# Time Value of Money Dallas Brozik, Marshall University

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1 Time Value of Money Dallas Brozik, Marshall University There are few times in any discipline when one topic is so important that it is absolutely fundamental in the understanding of the discipline. The concept of time value of money bears that relationship to finance. The notion that money has different value at different times is really special and leads to many different applications and management practices. The concept of time value of money is also insidious in finance; it pops up everywhere from investments to corporate finance to personal finance. Quite simply, no one will be able to have a good working knowledge of finance without having a thorough knowledge of the time value of money. The neat part about time value of money is that it is easy. There is a little math involved, but none of it is really difficult. There are really only a few basic definitions, and everything else is application. Future Value The best way to learn the time value of money is by doing. Assume that Bubba and Charlie have previously negotiated a forward contract and that the terms of that contract call for Charlie to pay Bubba \$100 one year from today. As it turns out, Charlie has unexpectedly come into some money, and he could pay Bubba the money today. How much would Charlie have to pay Bubba to have Bubba mark the contract "Paid in Full"? The easy answer is that Bubba would require Charlie to pay the entire \$100. But since there is nothing in the deal for Charlie, he would keep the money. He could put it under a rock or carry it in his wallet, or he could even put it in the bank and draw some interest. If Charlie put the money in the bank, at the end of the year he would withdraw the money, pay Bubba \$100, and close out the contract. Then Charlie would use the interest he had earned to buy a hamburger. This way, the contract is paid on time, and Charlie gets one more hamburger than he would have had. Bubba could make Charlie another offer, though. Since Bubba could also put the money in his bank, he could offer to sell the contract back to Charlie for \$95. That way

2 Bubba would have control of the money and not have to wonder if Charlie would be around next year. Bubba could use the money right now or save it until next year or put it under a rock, the choice would be his. But the money would be in Bubba's hands. Money has value through time due to its ability to earn interest. The question is how much value does it have? If Bubba had offered to sell Charlie the contract for \$25, Charlie would most certainly have taken the opportunity to get rid of his future \$100 obligation. If Bubba had offered to sell the contract for \$200, Charlie would have walked away from the deal. There are obviously extremes which would be unfair to one of the parties. No rational debtor would pay extra to pay off a future obligation, so it would be unreasonable for Bubba even to suggest that Charlie pay anything over \$100. The question is how much discount should Bubba offer for immediate payment of a future obligation. The discount offered should have some relationship to the use that Bubba could make of the money if he had it a year early. At the simplest level, he could always put the money in a bank account. Since \$100 is due in one year, the bank account should provide Bubba with at least \$100 in one year so that he is not out anything on the deal. Further analysis of this problem will use a technique called a "time line". This is a graphical method of watching money move through time. Other techniques exist that will accomplish the same end, but the time line is by far the most important and most robust. Other methods that rely on tabular arrangements of data will provide correct answers if they are set up properly, but in a messy problem it is easy to miss something that should be on the table. Since the time line is graphical, it presents a picture of the money moving through time, and it can easily be seen when something is missing from the picture. The use of time line analysis is not just for the fun of it; this is an extremely powerful analytical technique that will be used throughout this book. The reader is strongly urged to make the effort to understand time lines and the type of work they can do. As money moves forward through time it collects interest. The amount of interest collected depends on the amount of time the money is allowed to roll forward and the rate

3 at which it collects interest. Assume that the \$100 were allowed to sit in a bank account at 10% annual interest for one year. At the end of the year, the principal of \$100 would have earned \$10 in interest. On the time line it looks like this. The same problem can be done mathematically as: Future Value = Principal + Interest = Principal + (Principal) (Interest Rate) = (Principal) ( 1 + Interest Rate) = (\$100) ( ) = \$110 While this may seem like a long, drawn out way to do something that seems obvious, rest assured that this simple example is indeed the cornerstone on which all of time value of money is built. When money is left in the bank for longer periods, the interest earned can itself act as principal in the next period and earn interest on itself. This "interest on interest" effect is sometimes called the "magic of compounding" and used to impress financially unsophisticated folks. If the money in the previous example were allowed to sit in the bank for three years (three compounding periods), the time line would look like:

