that is the security's "fair value." If the price is less than $100, GroupWest should buy it, while if the price is greater than $100, GroupWest should decline the offer.
Conceptually, the present value of a cash flow due N years in the future is the amount which, if it were on hand today, would grow to equal the future amount when compounded at the opportunity cost rate. Because $100 would grow to $127.63 in five years at a 5 percent interest rate, $100 is the present value of $127.63 due five years in the future when the opportunity cost rate is 5 percent. In effect, the present value tells us what amount would have to be invested to earn the opportunity cost rate. If the investment can be obtained for a lesser amount, a higher rate will be earned. If the investment costs more than the present value, the rate earned will be less than the opportunity cost rate.
Finding present values is called discounting, and it is simply the reverse of compounding: if the PV is known, compound to find the FV; if the FV is known, discount to find the PV. Here are the solution techniques used to solve this discounting problem.
To develop the discounting equation, solve the compounding equation for PV:
The equations show us that compounding problems are solved by multiplication, while discounting problems are solved by division.
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Regular Calculator Solution |
Enter $127.63 and divide it five times by 1.05:
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As shown by the arrows, discounting is moving left along a time line. As with compounding, the term discounting is descriptive. As we move left along a time line, values get smaller, or discount, over time.
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Financial Calculator Solution |
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Spreadsheet Solution |
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One solution would be to enter the applicable formula, as shown to the right of Cell A6: =A3/(1+a4)^a2. Here, the future value ($127.63) is contained in Cell A3, the interest rate (0.05, which is displayed as 5%) in Cell A4, and the number of periods (5) in Cell A2. With this formula, present values easily can be calculated with different starting future amounts, interest rates, or number of years.
The function approach is illustrated in Cell A8. First, move the cursor to that cell (the cell where you want the answer to appear). Then, click on the function wizard, select Financial for the function category and Pv (present value) for the function name, and enter A4 for Rate, A2 for Nper (number of periods), and —A3 for Fv. (Note that the Pmt and Type entries are left blank for this problem. Also, note that the cell address entered for Fv has a minus sign. This is necessary for the answer to be displayed as a positive number.) Finally, press OK and the result, $100.00, appears in Cell A8.
Discounting at Work
At relatively high interest rates, funds due in the future are worth very little today, and even at moderate discount rates, the present value of a sum due in the distant future is quite small. To illustrate discounting at work, consider 100-year bonds. A bond is a type of debt security in which an investor loans some amount of principal—say, $1,000—to a company (borrower), which in turn promises to pay interest over the life of the bond and to return the principal amount at maturity. Typically, the longest maturities for bonds are 30-40 years, but in the early 1990s, several companies, including Columbia/HCA Healthcare (now HCA), issued 100-year bonds.
At first blush, it might appear that anyone who would buy a 100-year bond must be irrational because there is little assurance that the borrower will even be around in 100 years to repay the amount borrowed. However, consider the present value of $1,000 to be received in 100 years. If the discount rate is 7.5 percent, which is roughly the interest rate that was set on the bond, the present value is a mere $0.72. Thus, the time value of money eroded the value of the bond's principal repayment to the point that it was worth less than $1 at the time the bond was issued. This tells us that the value of the bond when it was sold was based primarily on the interest stream received in the early years of ownership, and that the payments expected during the later years contributed little to the bond's initial $1,000 value. Thus, the risk of not recovering the initial $1,000 principal amount did not have a large impact on investors' willingness to buy the bond.
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Opportunity Costs
In the last section, the opportunity cost concept was used to set the discount rate on the time value analysis of Group West's investment offer. The opportunity cost concept plays a critical role in time value analysis. To illustrate, suppose an individual found the winning ticket for the Florida lottery and now has $1 million to invest. Should the individual assign a cost to these funds? At first blush it might appear that this money has zero cost because its acquisition was purely a matter of luck. However, as soon as the lucky individual thinks about what to do with the $ 1 million, he or she has to think in terms of the opportunity costs involved. By using the funds to invest in one alternative—for example, in the stock of Health Management Associates (HMA)—the individual forgoes the opportunity to make some other investment—for example, buying U.S. Treasury bonds. Thus, there is an opportunity cost associated with any investment planned for the $1 million, even though the lottery winnings were "free."
