Return on Investments

Previously, we built a path for estimating costs of equity, debt, and capital and presented an argument that the cost of capital is the minimum acceptable hurdle rate when considering new investments. We also argued that an investment has to earn a return greater than this hurdle rate to create value for the owners of a business. In this chapter, we turn to the question of how best to measure the return on a project. In doing so, we will attempt to answer the following questions:

• What is a project? In particular, how general is the definition of an investment and what are the different types of investment decisions that firms have to make?
• In measuring the return on a project, should we look at the cash flows generated by the project or at the accounting earnings?
• If the returns on a project are unevenly spread over time, how do we consider (or should we not consider) differences in returns across time?

We will provide an overview of project analysis using two hypothetical projects: an door company for Capital Construction and a quick-charge construction battery charge company for Granite Construction.

What Is a Project?

Project analysis concerns which projects a company should accept and which it should reject; accordingly, the question of what makes up a project is central to this. The traditional project reviewed in capital budgeting has three criteria: (1) a large up-front cost, (2) cash flows for a specific time period, and (3) a salvage value at the end, which captures the value of the assets of the project when the project ends. Although such projects unquestionable form a substantial percentage of project decisions, particularly for construction companies, it would be foolish to think that project analysis stops there. If a project is defined more broadly to include any decision that results in using the limited capital of a business, then everything from strategic decisions and acquisitions to decisions about which air conditioning system to install in a building would fit within its reach.

Defined broadly then, any of the following decisions would meet the requirements as projects:

• Key planned decisions to enter new areas of business (such as Granite Construction's foray into the water business or Capital Construction's into the retail door business) or new markets (such as Granite Construction expansion into water-related construction).
• Acquisitions of other companies are projects as well, despite efforts to invent separate sets of rules for them.
• Decisions on new undertakings within existing businesses or markets, such as the one made by Granite Construction to wind down its Specialty division or to expand its water division.


• Decisions that may alter the way existing divisions and projects are managed, including new Federal construction grant programs or prevailing wage requirements at Capital Construction.

Decisions on the best way to deliver an essential service for the business to function profitable. A perfect example would be Granite Construction's choice of what type of CAD system to buy to allow engineers and construction managers to do their jobs. Given, the CAD system itself generally won't generate revenues and profits, it is an absolute must for other revenue generating projects.

Salvage Value : The projected cash value of the assets invested in the projects at the end of the project life.

Mutually Exclusive Projects: A group of projects is said to be mutually exclusive when acceptance of one of the projects implies that the rest have to be rejected.

Project decisions can be placed into a number of categories. The first is how one project impacts other projects the company is considering and reviewing. Some projects are independent of other projects, and therefore can be reviewd independently, whereas other projects are mutually exclusive-that is, taking one project will mean rejecting other projects. At the other extreme, some projects are linked to other projects down the road and others are complementary. In general, projects can be categorized as fitting somewhere on the scale between prerequisites and mutually exclusive, as depicted in Image 3.1.

Image 3.1 The Project Scale

The second measurement that can be drawn on to identify a project is its ability to generate revenues or lower costs. What we are trying to figure out is if the earnings or cash flows from the projects justify the capital contribution needed to move forward. When it comes to cost-reduction projects, what we are trying to figure out is if the reduction in costs justifies the up-front capital contribution needed for the projects.

Hurdle Rates for Companies versus Hurdle Rates for Projects

In the previous section we developed a way to calculate the costs of equity and capital for companies. In this section, we will broaden the discussion to hurdle rates in the context of new or individual projects.

Using the Company's Hurdle Rate for Individual Projects

Can we use the costs of equity and capital that we have calculated for the companies for these projects? In some cases we can, but only if all projects made by a company are similar in terms of their risk exposure. As a company's projects become more diverse, the company will no longer be able to use its cost of equity and capital to review these projects. Projects that are riskier have to be reviewed using a higher cost of equity and capital than projects that are safer. In this section, we consider how to estimate project costs of equity and capital.

What would happen if a company elects to use a single, unchanging, cost of equity and capital to review all projects? This company would find itself overinvesting in risky projects and under investing in safe projects. Over time, the company will become riskier, as its safer businesses find themselves powerless to compete with riskier businesses.

Cost of Equity for Projects

In estimating the beta for a project, we will look at three possible outcomes. The first outcome is the one where all the projects considered by a company are identical in their exposure to risk; this plain vanilla approach makes risk assessment simple. The second outcome is one where a company is in many businesses with broad exposures to risk, but projects within each business have the same risk exposure. The third outcome is the most challenging where each project considered by a company has different risk exposure.

Single Business; Project Risk Similar within Business

When a company works in only a single type of business and all projects within that business share the same risk profile, the company can use its company cost of equity as the cost of equity for the project. Since we determined the cost of equity using a company beta previously, we can use this same beta to calculate the cost of equity for each project the company is currently reviewing. The benefit of this approach is that it does not require risk estimation prior to every project, providing managers with a fixed target for their project analysis. The approach is limited, though, because it can only be used for companies who work a single line of business and take on the same type of projects repeatedly.

Multiple Businesses with Different Risk Profiles: Project Risk Similar within Each Business

When companies operate in more than one line of business, the risk profiles are likely to be unique across different businesses. Making the assumption that projects taken within each business have the same risk characteristic, we can calculate the cost of equity for each business individually and use that cost of equity for all projects within that business. Riskier businesses will have higher costs of equity than safer businesses, and projects taken by riskier businesses will have to pay these higher costs. Inserting the company's cost of equity on all projects in all businesses will lead to overinvesting in risky businesses (because the cost of equity will be set too low) and under investing in safe businesses (because the cost of equity will be set too high).

How do we determine the cost of equity for individual businesses?

When the evaluation requires using equity betas, we cannot use the traditional regression method (in the CAPM) because these methods require past prices. Instead, we have to use one of the two methods that we described in the last section as substitutes to regression betas-bottom-up betas based on other publicly traded company's in the same business, or accounting betas, calculated based on the accounting earnings for the division.

Projects with Different Risk Profiles

One could contend that each project's risk profile is, in fact, unique and that it is inappropriate to use either the company's cost of equity or divisional costs of equity to evaluate projects. Even though this may have some merit, we have to think of the trade-off. Given that small differences in the cost of equity should not make a meaningful difference in our project estimates, we need to judge if the additional benefits of reviewing each project individually is greater then the costs of doing so.

When does it make sense to examine a project's risk independently? If a project is large in terms of its capital contribution requirements relative to the company's experience, and the project has a number of unusual risk characteristics, it would seem right to calculate the cost of equity for the project independently. The only sensible way of determining betas and costs of equity for individual projects is the bottom-up beta approach.

