CH 7

CH 7 7.1 Net present value In Chapter 1, we argued that the goal of financial management is to create value for the shareholders. The financial manager must thus examine a potential investment in light of its likely effect on the price of the firm's shares. In this section, we describe a widely used procedure for doing this, the net present value (NPV)the difference between an investment’s market value and its cost approach.   An investment is worth undertaking if it creates value for its owners. In the most general sense, we create value by identifying an investment that is worth more in the marketplace than it costs us to acquire. How can something be worth more than it costs? It's a case of the whole being worth more than the cost of the parts.     For example, suppose you buy a dilapidated house for $380 000 and spend another $65 000 on painters, plumbers, and so on to get it fixed up. Your total investment is $445 000. When the work is completed, you place the house back on the market and find that it’s worth $495 000. The market value ($495 000) exceeds the cost ($445 000) by $50 000. What you have done here is to act as a manager and bring together some fixed assets (a house), some labour (plumbers, carpenters and others), and some materials (carpeting, paint and so on). The net result is that you have created $50 000 in value. Put another way, this $50 000 is the value added by management.     With our house example, it turned out after the fact that $50 000 in value was created. Things thus worked out very nicely. The real challenge, of course, was to somehow identify ahead of time whether or not investing the necessary $245 000 was a good idea in the first place. This is what capital budgeting is all about, namely, trying to determine whether a proposed investment or project will be worth more than it costs once it is in place.     For reasons that will be obvious in a moment, the difference between an investment's market value (in today's dollars) and its cost (also in today's dollars) is called the net present value of the investment, abbreviated as NPV. In other words, net present value is a measure of how much value is created or added today by undertaking an investment. Given our goal of creating value for the shareholders, the capital budgeting process can be viewed as a search for investments with positive net present values.     With our dilapidated house, you can probably imagine how we would go about making the capital budgeting decision. We would first look at what comparable, renovated properties were selling for in the market. We would then get estimates of the cost of buying a particular property and bringing it up to market. At this point, we have an estimated total cost and an estimated market value. If the difference is positive, then this investment is worth undertaking because it has a positive estimated net present value. There is risk, of course, because there is no guarantee that our estimates will turn out to be correct.     As our example illustrates, investment decisions are greatly simplified when there is a market for assets similar to the investment we are considering. Capital budgeting becomes much more difficult when we cannot observe the market price for at least roughly comparable investments. The reason is that we are then faced with the problem of estimating the value of an investment using only indirect market information. Unfortunately, this is precisely the situation that the financial manager usually encounters. We examine this issue next.   p. 213 Estimating net present value Imagine that we are thinking of starting a business to produce and sell a new product, say, organic fertiliser. We can estimate the start-up costs with reasonable accuracy because we know what we will need to buy to begin production. Would this be a good investment? Based on our discussion, you know that the answer depends on whether or not the value of the new business exceeds the cost of starting it. In other words, does this investment have a positive NPV?     This problem is much more difficult than our 'fixed-up' house example, because entire fertiliser companies are not routinely bought and sold in the marketplace; so it is essentially impossible to observe the market value of a similar investment. As a result, we must somehow estimate this value by other means.     Based on our work in Chapters 5 and 6 , you may be able to guess how we will go about estimating the value of our fertiliser business. We will first try to estimate the future cash flows that we expect the new business to produce. We will then apply our basic discounted cash flow procedure to estimate the present value of those cash flows. Once we have this estimate, we then estimate NPV as the difference between the present value of the future cash flows and the cost of the investment. As we mentioned in Chapter 5 , this procedure is often called discounted cash flow (DCF) valuationthe process of valuing an investment by discounting its future cash flows     To see how we might go about estimating NPV, suppose we believe that the cash revenues from our fertiliser business will be $200 000 per year, assuming that everything goes as expected. Cash costs (including taxes) will be $140 000 per year. We will wind down the business in eight years. The plant, property and equipment will be worth $20 000 as salvage at that time. The project costs $300 000 to launch. We use a 15 per cent discount rate on new projects such as this one. Is this a good investment? If there are 10 000 shares on issue, what will be the effect on the price per share from taking it? p. 214 Table 7.1   Projected cash flows ($000)    From a purely mechanical perspective, we need to calculate the present value of the future cash flows at 15 per cent. The net cash inflow will be $200 000 cash profit less $140 000 in costs per year for eight years. These cash flows are illustrated in Table 7.1. As Table 7.1 suggests, we effectively have an eight-year annuity of $200 000 - $140 000 = $60 000 per year along with a single lump-sum inflow of $20 000 in eight years time. Calculating the present value of the future cash flows thus comes down to the same type of problem we considered in Chapter 5 .    The present value is provided as follows:     When we compare this to the $300 000 estimated cost, the NPV is:        Therefore, this is not a good investment. Based on our estimates, taking it would decrease the total value of the shares by $24 223. With 10 000 shares on issue, our best estimate of the impact of taking this project is a loss of value of $24 223/10 000 = $2.42 per share.    Our fertiliser example illustrates how NPV estimates can be used to determine whether or not an investment is desirable. From our example, notice that, if the NPV is negative, the effect on share value will be unfavourable. If the NPV were positive, the effect would be favourable. As a consequence, all we need to know about a particular proposal for the purpose of making an accept/reject decision is whether the NPV is positive or negative.    Given that the goal of financial management is to increase share value, our discussion in this section leads us to the net present value rule:     An investment should be accepted if the net present value is positive and rejected if it is negative.        In the unlikely event that the net present value turned out to be exactly zero, we would be indifferent to taking the investment or not taking it.    