Because one investment decision automatically negates all other possible investments with the same funds, the cash flows expected to be earned from any investment must be discounted at a rate that reflects the return that could be earned on forgone investment opportunities. The problem is that the number of forgone investment opportunities is virtually infinite, so which one should be chosen to establish the opportunity cost rate? The opportunity cost rate to be applied in time value analysis is the rate that could be earned on alternative investments of similar risk. It would not be logical to assign a very low opportunity cost rate to a series of very risky cash flows, or vice versa. This concept is one of the cornerstones of healthcare finance, so it is worth repeating. The opportunity cost rate (i.e., the discount rate) applied to investment cash flows is the rate that could be earned on alternative investments of similar risk.
It is very important to recognize that the discounting process itself accounts for the opportunity cost of capital (i.e., the loss of use of the funds for other purposes). In effect, discounting a potential investment at, say, 10 percent, produces a present value that provides a 10 percent return. Thus, if the investment can be obtained for less than its present value, it will earn more than its opportunity cost of capital and hence is a good investment. Alternatively, if the cost of the investment is greater than its present value, it will earn less than its opportunity cost of capital and hence is a bad investment. It is also important to note that the opportunity cost rate does not depend on the source of the funds to be invested. Rather, the primary determinant of this rate is the riskiness of the cash flows being discounted. Thus, the same 10 percent opportunity cost rate would be applied to this potential investment regardless of whether the funds to be used for the investment were won in a lottery, taken out of petty cash, or obtained by selling some securities.
Generally, opportunity cost rates are obtained by looking at rates that could be earned, or more precisely, rates that are expected to be earned, on securities such as stocks or bonds. Securities are usually chosen to set opportunity cost rates because their expected returns are more easily estimated than rates of return on real assets such as hospital beds, MRI machines, and the like. Furthermore, as discussed in Chapter 12, securities generally provide the minimum return appropriate for the amount of risk assumed, so securities returns provide a good benchmark for other investments.
To illustrate the opportunity cost concept, assume that Oakdale Community Hospital is considering building a nursing home. The first step in the financial analysis is to forecast the cash flows that the nursing home is expected to produce. These cash flows, then, must be discounted at some opportunity cost rate to determine their present value. Would the hospital's opportunity cost rate be (1) the expected rate of return on a bank CD; (2) the expected rate of return on the stock of Manor Care, which operates a large number of nursing homes and assisted living centers; or (3) the expected rate of return on pork belly futures? (Pork belly futures are investments that involve commodity contracts for delivery at some future time.) The answer is the expected rate of return on Manor Care's stock because that is the rate of return available to the hospital on alternative investments of similar risk. Bank CDs are very low-risk investments, so they would understate the opportunity cost rate in owning a nursing home. Conversely, pork belly futures are very high-risk investments, so that rate of return is probably too high to apply to Oakdale's nursing home investment.[4]
The source of the funds used for the nursing home investment is not relevant to the analysis. Oakdale may obtain the needed funds by borrowing, by soliciting contributions, or by using excess cash accumulated from profit retention. The discount rate applied to the nursing home cash flows depends only on the riskiness of those cash flows and the returns available on alternative investments of similar risk, not on the source of the investment funds.
At this point, you may question the ability of real-world analysts to assess the riskiness of a cash flow stream or to choose an opportunity cost rate with any confidence. Fortunately, the process is not as difficult as it may appear here because businesses have benchmarks that can be used as starting points. (Chapter 13 contains a discussion of how benchmark opportunity cost rates are established for capital investments, while Chapter 15 presents a detailed discussion on how the riskiness of a cash flow stream can be assessed.)
Solving for Interest Rate and Time
At this point, it should be obvious that compounding and discounting are reciprocal processes. Furthermore, four time value analysis variables have been presented: PV, FV, I, and N. If the values of three of the variables are known, the value of the fourth can be found with the help of a financial calculator or spreadsheet. Thus far, the interest rate, I, and the number of years, N, plus either PV or FV have been given in the illustrations. In some situations, however, the analysis may require solving for either I or N.[5]
Solving for Interest Rate (I)
Suppose that Family Practice Associates (FPA), a primary care group practice, can buy a bank CD for $78.35 that will return $100 after five years. In this case, PV, FV, and N are known, but I, the interest rate that the bank is paying, is not known.
