Notes on the Investment Decision

Transcription

1 Notes on the Investment Decision. Introduction How does a business decide to make an investment in a new plant, new product line, new store opening, or additional machinery? A small retail-business owner might forecast the expected profits from opening a new store. However, the owner will then have to weigh these potential profits against the cost of the new store. How does the owner compare the current cost of the new store to the expected future profits to be generated from the store? Maybe things would be easier if the owner took out a loan to buy the new store then compared the expected annual profits with the annual payments on the loan (interest and principal). Assuming the expected annual profits exceeded the annual payments on the loan, would this mean the owner should make the new investment? The expected annual profits are uncertain, while the annual payments on the loan must be made. Thus, maybe the owner should allow for the risk that the profits will not all be realized and therefore make the investment only if the expected profits far exceed the annual payment. Now the question becomes just how far is far? The investment decision made by corporations is even more complicated. At least with the small business owner (assuming a proprietorship), the owner can decide for him or herself whether the investment is too risky or the return not high enough. The executives making the decision at a corporation must make the decision in the best interest of the ultimate owners (i.e., shareholders). How would the executives know what the multitude of shareholders think would be too risky or an expected return not high enough? Thus, once the expected profits from an investment (e.g., opening a new product line, constructing a new manufacturing plant, opening up additional retail stores, opening up a web-based store, etc.) are forecasted, what should the executives do next? They must find a way to discover how their shareholders and stakeholders (e.g., shareholders and credit-holders) view risk. Corporate finance develops ways to evaluate investment projects. In addition, corporate finance develops ways to view the financing decisions (e.g., debt-equity on the balance sheet) of the decisions and the corporation as a whole. In order to do so, there must be a clear objective in the minds of decision makers. Following corporate finance, we will assume the objective is the following. Objective: MAXIMIZE THE PRESENT VALUE OF THE FIRM Now, this objective will need to be unpacked a bit to grasp the full implications, and we should note from the outset that there are disagreements about this objective along with difficulties of applying it.. Future Value and Present Value What is the value of a firm? Here, the term firm can refer to a proprietorship, partnership, or corporation. We might consider the value of a firm to be simply its net

2 worth, hence the value of its assets minus the value of its liabilities. For a corporation, this is often referred to as the book value, thus the shareholders equity section of the balance sheet. Alternatively, we might think of value as the total assets as listed on the firm s balance sheet. Both of these definitions, though, equate value with the historical costs of assets. However, finance conceptualizes economic activity by looking into the future and discounting it back to the present. That is, finance views value in terms of the future benefits from the assets rather than what those assets costs to obtain. As a brief example, imagine that you are considering investing in either asset A or asset B. Asset A is a brand new factory that took years to construct at great cost (money and resources), though it will produce typewriters. Asset B is an internet dating company run by one person from his basement with a few computers. From an investment standpoint, it probably does not matter much how much it cost to produce either asset. What really matters is the potential profit each asset may generate in the future. Thus, we would not necessarily just look at the monetary value of the assets on the balance sheet to assess the true value of the firm. If value is thought of as being based on the future, then the present value of an asset can change for all sorts of reasons none of which has much to do with what happened in the past. A generic, or superficial, definition of value can be stated as the following: the value of an asset is the risk adjusted present value of all future net benefits. Beginning with the last, we will discuss each underlined portion of this definition separately. The future net benefit of an asset is surprisingly difficult to pin down. On the one hand, there may be intangible (or, non-quantifiable) net benefits of an asset. For example, a house is an asset with part of the net benefit being the price appreciation and another part being the enjoyment of living in it. The price appreciation would have to be estimated, but so would the enjoyment aspect (e.g., maybe you didn t realize your neighbors would call the police every time you held a party). On the other hand, limiting the analysis to assets where the net benefit is always quantifiable holds out problems as well. Take the case of investing in a share of a corporation (i.e., stock). As part owner of the corporation, what are your future net benefits? Note we are not attempting to quantify anything, just arrive at what should be quantified. One answer might be the profits (or, earnings, net income, etc.) of the corporation. Profits are simply revenues minus expenses, what could be simpler? According to accounting principles, revenues include sales made on credit, which may turn out to be bad credit and never paid. Accounting principles do not count capital expenditures (e.g., new plant and long-lived equipment) as expenses at the time of purchase. Rather, the capital expenditures are expensed as they depreciate over time. At this point, maybe you are thinking that profits should mean cash coming in minus cash going out. This would seem to be very reasonable. However, consider that huge capital expenditure again. Would you really like to say that because the corporation invested $00 million in a new plant with the latest technology in order to drive out its competitors, they were making less profits this year (i.e., the net benefit of the asset was lower)? Would we not be better off attempting to match expenses to the revenues generated from them? This is what the accounting principals attempt to do. Which is correct?