4 During the first year the interest is earned on \$100 principal. When the first year's interest of \$10 is allowed to accumulate in the account, it becomes part of the principal, so the principal for the second year is \$110. The interest earned during the second year is \$11 which becomes part of the principal for the third year. By the end of three years the account has grown to a value of \$ If the interest had not been compounded, the account would have earned \$10 interest for three years for a total of \$30. The extra \$3.10 is the effect of interest being earned on interest. The mathematics for this problem is merely an extension of what was done before. Future Value (end of year 1) = Principal (time = 0) + Interest = Principal time=0 + (Principal time=0 ) (Interest Rate) = (Principal time=0 ) ( 1 + Interest Rate) = (\$100) ( ) = \$110 Future Value (end of year 2) = Principal time=1 + Interest = Principal time=1 + (Principal time=1 ) (Interest Rate) = (Principal time=1 ) ( 1 + Interest Rate) = (\$110) ( ) = \$121 Future Value (end of year 3) = Principal time=2 + Interest = Principal time=2 + (Principal time=2 ) (Interest Rate) = (Principal time=2 ) ( 1 + Interest Rate) = (\$121) ( ) = \$ Since writing out the problem this way is rather tedious, mathematical notation can be used to provide an appropriate shortcut and even speed calculations. The value at the end of the year is merely the sum of the principal and interest which then becomes the principal for the next year. Principal time=1 = (Principal time=0 ) ( 1 + Interest Rate) Principal time=2 = (Principal time=1 ) ( 1 + Interest Rate) Principal time=3 = (Principal time=2 ) ( 1 + Interest Rate)

5 The question is how much money will there be after three years compounding, so of the three equations the last one is the answer. But part of the last equation is the second equation, and part of the second equation is the first equation. Using algebraic substitution, the third equation can be "simplified" as follows: Principal time=3 = (Principal time=2 ) ( 1 + Interest Rate) Principal time=3 = (Principal time=1 ) ( 1 + Interest Rate) ( 1 + Interest Rate) Principal time=3 = (Principal time=0 ) ( 1 + Interest Rate) ( 1 + Interest Rate) ( 1 + Interest Rate) or Principal time=3 = (Principal time=0 ) ( 1 + Interest Rate) 3 The principal at time=3 is called the Future Value since it is the amount of money that will be available in the future. The principal at time=0 is called the Present Value since it is the amount of money here right now in the present. By changing these terms the equation becomes: Future Value ( time=3 ) = ( Present Value ) ( 1 + Interest Rate ) 3 which is often abbreviated to be: FV(3) = PV ( 1 + i ) 3 There is even a more general notation that can be used. The previous equation shows that the Future Value (FV) three years from now is just the Present Value (PV) multiplied by one plus the Interest Rate three times. This interest multiplier can be written out three times or it can be shown as a cubed value as was done above. It does not take a giant leap of the understanding to recognize that if a certain amount of money (the Present Value) is left to compound for seven periods that this amount of money is multiplied by the compounding factor raised to the seventh power, and so on. The formula is usually written: FV(N) = PV ( 1 + i ) N where: i = interest rate and N = number of compounding periods