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Financial Calculator Solution |
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Spreadsheet Solution |
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Here, the spreadsheet function named RATE is used to solve for I, as illustrated to the right of Cell A8. First, click on the function wizard, select Financial for the function category and RATE for the function name, and enter A2 for Nper (number of periods), A3 for Pv (present value), and A4 for Fv (future value). (Note that the Pmt and Type entries are left blank for this problem. Also note that the Pv was entered as a negative number, as shown on the time line.) Finally, press OK and the result, 5%, appears in Cell A8. (Note that some spreadsheet programs display the answer in decimal form unless the cell is formatted to display in percent.)
Solving for Time (N)
Suppose that the bank told FPA that a certificate of deposit pays 5 percent interest each year, that it costs $78.35, and that at maturity the group would receive $100. How long must the funds be invested in the CD? In this case, Pv, Fv, and I are known, but N, the number of periods, is not known.
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Financial Calculator Solution |
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Spreadsheet Solution |
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To solve for time, the spreadsheet function named NPER (number of periods) is used. To begin, place the cursor in Cell A8 and click on the function wizard. Then, select Financial for the function category and NPER for the function name, and enter A2 for Rate, A3 for Pv, and A4 for Fv. (Note that the Pmt and Type entries are left blank for this problem. Also, note that the Pv was entered as a negative number, as shown on the time line.) Finally, press OK and the result, 5.0, appears in Cell A8.
Annuities
Whereas lump sums are single values, an annuity is a series of equal payments at fixed intervals for a specified number of periods. Annuity payments, which are given the symbol PMT or Pmt, can occur at the beginning or end of each period. If the payments occur at the end of each period, as they typically do, the annuity is an ordinary, or deferred, annuity. If payments are made at the beginning of each period, the annuity is an annuity due. Because ordinary annuities are far more common in time value problems, when the term annuity is used in this book (or in general), payments are assumed to occur at the end of each period. Furthermore, we begin our discussion of annuities by focusing on ordinary annuities.
Ordinary Annuities
If Meridian Clinics were to deposit $100 at the end of each year for three years in an account that paid 5 percent interest per year, how much would Meridian accumulate at the end of three years? The answer to this question is the future value of the annuity, which for ordinary annuities coincides with the final payment.
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Regular Calculator Solution |
One approach to the problem is to compound each individual cash flow to Year 3.
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Financial Calculator Solution |
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In annuity problems, the PMT key is used in conjunction with either the PV or FV key.
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Spreadsheet Solution |
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Here, we again use the future value function, but now we will use the payment (Pmt) entry in the function wizard to recognize that the problem involves annuities. Place the cursor in Cell A8. Then, click on the function wizard, select Financial for the function category and FV (future value) for the function name, and enter A4 for Rate, A2 for Nper (number of periods), and A3 for Pmt. (Note that the Pv and Type entries are left blank for this problem.) Finally, press OK and the result, $315.25, appears in Cell A8.
Suppose that Meridian Clinics was offered the following alternatives: a three-year annuity with payments of $100 at the end of each year or a lump sum payment today. Meridian has no need for the money during the next three years. If it accepts the annuity, it would deposit the payments in an account that pays 5 percent interest per year. Similarly, the lump sum payment would be deposited into the same account. How large must the lump sum payment be today to make it equivalent to the annuity? The answer to this question is the present value of the annuity, which for ordinary annuities occurs one period prior to the first payment.
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Regular Calculator Solution |
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Financial Calculator Solution |
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Spreadsheet Solution |
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Here we use the present value function, but again with a payment entry to recognize that the problem involves annuities. Place the cursor in Cell A8. Then click on the function wizard, select Financial for the function category and PV for the function name, and enter A4 for Rate, A2 for Nper (number of periods), and A3 for Pmt. (Note that the Fv and Type entries are left blank for this problem.) Finally, press OK and the result, $272.32, appears in Cell A8.