3 The present value concept underlies just about everything in finance. The basic idea of present value is not very difficult. The simplest case is to think of what happens with a bank account. Suppose you put $00 in a bank account that paid 0% interest for one year. At the end of the one year, you could withdraw $0 the $00 investment and the $0 interest (or, net benefit). We can translate this into simple math. $00(+.0) $00 + $0 $0 Now, if this bank account existed, what would receiving $0 in a year be worth to you now? In other words, supposed you were offered a contract that would pay you $0 in one year, how much would you pay for the contract? The answer of course is no more than $00. Thus, the present value of $0 a year from now is $00 given the 0% interest rate. Again, we can translate this into math by a simple rearrangement of what we did before. $00 $0 If you had deposited the money for two years, your $00 would have grown to $. $00 $00 $ Flipping the question, we could ask how much would you pay for a contract that would deliver $ two years from today? Given the 0% interest rate, we would pay no more than $00 for the contract. Thus, the present value of $ two years from now is $00. $00 $ We can extend this thinking to however many years you like. Depositing $00 into a bank account for years, with the interest rate at 0%, would yield the following future sum of money. $00 $00 $.0 The present value calculation just reverses the question. What is the present value of $.0 three years from now worth today (when the interest rate is 0%)? $.0 $00 In order to generalize the above ideas, suppose you could purchase a contract that would pay $0 after year, $ after years, and $.0 after three years. Again, assume the interest rate to be 0%, what would you pay for this contract today? In other words, what is the present value of those future payments?

4 0.0 Present Value $ 00 Notice how it takes a larger future payment further into the future (e.g., $.0) in order to get the same present value for that sum (e.g., $00) as an earlier payment. It might be easier to see this with a constant future payment. Suppose you would receive a $00 payment each year for the following three years. What is the present value of this future flow of payments? Present Value $ The reasoning is quite simple. Money today is worth more than money tomorrow. That is, receiving $00 in year is worth (i.e., $90.9) more today than receiving $00 in years (i.e., $75.). In a narrow sense, the concept of present value illustrates the time value of money. As long as we can earn something (e.g., interest) on money today, then money in the future will be worth less at the present time. We can state present value more formally in the following general way. () NB NB NB L ( + k ) ( + k ) ( + k ) where, is Present Value NB is Net Benefit k is the discount rate (much more on this later) superscripts denote the time dimension (e.g., year, year, etc.) In our previous examples, the net benefit was the payment made to us and the discount rate was merely the interest rate. Equation () is the general form of our previous examples. Notice a couple of implications. First, as the future net benefits increase, the present value will also increase. Compare the results of our previous two numerical examples. Second, as the discount rate (in our examples this was the interest rate) increases, the present value will decrease and vice versa. Consider changing the interest rate to 5% and 0% for the example of receiving $00 annual payments for the next three years $7. ( +.05) ( +.05) ( +.05) 4