Two comments about our example are in order. First and foremost, it is not the rather mechanical process of discounting the cash flows that is important. Once we have the cash flows and the appropriate discount rate, the required calculations are fairly straightforward. The task of coming up with the cash flows and the discount rate in the first place is much more challenging. We will have much more to say about this in the next several chapters. For the remainder of this chapter, we take it as given that we have estimates of the cash revenues and costs and, where needed, an appropriate discount rate. p. 215    The second thing to keep in mind about our example is that the -$24 223 NPV is an estimate. Like any estimate, it can be high or low. The only way to find out the true NPV would be to place the investment up for sale and see what we could get for it. We generally won't be doing this, so it is important that our estimates be reliable. Once again, we will have more to say about this later. For the rest of this chapter, we will assume the estimates are accurate.    Example 7.1    Using the NPV rule    Suppose we are asked to decide whether or not a new consumer product should be launched. Based on projected sales and costs, we expect that the cash flows over the five-year life of the project will be $2000 in the first two years, $4000 in the next two, and $5000 in the last year. It will cost about $10 000 to begin production. We use a 10 per cent discount rate to evaluate new products of this type. What should we do here?    Given the cash flows and discount rate, we can calculate the total value of the product by discounting the cash flows back to the present:     The present value of the expected cash flows is $12 313, but the cost of getting those cash flows is only $10 000, so the NPV is $12 313 - $10 000 = $2313. This is positive; so, based on the net present value rule, we should take it.       As we have seen in this section, estimating NPV is one way of assessing the profitability of a proposed investment. It is certainly not the only way that profitability is assessed, and we now turn to some alternatives. As we will see, when compared to NPV, each of the ways of assessing profitability that we examine is flawed in some key way, so NPV is the preferred approach in principle, if not always in practice.     Question:    For a loan of $5000, Mark will give you $5250 in a year, however Paul will give you $6000 in a week. Which is preferred? Answer:    The missing element here is the risk of the cash flows. Suppose Mark is going to put his $5000 into a major bank and Paul is going to do some major gambling at the casino.Now make your decision.           Remember the important elements in making financial decisions are the cash flows, the timing of the cash flows and the risk of the cash flows.       A financial calculator has features that allow you to do an NPV calculation very easily. The different types of calculators require slightly different keystrokes but the basic concepts are the same. What you need to do before any calculation is clear any stored cash flow data from previous calculations. After this you will obviously need to input the new cash flows and the discount rate. To input cash flows there will be a key like . The  means cash flow and the j is time. You need to input the cash flow in time order as the calculator will assume that the first cash flow is at time 0 or today and this will not be discounted, the next cash flow is time one and this will be discounted one period etc. With the short example the cash flows are -250, 100 and 200. Remember from Chapter 5, depending on what type of calculator you have, to input a negative cash flow you may have to use the +- key. The keystrokes will be -250 (and what will probably flash on the screen is  ), then 100  (  ) and finally 200 . Now all you have to do is input the discount rate of 15 per cent so 15 . If you press the key the answer is -11.81 and press the  key and the  is 11.65 per cent.     p. 216 Microsoft Excel® is great for adding up columns of numbers and it has a number of inbuilt functions. Excel has an inbuilt NPV and IRR function that we can use. We will show you the functions using the example of -250, 100 and 200 with a discount rate of 15 per cent. Set up an Excel spreadsheet as we have done and you can follow the example.    If you use the function fx, one of the choices is NPV. When you choose NPV the drop down screen requires you to identify the cash flows to be discounted and the discount rate.    Move the cursor to cell B3 before you use this function. When you use the function you need to input the discount rate expressed as a decimal. Take care with the cash flows to be discounted. It is only the $100 and the $200 that need to be discounted and the calculation has not included the outlay of $250. You will need to adjust the calculation as we have done below. To do this press the OK button on the drop down menu and then go to cell B3 and adjust the formula by +B2.    Now you can use the NPV function. The IRR function is very similar except you use all of the cash flows now. Source: Excel spreadsheets used with permission from Microsoft.    And the IRR rounded is 12%.   p. 217     Concept questions 7.1a    What is the net present value rule? 7.1b    What is a call provision?   7.2 The payback rule It is very common in practice to talk of the payback on a proposed investment. Loosely, the payback is the length of time it takes to recover our initial investment or 'get our money back'. Because this idea is widely understood and used, we will examine it in some detail. p. 218 Defining the rule We can illustrate how to calculate a payback with an example. Table 7.2 shows the cash flows from a proposed investment. How many years do we have to wait until the accumulated cash flows from this investment equal or exceed the cost of the investment? As Table 7.2 indicates, the initial investment is $50 000. After the first year, the firm has recovered $30 000, leaving $20 000 to be recovered. The cash flow in the second year is exactly $20 000, so this investment 'pays for itself' in exactly two years. Put another way, the  payback periodthe amount of time required for an investment to generate cash flows to recover its initial cost is two years. If we require a payback of, say, three years or less, then this investment is acceptable. This illustrates the payback period rule: Table 7.2   Net project cash flows      Based on the payback rule, an investment is acceptable if its calculated payback is less than some prescribed number of years.         In our example, the payback works out to be exactly two years. This won't usually happen, of course. When the numbers don't work out exactly, it is customary to work with fractional years. For example, suppose the initial investment is $60 000, and the cash flows are $20 000 in the first year and $80 000 in the second. The cash flows over the first two years are $100 000, so the project obviously pays back sometime in the second year. After the first year, the project has paid back $20 000, leaving $40 000 to be recovered. To figure out the fractional year, note that this $40 000 is $40 000/$80 000 = 1/2 of the second year's cash flow. Assuming that the $80 000 cash flow is paid uniformly throughout the year, the payback would thus be 1.5 years.    Example 7.2    Calculating payback    The projected cash flows from a proposed investment are: This project costs $500. What is the payback period for this investment?    