One especially important application of the annuity concept relates to loans with constant payments, such as mortgages, auto loans, and many bank loans to businesses. Such loans are examined in more depth in a later section on amortization.
Annuities Due
If the three $100 payments in the previous example had been made at the beginning of each year, the annuity would have been an annuity due. The future value of an annuity due occurs one period after the final payment, while the future value of a regular annuity coincides with the final payment. Here are the solution techniques for the future value of an annuity due.
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Regular Calculator Solution |
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In the case of an annuity due, as compared with an ordinary annuity, all the cash flows are compounded for one additional period, and hence the future value of an annuity due is greater than the future value of a similar ordinary annuity by (1 + I). Thus, the future value of an annuity due also can be found as follows:
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Financial Calculator Solution |
Most financial calculators have a switch or key marked DUE or BEGIN that permits the switching of the mode from end-of-period payments (ordinary annuity) to beginning-of-period payments (annuity due). When the beginning-of-period mode is activated, the calculator will normally indicate the changed mode by displaying the word BEGIN or some other symbol. To deal with annuities due, change the mode to the beginning of period and proceed as before. Because most problems will deal with end-of-period cash flows, do not forget to switch the calculator back to the END mode. |
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Spreadsheet Solution |
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One approach (as shown in Cell A6) is to use the spreadsheet future value (FV) function but with a "1" entered for Type (as opposed to a blank). Now the spreadsheet treats the entries as an annuity due, and $331.01 is displayed as the answer.
As an alternative, note that the solution is the same as for an ordinary annuity, except the result must be multiplied by (1 + Rate), which is (1 + A4) in this example. This solution approach is given in Cell A8. The result, $331.01, is the future value of the annuity due.
Here are the solution techniques for the present value of an annuity due.
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Regular Calculator Solution |
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Because the payments are shifted to the left, each one is discounted for one less year. Thus, the present value of an annuity due is larger than that of a similar regular annuity.
Note that the present value of an annuity due can be thought of as the present value of an ordinary annuity that is compounded for one additional period, so it also can be found as follows:
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Financial Calculator Solution |
Activate the beginning of period mode (i.e., the BEGIN mode), and then proceed as before. Again, because most problems will deal with end-of-period cash flows, do not forget to switch the calculator back to the END mode. |
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Spreadsheet Solution |
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As with future value, one approach (as shown in Cell A6) is to use the spreadsheet present value (PV) function but with a "1" entered for Type (as opposed to a blank). Now the spreadsheet treats the entries as an annuity due, and $285.94 is displayed as the answer.
Note that the alternative solution is the same as for an ordinary annuity, except the function in Cell A8 is multiplied by (1 + A4). The result, $285.94, is the present value of the annuity due.
Perpetuities
Most annuities call for payments to be made over some finite period of time— for example, $100 per year for three years. However, some annuities go on indefinitely, or perpetually, and hence are called perpetuities. The present value of a perpetuity is found as follows:
Perpetuities can be illustrated by some securities issued by the Canadian Healthcare Board. Each security promises to pay $100 annually in perpetuity (forever). What would each security be worth if the opportunity cost rate, or discount rate, is 10 percent? The answer is $1,000:
Or, using a spreadsheet, merely enter the perpetuity formula into a cell, as shown here in Cell A8.
Suppose interest rates, and hence the opportunity cost rate, rose to 15 percent. What would happen to the security's value? The interest rate increase would lower its value to $666.67:
Assume that interest rates fell to 5 percent. The rate decrease would increase the perpetuity's value to $2,000:
The value of a perpetuity changes dramatically when opportunity costs (interest rates) change. All securities' values are affected by interest rate changes, but some, like perpetuities, are more sensitive to interest rate changes than others, such as short-term government bonds. The risks associated with interest rate changes are discussed in more detail in Chapter 11.
Uneven Cash Flow Streams
The definition of an annuity (or perpetuity) includes the words "constant amount," so annuities involve payments that are the same in every period. Although some financial decisions, such as bond valuation, do involve constant payments, most important healthcare time value analyses involve uneven, or nonconstant, cash flows. For example, the financial evaluation of a proposed outpatient clinic or MRI facility rarely involves constant cash flows.