5 ( +.0) ( +.0) ( +.0) $0. 65 When you think about it, this relationship should make perfect sense. When you earn a higher interest rate on a bank account for example, you need to deposit less money today in order to get the same value in the future compared with a lower interest rate. The present value calculation is just the flip side of looking at computing future value. One final comment is in order on the technical aspect of present value. When actually calculating present value by hand or with a cheap calculator, the calculations can become quite tedious. We will do much of these calculations with Excel. However, when we do need to do these calculations by hand (e.g., on an exam or quiz) or when we want to illustrate a basic idea, it will often be useful to assume a constant payment which lasts forever. Why would this assumption be helpful? First consider the following example of receiving payment of $00 under various assumptions about when the payment will be received. 00 Payment in Year: $ Payment in 50 Years: $ Payment in 00 Years: $ Thus, put a penny into a bank account paying 0% annual interest, and after 00 years it will grow to $00. Anyway, notice how small the present value of a sum of money gets the much further out into the future we receive it. To reinforce the point, what is the present value of receiving $million 00 years from today? $,000, $7.57 You could deposit $7.57 into a bank account paying 0% and your great grandchild will be a millionaire! But all of this is just to get to the implications of assuming that a constant payment is received annually forever (not just one payment, but a payment each year). So, suppose that you will receive $00 every year forever (your grandchildren get it when you die). Thus, we have the following present value calculation to make L 5

6 This would be a tedious calculation (the three dots mean that you keep extending the calculation without end). You would spend now until forever actually doing the calculation. Fortunately, the above calculation simplifies very nicely L 4.0 $,000 That is, in general terms, the present value of receiving a constant annual net benefit (the $00 in our example) forever is NB k where k again refers to the discount rate (or, in our example, the interest rate). You may have thought that the present value of receiving $00 every year forever would be more than $,000. However, recall that once the payment of $00 got out to 00 years into the future, then the present value was nearly worthless (ok, maybe a penny isn t worthless, but its close). At any rate, the forever assumption will often make calculations easier, and won t be far off if we re talking about something that lasts well into the future (0 years will probably do it). Also, we can readily see how an increase in the net benefit (NB) increases the present value, while an increase in the discount rate (k) lowers it. The present value formula appears very simple. However, we have already seen that defining the future net benefit can be tricky when it comes to applications. Even trickier is the mysterious discount rate. Before going too far, we should note that the discount rate can be subjective. Consider a friend of yours offering to make a pie that you just love. The twist, though, is that the offer is to make you one pie today or one pie a year from today. Whether human nature or socialization, most people seem to require a reward (or, bribe) in order to forgo present enjoyment for future enjoyment. In other words, most people require more of a good thing in the future to give it up today. Suppose then, you counter offer by agreeing to one and a half pies a year from today. What is your subjective discount rate (i.e., how do you discount the future)?.5pies pie k.5 50% ( + k) The subjective nature of the discount rate can be very important for certain types of analysis. It might help understand how the small business owner from the introduction would determine investment decisions. However, for a corporation, we will need to discover the discount rate to be used. Notice, your very good friend is not charging you for the pie, thus we have gotten rid of the time value of money basis of the discount rate. Also, you might think of the offer being either today or tomorrow, or either today or in a week, to make it more realistic we use one year just for ease of calculation. 6

