9 Valuation of investments and projects
,Let us now look at investments in real assets such as real estate and machinery. Real estate will be treated in more detail in the next chapter. The main difference between real and financial assets is depreciation, which is tax deductible. We therefore have to adjust for the tax effect arising from depreciation in order to find the correct after-tax cash flow. Let us start with a very simple valuation of an investment. Later, we will discuss the size and calculation of the cost of capital (also known as the discount rate). We have to consider both tax and inflation to make accurate investment decisions.
Example 9.1
The company Joe Louis & Sons is considering a seven-year investment in a machine that produces aeroplane parts. The only investment is a machine worth 14 million. The machine is depreciated by 20 per cent each year, but is worthless at the end of the project. The tax rate is 28 per cent.
Annual cash flow (in thousands) is estimated below. An equally risky project offers 12 per cent annual return. Is the project profitable?
Figure 9.1 Cash flow from machinery investment.
As shown in section 7.1, we always use the cash flows to value a project. In order to calculate the tax, we have to find the taxable profit (loss) for each year. In this example we use the declining balance method, also known as reducing balance method. In the spreadsheet in figure 9.2, the yearly depreciation and book value are shown in cells C11:I13. The first year’s depreciation will be 14,000 · 0.2 = 2,800. Book value at the start of year 2 will therefore be 14,000 – 2,800 = 11,200. The next year’s depreciation will then be 20 per cent of the balance: 11,200 · 0.2 = 2,240 etc.
Reportable income in this example is cash flow in minus cash flow out minus depreciation in cells C15:I15. Tax rate is 28 per cent and the tax is calculated in cells C16:I16. Note that a high level of depreciation is favourable as it reduces the reportable income and thus lowers the payable tax. Depreciation is nevertheless a true cost as it reflects the estimated reduction in the value of an asset. The depreciation rate should mirror the asset’s actual decline in value. However, at the end of a project, the asset can be sold at a price below or above the depreciated value (book value), returning a taxable loss or profit respectively. Here we assume the machine is worthless at the end of year 7.
Figure 9.2 Cash flow projections for investment in machinery.
The cash flows in cells C18:I18 are the after-tax cash flows for years 1–7. The net present value of these cash flows is shown in C19:I19. For instance, net present value in year 2 (cell D19) is calculated by using the formula “= D18/(1+B23)^D7” = 2,787/(1+0.12)2 = 2,222.
The book value of the machine is 2,936 at the end of year 7. There is still some depreciation left when the project is terminated (587 in year 8, 470 in year 9, etc.). This will continue to give tax deduction of 28 per cent each year (also known as tax shield) and will therefore represent some value. The deduction is reduced by 20 per cent per year as the balance diminishes. The net present value of the remaining tax shield at year 7 is therefore an infinite geometric series with a growth factor of g = –0.2.
Depreciation in year 8 gives us a tax shield of 587 · 0.28. The net present value of the remaining tax shield at the end of year 7 will then be:
The net present value of the tax shield is
This is calculated in cell J19 with the formula 
Now we can find the net present value for the project in cell B21 as the sum of all net present values in cells B19:J19. The NPV is negative and the project consequently not profitable at a cost of capital of 12 per cent:
It would be interesting to know at what discount rate the project is profitable. This can be easily found by using the goal seek function in Excel. We know from chapter 7 that a project is profitable when the net present value is zero or higher. We can therefore ask Excel to find at which discount rate the project is profitable. You will find the Goal Seek function under the Data tab and “What-if” button.
Figure 9.3 Using Goal Seek function in Excel to find cost of capital that returns NPV = 0.
And then choose cell B21, which returns the net present value, to be 0, by changing the discount rate in cell B23.
Figure 9.4 Goal Seek function in Excel (Data →, What-If → Goal Seek).
This returns a discount rate of 9.6 per cent.
We can also use the Goal Seek function to find at which tax rate the project becomes profitable at 12 per cent discount rate. Instead of asking the Goal Seek function to change cell B23, we ask it to change cell C4. The result is a tax rate of 9.8 per cent.