In general, the term lump sum is used with a single cash flow; the term payment (PMT) is reserved for annuity situations in which there are multiple constant lump sums; and the term cash flow (CF) is used when there is a series of uneven lump sums. Financial calculators are set up to follow this convention. When dealing with uneven cash flows, the CF function, rather than the PMT key, is used.
Present Value
The present value of an uneven cash flow stream is found as the sum of the present values of the individual cash flows of the stream. For example, suppose that Wilson Memorial Hospital is considering the purchase of a new x-ray machine. The hospital's managers forecast that the operation of the new machine would produce the following stream of cash inflows (in thousands of dollars):
What is the present value of the new x-ray machine investment if the appropriate discount rate (i.e., the opportunity cost rate) is 10 percent?
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Regular Calculator Solution |
The PV of each lump sum cash flow can be found using a regular calculator, and then these values are summed to find the present value of the stream, $580,950:
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Financial Calculator Solution |
The present value of an uneven cash flow stream can be solved with most financial calculators by using the following steps: · Input the individual cash flows, in chronological order, into the cash flow register, where they usually are designated as CF0 and CFj (CF1, CF2, CF3, and so on) or just CFj (CF0, CF1, CF2, CF3, and so on). · Enter the discount rate. · Push the NPV key. |
For this problem, enter 0, 100, 120, 150, 180, and 250 in that order into the calculator's cash flow register; enter 1 = 10; then push NPV to obtain the answer, 580.95. Note that an implied cash flow of zero is entered for CF0.
Three points should be noted about the calculator solution. First, when dealing with the cash flow register, the term NPV, rather than PV, is used to represent present value. The letter N in NPV stands for the word net, so NPV is the abbreviation for net present value. Net present value means the sum or net of the present values of a cash flow stream. Often, the stream will consist of both inflows and outflows, but the stream here contains all inflows.
Second, annuity cash flows within any uneven cash flow stream can be entered into the cash flow register most efficiently on most calculators by using the Nj key. This key allows the user to specify the number of times a constant payment occurs within the stream. (Some calculators prompt the user to enter the number of times each cash flow occurs.)
Finally, amounts entered into the cash flow register remain there until the register is cleared. Thus, if a problem had been previously worked with eight cash flows, and a problem is worked with only four cash flows, the calculator assumes that the final four cash flows from the first calculation belong to the second calculation. Be sure to clear the register before starting a new time value analysis.
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Spreadsheet Solution |
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The NPV function calculates the present value of a stream, called a spreadsheet range, of cash flows. First, the cash flow values must be entered into consecutive cells in the spreadsheet, as shown above in Cells A4 through A8. Next, the discount (opportunity cost) rate must be placed into a cell (as in Cell A2 above). Then, place the cursor in Cell A10, use the function wizard to select Financial and NPV, and then enter A2 as Rate and A4 A8 as Value 1. Press OK and the value $580.95 is displayed in the cell. (Note that the Value 1 entry is the range of cash flows contained in Cells A4 through A8. Also, note that NPV stands for net -present value, which indicates that the resulting present value is the net of the present values of two or more cash flows.)
The NPV function assumes that cash flows occur at the end of each period, so NPV is calculated as of the beginning of the period of the first cash flow specified in the range, which is one period before that cash flow occurs. Because the cash flow specified as the first flow in the range is a Year 1 value, the calculated NPV occurs at the beginning of Year 1, or the end of Year 0, which is correct for this illustration. However, if a Year 0 cash flow is included in the range, the NPV would be calculated at the beginning of Year 0, or the end of Year —1, which typically is incorrect. This problem will be addressed in the next major section.
Future Value
The future value of an uneven cash flow stream is found by compounding each payment to the end of the stream and then summing the future values.