7 The determination of the appropriate discount rate to use in present value calculations is intimately connected with adjustments for risk. It is possible to make risk adjustments to the net benefit component of the present value formula. For instance, if you were not quite sure whether you were going to get $0 next year from your bank deposit, then you could use an expected net benefit calculation based on probabilities. However, it is more common to make risk adjustments by changing the discount rate. Suppose, for instance, a friend asks for a loan with the agreement of paying you back $0 in one year. Now, as before, you could loan your bank $00 today and receive repayment of $0 a year from now (after all, a bank deposit is simply a loan to the bank). Would you be willing to loan your friend $00 today in exchange for $0 in a year? The answer is probably a big fat NO. After all, your friend is probably not as credit worthy as the bank (though maybe in the current environment he is). You may want to raise the discount rate you use to lend to your friend above what you require from the bank. Suppose, after analyzing your friend s finances, you require a 6% discount rate in order to make the loan to your friend, then you would only lend your friend about $95. $95 $0 ( +.6) By raising the discount rate, you are lowering the present value of the $0 from your friend as compared to the bank. This is an important result in any present value calculation. The discount rate and present value always move in opposite directions for any given net benefit. The tricky business of determining what discount rate should be used in a present value calculation can now be seen. How is one supposed to come up with a discount rate to use in any given situation? It is easy enough to say that one should use a higher discount rate for a loan to a friend than a loan to a bank. However, how much higher should it be? What discount rate should be used when valuing a corporation? Again, we may think that it should be higher for a corporation than for say a loan to the federal government, but how much higher? We arrive back at our conclusion that applying our generic definition of the value of an asset is difficult. We will need to apply the present value calculation to two situations. First, when deciding upon whether to make an investment (e.g., purchase additional machine, construct a new plant, enter a new market, or open a new line of product) requires us to find the present value of the expected net benefit. This requires estimating the expected net benefit and also determining the discount rate to be used. Second, as stated previously, our objective is to maximize the present value of the business. Again, this requires estimating the future profits or cash flows to be generated by the assets of the Of course, all of this could be said in much simpler terms. Really, you are just charging a higher interest rate (6%) to your friend than you do for the bank (0%). The apparent awkward wording in the text anticipates more sophisticated financial instruments. In addition, the wording emphasizes possibly more than necessary - the distinction between discount rates and interest rates. Discount rates are like interest rates in terms of being pure numbers. However, the two concepts have different interpretations. 7

8 business and also the appropriate discount rate to be used in the present value calculation. Notice that both purposes require determining the proper discount rate to be used. In corporate finance, the appropriate discount rate goes under various names: cost of capital, weighted average cost of capital (or, WACC for short), required rate of return, and hurdle rate. This terminology is used interchangeably. For now, we will simply assume the cost of capital to be given to us. Thus, we begin by side-stepping the issue of how to arrive at the appropriate discount rate to be used in the present value calculation. This will allow us to focus on the two most popular methods for evaluating an investment project.. The Net Present Value (N) Method Evaluating a potential investment in new equipment, entry into a new market, or introduction of a new product can be accomplished like any sort of investment decision (e.g., purchase of bonds, making a loan, buying stock). The decision requires evaluating the present value of the expected benefits from the investment. The N method compares the calculated present value of the investment with the initial cost of it. T CFt CF CF CFT () N Cost Cost t + + L + T t ( + k) ( + k) ( + k) ( + k) where, CF expected Cash Flow generate by the investment in period t k cost of capital (thus, the appropriate discount rate) T number of periods (how long the investment will generate a cash flow) Cost Initial cost of the project (e.g., price of machine, etc.) We should note that the first term in the equation is simply the present value of the expected future cash flows from the investment. Notice as well, we are defining the net benefit here as the cash flows. We could have defined net benefit as net income, gross profit, EBIT, or something else. We have chosen to use cash flows because it tends to be most common, though there are issues with this. Consider the difference between net income and cash flows (think of accrual accounting vs. the cash basis of accounting). We will have more to say about the various ways to define the cash flows, especially the expected cash flows. Once you have calculated the N, what do you do with it? How should it be used to make a decision? The decision rule is actually quite straightforward. N > 0 Invest N < 0 Do NOT Invest However, in practice, you may want to do a sensitivity analysis as well. That is, how sensitive is the decision to various assumptions about the expected cash flows? How sensitive is the decision to assumptions about the cost of capital (i.e., the appropriate discount rate)? We will be doing some of this sensitivity analysis, but first take a look at a few examples. 8