The initial cost is $500. After the first two years, the cash flows total $300. After the third year, the total cash flow is $800, so the project pays back sometime between the end of year 2 and the end of year 3. Since the accumulated cash flows for the first two years are $300, we need to recover $200 in the third year. The third year cash flow is $500, so we will have to wait $200/500 = 2/5 of the year to do this. The payback period is thus 2.4 years, or about two years and five months.       Now that we know how to calculate the payback period on an investment, using the payback period rule for making decisions is straightforward. A particular cut-off time is selected, say two years, and all investment projects that have payback periods of two years or less are accepted, and all of those that pay off in more than two years are rejected.    Table 7.3 illustrates cash flows for five different projects. The figures shown as the year 0 cash flows are the cost of the investment. We examine these to indicate some peculiarities that can, in principle, arise with payback periods. p. 219 Table 7.3   Expected cash flows for projects A to E    The payback for the first project, A, is easily calculated. The sum of the cash flows for the first two years is $70, leaving us with $100 - $70 = $30 to go. Since the cash flow in the third year is $50, the payback occurs sometime in that year. When we compare the $30 we need to the $50 that will be coming in, we get $30/$50 = 0.60, so payback will occur 60 per cent of the way into the year. The payback period is thus 2.6 years.    Project B's payback is also easy to calculate: it never pays back because the cash flows never total up to the original investment. Project C has a payback of exactly 4 years because it supplies the $130 that B is missing in year 4. Project D is a little strange. Because of the negative cash flow in year 3, you can easily verify that it has two different payback periods, two years and four years. Which of these is correct? Both of them; the way the payback period is calculated doesn't guarantee a single answer. Finally, Project E is obviously unrealistic, but it does pay back in 6 months, thereby illustrating the point that a rapid payback does not guarantee a good investment.      Question:    How does payback stand in relation to its consideration of the timing of the cash flows? Answer:    Because payback adds and subtracts cash flows at dollar face value it does not consider the fact that a dollar now is more valuable than a dollar in a year. It ignores the timing of cash flows.        Analysing the payback period rule When compared to the NPV rule, the payback period rule has some rather severe shortcomings. First of all, the payback period is calculated by simply adding up the future cash flows. There is no discounting involved, so the time value of money is completely ignored. A payback rule also does not consider risk differences at all. The payback would be calculated the same way for both very risky and very safe projects.    Perhaps the biggest problem with the payback period rule is coming up with the right cut-off period, because we don't really have an objective basis for choosing a particular number. Put another way, there is no economic rationale for looking at payback in the first place, so we have no guide as to how to pick the cut-off. As a result, we end up using a number that is arbitrarily chosen.    Suppose we have somehow decided on an appropriate payback period, say two years or less. As we have seen, the payback period rule ignores the time value of money for the first two years. More seriously, cash flows after the second year are ignored entirely. To see this, consider the two investments, Long and Short in Table 7.4. Both projects cost $250. Based on our discussion, the payback on Long is 2 + $50/$100 = 2.5 years, and the payback on Short is 1 + $150/$200 = 1.75 years. With a cut-off of two years, Short is acceptable and Long is not. Table 7.4   Investment projected cash flows    Is the payback period rule giving us the right decisions? Maybe not. Suppose again that we require a 15 per cent return on this type of investment. We can calculate the NPV for these two investments as: p. 220        Now we have a problem. The NPV of the shorter term investment is actually negative, meaning that taking it diminishes the value of the shareholders' equity. The opposite is true for the longer term investment-it increases share value.    Our example illustrates two primary shortcomings of the payback period rule. First, by ignoring time value, we may be led to take investments (like Short) that actually are worth less than they cost. Second, by ignoring cash flows beyond the cut-off, we may be led to reject profitable long-term investments (like Long). More generally, using a payback period rule will tend to bias us towards shorter term investments.    Table 7.5 illustrates the shortcomings of the payback period rule. All of the projects have a payback of three years but clearly some are preferred to others. Project B is preferred to all other projects ignoring time value because its cash flows add to $1200 and the others are less. E is preferred to D because it has the same pattern of cash flow in the first three years plus an additional $100 in year 4. D is preferred to C if time value is considered. D provides larger cash flows earlier in the life of the project. Obviously B is preferred to A because again it has the same pattern of cash flow plus an additional $300 in year four. Table 7.5   Projects with different cash flows      Question:    Consider A and F above. Both have the same payback period and cash flow pattern. Which one is preferred? Answer:    If A was a relatively safe, low-risk project and F was an extremely high-risk project then A would be preferred to F, but payback does not allow for this.             Remember the important elements in making financial decisions are the cash flows, the timing of the cash flows and the risk of the cash flows.      Redeeming qualities Despite its shortcomings, the payback period rule is often used by large and sophisticated companies when making relatively small decisions. There are several reasons for this. The primary reason is that many decisions simply do not warrant detailed analysis because the cost of the analysis would exceed the possible loss from a mistake. As a practical matter, an investment that pays back rapidly and has benefits extending beyond the cut-off period probably has a positive NPV.    Small investment decisions are made by the hundreds every day in large organisations. Moreover, they are made at all levels. As a result, it would not be uncommon for a corporation to require, for example, a two-year payback on all investments of less than $10 000. Investments larger than this are subjected to greater scrutiny. The requirement of a two-year payback is not perfect for reasons we have seen, but it does exercise some control over expenditures and thus has the effect of limiting possible losses. p. 221    In addition to its simplicity, the payback rule has two other features to recommend it. First, because it is biased towards short-term projects, it is biased towards liquidity. In other words, a payback rule tends to favour investments that free up cash for other uses more quickly. This could be very important for a small business; it would be less so for a large corporation. Second, the cash flows that are expected to occur later in a project's life are probably more uncertain. Arguably, a payback period rule adjusts for the extra riskiness of later cash flows, but it does so in a rather draconian fashion-by ignoring them altogether. We should note here that some of the apparent simplicity of the payback rule is an illusion. The reason is that we still must come up with the cash flows first, and, as we discussed above, this is not at all easy to do. Thus, it would probably be more accurate to say that the concept of a payback period is both intuitive and easy to understand. Summary of the payback period rule To summarise, the payback period is a kind of 'break-even' measure. Because time value is ignored, you can think of the payback period as the length of time it takes to break even in an accounting sense, but not in an economic sense. The biggest drawback to the payback period rule is that it doesn't ask the right question. The relevant issue is the impact an investment will have on the value of our shares, not how long it takes to recover the initial investment.    