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Regular Calculator Solution |
The future value of each lump sum cash flow can be found, using a regular calculator, by summing these values to find the future value of the stream, $935,630:
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Financial Calculator Solution |
Some financial calculators have a net future value key (NFV) that, after the cash flows have been entered into the cash flow register, can be used to obtain the future value of an uneven cash flow stream. However, analysts generally are more concerned with the present value of a cash flow stream than with its future value. The reason, of course, is that the present value represents the value of the investment today, which then can be compared to the cost of the investment—whether a stock, bond, x-ray machine, or new clinic—to make the investment decision. |
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Spreadsheet Solution |
Most spreadsheet programs do not have a function that computes the future value of an uneven cash flow stream. However, future values can be found by building a formula in a cell that replicates the regular calculator solution. |
Using Time Value Analysis to Measure Return on Investment (Roi)
In most investments, an individual or a business spends cash today with the expectation of receiving cash in the future. The financial attractiveness of such investments is measured by return on investment (ROI), or just return. There are two basic ways of expressing ROI: in dollar terms and in percentage terms.
To illustrate the concept, let's reexamine the cash flows expected to be received if Wilson Memorial Hospital buys its new x-ray machine (shown on the time line in thousands of dollars). In the last section, we determined that the present value of these flows, when discounted at a 10 percent rate, is $580,950:
Dollar Return
The $580,950 calculated above represents the present value of the cash flows that the x-ray machine is expected to provide to Wilson Memorial Hospital, assuming a 10 percent discount rate (opportunity cost of capital). This result tells us that a 10 percent return on a $580,950 investment would produce a cash flow stream that is identical to one being discounted.
To measure the dollar return on the investment, the cost of the x-ray machine must be compared to the present value of the expected benefits (the cash inflows). If the machine will cost $500,000 and the present value of the inflows is $580,950, then the expected dollar return on the machine is $580,950 − $500,000 = $80,950. Note that this measure of dollar return incorporates time value, and hence opportunity costs, through the discounting process. The opportunity cost inherent in the use of the $500,000 is accounted for because the 10 percent discount rate reflects the return that could be earned on alternative investments of similar risk. By virtue of the $80,950 excess, the x-ray machine has an expected present value that is $80,950 more than would occur if it had only a 10 percent return, which is the opportunity cost rate. Thus, the x-ray machine makes sense financially because it creates an excess dollar return for the hospital.
The dollar return process can be combined into a single calculation by adding the cost of the x-ray machine to the time line:
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Financial Calculator Solution |
Now, with the investment outlay (cost) added to the time line, the following cash flows would be entered into the cash flow register: —500, 100, 120, 150, 180, and 250, in that order. Then, enter 1 = 10 and push NPV to obtain the answer, 80.95. |
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Spreadsheet Solution |
As in the financial calculator solution, the cost of the machine must be added to the cash flow data. Here, it is added to the spreadsheet range:
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Note that the situation here is the same as in the previous cash flow stream, except that there is an initial investment outlay of $500 added in Cell A3. Because the NPV of the cash inflows in Cells A4 through A8 represents the value one period before the first (A4) cash flow, all that must be done is to add the investment outlay to the calculated NPV. This is done in Cell A10 above by adding A3 to the NPV function, and the result, $80.95, appears in that cell.
Rate of Return
The second way to measure the ROI of an investment is by rate of return, or percentage return. This measures the interest rate that must be earned on the investment outlay to generate the expected cash inflows. In other words, this measure provides the expected periodic rate of return on the investment. If the cash flows are annual, as in this example, the rate of return is an annual rate. In effect, we are solving for I—the interest rate that equates the sum of the present values of the cash inflows to the dollar amount of the cash outlay.
Mathematically, if the sum of the present values of the cash inflows equals the investment outlay, then the NPV of the investment is forced to $0. This relationship is shown here:
Note that the rate of return on an investment, particularly an investment in plant or equipment, typically is called the internal rate of return (IRR), a somewhat archaic term that often is used instead of ROI, or just rate of return. Although a trial-and-error procedure could be used on a regular calculator to determine the rate of return, it is better to use a financial calculator or spreadsheet.
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Financial Calculator Solution |
Use the same cash flows that were entered to solve for NPV: —500, 100, 120, 150, 180, and 250. However, now push the IRR button to obtain the answer—15.3 percent. |
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Spreadsheet Solution |
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The IRR function is used to calculate rate of return. Choose Financial and IRR on the function wizard, then enter A3:A8 as Values and A2 as Guess. The result, 15.3%, appears in Cell A10, the cell that has the IRR function in it. (Note that IRR stands for internal rate of return. Also, note that a starting guess is required to calculate the IRR because the methodology used by the spreadsheet IRR function is actually a trial-and-error process that requires a starting point.)