Cost of Debt for Projects

In the previous section, we stated that the cost of debt for a company should explain its default risk. With individual projects, determining the default risk becomes much more complicated, because projects seldom borrow on their own; most companies borrow money for all the projects that they undertake.

There are three ways to estimating Granite Construction cost of debt for a project:

• One way is based on the claim that since the borrowing is done by the company rather than by individual projects, the cost of debt for a project should be the cost of debt for the company doing the project. This approach makes the most sense when the projects being reviewed are small relative to the company taking them and thus have little or no noticeable effect on the company's default risk.
• Determine the project's ability to generate cash flows relative to its financing costs and calculate default risk and cost of debt for the project, You can also calculate this default risk by looking at other companies in the same sector taking on similar projects, and use the average default risk and cost of debt for these companies. This choice usually works when the project is large in terms of its capital needs relative to the company and has different cash flow characteristics (both in terms of size and volatility) from other projects taken by the company and can borrow money against its own cash flows.
• The third approach applies when a project actually borrows its own funds, with lenders having no recourse against the parent company, in case the project defaults. This is unusual, but it can happen when projects have significant value of their own and the project is large relative to the company considering it. In this case, the cost of debt for the project can be determined based on the cash flows relative to its financing obligations. In the last section, we used the bond rating of a company to figure out the cost of debt for the company. Even though projects may not be rated, we can still compute a rating for a project based on financial ratios, and this can be used to calculate default risk and the cost of debt.

Financing Mix and Cost of Capital for Projects

To get from the costs of debt and equity to the cost of capital, we have to weight each by their relative proportions in financing. Again, the task is much easier at the company level, where we use the current market values of debt and equity to arrive at these weights. We may borrow money to fund a project, but it is often not clear whether we are using the debt capacity of the project or the company's debt capacity. The answer to this problem will again change depending on the conditions we face.

• When we are establishing the financing weights for small projects that do not involve a company's debt capacity, the financing weights should be those of the company before the project.
• When determining the financing weights of large projects, with risk characteristics uncommon from that of the company, we have to be more careful. Employing the company's financing mix to determine the cost of capital for these projects can be wrong, because the project being reviewed could be riskier than the company as a whole and thus unable of carrying the company's debt ratio. In this case, we would press for the use of the average debt ratio of the other companies in the business in calculating the cost of capital of the project.
• The financing weights for stand-alone projects that are large enough to issue their own debt should be based on the actual amounts borrowed by the projects. For companies with such projects, the financing weights can change from project to projects, as will the cost of debt.

In summary, the cost of debt and debt ratio for a project will show the size of the project relative to the company, and its risk profile, again relative to the company. Worksheet 3.1 summarizes our analyses.

Worksheet 3.1 Cost of Debt and Debt Ratio: Project Analyses

Example 3.2: Estimating Hurdle Rates for Individual Projects

Using the estimation approach that we just laid out, we can calculate the hurdles rates for the projects that we are reviewing in this section.

Capital Construction Door Company : Because the beta and cost of equity that we calculated for Capital Construction as a company explain its status as a construction company, we will re-calculate the beta for this retail door business by looking at publicly traded Construction Material retailers. The unlevered total beta for construction material retailers is 4.51,¹ and we imagine that this project will be funded with the same mix of debt and equity (D/E = 40.78%, Debt/Capital = 15.53%) that Capital Construction uses in the rest of the business. We will imagine that Capital Construction's tax rate (25%) and pretax cost of debt (4.31%) apply to this project.

Levered Beta _{Construction Materials Service} = 4.51 [1 + (1 - 0.25) (0.4078)] = 5.88

Cost of Equity _{Construction Materials} = 1.25% + 5.88 (4.31%) = 26.59%

Cost of Capital_{Construction Materials}= 26.59% (0.4078) + 4.31% (1 - 0.25) (0.5922) = 12.76%

This is much higher than the cost of capital we computed for Capital Construction in previously, but it shows the higher risk of the retail venture.

From Accounting Earnings to Cash Flows

The three factors outlined can cause accounting earnings to deviate significantly from the cash flows. To get from after-tax operating earnings, which measures the earnings to the company, to cash flows to all investors in the company, we have to:

Add back all noncash charges , such as depreciation and amortization, to the operating earnings.

Subtract out all cash outflows that represent capital expenditures. Net out the effect of changes in noncash working capital , that is, changes in accounts receivable, inventory, and accounts payable. If noncash working capital increased, the cash flows will be reduced by the change, whereas if it decreased, there is a cash inflow.

The first two adjustments change operating earnings to account for the distinction drawn by accountants between operating, financing and capital expenditures, whereas the last adjustment converts accrual revenues and expenses into cash revenues and expenses.

Cash Flow to Company = Earnings before interest and taxes (1 - t) + Depreciation & Amortization - Change in Noncash Working Capital - Capital Expenditures

The cash flow to the company is a pre-debt, after-tax cash flow that measures the cash generated by a project for all claim holders in the company after reinvestment needs have been met.

To get from net income, which measures the earnings of equity investors in the company, to cash flows to equity investors requires the added step of looking at the net cash flow created by repaying old debt and taking on new debt. The difference between new debt issues and debt repayments is called the net debt, and it must be added back to arrive at cash flows to equity. Also, other cash flows to nonequity claim holders in the company, such as preferred dividends, have to be netted from cash flows.

Cash Flow to Equity = Net Income + Depreciation & Amortization - Change in Noncash Working Capital - Capital Expenditures + (New Debt Issues - Debt Repayments) - Preferred Dividends

The cash flow to equity measures the cash flows generated by a project for equity investors in the company, after taxes, debt payments, and reinvestment needs.

The Case for Cash Flows

When earnings and cash flows are different, as they are for many projects, we must determine which one provides a more reliable measure of performance. Accounting earnings, especially at the equity level (net income), can be gamed at least for individual periods, through the use of innovative accounting practices. A book titled Accounting for Growth, which garnered national headlines in the United Kingdom and cost the author, Terry Smith, his job as an analyst at UBS Phillips & Drew, examined twelve legal accounting schemes frequently used to misinform investors about the profitability of individual companies. To show how innovative accounting practices can increase reported profits, Smith highlighted such companies as Maxwell Communications and Polly Peck, both of which eventually succumbed to bankruptcy.

The second reason for using cash flow is much more direct. No business that we know of accepts earnings as payment for goods and services delivered; all of them require cash. Thus, a project with positive earnings and negative cash flows will drain cash from the business undertaking it.

On the other hand, a project with negative earnings and positive cash flows might make the accounting bottom line look worse but will generate cash for the business undertaking it.