Nevertheless, because it is so simple, companies often use it as a screen for dealing with the myriad of minor investment decisions they have to make. There is certainly nothing wrong with this practice. Like any simple rule of thumb, there will be some errors in using it, but it wouldn't have survived all this time if it weren't useful. Now that you understand the rule, you can be on the alert for those circumstances under which it might lead to problems. To help you remember, Table 7.6 lists the pros and cons of the payback period rule. Table 7.6   Advantages and disadvantages of the payback period rule      Concept questions 7.2a    In words, what is the payback period? What is the payback period rule? 7.2b    Why do we say that the payback period is, in a sense, an accounting break-even?   7.3 The discounted payback rule We saw that one of the shortcomings of the payback period rule was that it ignored time value. There is a variation of the payback period, the discounted payback periodthe length of time required for an investment’s discounted cash flows to equal its initial cost , that fixes this problem. The discounted payback period is the length of time until the sum of the discounted cash flows is equal to the initial investment. The discounted payback rule would be: p. 222      Based on the discounted payback rule, an investment is acceptable if its discounted payback is less than some prescribed number of years.         To see how we might calculate the discounted payback period, suppose that we require a 12.5 per cent return on new investments. We have an investment that costs $300 and has cash flows of $100 per year for five years. To get the discounted payback, we have to discount each cash flow at 12.5 per cent and then start adding them. We do this in Table 7.7. In Table 7.7, we have both the discounted and the non-discounted cash flows. Looking at the accumulated cash flows, the regular payback is exactly three years (look for the => in year 3). The discounted cash flows total $300 only after four years, however, so the discounted payback is four years as shown. (In this case, the discounted payback is an exact number of years. This will not ordinarily happen. Calculating a fractional year for the discounted payback period is more involved than it is for the ordinary payback, but it is not commonly done.) Table 7.7   Ordinary and discounted payback      Question:    Refer to Table 7.7. Why does the present value of the year 5 cash flow equal the NPV? Answer:    The sum of the present values of the first four years equals the outlay, so the extra fifth year present value will give a present value greater than the outlay, which will create the NPV.           How do we interpret the discounted payback? Recall that the ordinary payback is the time it takes to break even in an accounting sense. Since it includes the time value of money, the discounted payback is the time it takes to break even in an economic or financial sense. Loosely speaking, in our example, we get our money back along with the interest we could have earned elsewhere in four years.    Figure 7.1 illustrates this idea by comparing the future value at 12.5 per cent of the $300 investment versus the future value of the $100 annual cash flows at 12.5 per cent. Notice that the two lines cross at exactly four years. This tells us that the value of the project’s cash flows catches up and then passes the original investment in four years.    Table 7.7 and Figure 7.1 illustrate another interesting feature of the discounted payback period. If a project ever pays back on a discounted basis, then it must have a positive NPV. (This argument assumes that cash flows other than the first are all positive. If they are not, then these statements are not necessarily correct. Also, there may be more than one discounted payback.)    This is true because, by definition, the NPV is zero when the sum of the discounted cash flows equals the initial investment. For example, the present value of all the cash flows in Table 7.7 is $355. The cost of the project was $300, so the NPV is obviously $55. This $55 is the present value of the cash flow that occurs after the discounted payback (see the last line in Table 7.7). In general, if we use a discounted payback rule, we won’t accidentally take on any projects with a negative estimated NPV. The problem though is the rejection of positive NPV projects. If management had set a discounted payback period of three years in the above example, the project would have been rejected. p. 223     Figure 7.1         Future value of project cash flows       Based on our example, the discounted payback would seem to have much to recommend it. You may be surprised to find out that it is rarely used in practice. Why? Probably because it really isn't any simpler than NPV. To calculate a discounted payback, you have to discount cash flows, add them up, and compare them to the cost, just as you do with NPV. So, unlike an ordinary payback, the discounted payback is not especially simple to calculate.    A discounted payback period rule still has a couple of significant drawbacks. The biggest one is that the cut-off still has to be arbitrarily set and cash flows beyond that point are ignored. (If the cut-off were forever, then the discounted payback rule would be the same as the NPV rule. It would also be the same as the profitability index rule considered in a later section.)    As a result, a project with a positive NPV may not be acceptable because the cut-off is too short. Also, just because one project has a shorter discounted payback than another does not mean it has a larger NPV.    All things considered, the discounted payback is a compromise between a regular payback and NPV that lacks the simplicity of the first and the conceptual rigour of the second. Nonetheless, if we need to assess the time it will take to recover the investment required by a project, then the discounted payback is better than the ordinary payback because it considers time value. In other words, the discounted payback recognises that we could have invested the money elsewhere and earned a return on it. The ordinary payback does not take this into account.      Concept questions 7.3a    In words, what is the discounted payback period? Why do we say it is, in a sense, a financial or economic break-even measure? 7.3b    What advantage(s) does the discounted payback have over the ordinary payback?   p. 224    Example 7.3    Calculating the discounted payback    Consider an investment that costs $400 and pays $100 per year forever. We use a 20 per cent discount rate on this type of investment. What is the ordinary payback? What is the discounted payback? What is the NPV?    