9 Example : A corporation is evaluating possible investments (A, B, C). The cost and expected net cash flows (CF) for each are given below. Year CF on A CF on B CF on C 0-0,000-0,000 -,000,000 5,000,000,000 4,000,000 4,000 4,000 4, ,000,000 5, ,000,000 6,000 Assume the corporation uses a 0% discount rate. Calculate the N for each project. N A,000,000 4,000 4,000 5, ,000 $, 4 5 (.0).0 (.0) (.0) (.0) Similar calculations can be done for B and C. N B $,464 N C $,67 Note, the calculations are tedious especially if the investment will last many years. You can plug the numbers into Excel in order to solve. In order to do this properly, you will use a N function. However, you must be careful. The N function in Excel actually just calculates the present value of a stream of payments. Thus, you should use the following: N(cell for discount rate, cells for Cash Flows) + Cost (which should be negative) Points to Note:. All projects sum to the same amount (i.e., $7,000). Project B has the highest N, why?. Project C has the lowest N, why? 9

10 Example : Assume the corporation is just considering project A. However, it is uncertain about the appropriate discount rate. Hence, it calculates the N for project A under different assumptions about the discount rate. The results are the following. Discount Rate N-A 5% $4,8 6% $,90 7% $,498 8% $,085 9% $,690 0% $, % $,95 % $,606 % $,75 4% $958 5% $654 6% $6 7% $8 8% -$85 9% -$44 0% -$69 % -$99 % -$,59 % -$,79 4% -$,59 5% -$,795 0

11 We can depict this graphically. Project A $5,000 $4,000 Net Present Value $,000 $,000 $,000 $0 -$,000 5% 6% 7% 8% 9% 0% % % % 4% 5% 6% 7% 8% 9% 0% % % % 4% 5% -$,000 -$,000 Discount Rate Is there anything special about where the N line crosses the x-axis? YES!

12 4. The Internal Rate of Return (IRR) Method The internal rate of return method is not very different from the net present value method. Both methods utilize a present value calculation, and therefore account for the so-called time-value of money (i.e., money today is worth more than money tomorrow). The basic formula is straightforward. T CF t () Cost t ( + IRR t ) If you compare this to the N, you see that it appears that all we ve done is move the cost over to the right-hand side. We have done this, but there is something more. The IRR (called the internal rate of return ) is a rate (stated in %), but a special rate. The IRR is the discount rate that makes the present value of the future cash flows from the investment project just equal to the cost of the project. It is very tedious to calculate without a financial calculator or a spreadsheet, unless of course we make the simplifying forever assumption. Once you have determined the internal rate of return (IRR), then what? Like the N, the IRR method uses a simple decision rule. IRR > Cost of Capital Invest IRR < Cost of Capital Do NOT Invest Thus, the IRR method does not escape from the need to determine a cost of capital. Remember, the cost of capital is the appropriate discount rate to be used and goes by other names such as required rate of return or hurdle rate. Thus, if the IRR for a project gets over the hurdle rate (i.e., cost of capital) then the project should be done. A couple of examples might help. Example : Suppose we return to the corporation in example. We continue to use the 0% as the required rate of return (or, cost of capital). Though, this time the corporation uses the internal rate of return method to decide on whether to invest. Year CF on A CF on B CF on C 0-0,000-0,000 -,000,000 5,000,000,000 4,000,000 4,000 4,000 4, ,000,000 5, ,000,000 6,000

13 For Project A:,000,000 4,000 4,000 5, , % 4 5 IRR A ( + IRR) ( + IRR) ( + IRR) ( + IRR) ( + IRR) IRR for B 5% IRR for C 4% Calculating the internal rate of return in Excel is straightforward. IRR(Cells containing Costs and Cash Flows, guess) As stated previously, calculating the IRR requires a trial-and-error procedure. Excel allows you to input a best guess in order to get started and hopefully converge on the correct answer quicker. If you leave this blank, Excel automatically uses 0% as the initial guess. Notice, the internal rates of returns for the projects would seem to be consistent with our net present value results (e.g., B has higher N and IRR, C has lowest N and IRR). Though, this may not be true in general there are some problems with IRR. Example 4: A project with unusual cash flows. Suppose a corporation is considering an investment project with the following characteristics. Year CF 0-504,86-6,070 5, ,000 We will need to use the trial-and-error procedure to calculate the IRR for this project. However, in this case, it matters where we begin (e.g., initial guess) the procedure. Guess IRR 0% 5% 0% % 40% 4% 70% 67% What is happening? It may help to recall that the IRR is simply the break-even for N. Let s graph the N versus various discount rates.