Total versus Incremental Cash Flows

The goal when reviewing a project is to answer the question: Will investing in this project make the entire company or business more valuable? So, the cash flows we need to look at in project analysis are the cash flows the project generates for the company or business doing it. We will call these incremental cash flows.

Differences between Incremental and Total Cash Flows

The total and the incremental cash flows on a project will normally be different for two reasons. First, some of the cash flows on a project might have happened previously and so are unaffected whether we accept the project or not. Such cash flows are called sunk costs and should be removed from the evaluation. The second is that some of the projected cash flows on a project will be generated by yhe company, whether this project is accepted or rejected. Splitting up fixed expenses, such as general and administrative costs, usually fit into this category. These cash flows are not incremental, and the analyst needs to figure this out and remove them.

There are some expenses tied to a project that are spent before the project review is complete. One example would be expenses linked to a test market done to determine the potential market for a product before doing a full-blown project analysis. Such expenses are called sunk costs. Because they will not be recovered if the project is rejected. Sunk costs are not incremental and so should not be included as part of the project analysis. This goes completely opposite of how sunk costs are looked at in terms of accounting statements, which do not differentiate between expenses that have already been spent and expenses.

One type of expenses that always falls into the sunk cost column in project analysis is research and development (R&D), which takes place well before a product is even close to production. Companies that invest large amounts on R&D, such as Merck and Intel, have struggled to come to terms with the fact that the analysis of these expenses generally occur after the fact, when little can be done about them.

Even though sunk costs should not be included as part of the project review, a company must make up for its sunk costs over time or it will eventually fail. Consider, for example, a company like McDonald's, who spend a ton of money test marketing fast-foods before introducing them. Imagine, on the ill-fated McLean Deluxe (a low-fat hamburger introduced in 1990), that the test market expenses amounted to $30 million and that the net present value of the project, determined after the test market, amounted to $20 million. The project should be taken. If this is the pattern for every project McDonald's takes on, however, it will collapse under the weight of its test marketing expenses. To be successful, the cumulative net present value of its successful projects will have to exceed the cumulative test marketing expenses on both its successful and unsuccessful products.

Allocated Costs

An accounting method manufactured to guarantee that every part of a business bears its fair share of costs is allocation, where expenses that are not directly traceable to revenues generated by individual products or divisions are allocated across these units, based on revenues, profits, or assets. Even if the purpose of such allocations might be fairness, their effect on project analyses have to be viewed in terms of whether they create incremental cash flows. An allocated cost that will exist with or without the project being reviewed does not belong in the project analysis.

Any increase in administrative or staff costs that can be traced to the project is an incremental cost and belongs in the analysis. One way to calculate the incremental component of these costs is to break them down on the basis of whether they are fixed or variable and, if variable, where did they come from. So, a slice of administrative costs could be linked to revenue, and the revenue projections of a new project can be used to estimate the administrative costs to be assigned to it.

Example 3.3: Dealing with Allocated Costs

Case 1 : Imagine that you are looking at a retail company with general and administrative (G&A) costs currently of $1,000,000 a year. The company currently has five stores and the G&A costs are allocated evenly across the stores; the allocation to each store is $200,000. The company is considering opening a new store; with six stores, the allocation of G&A expenses to each store will be $166,6700.

Here, assigning an expense of $166,670 for G&A costs to the new store in the project analysis would be a mistake, because it is not an incremental cost-the total G&A cost will be $1,000,000, whether the project is taken or not.

Case 2: In the previous analysis, suppose that all the facts remain unchanged except for one. The total G&A costs are expected to increase from $1,000,000 to $1,200,000 as a consequence of the new store. Each store is still allocated an equal amount; the new store will be allocated one-sixth of the total costs, or $200,000.

In this case, the allocated cost of $250,000 should not be considered in the project analysis for the new store. The incremental cost of $200,000 ($1,200,000 - $1,000,000), however, should be considered as part of the analysis.

In 2021, Granite Construction's corporate shared costs amounted to $353 million. Assuming that these general administrative costs serve a purpose, which otherwise would have to be borne by each of Granite Construction's business, and that there is a positive relationship between the magnitude of these costs and revenues, it seems reasonable to argue that the company should compute a fixed charge for these costs that every new project has to cover, even though this cost may not occur immediately or as a direct consequence of the new project.

Who Will Pay for Headquarters?

As in the case of sunk costs, the right thing to do in project analysis (i.e., considering only direct incremental costs) may not add up to create a company that is financially healthy. Thus, if a company like Granite Construction does not require individual projects that it reviews to cover the allocated costs of general administrative expenses of the transportation division, it is difficult to see how these costs will be covered at the level of the company.

The Case for Incremental Cash Flows

When reviewing projects it is easy to get tunnel vision and focus on the project or investment at hand, acting as if the goal of the exercise is to maximize the value of the individual project. There is also the tendency, with perfect hindsight, to require projects to cover all costs that they have generated for the company, even if such costs will not be recovered by rejecting the project. The objective in project analysis is to maximize the value of the business or company taking the project. Consequently, it is the cash flows that a project will add on in the future to the business, that is, the incremental cash flows, that we should focus on.

Example 3.4: Estimating Cash Flows for a retail door company: Capital Construction

As described in Example 3.1, Capital Construction is considering starting a door company, which will be staffed by one part-time employees. The following estimates relate to the costs of starting the service and the subsequent revenues from it.

• The initial funding required to start the door company, including adding high-speed internet, software, computers and office equipment, will be $50,000. These assets are expected to have a life of four years, at which point they will have no salvage value. The assets will be depreciated straight line over the four-year life.
• The revenues in the first year are expected to be $60,000, growing 20% in year two, and 10% in the two years following.
• The salaries and other benefits for the employee are estimated to be $10,000 in year one, and grow 10% a year for the following three years.
• The cost of the doors is supposed to be 60% of the revenues in each of the four years.
• The working capital, which includes the inventory of doors needed for the company and the accounts receivable (associated with selling doors on credit) is expected to amount to 10% of the revenues; the cash for working capital have to be made at the beginning of each year. At the end of year four, the entire working capital is supposed to be salvaged.
• The tax rate on income is expected to be 25%, which is also the marginal tax rate for Capital Construction.

Based on this information, we calculate the operating income for Capital Construction Door Company in Worksheet 3.3:

Worksheet 3.3 Expected Operating Income on Capital Construction Door Company

To get from operating income to cash flows, we add back the depreciation charges and subtract out the working capital requirements (which are the changes in working capital from year to year) in Worksheet 3.4. We also show the initial funding of $50,000 as a cash outflow right now (year zero) and the salvage value of the entire working capital requirement in year four.