9.1 Finding the cost of capital
All investment decisions must take account of the cost of capital, also known as the discount rate. If a project is very risky, it should return a high profit to compensate for the risk involved. A useful definition of risk is how much the actual return can deviate from the expected return. In section 8.6 we looked at the Capital Asset Pricing Model (CAPM) as one way of determining the discount rate. When we are considering investing in specific projects we should consider the funding cost: what is the cost of raising money to this project?
In chapter 7 we looked at two different methods for estimating the profitability of a project: the net present value (NPV) and the internal rate of return (IRR). So far we have taken the discount rate as given without any deeper discussion of what the appropriate discount rate should be. Finding the right discount rate is very important, but not straightforward.
Having found the appropriate discount rate we can calculate the NPV. We will accept the project if the NPV is bigger than zero or if the IRR is equal to or exceeds the discount rate.
When we are evaluating investments within a firm, the appropriate cost of capital is the average funding cost for the firm. Since firms are financed with different mixes of debt and equity we have to calculate the weighted average cost of capital (WACC).
In order to find the firm’s total cost of capital we have to weigh the mix of debt and equity:
The after-tax cost of capital will be somewhat lower due to the tax deductibility (t)of interest costs:
Consider a company with a debt of 2 million and equity of 6 million. Ideally, we should use the market value of the equity (share price times the number of shares outstanding). This company is not listed on any stock exchange, however, and in the absence of any market prices we have to use some other measure to find the weights. A common but not recommended proxy is to use the book value of the equity, which is readily available in the balance sheet.
Finding the weights is then fairly straightforward:
The weights may change over time, especially if we use the market value of equity. A remedy is to user the target levels of equity and debt, i.e. the company’s long-term target of mix of equity and debt.
Now let us look closer at the cost of capital for equity, rE, and the cost of debt, rD, respectively.
Cost of equity
The cost of equity should equal the cost of issuing new shares or the cost of using retained earnings to finance a project. This is the same as the required return by the shareholders.
How do we estimate the cost of equity? We have basically two methods:
1 Capital Asset Pricing Model
2 Dividend Growth Model (Gordon’s formula).
Using the Capital Asset Pricing Model
The CAPM in section 8.6 gives us the cost of equity under some specific and rather strict conditions. The model asserts that the expected return for a share i can be found by using this formula:
In equilibrium, the expected return of share i must be the same as the cost of equity. We can therefore use CAPM to find the cost of equity, rE. Let us assume that the risk-free rate is 5 per cent and that the risk premium is 7 per cent after tax. The tax rate (t) is 28 per cent and the beta is 1.5. We then get the cost of equity:
This is the required rate of return on equity and mirrors the company’s risk. However, the risk in a specific project can be higher or lower than for the whole company. If the risk of the project is much higher than the risk of the overall company, a higher discount rate should be applied. But how? If the new project is somewhat remote from the usual business, an industry WACC might be more relevant. If you can find companies that are “pure play”, i.e. comparable to the project in question, an average of these companies’ WACC could be more appropriate.
Example 9.2
In figure 9.5, we have entered some mock numbers for the period 1981–2010 showing yearly returns of the market portfolio and risk-free government bonds. All returns are pre-tax.
Estimate the return on equity as of 31.12.2010 for a share that has a beta, β, of 1.35.
Figure 9.5 Market returns and risk-free rate, 1981–2010 (mock numbers).
In the spreadsheet below we have computed E(rm) – rf for all years. The average is calculated in cell D16 and can be used as the estimated market risk premium in the CAPM. As shown in cell H17, the pre-tax return on equity as of 31.12.2010 is:
Figure 9.6 Return on equity for a company with a beta of 1.35.
Of course, the usefulness of CAPM depends critically on the inputs. We have to find the best estimates for:
• Risk-free rate: Should we use short-term or long-term government bonds?
• Beta: How long a period should we use to compute the beta? Two, five or ten years?
• Equity risk premium: Should we just extrapolate long-term historical returns or use current market ratios implied by the dividend discount model etc.?
There do not seem to be any definite answers, and discussing these issues is beyond the scope of this book.
Dividend Discount Model
Another way of computing the firm’s cost of equity is the Dividend Discount Model. The formula says that the value of a share is the net present value of all future anticipated dividends discounted at the appropriate risk-adjusted cost of equity.