14 % 6% 64% 66% 68% 70% 0% % 4% 6% 8% 0% % 4% 6% 8% 40% 4% 44% 46% 48% 50% 5% 54% 56% 58% Net P resent V alue (NP V ) -0.0 Discount Rate The Internal Rate of Return is the discount rate that makes the Net Present Value zero. So, which one is it? 4

15 Example 5: A second problem that arises with the IRR method is when projects are mutually exclusive. That is, undertaking one project implies that the other project cannot be done. A corporation owns a small plot of land in an excellent location. The corporation is considering whether to put up a coffee stand or ice-cream stand. Year Coffee Ice-Cream IRR of Coffee Stand 4% IRR of Ice-Cream Stand % Clearly, the Coffee Stand is the superior investment. Or, is it? Consider the N for each project at various discount rates. N Discount Rate Coffee Ice-Cream 5% $4. $ % $9.06 $9.79 5% $7.8 $4.8 0% $7.06 $. 5% -$.6 -$8. Now, which is the better investment project? Remember, the corporation has just the one plot of prime commercial land, so they can only do one of the projects. Another graph may help to see what s happening. 5

16 $80.00 $60.00 Net P resent V alue $40.00 $0.00 $0.00 -$0.00 % % 5% 7% 9% % % 5% 7% 9% % % 5% 7% 9% % % 5% 7% 9% Coffee Ice-Cream -$40.00 Discount Rates At discount rates below about % (.% to be exact), the Ice-Cream stand is superior. Discount rates greater than about % lead the corporation to choose the coffee stand. 6

17 Appendix (Optional). Details of Present Value In this appendix we look at how more of the details of present value. We will begin with compounding (or, future value) prior to present value. In addition to just more practice, the appendix lays out how to deal with various assumptions concerning growth and payment periods.. Compounding Suppose the rabbit population of a zoo grows by 40% per year. If the zoo initially has rabbits (hopefully one male and one female), how many rabbits will the zoo have in 0 years? The following begins the procedure. End of Year ( +.40). 8 End of Year.8( +.40). 9 End of Year.9( +.40) Etc. etc. We can write the above in the following way. End of Year ( +.40). 8 End of Year ( +.40). 9 End of Year ( +.40) Etc. etc. Writing things in this way, we can easily arrive at the rabbit population at the end of 0 years. ( +.40) rabbits A general expression for growth would be the following, X ( + k) t where X stands for the beginning amount, k the rate of growth, and t the time. As another example, suppose you deposited $00 in a bank account that pays 0% interest per year. How much would you have at the end of year, years, and 40 years? year $00 $ 0 years $00 $ 40 years $00 40 $4, 56 7

18 We might extend things a bit to see the power of compounding (we see it already when leaving the $00 in the bank account for 40 years). Suppose that the bank account pays interest semiannually. Previously, we were assuming that the bank paid 0% interest once per year. In the case of the year deposit, we saw that the second year began earning interest not on $00 but rather on $0. Now, the same thing will occur every 6 months implying that after the first 6 months interest will begin to be earned on more than the $00. Let s look at the year deposit in detail. End of 6 months $00(+.05) $05 End of year $05(+.05) $0.5 Notice, the interest rate is stated in annual terms as 0% - more generally, we think of this as the annual growth rate of what we re interested in. Half of the interest gets paid every 6 months. The important thing to note is that paying semiannually has meant a larger sum at the end of the year. Suppose the bank paid the interest quarterly. End of months $00(+.05) $0.50 End of 6 months $0.50(+.05) $05.06 End of 9 months $05.06(+.05) $07.69 End of the year $07.69(+.05) $0.8 We end up with even more at the end of the year! Now, we can write a general expression for all of this as, X + k n nt where the n represents the number of times interest is paid during a year. Take a year, quarterly compounding example. nt 4 k.0 X + $00 + n 4 $0.8 For the year and 40 year deposits, we have the following. nt 4 k.0 X + $00 + n 4 $.84 nt 4 k.0 X + $00 + n 4 40 $5,97.79 We could replace dollars with rabbits and the interest rate with the population growth rate to discover how many rabbits we ll have at the end of any number of years. 8