Worksheet 3.4 From Operating Income to After-Tax Cash Flows

Note that there is an initial working capital cash requirement, which is 10% of the first year's revenues, funded at the beginning of the year. Each subsequent year has a change in working capital that represents 10% of the revenue change from that year to the next. In year 4, the cumulative working capital cash requirement over the four years ($ 8,712) is salvaged, resulting in a positive cash flow.²

Salvaging working capital is basically the same as having a going out of business sale, where all the inventory is sold at cost and all accounts receivable are collected.

Example 3.5: Estimating Earnings, Incremental Earnings and Incremental Cash Flows: Granite Quick-Charge

Granite Construction thinks the future is quick-charge battery powered construction equipment and believes there is a strong market for quick-charge construction battery charges. Granite will sell the quick-charge construction battery chargers to retail and its US construction partners. It will purchase the chargers from three recognized US quick-charge construction battery charger manufacturers. Granite plans to build a 100,000 square foot warehouse, beginning immediately, the project will take one year to plan and construct. A second warehouse (Warehouse II) will be constructed in the second and third year and become operational at the beginning of year four. A third warehouse (Warehouse III) will be constructed in the third and fourth year and become operational at the beginning of year five.

• The cash flows will be calculated in nominal dollars, even thought the actual cash flows will be in Mexican Peso (MXN$).
• The cost of building warehouse I will be $3 million, with $2.5 million to be spent right now and $.5 million to be spent a year from now. Granite has already spent $0.1 million researching the project and getting the necessary licenses for the warehouse; none of this cash can be recovered if the warehouse is not built. This amount will be capitalized and depreciated straight line over the next 10 years to a salvage value of zero.
• The cost of building a second warehouse, Warehouse II, will be $5 million, with $3.5 million spent at the end of the second year and $0.5 million at the end of the third year.
• The revenues for the two warehouses are assumed to be the following, based on projected demand figures until the tenth year and an expected inflation rate of 1.29% (in U.S. dollars). Starting in year ten, the revenues are expected to grow at the inflation rate. Worksheet 3.5 summarizes the revenue projections:

Worksheet 3.5 Revenue Projections: Granite Construction Quick-Charge

Note that the revenues at Warehouse III property is set at 25% of the revenues at Warehouse I.

• The direct operating expenses are supposed to be 80% of the revenues at Warehouse I and 85% of revenues at Warehouse III.
• The depreciation on fixed assets will be calculated as a percent of the remaining book value of these assets at the end of the previous year. Additionally, the warehouses will require capital maintenance cash contributions each year, specified as a percent of the depreciation that year. Worksheet 3.6 lists both these statistics by year:³

Worksheet 3.6 Depreciation and Capital Maintenance Percentages

The capital maintenance expenses are low in the early years, when the warehouses are still new but increase as the warehouses age since old warehouses have to go through either major renovations or be replaced with new materials. After year ten, both depreciation and capital expenditures are supposed to grow at the inflation rate (1.29%).

Granite Quick-Charge will also allocate corporate G&A costs to this project, based on revenues; the G&A allocation will be 15% of the revenues each year. It is worth noting that a recent analysis of these expenses found that only one-third of these expenses are variable (and a function of total revenue) and that two-thirds are fixed. After year ten, these expenses are also supposed to grow at the inflation rate of 1.29%.

• Granite Quick-Charge will have to maintain noncash working capital (primarily consisting of inventory at the warehouses, netted against accounts payable) of 5% of revenues, with the cash contributions being made at the end of each year.
• The income from the project will be taxed at Granite Construction's marginal tax rate of 25%. The projected operating earnings at the warehouses, starting in the first year of operation (which is the second year) are summarized in Worksheet 3.1. Note that the project has no revenues until year two, when the first warehouse becomes operational and that the project is expected to have an operating loss of $2.5 million in that year. We have assumed that the company will have enough income in its other businesses to claim the tax benefits from these losses (25% of the loss) in the same year. If this had been a stand-alone project, we would have had to carry the losses forward into future years and reduce taxes in those years.

The estimates of operating earnings in Worksheet 3.1 are distorted because they do mix together expenses that are incremental with expenses that are not. Specifically, there are two points of disagreement:

Pre-project funding : We included the depreciation on the pre-project funding of $100,000 in the total depreciation for the project. This depreciation, though, can be used by Granite Construction, regardless if it goes ahead with the new quick-charge project.

Allocated G&A Expenses : While we considered the total allocated expense in computing earnings, only one-third of this expense is incremental. As a result, we are understating the earnings on this project.

In Worksheet 3.2a, we compute the incremental earnings for Granite Quick-Charge, using only the incremental depreciation and G&A expenses. Note that the incremental earnings are more positive than the unadjusted earnings in Worksheet 3.1. In Worksheet 3.2, we also determine the incremental after-tax cash flow to Granite Construction, prior to debt payments by:

• Adding back the incremental depreciation each year, because it is a noncash charge.
• Subtracting out the maintenance capital expenditures in addition to the primary capital expenditures because these are cash outflows.
• Subtracting out the incremental cash contribution in working capital each year, which represent the change in working capital from the prior year. Here, we have assumed that the working capital cash contributions are made at the end of each year.

The funding of $3 million in Warehouse I is shown at time 0 (as $2.5 million) and in year one (as $.5 million). The expensed $0.1 million pre-project funding is not considered because it has already been made (sunk cost). Note that we could have arrived at the same estimates of incremental cash flows, starting with the unadjusted operating income and correcting for the non-incremental items (adding back the fixed portion of G&A costs and subtracting out the tax benefits from non-incremental depreciation). Worksheet 3.2b provides the proof.

Book value of fixed assets_t= Book value of fixed assets_t-1+ New Investment_t + Capital Maintenance_t – Depreciation_t Depreciation on fixed assets_t = Book value of fixed assets_t-1* Depreciation as % of prior year’s book value of fixed assets
Depreciation & Amortization_t = Depreciation: Pre-project investment_t + Depreciation on fixed assets

Project Decision Rules

Having calculated the accounting earnings, cash flows, and time-weighted cash flows on a project, we are still faced with the important decision: should we take the project or not. In this section, we will look at a number of project decision rules and put them to the test.

What Is a Project Decision Rule?

When dealing with new investments and projects, companies have to decide whether to invest in them or not. We have been leading up to this decision over the last few sections, but project decision rules allow us to formalize the process and identify what conditions need to be satisfied for a project to be acceptable. Although we will be looking at a variety of project decision rules in this section, it is worth keeping in mind what characteristics we would like a good project decision rule to have.