The NPV and ordinary payback are easy to calculate in this case because the investment is a perpetuity. The present value of the cash flows is $100/0.20 = $500, so the NPV is $500 - $400 = $100. The ordinary payback is obviously four years.    To get the discounted payback, we need to find the number of years such that a $100 annuity has a present value of $400 at 20 per cent. In other words, the present value annuity factor is $400/$100 = 4, and the interest rate is 20 per cent per period; so what’s the number of periods? From Table A.3 the annuity factor for 8 years is 3.8372 and for 9 years it is 4.0310. Therefore the answer is a little less than nine years, so this is the discounted payback. 7.4 The accounting rate of return Another attractive, but flawed, approach to making capital budgeting decisions is the  accounting rate of return (ARR)an investment’s average net income divided by its average book value. There are many different definitions of the ARR. However, in one form or another, the ARR is always defined as: To see how we might calculate this number, suppose we are deciding whether or not to open a store in a new shopping mall. The required investment in improvements is $500 000. The store would have a five-year life because everything reverts to the mall owners after that time. The required investment would be 100 per cent depreciated (straight-line) over five years, so the depreciation would be $500 000/5 = $100 000 per year. The tax rate is 25 per cent. Table 7.8 contains the projected revenues and expenses. Based on these figures, net profit in each year is also shown. Table 7.8   Projected yearly revenue and costs for average accounting return p. 225    To calculate the average book value for this investment, we note that we started out with a book value of $500 000 (the initial cost) and ended up at $0. The average book value during the life of the investment is thus ($500 000 + 0)/2 = $250 000. As long as we use straight-line depreciation, the average investment will always be half of the initial investment. (We would of course calculate the average of the six book values directly. In thousands, we would have ($500 + $400 + $300 + $200 + $100 + 0)/6 = $250.)    Looking at Table 7.8 , net profit is $100 000 in the first year, $150 000 in the second year, $50 000 in the third year, $0 in year 4, and -$50 000 in year 5. The average net profit, then, is:        The average accounting return is:        If the firm has a target ARR less than 20 per cent, then this investment is acceptable; otherwise not. The accounting rate of return rule is thus:      Based on the average accounting return rule, a project is acceptable if its average accounting return exceeds a target average accounting return.      As we will see in the next section, this rule has a number of problems. Analysing the accounting rate of return method You recognise the chief drawback to the ARR immediately. Above all else, the ARR is not a rate of return in any meaningful economic sense. Instead, it is the ratio of two accounting numbers, and it is not comparable to the returns offered, for example, in financial markets.    One of the reasons that the ARR is not a true rate of return is that it ignores time value. When we average figures that occur at different times, we are treating the near future and the more distant future the same way. For example, there was no discounting involved when we computed the average net profit.    The second problem with the ARR is similar to the problem we had with the payback period rule concerning the lack of an objective cut-off period. Since a calculated ARR is really not comparable to a market return, the target ARR must somehow be specified. There is no generally agreed-upon way to do this. One way of doing it is to calculate the ARR for the firm as a whole and use this for a benchmark, but there are lots of other ways as well.    The third, and perhaps worst, flaw in the ARR is that it doesn't even look at the right things. Instead of cash flow and market value, it uses net profit and book value. These are both poor substitutes. As a result, an ARR doesn't tell us what the effect on share price will be from taking an investment, so it doesn't tell us what we really want to know.    Does the ARR have any redeeming features? About the only one is that it can almost always be computed. The reason is that accounting information will almost always be available, both for the project under consideration and for the firm as a whole through the accounting budget estimates. We hasten to add that once the accounting information is available, we can always convert it to cash flows, so even this is not a particularly important fact.      Concept questions 7.4a    What is an accounting rate of return (ARR)? 7.4b    What are the weaknesses of the ARR rule?   7.5 The Internal rate of Return p. 226 We now come to the most important alternative to NPV, the internal rate of returnthe discount rate that makes the NPV of an investment zero , universally known as the IRR. As we will see, the IRR is closely related to NPV. With the IRR, we try to find a single rate of return that summarises the merits of a project. Furthermore, we want this rate to be an 'internal' rate in the sense that it only depends on the cash flows of a particular investment, not on rates offered elsewhere.    To illustrate the idea behind the IRR, consider a project that costs $100 today and pays $110 in one year. Suppose you were asked '˜What is the return on this investment?' What would you say? It seems both natural and obvious to say that the return is 10 per cent because, for every dollar we put in, we get $1.10 back. In fact, as we will see in a moment, 10 per cent is the internal rate of return or IRR on this investment.    Is this project with its 10 per cent IRR a good investment? Once again, it would seem apparent that this is a good investment only if our required return is less than 10 per cent. This intuition is also correct and illustrates the IRR rule:      Based on the IRR rule, an investment is acceptable if the IRR exceeds the required return. It should be rejected otherwise.         Imagine that we wanted to calculate the NPV for our simple investment. At a discount rate of r, the NPV is:        Now, suppose we didn't know the discount rate. This presents a problem, but we could still ask how high the discount rate would have to be before this project was unacceptable. We know that we are indifferent to taking or not taking this investment when its NPV is just equal to zero. In other words, this investment is economically a break-even proposition when the NPV is zero because value is neither created nor destroyed. To find the break-even discount rate, we set NPV equal to zero and solve for r:     This 10 per cent is what we have already called the return on this investment. What we have now illustrated is that the internal rate of return on an investment (or just 'return' for short) is the discount rate that makes the NPV equal to zero. This is an important observation, so it bears repeating:      The IRR on an investment is the required return that results in a zero NPV when it is used as the discount rate.         