The IRR of 15.3 percent tells the hospital's managers that the expected rate of return on the x-ray machine exceeds the opportunity cost rate by 15.3 — 10.0 = 5.3 percentage points. Thus, the expected rate of return is higher than that available on alternative investments of similar risk (the required rate of return), and hence the x-ray machine makes financial sense. Note that both the dollar (NPV) return and the percentage (IRR) return indicate that the x-ray machine should be acquired. In general, the two methods lead to the same conclusion regarding the financial attractiveness of an investment.
We will have much more to say about financial returns in Chapters 11, 12, and 14. For now, an understanding of the basic concept is sufficient.
Semiannual and other Compounding Periods
In all the examples thus far, we assumed that interest is earned (compounded) once a year, or annually. This is called annual compounding. Suppose, however, that Meridian Clinics puts $100 into a bank account that pays 6 percent annual interest, but it is compounded semiannually. How much would the clinic accumulate at the end of one year, two years, or some other period? Semiannual compounding means that interest is paid each six months, so interest is earned more often than under annual compounding.
The Effect of Semiannual Compounding
To illustrate semiannual compounding, assume that the $100 is placed into the account for three years. The following situation occurs under annual compounding:
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Regular Calculator Solution |
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Financial Calculator Solution |
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Spreadsheet Solution |
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Now, consider what happens under semiannual compounding. Because interest rates usually are stated as annual rates, this situation would be described as 6 percent interest, compounded semiannually. With semiannual compounding, N = 2 × 3 = 6 semiannual periods, and I = 6/2 = 3% per semiannual period. Here is the solution.
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Regular Calculator Solution |
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Financial Calculator Solution |
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Spreadsheet Solution |
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The $100 deposit grows to $119.41 under semiannual compounding, but it grows only to $119.10 under annual compounding. This result occurs because interest on interest is being earned more frequently under semiannual compounding.
Stated versus Effective Interest Rates
Throughout the economy, different compounding periods are used for different types of investments. For example, bank accounts often compound interest monthly or daily, most bonds pay interest semiannually, and stocks generally pay quarterly dividends.[6] Furthermore, the cash flows that stem from capital investments such as hospital wings or diagnostic equipment can be analyzed in monthly, quarterly, or annual periods or even some other interval. To properly compare time value analyses with different compounding periods, they need to be put on a common basis, which leads to a discussion of stated interest rates versus effective annual rates.
The stated interest rate in the Meridian Clinics's semiannual compounding example is 6 percent. The effective annual rate is the rate that produces the same ending (i.e., future) value under annual compounding. In the example, the effective annual rate is the rate that would produce a future value of $119.41 at the end of Year 3 under annual compounding. The solution is 6.09 percent, as found here using a financial calculator and a spreadsheet:
Thus, if one bank offered to pay 6 percent interest with semiannual compounding on a savings account, while another offered 6.09 percent with annual compounding, they both would be paying the same effective annual rate because the ending value is the same under both sets of terms:
In general, the effective annual rate (EAR) can be determined, given the stated rate and number of compounding periods per year, by using this equation:[7]
where IStated is the stated (i.e., the annual) interest rate and M is the number of compounding periods per year. The term IStated / M is the periodic interest rate, so the EAR equation can be recast as:
To illustrate the use of the EAR equation, the effective annual rate when the stated rate is 6 percent and semiannual compounding occurs is 6.09 percent:
which confirms the answer that we obtained previously.
As shown in the preceding calculations, semiannual compounding, or for that matter any compounding that occurs more than once a year, can be handled in two ways. First, the input variables can be expressed as periodic variables rather than annual variables. In the Meridian Clinics example, use N = 6 periods rather than N = 3 years, and 1 = 3% per period rather than I = 6% per year. Second, find the effective annual rate and then use this rate as an annual rate over the number of years. In the example, use I = 6.09% and N = 3 years.