• First, a good project decision rule has to maintain a fair balance between allowing a manager reviewing a new project to bring in his or her subjective opinions into the decision making and ensuring that different projects are judged consistently. So, an investment decision rule that is too mechanical (by not allowing for subjective inputs) or too malleable (where managers can bend the rule to match their biases) is not a good rule.
• Second, a good project decision rule will allow the company to further the stated objective in corporate finance, which is to maximize the value of the company. Projects that are acceptable using the decision rule should increase the value of the company accepting them, whereas projects that do not meet the requirements would destroy value if the company invested in them.
• Third, a good project decision rule should work across a variety of investments.

Projects can be revenue-generating projects (such as Home Depot opening a new store) or they can be cost-saving projects (as would be the case if Boeing adopted a new inventory management system). Some projects have large up-front costs (as is the case with the Boeing 777 aircraft), whereas other projects may have costs spread out across time. A good project decision rule will give an answer to all of these special types of projects.

Does there have to be only one project decision rule? While many companies review projects using any number of different project decision rules, one rule has to govern. In other words, when the project decision rules lead to different conclusions on if the project should be accepted or rejected, one decision rule has to be the tie- breaker and can be viewed as the governing rule.

Accounting Income-Based Decision Rules

Many of the oldest and most established project decision rules have been drawn from the accounting statements and, in particular, from accounting measures of income. Some of these rules are based on income to equity investors (i.e., net income), and others are based on operating income.

Return on Capital

The return on capital on a project measures the returns earned by the company based on it is total capital contribution in the project. Consequently, it is a return to all claimholders in the company on their collective contribution in a project. Defined generally,

Return on Capital (Pretax) =Earnings before interest and taxes/Average Book Value of Capital Invested in Project

Return on Capital (After-tax) = Earnings before interest and taxes (1- tax rate)/Average Book Value of Capital Invested in Project

To illustrate, consider a one-year project, with an initial capital contribution of $1 million, and earnings before interest and taxes (EBIT) of $300,000. Imagine that the project has a salvage value at the end of the year of $800,000, and that the tax rate is 25%. In terms of a time line, the project has the following inputs:

Earnings before interest & taxes (EBIT)= $ 300,000

Book Value (BV)= $ 1,000,000
Salvage Value = $ 800,000

Average Book Value of Capital = $(1,000,000+$800,000)/2 = $ 900,000

The pretax and after-tax returns on capital can be calculated as follows:

Return on Capital (Pre-tax) = $ 300,000/$900,000 = 33.33%

Return on Capital (After-tax) = $300,000 * (1-0.25)/$900,000 = 25%

Even though this calculation is rather straightforward for a one-year project, it becomes more complicated for multiyear projects, where both the operating income and the book value of the project change over time. In these cases, the return on capital can either be calculated each year and then averaged over time or the average operating income over the life of the project can be used in conjunction with the average capital contribution during the period to determine the average return on capital.

The after-tax return on capital on a project has to be compared to a hurdle rate that is defined consistently. The return on capital is calculated using the earnings before debt payments and the total capital invested in a project. Thus, it can be viewed as return to the company, rather than just to equity investors. Therefore, the cost of capital should be used as the hurdle rate.

If the after-tax return on capital > Cost of Capital → Accept the project

If the after-tax return on capital < Cost of Capital → Reject the project

For example, if the company considering this project had a cost of capital of 10%, it would view the investment in the new project as a good one.

Example 3.8: Estimating and Using Return on Capital in Decision Making: Granite Quick-Charge and Capital Construction Door Company projects

In Examples 5.4 and 5.5, we computed the operating income from two projects-a retail door company start-up by Capital Construction and an start-up construction battery quick-charge company by Granite Construction. We will calculate the return on capital on each of these projects using our earlier estimates of operating income. Worksheet 3.8 summarizes the estimates of operating income and the book value of capital at Capital Construction.

Worksheet 3.8 Return on Capital on Capital Construction Door Company

The book value of capital each year includes the capital contribution in fixed assets and the noncash working capital. If we average the year-specific returns on capital, the average return on capital is 44.82%, but this number is pushed up by the extremely high return in year four. A better estimate of the return on capital is computed by dividing the average after-tax operating income ($193,058) over the four years by the average capital invested ($667,700) over this time, which yields a return on capital of 28.91%. Because this number exceeds the cost of capital of 25.42% that we calculated in Example 3.2 for this project, the return on capital approach would suggest that this is a good project.

In Worksheet 3.9, we compute operating income, book value of capital, and return on capital (ROC) for Granite Quick-Charge. The operating income numbers are from Worksheet 3.1.

Worksheet 3.9 Return on Capital for Granite Quick-Charge (Income and capital in millions)

Average BV of Capital_t = (Capital_t-1 + Capital _t)/ 2

ROC (a) = After-tax Operating Income/ Average BV of Capital

ROC (b) = After-tax Operating Income/ BV of Capital at end of prior year

The book value of capital includes the capital contribution in fixed assets (capital expenditures), net of depreciation, and the capital contribution in working capital that year. It also includes the capitalized pre-project funding and the return on capital each year is computed based on the average book value of capital invested during the year. The average after-tax return on capital, computed using the average capital invested, over the ten-year period is 4.05%; it is slightly lower if we use capital at the end of the prior year. Here, the return on capital is lower than the cost of capital that we calculated in Example 3.2 to be 6.35%, and this suggests that Granite Construction should not make this investment.

Return on Equity

The return on equity looks at the return to equity investors, using the accounting net income as a measure of this return. Again, defined generally,

Return on Equity = Net Income /Average Book Value of Equity in Project

To illustrate, consider a four-year project with an initial equity contribution of $800, and the following estimates of net income in each of the four years:

Net Income $ 140/($800 + $700)/2 = 18.67%
Net Income $ 170/($700 + $600)/2 = 26.15%
Net Income $ 210/($600 + $500)/2 = 38.18%
Net Income $ 250/($500 + $400)/2 = 55.56%

Like the return on capital, the return on equity tends to increase over the life of the project, as the book value of equity in the project is depreciated.

Just as the correct comparison for the return on capital is the cost of capital, the correct comparison for the return on equity is the cost of equity, which is the rate of return equity investors demand.

The cost of equity should explain the riskiness of the project being considered and the financial debt taken on by the company. When choosing between mutually exclusive projects of similar risk, the project with the higher return on equity will be viewed as the better project.

Net Present Value (NPV): The sum of the present values of the expected cash flows on the project, net of the initial funding.

The full estimate of cash flows, described earlier in the section, requires subtracting out capital expenditures and changes in non-cash working capital but it is far too volatile on a year-to-year basis to yield reliable measures of returns on equity or capital.

Discounted Cash Flow Measures

Project decision rules based on discounted cash flows not only replace accounting income with cash flows but explicitly factor in the time value of money. The two most widely used discounted cash flows rules are net present value and the internal rate of return.