6 It is very important to note that this relationship between money at different points in time has only four components, the Future Value, the Present Value, the Interest Rate, and the Number of Periods. These are the only four pieces. Since the relationship includes a mathematical equality, it is only necessary to have any three pieces to be able to find the fourth. In this example, the Present Value, Interest Rate, and Number of Periods were used to find the Future Value. In other applications of this relationship, it is possible to use three different components to find the fourth. Using the data from the original example, the solution would be: FV(t=3, i=10%) = PV ( 1 + i ) N = (\$100)( ) 3 = (\$100)(1.331) = \$ If the original principal had been \$500 instead of \$100, the value of \$500 compounded at 10% annual interest for three years would be: FV(t=3, i=10%) = (\$500)( ) 3 = (\$500)(1.331) = \$ It is worthwhile to note that once the interest rate and number of periods are set, the compounding factor does not change. It really does not matter if the original principal was one dollar or one million dollars, the compounding effect works exactly the same on each and every individual dollar. Since these compounding factors are constant, it is possible to store them away on paper or electronically so that it is not necessary to do the calculations all the time. Many textbooks include various compounding and discount tables. The tables that compound into the future give the Future Value Factors while the tables that discount future money to the present give the Present Value Factors. This was the preferred method of storing numbers in the years B.C. (Before Calculators). Most financial calculators today can calculate any compounding factor ever needed with just a few keystrokes, and anyone who does this kind of work must know how to use these types of calculators and even their big brothers, the computers.

7 The original question still has not been answered. Charlie still owes Bubba \$100 due in one year, and Charlie is willing to pay Bubba today if the deal is good enough. And Charlie will not pay a premium to get the note back, so Bubba will have to offer a discounted price. This problem is amazingly similar to the future value problem that was just solved, except for one detail. In the future value problem, the amount of money at time=0 (the present) was known. That question dealt with how large the initial amount would grow to if left alone and allowed to compound the interest. In the problem at hand, the \$100 Charlie owes Bubba is a future value, and it is necessary to relate that amount of money in the future to some amount in the present (time=0). On a time line the problem looks like this: The easy part is that the problem has already been solved. The solution is just a little algebra away. Recall that there is already a firm relationship between future and present value. FV(N) = PV ( 1 + i ) N where: i = interest rate and N = number of compounding periods This relationship is stated in a way that allows the calculation of the future value once the other three factors are known, but there is no reason that this mathematical relationship must be used solely to solve future value problems. Remember, any three of the four factors provides enough information to calculate the fourth. By applying some basic algebra, this future value equation can be transformed into a present value equation.

8 or: By rearranging the components of the equation, the relationship between the Present Value and the Future Value is again visible. Just as the Future Value is the Present Value compounded for a Number of Periods, the Present Value is the Future Value discounted for a Number of Periods. In both cases the Interest Rate and the Number of Periods defines a factor that relates the Future Value and the Present Value. This relationship of monetary values across time depends solely on the Interest Rate and the Number of Periods, so it is possible to calculate Future Value Factors (FVF) and Present Value Factors (PVF) for a single dollar as follows: where: i = interest rate and N = number of compounding periods These factors are based on one dollar, so it is easy to scale them to other amounts. For example, the FVF(1,10%) = 1.10 which means that one dollar will become \$1.10 after one period of compounding at 10%. If the initial amount (the Present Value) had been \$1,000, the ending amount (the Future Value) would have been \$1,100. Each of the original dollars is now worth \$1.10, so the final answer is simply the initial amount times the Future Value Factor. A similar argument holds for using the Present Value Factor.

9 Inserting the numerical values for the Future Value, the Number of Periods, and the Interest Rate, allows the Present Value to be calculated. The Present Value of \$100 to be received one year from now is \$90.91 if the money could be invested at 10% per year. This can be shown by calculating the Future Value of \$90.91 in one year at 10% interest. The time line for the Present Value can be drawn as: The related Future Value time line is drawn as: These two time lines show that the Present Value calculation is really just the inverse of the Future Value calculation, and vice versa. The two monetary values are joined through time by the Number of Periods and the Interest Rate. This whole relationship between the four variables is simple and straightforward. All Present Value and Future Value problems can be broken down into simple steps like this, no matter how