19 Clearly, compounding matters. For example, the longer you leave money in a deposit with a positive interest rate, the more you will have. The more times interest is paid during the course of a year, the greater the sum will be at the end of any given time period. A natural question arises now as to what would happen if interest was paid continuously (every nanosecond). Although the derivation of the solution takes a bit of work, the solution itself is quite straightforward. kt Xe The e is our old, irrational friend of.788 Thus, if our bank pays an annual interest rate of 0% compounded continuously, we ll have the following amounts. End of year Xe kt.0 $00e $0. 5 End of years Xe kt.0 $00e $. 4 End of 40 years Xe kt.0 40 $00e $5, Of course, all of these amounts exceed the annual, semiannual, and quarterly compounding. See what fun the e can be. Keep in mind, we have been mainly discussing the growth of a bank account. However, the above discussion can be applied to growth rates of anything (e.g., bacteria, the plague, etc.) so long as we have a constant growth rate.. Present Value Mechanically, calculating present value is simply the reverse of calculating future value. The previous section took some present amount ( rabbits, $00), assumed it grew at a particular rate then calculated the future amount at a particular time. We can quickly see the relation between present value and future value by returning to our bank account. Recall, we deposited the present value (labeled X before, now we ll label it ) of $00 into a bank deposit that paid 0% interest. At the end one year (assuming interest paid once), the amount grew to $00 (we ll label this future value or FV). FV ( + k) t $0 $00 We simply rearrange the above, solving for, in order to arrive at the present value. FV ( + k) t $0 $00 You might think of present value () as stating the current worth of receiving something in the future. Alternatively, you can think of present value () as the amount needed today in order to have some future value (FV) in some later time. 9

20 Just as we did for future value, we could state the present value under various compounding assumptions (e.g., semi-annually, quarterly). To do so, simply requires that we rearrange the future value formula and solve for the present value. It might be more interesting, in the limited space here, if we take up a particular issue arising from present value. Suppose, for example, you received a court settlement (e.g., car accident leaving you with neck pain) that paid you $0,000 over the course of the following years. Now, in response to this type of settlement, businesses have arisen that will buy you out of the settlement. That is, the business will give you a lump sum amount today in exchange for receiving your future payments. Suppose a business such as this offers to pay you $5,000 today in exchange for receiving your $0,000 over the course of the following years. Is this a good deal? Should you take the money and run or be patient and receive your yearly settlement checks? Let s see. Suppose the interest rate on a typical bank deposit is 5%. We need to calculate the present value of those settlement checks. of receiving $0,000 at the end of the first year + of receiving $0,000 at the end of the second year + of receiving $0,000 at the end of the third year Now, we can just apply our general present value formula. $0,000 $0,000 $0, $9,54 + $9,070 + $8,68 $7, ( +.05) ( +.05) ( +.05) The present value of those settlement checks actually exceeds what the business is offering (why else would they offer it?). Notice how the present value of the same $0,000 gets smaller as the payment is received further into the future. But, this merely says that in order to have $0,000 in a bank account paying 5% annual interest (compounded annually) three years from today requires putting $8,68 into the bank today. We can use summation notation to write out a general expression for the present value of any future values. FV FV FV FV N i + + L i i ( + r) ( + r) ( + r) ( + N N r) Before turning to continuous compounding, we might consider what happens when a periodic future value is received forever (thus, N is infinity). We have seen that the further into the future the payment is received, the smaller the present value. Consider receiving a single $,000 payment in the following future time periods assume the interest rate is 0%. $,000 One year from now: $ $,000 0 years from now: $