Net Present Value (NPV)

The net present value of a project is the sum of the present values of each of the cash flows-positive as well as negative-that occurs over the life of the project. The general formulation of the NPV rule is as follows:

NPV of Project =∑ CF_t/(1 + r)^t - Initial Funding

where

CF_t = Cash flow in period t
r = Discount rate
N = Life of the project.

Consider a simple project, with an initial funding of $ 1 billion and expected cash flows of $300 million in year 1, $ 400 million in year 2, $ 500 million in year 3 and $ 600 million in year 4. Assuming a discount rate of 12%, the NPV of a project is shown in Image 3.3:

Image 3.3: NPV of a Project

Once the NPV is computed, the decision rule is extremely simple because the hurdle rate is already factored in the present value.

Decision Rule for NPV for Independent Projects

If the NPV > 0 → Accept the project
If the NPV < 0 → Reject the project

Note that an NPV that is greater than zero means that the project makes a return greater than the hurdle rate.

Example 3.11: NPV from the Company's Point of View: Capital Construction Door Company

Worksheet 3.13 we calculated the present value of the cash flows to Capital Construction as a company from the proposed door company using the cost of capital of 25.48% as the discount rate on the cash flows. (The cash flows are shown in Example 3.4 and the cost of capital is shown in Example 3.2.)

Worksheet 3.13 Cashflow to the Company for Capital Construction Door Company

This project has a net present value of -$98,775, suggesting that it is a project that should not be accepted based on the projected cash flows and the cost of capital of 12.26%.

Example 3.12: NPV from the Company's Point of View: Granite Quick-Charge

In estimating the cash flows to discount for Granite Quick-Charge business, the first point to note when computing the NPV of the proposed quick-charge business is the fact that it has a life far longer than the ten years shown in Worksheet 3.2. To bring in the cash flows that occur after year ten, when cash flows start growing at 1.29%, the inflation rate forever, we draw on a present value formula for a growing perpetuity :

Present Value of Cash Flows after Year 10 = Cashflow₁₁/(Cost of Capital - Perpetual growth rate)

=($300 * (1.0129))/(.0635 - .0129) =$5,926 million

The cost of capital of 6.35% is the cost of capital for Granite Quick-Charge that we calculated in Example 3.2. This present value is called the terminal value and occurs at the end of year ten.

Worksheet 3.14 shows the NPV of the proposed project computed using the cash flows in millions of U.S. dollars from Worksheet 3.2 and Granite Construction's cost of capital, in dollar terms, of 6.35%.

Worksheet 3.14 NPV of Granite Quick-Charge

The NPV of this project is positive. This suggests that it is a bad project that will lose -$474 thousands, in lost value for Granite Construction.

NPV and Currency Choices

When reviewing a project, the cash flows and discount rates can often be calculated in one of several currencies. For a project like the Granite Quick-Charge business, the obvious choices are the project's local currency (Mexican Peso- MXN$) and the company's home currency (U.S. dollars), but we can in fact use any currency to review the project. When switching from one currency to another, we have to follow the following steps:

• Calculate the expected exchange rate for each period of the analysis: For some currencies (Euro, yen, or British pound), we can calculate expected exchange rates from the financial markets in the form of forward rates. For other currencies, we have to calculate the exchange rate, and the best way to do so is to use the expected inflation rates in the two currencies in question. For instance, we can calculate the expected MXN$/$ exchange rate in n years:

Expected Rate (MXN$/$) = $MXN/$ (Today) * [ (1 + Expected Inflation_US )/(1 + Expected Inflation_Mexico]ⁿ

We are assuming that purchasing power ultimately drives exchange rates-this is called purchasing power parity.

• Convert the expected cash flows from one currency to another in future periods, using these exchange rates: Multiplying the expected cash flows in one currency to another will accomplish this.
• Convert the discount rate from one currency to another: We cannot discount cash flows in one currency using discount rates estimated in another. To convert a discount rate from one currency to another, we will again use expected inflation rates in the two currencies. A US dollar cost of capital can be converted into MXN$ cost of capital as follows:



Cost of Capital(MXN$) = (1 + Cost of Capital ($)) * [(1+ Exp Inflation_Mexico)/(1+ Exp Inflation_US)]-1

• Compute the NPV by discounting the converted cash flows (from step 2) at the converted discount rate (from step 3): The NPV should be identical in both currencies but only because the expected inflation rate was used to calculate the exchange rates. If the forecasted exchange rates diverge from purchasing power parity, we can get different NPVs, but our currency views are then contaminating our project analysis.

Example 3.13: NPV in MXN$: Granite Quick-Charge

In Example 3.12, we computed the NPV for Granite Quick-Charge in dollar terms to be -$447 thousands. The entire analysis could have been done in Mexician Peso (MXN$) terms. To do this, the cash flows would have to be converted from dollars to MXN$, and the discount rate would then have been a MXN$ discount rate. To calculate the expected exchange rate, we will imagine that the expected inflation rate will be 3.5% in Mexico and 1.29% in the United States and use the exchange rate of 20.04 MXN$ per U.S. dollar in August 2021 as the current exchange rate. The projected exchange rate in one year will be:

Expected Exchange Rate in Year 1 = 20.4 MXN$ * (1.035/1.0129) = 20.48 MXN$/$

Similar analyses will yield exchange rates for each of the next ten years.

The dollar cost of capital of 8.67%, calculated in Example 3.1, is converted to a MXN$ cost of capital using the expected inflation rates:

Cost of Capital (MXN$) = (1 + Cost of Capital ($)) * [(1+ Exp Inflation_Mexico/(1+ Exp Inflation_US)] ) -1
= (1.0635) (1.035/1.0129) - 1 = 8.67%

Worksheet 3.15 summarizes exchange rates, cash flows, and the present value for the proposed Granite Quick-Charge business, with the analysis done entirely in Mexico Pesos.

Worksheet 3.15 Expected Cash Flows from Granite Quick-Charge in MXN$

Note that the NPV of MXN$ $-447 thousands is exactly equal to the dollar NPV computed in Example 3.12, converted at the current exchange rate of 20.04 MXN$ per dollar.

NPV in dollars = NPV in MXN$/Current Exchange Rate = -$8,981/20.04 = $-447 million

Terminal Value, Salvage Value, and Net Present Value

When estimating cash flows for an individual project, practicality constrains us to estimate cash flows for a finite period-three, five, or ten years, for instance. At the end of that finite period, we can make one of three assumptions.