10 messy they might look at the beginning. Use of the time line makes it easy to see money flowing back and forth through time. The simplicity of the future value and present value concepts can be illustrated with a few examples. These examples may seem to be almost too easy, but the whole business of moving money through time is easy. Do not try to make present value and future value more complex than it really is. An Example of Future Value Q1. How much money would you have after four years if you deposited \$100 today in a savings account at a bank that paid 6% compounded annually? The first step to any time value of money problem is to draw the time line and identify which pieces of information are available. While it is possible and correct to draw out a detailed time line that shows the compounding for each year, that process will get tedious very quickly. It really is not necessary to draw all those lines since the compounding has already been captured in the Future Value Factor which is calculated as: FVF(N=4, i=6%) = ( ) 4 = Using this Future Value Factor in the time line

11 A More Interesting Example of Future Value The previous example shows that calculating the future value of single amounts of money is really easy. Things get more interesting when multiple amounts are used at different times. Not harder, just more interesting. Q2. How much money would you have after four years if you deposited \$100 today and \$50 in two years in a savings account at a bank that paid 6% compounded annually? This problem has no solution in its current form. There are no tables that provide a factor for two different sums at different points of time. The trick is to think of each of the amounts deposited as separate problems. The time line then becomes This is really two simple problems put together. The solution for the complicated problem cannot be done easily, but the solution for the two simple problems is straightforward. All time value of money problems, no matter how complex they look, are really collections of simple problems. The time line allows the analyst to break the messy problems into simple problems; that is the great strength of time line analysis. The solution for this two part problem is shown below.

12 Though this solution is fairly simple, the technique used is VERY important. The first step in any time value of money problem is to lay out the time line and identify all the information available, usually the cash flows and the times they occur. If there are several cash flows, the trick is to break the big problem into a bunch of smaller problems. That simple trick makes it possible to solve the messiest problem and get the correct answer. An Even More Interesting Future Value Problem Sometimes contracts are written that provide for the payment of equal sums of money equally spaced in time. The amount can be any value, and the length of the contract can be anything agreeable to both parties. This type of contract is called an annuity because it usually represents annual payments. There is the question of exactly when the payment is made, at the beginning of the year or the end of the year. For an ordinary annuity it is assumed that the payments are made at the end of the year. Q3. How much money would you have after four years if you deposited \$100 at the end of each year in a savings account at a bank that paid 6% compounded annually? This example looks a bit different than previous examples because the first cash flow does not occur until one year from the beginning of the problem. This is a characteristic of an ordinary annuity, by definition. Another aspect of the ordinary annuity is that the last payment occurs at the very end. In this case, the last \$100 will not draw any interest. With these characteristics in mind, the solution to the problem is just the same as before. The large problem is broken into a group of small problems.

13 The annuity problem really is not much of a problem. The annuity contract is so common in business dealings, that tables have been developed to give the answer a bit more directly than was shown above. Some annuity contracts can run for extremely long periods of time, and it just gets cumbersome to deal with all sorts of little pieces all the time. The development of the annuity table is simple. Every payment in the annuity contract must be the same size (or else it is not an annuity). The math for the problem shown above looks like this: Future Value of the Annuity = (\$100)(1.1910) + (\$100)(1.1236) + (\$100)(1.0600) + \$100 By factoring out the common component of \$100 from each of the terms the equation becomes Future Value of the Annuity = (\$100) [(1.1910) + (1.1236) + (1.0600) + 1] = (\$100) [ ] The value is called the Future Value Annuity Factor (FVAF) is based on the value of an annuity of one dollar per period and is written as FVAF (4,6%) = Remember that these factors are for ordinary annuities where the payments are made at the end of the year. The time line for an ordinary annuity would look like

14 Annuity problems are easy. In fact, some problems that do not look like annuities actually contain annuities as one component of the solution. The tables presented in various books and available from other sources have the distinct disadvantage of being incomplete. It is simply impossible to have every possible combination listed. Financial calculators can find the annuity factor for any combination of interest rate and number of periods, but a financial calculator is only as smart as its program. The mathematical formula for the future value factor for an ordinary annuity is where: i = interest rate and N = number of compounding periods This formula can be used in those cases where the tables are incomplete. Another Extremely Interesting Future Value Problem Q4. How much money would you have after four years if you deposited \$100 at the end of year one, \$150 at the end of year two, and \$100 at the end of years three and four in a savings account at a bank that paid 6% compounded annually? The first step is to draw the time line to illustrate the cash flows.