21 $, years from now: $ Thus, as the time of payment gets pushed further into the future, the present value of that payment can become exceedingly small. Once we get passed 00 years, in our example, the present value becomes nearly zero. Keeping this in mind, calculate the present value of a $00 yearly payment forever. We ll assume a 0% interest rate. i $00 i $00 $ L??? Previously, we might have thought that the present value of this stream of income would be enormous. However, now we recognize that the present value of $00 received 00 years from now would be quite small (just less than a penny). But, what is the answer? We cannot expect to actually do the calculation it would take forever. We can employ, however, note that the above is a geometric series with a common ratio of /(+k). First, note that we have a constant within the summation that can be pulled to the left hand side. $00 $00 + i i + L We are left with one of those pesky expressions to solve for what does it sum to? Let s just set the sum to S and do some manipulation. S L S + + L S S Now we can just solve for S. It may look tricky to solve, but we really just have one equation and one unknown. S 0.0 We should now be able to calculate the present value of a yearly $00 payment forever. This is not an example without relevance to the real world. There are some bonds that are issued (not in the U.S.) that promise to pay a set amount forever.

22 $ 00 $00 (.0) + i i + + L $00 0 $,000 The present value is probably nowhere near what one thought it might be. The general statement of the present value of a constant periodic payment made forever is the following. FV i i ( + k) FV k Our final application for present value deals with the issue of continuous compounding. In the above, we have assumed that the interest rate referred to interest paid annually. We left the applications to semiannual and quarterly compounding for sample questions. Continuous compounding tends to be intimidating because it utilizes the mystical e. 4 We saw that to calculate the future value of a thing growing at a particular rate and for a particular time, required the following calculation (using the and FV notation now). kt FV e Recall what happens when we raise a number or variable to a negative power. x x In fact, we could have stated our previous present value formulas by raising the (+k) terms to the negative time period. This is what we do when calculating the present value of future payments with continuous compounding. FVe kt For example, assume a 5% annual interest rate compounded continuously for a $,000 payment received 0 years from today. FVe kt $,000e.05 0 $606.5 Compare this to interest paid annually. $,000 ( +.05) 0 $6.9 4 In fact, e should no longer be that mystical to us (though, I have to admit, there still remains a certain mystic about it to me). It is simply a number (an irrational number to be sure, but still a number) that is useful in many different contexts, much like the number pie (by the way, there exists a fascinating book on the history of pie for anyone interested). Mathematicians may disagree about this, but it might be a useful way for us to conceptualize e in order to get over any phobias concerning it.

23 Continuous compounding implies a small present value when other things (i.e., FV, r, t) are the same. Again, one way to interpret this is that a smaller amount today would need to be deposited in an account paying a 5% interest rate in order to grow to $,000 in ten years with continuous compounding versus annual compounding. Sample Problems for Compounding and Present Value. Alice is beginning to plan for retirement. He has put $5,000 in an account paying % interest annually. She plans to retire in 0 years. a. How much will Alice have for retirement in 0 years? b. How much would Alice have if interest was paid semiannually? c. How much would Alice have if interest was paid quarterly? d. How much would Alice have if interest was compounded continuously?. Suppose Alice from the previous question was not happy with her retirement amounts under any of the interest plans. Alice believes that she will need to have $00,000 in the account when she retires. Assuming the annual interest rate is %, calculate how much Alice will need to deposit today in order to have $00,000 in 0 years under the various assumptions. a. Interest compounded annually. b. Interest compounded semiannually. c. Interest compounded quarterly. d. Interest compounded continuously.. Suppose you win the lottery. Upon winning, you are given two payment options. Under the first option, you are given a lump sum payment today of $800,000. Under the second option, you will receive a check in the amount of $50,000 at the end of each year for the next 0 years. a. Assuming an interest rate of 0% (compounded annually), which option would you prefer? b. Assuming an interest rate of % (compounded annually), which option would you prefer? c. At what interest rate would you be indifferent between the two plans? d. Repeat each of the above with the modification that you will receive a $50,000 yearly payment forever.