• The most conservative one is that the project ceases to exist and its assets are worthless. In that case, the final year of operation will show only the operating cash flows from that year.
• We can imagine that the project will end at the end of the analysis period and that the assets will be sold for salvage. Although we can try to calculate salvage value directly, a common assumption that is made is that salvage value is equal to the book value of the assets. For fixed assets, this will be the undepreciated portion of the initial capital contribution, whereas for working capital, it will be the aggregate value of the capital contributions made in working capital over the course of the project life.
• We can also imagine that the project will not end at the end of the analysis period and try to calculate the value of the project on an ongoing basis-this is the terminal value. In the Granite Quick-Charge analysis, for instance, we imagined that the cash flows will continue forever and grow at the inflation rate each year. If that seems too optimistic, we can imagine that the cash flows will continue with no growth for a finite period or even that they will drop by a constant rate each year.

The best approach to use will depend on the project being reviewed. For projects that are not expected to last for long periods, we can use either of the first two approaches; a zero salvage value should be used if the project assets are likely to become obsolete by the end of the project life (e.g., computer hardware), and salvage can be set to book value if the assets are likely to retain significant value (e.g., buildings).

For projects with long lives, the terminal value approach is likely to yield more reasonable results but with one caveat. The capital contribution and maintenance assumptions made in the analysis should explain its long life. In particular, capital maintenance expenses will be much higher for projects with terminal value because the assets have to retain their earning power. For the Granite Quick-Charge, the capital maintenance expenes climb over time and become larger than depreciation as we approach the terminal year.

NPV: Company versus Equity Analysis

In the previous analysis, the cash flows we discounted were prior to interest and principal payments, and the discount rate we used was the weighted average cost of capital. In NPV parlance, we were discounting cash flows to the entire company (rather than just its equity investors) at a discount rate that explained the costs to different claimholders in the company to arrive at an NPV. There is an alternative. We could have discounted the cash flows left over after debt payments for equity investors at the cost of equity and arrived at an NPV to equity investors.

Will the two approaches yield the same NPV? As a general rule, they will, but only if the following assumptions hold:

• The debt is correctly priced and the market interest rate to compute the cost of capital is the right one, given the default risk of the company. If not, it is possible that equity investors can gain (if interest rates are set too low) or lose (if interest rates are set too high) to bondholders. This in turn can result in the NPV to equity being different from the NPV to the company.

The same assumptions are made about the financing mix used in both calculations.

Note that the financing mix assumption affects the discount rate (cost of capital) in the firm approach and the cash flows (through the interest and principal payments) in the equity approach.

Given that the two approaches yield the same NPV, which one should we choose to use? Many practitioners prefer discounting cash flows to the firm at the cost of capital, it is easier to do because the cash flows are before debt payments and therefore we do not have to calculate interest and principal payments explicitly. Cash flows to equity are more intuitive, though, because most of us think of cash flows left over after interest and principal payments as residual cash flows.

Assets in Place: The assets already owned by a company or projects that it has already taken.

Properties of the NPV Rule

The NPV has several important properties that make it an attractive decision rule and the preferred rule, at least if corporate finance practitioner were doing the picking.

NPVs Are Additive

The NPVs of individual projects can be aggregated to arrive at a cumulative NPV for a business or a division. No other project decision rule has this property. The property itself has a number of implications.

The value of a company can be written in terms of the present values of the cash flows of the projects it has already taken on as well as the expected NPVs of prospective future projects:

Value of company = ∑Present Value of Projects in Place +∑NPV of Future Projects

The first term in this equation captures the value ofassets in place, whereas the second term measures the value of expected future growth. Note that the present value of projects in place is based on anticipated future cash flows on these projects.

• When a company terminates an existing project that has a negative present value based on anticipated future cash flows, the value of the company will increase by that amount. Similarly, with an investments in a new project, with an expected negative NPV, the value of the company will decrease by that amount.
• When a company divests itself of an existing asset, the price received for that asset will impact the value of the company. If the price received exceeds the present value of the anticipated cash flows on that project to the company, the value of the company will increase with the divestiture; otherwise, it will decrease.
• When a company invests in a new project with a positive NPV, the value of the company will be impacted depending on if the NPV meets expectations. For example, a company like Apple is expected to take on high positive NPV projects, and this expectation is built into value. Even if the new projects taken on by Apple have positive NPV, there may be a drop in value if the NPV does not meet the high expectations of financial markets.

When a company makes an acquisition and pays a price that exceeds the present value of the expected cash flows from the company being acquired, it is the equivalent of taking on a negative NPV project and will lead to a drop in value.

Intermediate Cash Flows Are Invested at the Hurdle Rate

Hurdle Rate: The minimum acceptable rate of return that a company will accept for taking a given project.

Implicit in all present value calculations are assumptions about the rate at which intermediate cash flows get reinvested. The NPV rule assumes that intermediate cash flows on a projects-that is, cash flows that occur between the initiation and the end of the project-get reinvested at the hurdle rate, which is the cost of capital if the cash flows are to the company and the cost of equity if the cash flows are to equity investors. Given that both the cost of equity and capital are based on the returns that can be made on alternative projects of equivalent risk, this assumption should be reasonable.

NPV Calculations Allow for Expected Term Structure and Interest Rate Shifts

In all the examples throughout in this section, we have imagined that the discount rate remains unchanged over time. This is not always the case, however; the NPV can be computed using time-varying discount rates. The general formulation for the NPV rule is as follows:

NPV of Project = ∑ CF_t/∏(1 + r_t ) - Initial Funding

where

CF_t = Cash flow in period t

r_t = One-period discount rate that applies to period t

N = Life of the project.

The discount rates may change for three reasons:

• The level of interest rates may change over time, and the term structure may provide some insight on expected rates in the future.
• The risk characteristics of the project may be expected to change in a predictable way over time, resulting in changes in the discount rate.


• The financing mix on the project may change over time, resulting in changes in both the cost of equity and the cost of capital.

Example 3.16: NPV Calculation with Time-Varying Discount Rates

Imagine that you are analyzing a four-year project investing in artificial intelligence software development. Furthermore, imagine that the technological uncertainty associated with the software industry leads to higher discount rates in future years.

The present value of each of the cash flows can be computed as follows.

PV of Cash Flow in year 1 = $400/1.12 = $357.14
PV of Cash Flow in year 2 = $500/(1.12 * 1.13) = $395.06
PV of Cash Flow in year 3 = $600/(1.12 * 1.13 * 1.14) =$415.86
PV of Cash Flow in year 4 = $700/(1.11 * 1.12 * 1.13 * 1.15) =$421.88
NPV of Project = $357.14 + $395.06 + $415.86 + $421.27 - $1000.00 = $589.94

Biases, Limitations, and Caveats

In spite of its advantages and its linkage to the objective of value maximization, the NPV rule continues to have its detractors, who point out several limitations.