15 This set of cash flows looks almost like an annuity, but not quite. If it were an annuity, it would only be necessary to look up one value in the Future Value Annuity Table and it would all be done. But there is an extra \$50 at the end of year two that must be accounted for, and that changes the whole problem. The problem does not change very much, though. If a person were paid \$150, it could be paid in bills of various denominations. The easy way to get around this problem is to think of the \$150 as two separate payments of \$100 and \$50 received simultaneously. The time line can be redrawn as The act of breaking the deposit at the end of the second year into two deposits does not in any way change the mathematics of the situation, it just makes it easier to use the tables. This is now a problem with one annuity and one lump sum. Had the analyst wanted to do things the hard way, the time line would have looked like

16 It really does not matter how the money is moved around, as lump sums or annuities. The rules of mathematics apply, and any problem must yield the same solution regardless of the method used. That is comforting to know. Moving money through time is really just a mechanical act, and once the technique is mastered it will not change. Present Value So far the fancy stuff has all had to do with Future Value. It should come as no surprise that the same techniques can apply when using Present Value; the only difference is that money is moved back through time instead of forward. In truth, Present Value calculations are more useful in decision making than Future Value calculations. The majority of decisions are of the nature "What is this contract (or asset) worth today? How much should I pay for it?" When you invest in any asset, physical or financial, you must give up dollars that you have today in exchange for dollars that should come in the future. You want to know the Present Value of the future cash flows when deciding whether or not to purchase the asset. An Example of Present Value Q5. Earl is owed \$100 payable in four years. If he had the money now, he could put it in the bank and earn 6% interest compounded annually. What is the Present Value of the money owed to Earl? The first step is to draw the time line.

17 The Present Value Factor can be calculated as follows: The time line would now look like One hundred dollars due four years from now is only worth \$79.21 in today's money If today's money can be invested at 6% compounded annually. This is equivalent to saying that if Earl had \$79.21 right now and put it in a bank account that paid 6% compounded annually, in four years he would have \$100. Another Present Value Problem Q6. Freda is owed \$50 payable in two years and \$100 payable in four years. If she had the money now, she could put it in the bank and earn 6% interest compounded annually. What is the present value of the money owed to Freda? By drawing the time line in this manner, the solution is obvious. Just treat each payment as a separate cash flow and calculate its present value separately.

18 It should be getting painfully obvious by now that present value works like future value and that future value is easy enough to do. To complete this section, it is only necessary to consider the present value of an annuity and of an unequal set of cash flows. The Present Value of an Annuity Q7. Grizwald has a contract that will pay him \$100 per year at the end of each of the next four years. Assuming he can earn 6% per year at your bank, what is the value today of this set of future cash flows? The time line would look like this: The value of this contract could be calculated as the sum of the values of each of the individual cash flows using the individual Present Value Factors.