The fact that the IRR is simply the discount rate that makes the NPV equal to zero is important because it tells us how to calculate the returns on more complicated investments. As we have seen, finding the IRR turns out to be relatively easy for a single period investment. However, suppose you were now looking at an investment with the cash flows shown in Table 7.9. As illustrated, this investment costs $100 and has a cash flow of $60 per year for two years, so it's only slightly more complicated than our single period example. However, if you were asked for the return on this investment, what would you say? There doesn't seem to be any obvious answer. However, based on what we now know, we can set the NPV equal to zero and solve for the discount rate:     p. 227 Table 7.9   Project cash flow    Unfortunately, the only way to find the IRR in general is by trial and error, either by hand or by calculator. This is precisely the same problem that came up in Chapter 5 when we found the unknown rate for an annuity and in Chapter 6 when we found the yield to maturity on a bond. In fact, we now see that in both of those cases we were finding an IRR.    In this particular case, the cash flows form a two-period, $60 annuity. To find the unknown rate, we can try various different rates until we get the answer. If we were to start with a 0 per cent rate, the NPV would obviously be $120 - $100 = $20. At a 10 per cent discount rate, we would have:     Now we're getting close. We can summarise these and some other possibilities as shown in Table 7.10. From our calculations, the NPV appears to be zero between 10 per cent and 15 per cent, so the IRR is somewhere in that range. With a little more effort, we can find that the IRR is about 13.1 per cent. (With a lot more effort-or a financial calculator-we can find that the IRR is 13.0662386291808 per cent, not that anybody would ever want this many decimal places.) So, if our required return is less than 13.1 per cent, we would take this investment. If our required return exceeds 13.1 per cent, we would reject it. Table 7.10   NPV at different discount rates    By now, you have probably noticed that the IRR rule and the NPV rule appear to be quite similar. In fact, the IRR is sometimes simply called the discounted cash flow or DCF return. The easiest way to illustrate the relationship between NPV and IRR is to plot the numbers we calculated in  Table 7.10. On the vertical or y-axis we put the different NPVs. We put discount rates on the horizontal or x-axis. If we had a very large number of points, the resulting picture would be a smooth curve called a net present value profilea graphical representation of the relationship between an investment’s NPVs and various discount rates . Figure 7.2 illustrates the NPV profile for this project. Beginning with a 0 per cent discount rate, we have $20 plotted directly on the y-axis. As the discount rate increases, the NPV declines smoothly. Where will the curve cut through the x-axis? This will occur where the NPV is just equal to zero, so it will happen right at the IRR of 13.1 per cent.     Figure 7.2         An NPV profile       In our example, the NPV rule and the IRR rule lead to identical accept/reject decisions. We will accept an investment using the IRR rule if the required return is less than 13.1 per cent. As Figure 7.2 illustrates, however, the NPV is positive at any discount rate less than 13.1 per cent, so we would accept the investment using the NPV rule as well. The two rules are equivalent in this case. p. 228    Example 7.4    Calculating the IRR    A project has a total up-front cost of $435.44. The cash flows are $100 in the first year, $200 in the second year, and $300 in the third year. What's the IRR? If we require an 18 per cent return, should we take this investment?    We will describe the NPV profile and find the IRR by calculating some NPVs at different discount rates. You should check our answers for practice. Beginning with 0 per cent, we have: The NPV is zero at 15 per cent, so 15 per cent is the IRR. If we require an 18 per cent return, then we should not take the investment. The reason is that the NPV is negative at 18 per cent (check that it is - $24.47). The IRR rule tells us the same thing in this case. We shouldn't take this investment because its 15 per cent return is below our required 18 per cent return.       At this point, you may be wondering whether the IRR and the NPV rules always lead to identical decisions. The answer is yes, as long as two very important conditions are met. 1  The project's cash flows must be conventional, meaning that the first cash flow (the initial investment) is negative and all the rest are positive. 2  The project must be independent, meaning that the decision to accept or reject this project does not affect the decision to accept or reject any other.   The first of these conditions is typically met, but the second often is not. In any case, when one or both of these conditions are not met, problems can arise. We discuss some of these next. Problems with the IRR The problems with the IRR come about when the cash flows are not conventional or when we are trying to compare two or more investments to see which is best. In the first case, surprisingly, the simple question 'What's the return' can become very difficult to answer. In the second case, the IRR can be a misleading guide. Non-conventional cash flows Suppose we have a strip-mining project that requires a $60 investment. Our cash flow in the first year will be $155. In the second year, the mine is depleted, but we have to spend $100 to restore the terrain. As Table 7.11 illustrates, both the first and third cash flows are negative. Table 7.11   Project cash flow p. 229 To find the IRR on this project, we can calculate the NPV at various rates:        The NPV appears to be behaving in a very peculiar fashion here. First, as the discount rate increases from 0 per cent to 30 per cent, the NPV starts out negative and becomes positive. This seems backwards because the NPV is rising as the discount rate rises. It then starts getting smaller and becomes negative again. What's the IRR? To find out, we draw the NPV profile in.     Figure 7.3         NPV profile    In Figure 7.3, notice that the NPV is zero when the discount rate is 25 per cent, so this is the IRR. Or is it? The NPV is also zero at 331⁄3 per cent. Which of these is correct? The answer is both or neither; more precisely, there is no unambiguously correct answer. This is the multiple rates of returnone potential problem in using the IRR method if more than one discount rate makes the NPV of an investment zero problem. Many financial computer packages are not aware of this problem and just report the first IRR that is found. Others report only the smallest positive IRR, even though this answer is no better than any other.    In our current example, the IRR rule breaks down completely. Suppose our required return were 10 per cent. Should we take this investment? Both IRRs are greater than 10 per cent, so, by the IRR rule, maybe we should. However, as Figure 7.3 shows, the NPV is negative at any discount rate less than 25 per cent, so this is not a good investment. When should we take it? Looking at  Figure 7.3 one last time, the NPV is positive only if our required return is between 25 per cent and 331 per cent.    The moral of the story is that when the cash flows aren't conventional, strange things can start to happen to the IRR. This is not anything to get upset about however, because the NPV rule, as always, works just fine. This illustrates that, oddly enough, the obvious question-What's the rate of return?-may not always have a good answer.    Example 7.5    What is IRR?    You are looking at an investment that requires you to invest $51 today. You'll get $100 in one year, but you must pay out $50 in two years. What is the IRR on this investment?    