For another illustration, consider the interest rate charged on credit cards. Many banks charge 1.5 percent per month and, in their advertising, state that the annual percentage rate (APR) is 18 percent.[8] However, the true cost rate to credit card users is the effective annual rate of 19.6 percent:
Amortized Loans
One important application of time value analysis involves loans that are to be paid off in equal installments over time, including automobile loans, home mortgage loans, and most business debt other than very short-term loans and bonds. If a loan is to be repaid in equal periodic amounts—monthly, quarterly, or annually—it is said to be an amortized loan. The word amortize comes from the Latin mors, meaning death, so an amortized loan is one that is killed off over time.
To illustrate, suppose Santa Fe Healthcare System borrows $1 million from the Bank of New Mexico, to be repaid in three equal installments at the end of each of the next three years. The bank is to receive 6 percent interest on the loan balance that is outstanding at the beginning of each year. The first task in analyzing the loan is to determine the amount that Santa Fe must repay each year, or the annual payment. To find this value, recognize that the loan amount represents the present value of an annuity of PMT dollars per year for three years, discounted at 6 percent.
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Financial Calculator Solution |
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Spreadsheet Solution |
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Therefore, if Santa Fe pays the bank $374,110 at the end of each of the next three years, the percentage cost to the borrower, and the rate of return to the lender, will be 6 percent.
Each payment made by Santa Fe consists partly of interest and partly of repayment of principal. This breakdown is given in the amortization schedule shown in Table 9.1. The interest component is largest in the first year, and it declines as the outstanding balance of the loan is reduced over time. For tax purposes, a taxable business borrower reports the interest payments in Column 3 as a deductible cost each year, while the lender reports these same amounts as taxable income.
Financial calculators are often programmed to calculate amortization schedules; simply key in the inputs and then press one button to get each entry in Table 9.1.
Table 9.1: Loan Amortization Schedule
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Year |
Beginning Amount (1) |
Payment (2) |
Interest[a] (3) |
Repayment of Principal[b] (4) |
Remaining Balance (5) |
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1 |
$1,000,000 |
$ 374.110 |
$ 60,000 |
$ 314,110 |
$685,890 |
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2 |
685,890 |
374,110 |
41.153 |
332,957 |
352,933 |
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3 |
352,933 |
374,110 |
21,177 |
352,933 |
0 |
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$1,122,330 |
$122,330 |
$1,000,000 |
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[a]Interest is calculated by multiplying the loan balance at the beginning of each year by the interest rate. Therefore, interest in Year 1 is $1,000,000 × 0.06 = $60,000; in Year 2 it is $685,890 × 0.06 = $41,153; and in Year 3 it is $352,933 × 0.06 = $21,177. [b]Repayment of principal is equal to the payment of $374,110 minus the interest charge for each year. |
Key Concepts
Financial decisions often involve situations in which future cash flows must be valued. The process of valuing future cash flows is called time value analysis. The key concepts of this chapter are:
· Compounding is the process of determining the future value (FV) of a lump sum or a series of payments.
· Discounting is the process of finding the present value (PV) of a future lump sum or series of payments.
· An annuity is a series of equal, periodic payments (PMT) for a specified number of periods.
· An annuity that has payments that occur at the end of each period is called an ordinary annuity.
· If each annuity payment occurs at the beginning of the period rather than at the end, the annuity is an annuity due.
· A perpetuity is an annuity that lasts forever.
· If an analysis that involves more than one lump sum does not meet the definition of an annuity, it is called an uneven cash flow stream.
· The financial consequence of an investment is measured by return on investment (ROI), or just return, which can be expressed either in dollar terms or in percentage (rate of return) terms.
· An amortized loan is one that is paid off in equal amounts over some specified number of periods. An amortization schedule shows how much of each payment represents interest, how much is used to reduce the principal, and how much of the principal balance remains on each payment date.
· The stated rate is the annual rate normally quoted in financial contracts.
· The periodic rate equals the stated rate divided by the number of compounding periods per year.
· If compounding occurs more frequently than once a year, it is often necessary to calculate the effective annual rate, which is the rate that produces the same results under annual compounding as obtained with more frequent compounding.