19 As might be expected, though, there is a table for the Present Value of an Annuity just as there was for the Future Value of an Annuity. Either way of doing the calculation is a correct way. The only advantage of using the annuity table is that it cuts down on the number of individual calculations needed to solve the problem. With a four period problem like this, there is not much difference in which way is chosen, but in a longer problem that could have 40 or 50 periods an annuity table can save a lot of work. There is another interpretation that can be used to explain the relationship between Present Value and a set of future cash flows. In this example, the Present Value of a four year annuity of \$100 per year is \$ when the discount rate is 6%. Assume that you started out with \$ and put it in your bank account at 6% and withdrew \$100 each year. The pattern of cash flows would look like this

20 Using this interpretation, the Present Value of a set of future cash flows is equivalent to the amount that it would be necessary to deposit in the bank at the given interest rate so that the future cash flows would be available from the bank account so that at the end of the period the account balance would be zero. This is how insurance companies price retirement annuities. Once the amount of the payment is given, the insurance company estimates how long the purchaser will live. The insurance company also knows at what interest rate it can invest funds. The minimum cost of the retirement annuity is then the present value of all expected future payouts. The same principal holds true for lotteries. Suppose that Helen wins the lottery and her prize is twenty million dollars. In some cases, Helen will not be allowed to take the money as a lump sum. The rules of this lottery require that she take the prize over twenty years, so Helen will get \$1,000,000 per year. This is still a nice prize, but the total value of it is less than \$20,000,000 since the money comes at different times, some of it nineteen years in the future. If Helen's appropriate discount rate is 5%, the Present Value of twenty payments of \$1,000,000 each received one year apart starting today is only \$13,085,000. One reason that state lotteries pay out over long periods of time is to reduce the value of the money the state has to pay out to the winner. That way the state gets to keep more of the money for its own purposes. These examples also illustrate that the procedure is indifferent to which side of the deal is being discussed. If the insurance company sold a four year, \$100 annuity for \$346.51, the purchaser of the annuity would pay out \$ and receive \$100 for four years. The purchaser would have a negative cash flow at time zero and positive cash flows for the next four years. From the insurance company's perspective, the \$ would be positive at time zero (a cash inflow) and the yearly payments would be negative (cash outflows). For those combinations of interest rate and number of periods that are not included on available tables, the formula for calculating the Present Value Annuity Factor (PVAF) for an ordinary annuity is

21 where: i = interest rate and N = number of compounding periods When dealing with time value of money problems, it is important to identify cash inflows and cash outflows. The easiest way to do this is to consider inflows to be positive and outflows to be negative. The mathematics does not care what the sign of the number is, so the analyst has to be careful to make sure each cash flow is identified and labeled correctly. The numerical values will be the same whichever side of the deal you are on; the signs of the numbers will change, however, to indicate the inflows and outflows. Present Value of Unequal Cash Flows Q8. Suppose you are scheduled to receive \$100 in one year, \$150 in two years, and \$100 after the third and fourth year. If you could earn 6% compounded annually, what amount of money would be equivalent to this set of future cash flows? As you no doubt expect, there are several ways to solve this problem. The present value of each cash flow could be calculated and summed.

22 Or the cash flow at Time=2 can be broken into two pieces so that the annuity table can be used. It makes absolutely no difference how the time line is evaluated. Once the cash flows are properly set up, the present value will be the same no matter how they are handled. Compounding Periods There is a small detail that sometimes appears in time value of money calculations that relates to the periods of compounding. Interest rates are always given in annualized terms, unless specifically stated otherwise. Sometimes this annual interest rate is not the applicable rate for the analysis, however, since it can be compounded within the period. Consider bank savings accounts that promise to pay 6% interest compounded semiannually. The semiannual compounding means that half of the interest is paid half way through the year so that it can then start compounding. In this situation, the year is really two periods long and the interest rate paid in each period is one half of the stated annual rate. Whenever a compounding period is given that is different than the term of the interest rate, it is necessary to adjust the interest rates to match the compounding period. All the Present and Future Value tables recognize that a year may not be the appropriate period for a certain interest rate. That is why the tables are set up in the format of interest rates versus number of periods rather than interest rates versus number of years. It will be necessary to state interest rates in annual terms since this is a convention