You are on the alert now to the non-conventional cash flow problem, so you probably wouldn't be surprised to see more than one IRR. However, if you start looking for an IRR by trial and error, it will take you a long time. The reason is that there is no IRR. The NPV is negative at every discount rate, so we shouldn't take this investment under any circumstances. For example, if the time value of money is zero and we simply add all the cash flows, the NPV is-$1. What's the return of this investment? There isn't any, as the investment will always be a loss.    p. 230 Mutually exclusive investments Even if there is a single IRR, another problem can arise concerning mutually exclusive investment decisionsone potential problem in using the IRR method if the acceptance of one project excludes that of another . If two investments, X and Y, are mutually exclusive, then taking one of them means that we cannot take the other. For example, if we own one corner block, then we can build a petrol station or a block of flats, but not both. These are mutually exclusive alternatives.    Thus far, we have asked whether or not a given investment is worth undertaking. There is a related question, however, that comes up very often: given two or more mutually exclusive investments, which one is the best? The answer is simple enough: The best one is the one with the largest NPV. Can we also say that the best one has the highest return? As we show, the answer is no.    Example 7.6    ‘I think; therefore, I know how many IRRs there can be’    We have seen that it is possible to get more than one IRR. If you wanted to make sure that you had found all of the possible IRRs, how could you tell? The answer comes from the great mathematician, philosopher and financial analyst Descartes (of 'I think; therefore I am' fame). Descartes' Rule of Sign says that the maximum number of IRRs that there can be is equal to the number of times that the cash flows change sign from positive to negative and/or negative to positive. To be more precise, the number of IRRs that are bigger than –100 per cent is equal to the number of sign changes, or it differs from the number of sign changes by an even number. Thus, for example, if there are five sign changes, there are either five, three or one IRRs. If there are two sign changes, there are either two IRRs or no IRRs.    In our example with the 25 and 33 1/3 per cent IRRs, could there be yet another IRR? The cash flows flip from negative to positive, then back to negative for a total of two sign changes. As a result, the maximum number of IRRs is two, and, from Descartes rule, we don't need to look for any more. Note that the actual number of IRRs can be less than the maximum (see Example 7.5).       To illustrate the problem with the IRR rule and mutually exclusive investments, consider the cash flows from the following two mutually exclusive investments:    Since these investments are mutually exclusive, we can only take one of them. Simple intuition suggests that investment A is better because of its higher return. Unfortunately, simple intuition is not always correct.    To see why investment A is not necessarily the better of the two investments, we’ve calculated the NPV of these investments for different required returns: p. 231    The IRR for A (24%) is larger than the IRR for B (21%). However, if you compare the NPVs, you'll see that the investment with the highest NPV depends on our required return. B has greater total cash flow, but it pays back more slowly than A. As a result, it has a higher NPV at lower discount rates.    In our example, the NPV and IRR rankings conflict for some discount rates. If our required return is 10 per cent, for instance, then B has the higher NPV and is thus the better of the two even though A has the higher IRR. If our required return is 15 per cent, then there is no ranking conflict: A is better.    The conflict between the IRR and NPV for mutually exclusive investments can be illustrated by plotting their NPV profiles as we have done in Figure 7.4. In Figure 7.4, notice that the NPV profiles cross at about 11.1 per cent. Notice also that at any discount rate less than 11.1 per cent, the NPV for B is higher. In this range, taking B benefits us more than taking A, even though A's IRR is higher. At any rate greater that 11.1 per cent, project A has the greater NPV.     Figure 7.4         NPV profiles for mutually exclusive investments       What this example illustrates is that whenever we have mutually exclusive projects, we shouldn't rank them based on their returns. More generally, whenever we are comparing investments to determine which is best, IRRs can be misleading. Instead, we need to look at the relative NPVs to avoid the possibility of choosing incorrectly. Remember, we are ultimately interested in creating value for the shareholders, so the option with the higher NPV is preferred, regardless of the relative returns.    If this does not seem intuitive, think of it this way. Suppose you have two investments. One has a 10 per cent return and makes you $100 richer immediately. The other has a 20 per cent return and makes you $50 richer immediately. Which one do you like better? We would rather have $100 than $50, regardless of the returns, so we like the first one better.    In general, you can find the crossover rate by taking the difference in the cash flows and calculating the IRR using the differences. It doesn't make any difference which one you subtract from which, so long as one of the investments has a positive NPV and the cash flows are conventional. To see this, find the IRR for (A-B); you'll see it's the same number. Also, for practice, you might want to find the exact crossover in Figure 7.4. (A big hint: it's 11.0704%.) Imagine analysing two investments, both with negative NPVs. If crossover is used solely, it is possible to select the investment with the smallest negative NPV. This simply means that the change from the larger negative NPV to the smaller negative NPV project is the same as going from 'very bad' to 'bad'. Remember that negative NPV investments should be rejected. p. 232    Example 7.7    Calculating the crossover rate    In Figure 7.4, the NPV profiles cross at about 11.1 per cent. How can we determine just what this crossover rate is? The crossover rate, by definition, is the discount rate that makes the NPVs of two projects equal. To illustrate, suppose we have the following two mutually exclusive investments: What’s the crossover rate?    To find the crossover, first consider moving out of investment A and into investment B. If you make the move, you will have to invest an extra $100 ($500 - $400). For this $100 investment, you will get an extra $70 ($320 - $250) in the first year and an extra $60 ($340 - $280) in the second year. Is this a good move? In other words, is it worth investing the extra $100?    Based on our discussion, the NPV of the switch, NPV(B - A) is:     We can calculate the return on this investment by setting the NPV equal to zero and solving for the IRR:     If you go through this calculation, you will find the IRR is exactly 20 per cent. What this tells us is that at a 20 per cent discount rate, we are indifferent between the two investments because the NPV of the difference in their cash flows is zero. As a consequence, the two investments have the same value, so this 20 per cent is the crossover rate. Check that the NPV at 20 per cent is $2.78 for both.    Redeeming qualities of the IRR Despite its flaws, the IRR is very popular in practice, more so than even the NPV. It probably survives because it fills a need that the NPV does not. In analysing investments, people in general, and financial analysts in particular, seem to prefer talking about rates of return rather than dollar values.    