• The NPV is stated in absolute rather than relative terms and does not therefore factor in the scale of the projects. Thus, project A may have an NPV of $200, whereas project B has an NPV of $100, but project A may require an initial capital contribution that is 10 or 100 times larger than project B. Proponents of the NPV rule argue that it is surplus value, over and above the hurdle rate, no matter what the project.
• The NPV rule does not control for the life of the project. Consequently, when comparing mutually exclusive projects with different lifetimes, the NPV rule is biased toward accepting longer-term projects.

Internal Rate of Return

Internal Rate of Return (IRR): The rate of return earned by the project based on cash flows, allowing for the time value of money.

The internal rate of return (IRR) is based on discounted cash flows. Unlike the NPV rule, however, it takes into account the project's scale. It is the discounted cash flow analog to the accounting rates of return. Again, in general terms, the IRR is that discount rate that makes the NPV of a project equal to zero. To illustrate, consider again the project described at the beginning of the NPV discussion.

Cash Flow = -$1,000 + $400 + $500 + $600 + $700
Internal Rate of Return = 30.35%

At the internal rate of return, the NPV of this project is zero. The linkage between the NPV and the IRR is most obvious when the NPV is graphed as a function of the discount rate in a net present value profile. An NPV profile for the project described is illustrated in Image 3.4.

Image 3.4: NPV Profile

NPV Profile: This measures the sensitivity of the NPV to changes in the discount rate.

The NPV profile provides several insights on the project's viability. First, the internal rate of return is clear from the graph-it is the point at which the profile crosses the x- axis. Second, it provides a measure of how sensitive the NPV-and, by extension, the project decision-is to changes in the discount rate. The slope of the NPV profile is a measure of the discount rate sensitivity of the project. Third, when mutually exclusive projects are being reviewed, graphing both NPV profiles together provides a measure of the break-even discount rate-the rate at which the decision maker will be indifferent between the two projects.

Using the IRR

One advantage of the IRR is that it can be used even in cases where the discount rate is unknown. While this is true for the calculation of the IRR, it is not true when the decision maker has to use the IRR to decide whether to take a project. At that stage in the process, the IRR has to be compared to the discount rate-f the IRR is greater than the discount rate, the project is a good one; alternatively, the project should be rejected.

Like the NPV, the IRR can be computed in one of two ways:

• The IRR can be calculated based on the free cash flows to the company and the total capital contribution in the project. In doing so, the IRR has to be compared to the cost of capital.
• The IRR can be calculated based on the free cash flows to equity and the equity contribution in the project. If it is calculated with these cash flows, it has to be compared to the cost of equity, which should explain the riskiness of the project.

Decision Rule for IRR for Independent Projects

A. IRR is computed on cash flows to the company

When choosing between projects of equivalent risk, the project with the higher IRR is viewed as the better project.

Example 3.17: Estimating the IRR Based on FCFF: Granite Quick-Charge

The cash flows to the company from Granite Quick-Charge, are used to arrive at a NPV profile for the project in Image 3.5.

Image 3.5: Granite Quick-Charge NPV

The IRR in dollar terms on this project is 5.03%, which is lower than the cost of capital of 6.35%. These results are consistent with the findings from the NPV rule, which also recommended not to invest in the battery quick-charge market.⁴

Biases, Limitations, and Caveats

The IRR is the most widely used discounted cash flow rule in project analysis, but it does have some serious limitations.

• Because the IRR is a scaled measure, it tends to bias decision makers toward smaller projects, which are much more likely to yield high%age returns, and away from larger ones.
• There are a number of scenarios in which the IRR cannot be computed or is not meaningful as a decision tool. The first is when there is no or only a very small initial capital contribution and the cashflows are spread over time. In such cases, the IRR cannot be computed or, if computed, is likely to be meaningless. The second is when there is more than one internal rate of return for a project, and it is not clear which one the decision maker should use.

Example 3.19: Multiple IRR Projects

Consider a project to manufacture and sell a consumer product, with a hurdle rate of 12%, that has a four-year life and the following cash flows over those four years. The project, which requires the licensing of a trademark, requires a large payment at the end of the fourth year. Image 5.7 shows the cash flows.

Image 3.7 Cash Flows on Project

The NPV profile for this project, shown in Image 3.8, explains the problems that arise with the IRR measure.

Image 3.8: NPV Profile for Multiple IRR Project

As you can see, this project has two IRRs: 6.60% and 36.55%. Because the hurdle rate falls between these two IRRs, the decision on whether to take the project will change depending on which IRR is used. To make the right decision in this case, the decision maker would have to look at the NPV profile. If, as in this case, the NPV is positive at the hurdle rate, the project should be accepted. If the NPV is negative at the hurdle rate, the project should be rejected.

Multiple IRRs: Why They Exist and What to Do about Them

The IRR can be viewed mathematically as a root to the present value equation for cash flows. In the conventional project, where there is an initial capital contribution and positive cash flows thereafter, there is only one sign change in the cash flows, and one root-that is, there is a unique IRR. When there is more than one sign change in the cash flows, there will be more than one IRR.⁵ In Image 3.7, for example, the cash flow changes sign from negative to positive in year one, and from positive to negative in year four, leading to two IRRs.

Lest this be viewed as some strange artifact that is unlikely to happen in the real world, note that many long-term projects require substantial reinvestment at intermediate points in the project and that these reinvestments may cause the cash flows in those years to become negative. When this happens, the IRR approach may run into trouble.

There are a number of solutions suggested to the multiple IRR problems. One is to use the hurdle rate to bring the negative cash flows from intermediate periods back to the present. Another is to construct an NPV profile. In either case, it is probably much simpler to calculate and use the NPV.

¹The unlevered market beta for internet retailers is 1.70, and the average correlation of these stocks with the market is 0.40. The unlevered total beta is therefore 1.70/0.4 = 4.25.

² Salvaging working capital is essentially the equivalent of having a going out of business sale, where all the inventory is sold at cost and all accounts receivable are collected.

³Capital maintenance expenditures are capital expenditures to replace fixed assets that break down or become obsolete. This is in addition to the regular maintenance expenses that will be necessary to keep the warehoues going, which are included in operating expenses.

⁴The terminal value of the project itself is a function of the discount rate used. That is why the IRR function in Excel will not yield the right answer. Instead, the NPV has to be recomputed at every discount rate and the IRR is the point at which the NPV = 0.

⁵Although the number of IRRs will be equal to the number of sign changes, some IRRs may be so far out of the realm of the ordinary (e.g. 10,000%) that they may not create the kinds of problems described here.

Excerpts (reworded) from Aswath Damodarn, Corporate Finance 2009.