23 in finance, but actual calculations must take into consideration the true length of the compounding period and the interest rate relevant during that period. As an example of the difference the number of compounding periods can make, consider the savings accounts offered by the following banks: Name of Bank 1st National Bank 2nd State Bank 3rd Regional Bank 4th Local Bank Interest on Savings 12%, compounded annually 12%, compounded semiannually 12%, compounded quarterly 12%, compounded monthly Each bank states that it pays 12% on a savings account, but the differences in the compounding periods means that the accounts really are different. The 1st Bank offers an account with one period at 12%, the 2nd Bank has an account with two periods of 6% each, the 3rd Bank pays 3% interest for four periods, and the 4th Bank offers twelve periods of 1% each. Each bank can legitimately say it pays 12% per year since the number of periods times the interest paid per period totals 12%. The effect of the different compounding periods can be seen by assuming that a depositor puts \$100 into each account at the beginning of the year and determining what the account balance will be after one year. The time lines for these four accounts would be:

24 The difference in the compounding period has created a difference in the final amount in the savings account. This means that even though the four accounts have the same nominal interest rate of 12% per year, the effective annual interest rate as shown below.

25 Name of Bank 1st National Bank 2nd State Bank 3rd Regional Bank 4th Local Bank Nominal Annual Interest Rate 12%, compounded annually 12%, compounded semiannually 12%, compounded quarterly 12%, compounded monthly Effective Annual Interest Rate 12.00% 12.36% 12.55% 12.68% The effective annual interest rate can also be calculated mathematically as follows: At first glance the differences in the effective annual interest rates may seem small, but if these differences are allowed to compound, the result can be substantial. If \$100 were allowed to compound for 25 years at an effective annual rate of 12%, the ending balance in the account would be \$170. If the \$100 had an effective annual compounding rate of 12.68%, the ending balance would be \$ If the initial deposit had been \$1,000,000 instead of \$100, the difference between the ending balances would exceed a quarter million dollars. Even seemingly small differences in effective interest rates can

26 make substantial long run differences in the final results. This is the reason that financial planners urge their clients to begin saving early, even if the amount is small. The compounding effect of interest works best if it is allowed to work over long periods of time. It is important to recognize when the compounding period effects the analysis of a time value of money problem. This is not just an academic exercise. Virtually all corporate bonds pay interest semiannually. A bond with 10 years to maturity a stated coupon rate of 12% paid semiannually is actually an investment of 20 periods (10 years of 2 semiannual periods each) with an interest rate of 6% for each period (12% per year divided by 2 periods per year) which would provide an annual effective interest rate of 12.36% if sold at par and held to maturity. It is vitally important that the compounding periods be considered when comparing investments with different compounding periods. The only way that an intelligent comparison of two such different investments can be made is if the compounding effects are considered and an effective annual interest rate is calculated. Terminology The examples given above have used an interest rate in the calculation of the time value of money. In practice, this interest rate has several different names. It can be referred to simply as an "interest rate", or as a "compounding rate" (future value problems), or a "discounting rate" (present value calculations). Another term used for the same concept is "opportunity cost". This terminology stems from the economic concept that if money is used for this investment, it misses the opportunity to be used in another investment. For example, if the money needed to purchase a security is taken out of a bank account, it does not have the opportunity to draw interest in the bank account. In practice, all these terms are interchangeable and refer to the interest rate used in the time value of money calculation. Summary Moving money through time is absolutely crucial to understanding finance and investments. Investments provide the investor with current and projected cash flows, and

27 the businessperson needs to know the value of these flows in current terms in order to make informed decisions about whether or not to buy or sell a specific asset or investment. The use of the Present Value and Future Value tables, for both lump sums and annuities, is arguably the most important skill a financial analyst can develop. Without a knowledge of how to move money through time, an analyst cannot understand the financial consequences of investments longer than one period, and that includes just about every asset and investment worth considering. The ability to perform time value of money calculations is merely a skill, though. The decisions that will be made concerning investments will also include other factors. But without the time value of money information, the decisions cannot be made intelligently.

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