In a similar vein, the IRR also appears to provide a simple way of communicating information about a proposal. One manager might say to another: 'Remodelling the clerical wing has a 20 per cent return'. This may somehow be simpler than saying: 'At a 10 per cent discount rate, the net present value is $4000'.    Finally, under certain circumstances, the IRR may have a practical advantage over NPV. We can't estimate the NPV unless we know the appropriate discount rate, but we can still estimate the IRR. Suppose we didn't know the required return on an investment, but we found, for example, that it had a 40 per cent return. We would probably be inclined to take it since it is very unlikely that the required return is that high. p. 233      Concept questions 7.5a    Under what circumstances will the IRR and NPV rules lead to the same accept/reject decisions? When might they conflict? 7.5b    Is it generally true that an advantage of the IRR rule over the NPV rule is that we don’t need to know the required return to use the IRR rule?   2012 McGraw-Hill Australia Any use is subject to the Terms of Use and Privacy Notice. McGraw-Hill Australia is one of the many fine businesses of The McGraw-Hill Companies. 7.6 The present value index Another method used to evaluate projects is called the present value index (PVI)the present value of an investment’s future cash flows divided by its initial cost. Also benefit/cost ratio or benefit/cost ratiothe profitability index of an investment project. Also present value index . This index is defined as the present value of the future cash flows divided by the initial investment. So, if a project costs $200 and the present value of its future cash flows is $220, the present value index value would be $220/$200 = 1.10. Notice that the NPV for this investment is $20, so it is a desirable investment.    More generally, if a project has a positive NPV, then the present value of the future cash flows must be bigger than the initial investment. The PVI would thus be bigger than 1.00 for a positive NPV investment and less than 1.00 for a negative NPV investment.    How do we interpret the PVI? In our example, the PVI was 1.10. This tells us that, per dollar invested, $1.10 in value or $0.10 in NPV results. The PVI thus measures the value created per dollar invested. For this reason, it is often proposed as a measure of performance for government or not-for-profit investments. Also, when capital is scarce, it may make sense to allocate it to those projects with the highest PVIs. We will return to this issue in a later chapter.    The PVI is obviously very similar to the NPV. However, consider an investment that costs $5 and has a $10 present value and an investment that costs $100 with a $150 present value. The first of these investments has an NPV of $5 and a PVI of 2. The second has an NPV of $50 and a PVI of 1.50. If these were mutually exclusive investments, then the second one is preferred even though it has a lower PVI. This ranking problem is very similar to the IRR ranking problem we saw in the previous section. In summary, there seems to be little reason to rely on the PVI instead of the NPV.    An index similar to the PVI is the net present value index (NPVI). This index is defined as the net present value of the future cash flows divided by the initial investment. Returning to our earlier example, if a project costs $200 and the present value of its future cash flows is $220, the PVI would be 1.1 and the NPVI would be $20/$200 = 0.1. The NPVI differs from the PVI by a scale of one. The ranking problems associated with the PVI are also applicable to the NPVI.      Concept questions 7.6a    What does the present value index measure? 7.6b    How would you state the PVI rule?   7.7 The practice of capital budgeting Given that NPV seems to be telling us directly what we want to know, you might be wondering why there are so many other procedures and why alternative procedures are commonly used. Recall that we are trying to make an investment decision and that we are frequently operating under considerable uncertainty about the future. We can only estimate the NPV of an investment in this case. The resulting estimate can be very 'soft', meaning that the true NPV might be quite different.    Because the true NPV is unknown, the astute financial manager seeks clues to assess whether the estimated NPV is reliable. For this reason, firms would typically use multiple criteria for evaluating a proposal. For example, suppose we have an investment with a positive estimated NPV. Based on our experience with other projects, this one appears to have a short payback and a very high ARR. In this case, the different indicators seem to agree that it's 'all systems go'. Put another way, the payback and the ARR are consistent with the conclusion that the NPV is positive. p. 234    On the other hand, suppose we had a positive estimated NPV, a long payback and a low ARR. This could still be a good investment, but it looks as if we need to be much more careful in making the decision since we are getting conflicting signals. If the estimated NPV is based on projections in which we have little confidence, then further analysis is probably in order. We will consider how to go about this analysis in more detail in the next two chapters. Table 7.12   Evaluation techniques Source: Adapted from an extract of Freeman M and Hobbs G (1991) ‘Capital Budgeting: Theory versus Practice’, Australian Accountant (Sept.edition), pp. 36–41. © 1991. Reproduced with the permission of CPA Australia Ltd. The information in the above table is accurate as at 1991.    There have been a number of surveys conducted asking large firms what types of investment criteria they actually use. Table 7.12 presents the results of one such survey. Based on the results, the most important capital budgeting technique is some form of discounted cash flow (such as NPV or IRR). Some 75 per cent of the firms rank it as the most important.    In practice, the payback period is a popular tool; about 44 per cent of the responding firms use it. Other surveys are consistent with these results. The most common practice is to look at NPV or IRR along with non-discounted cash flow criteria such as payback and ARR. Given our discussion, this is sound practice.   Most of the calculations that have been performed in this chapter can be done in Excel and we showed you near the start of the chapter how to use the NPV and IRR function. We draw your attention to one more IRR consideration and that is the 'guess'. If you do not guess, Excel assumes 10%. Excel asks you to guess because of Descartes' Rule of Signs. Recall the problem:    If you use a guess of 10%, the calculated IRR is 25%. Now try a guess of 40%. The calculated IRR is now 331 1/3%. This was the second IRR. Excel recognises the dual answer problem and that is why the program asks you for a guess.        Concept questions 7.7a    What are the most commonly used capital budgeting procedures? 7.7b    Since NPV is conceptually the best procedure for capital budgeting, why do you think that multiple measures are used in practice?   2012 McGraw-Hill Australia Any use is subject to the Terms of Use and Privacy Notice. McGraw-Hill Australia is one of the many fine businesses of The McGraw-Hill Companies. 2012 McGraw-Hill Australia Any use is subject to the Terms of Use and Privacy Notice. McGraw-Hill Australia is one of the many fine businesses of The McGraw-Hill Companies.