If future dividends are expected to grow at a constant rate, g, we can write:
When this series goes to infinity, we have a geometrical series that will converge to the much simpler expression:
which can be converted to:
The growth rate, g, must be smaller than the cost of equity, rE, or the company eventually will grow to become bigger than the economy itself, or, mathematically, the stock price, P, goes to infinity. If we happen to have a market price for the company’s share, P, we can derive the cost of equity from the dividend discount formula.
Consider a company that pays a dividend of 25 per share one year ahead (DIV1). This dividend is expected to grow at 3.5 per cent annually for ever (g). The share price is 300, which implies a cost of equity of
Sometimes companies buy back their own shares as a substitute or addition to dividend. This complicates the computation somewhat as there will be irregular cash flows and the simple dividend discount model cannot be applied.
Figure 9.7 Calculating the cost of equity from the dividend discount model.
We therefore have to use an alternative method for computing the cost of equity. If we have a period that can be considered fairly normal, we can use historical data from this period to find the cost of capital.
Example 9.3
For the past 10 years a company has paid dividends as shown in figure 9.8. In addition, the company has both issued new shares and bought back shares. All transactions have been carried out at the end of the year. At the end of 2010 the company has issued 3,673,000 shares. The share price is 83. Find a reasonable cost of equity (pre-tax) for the company.
Figure 9.8 Dividend and share buy-back, 2003–2012.
We have entered the cash flows in the spreadsheet shown in figure 9.9. In cells E4:E13 we have estimated the net cash flow for each year. In cell E6 we get the net cash flow (dividend + stock buy-back – share issue) = (13,000,000 + 9,000,000 – 7,000,000) = 15,000,000. In cells F5:F13 we have calculated the yearly change in cash flows to equity holders. The average is 11.7 per cent using the formula “=AVERAGE(F5:F13)”. At share price 83 and 3,673,900 shares issued, the market capitalization as of 31 December 2010 is 304,933,700.
Figure 9.9 Cost of equity with share buy-back.
We can now use formula 9.5 to find the cost of equity. Instead of share price, P0, we use the total market cap and total cash flow:
This may be used as the pre-tax cost of equity if the financial risk is held constant going forward.
Cost of debt
Before we can compute the weighted average cost of capital (WACC) we have to find the cost of the capital raised from debt. This cost should equal the marginal cost of additional borrowing. More borrowing can also change the cost of debt.
We have seen different methods for computing the cost of equity. Finding the cost of debt is less straightforward, and we often use approximations. The cost of debt is mainly determined by three factors:
1 Current interest rate level. Changes in the general interest level will also increase the firm’s cost of debt.
2 Default risk. The higher the probability of default, the higher the borrowing cost. This probability is stated either by bond rating agencies (for large corporations) or by the most recent interest rate level that the firm borrowed at.
3 Tax-shield. As interest is tax deductible, after-tax cost of debt will be (1 – tax rate) times lower than pre-tax cost.
The most recent borrowing cost is usually the most appropriate measure of the cost of debt. However, if the latest borrowing took place some time ago the actual capital cost could be different and we have to use other methods.
Most debt is not risk-free and the cost of debt should therefore be the risk-free rate plus some risk premium. For stocks, this can be done by using the Capital Asset Pricing Model. For debt, however, it is much more difficult to estimate because a firm may:
• have bonds issued that are infrequently traded (few prices) or
• have no bonds traded in regulated markets (no observable prices).
In chapter 11 we will show that a bond’s risk is basically determined by
• the time to maturity
• the risk of default.
The longer the maturity and/or default risk, the higher the beta. Very short debt will usually have a beta very close to 0. Using the CAPM, the cost of debt will therefore equal the risk-free rate:
In practice, companies have to pay a higher interest than the risk-free rate even for very short-term debt.
Example 9.4
The company Jamie Bond has risky debt. The company’s bonds are estimated to have a beta of 0.2. The expected return on the market portfolio is 5.0 per cent and the risk-free rate is 3.0 per cent. With a tax rate of 28 per cent, what is Jamie Bond’s cost of debt?
We use the tax-adjusted CAPM (also known as Benninga-Sarig cost of debt). First we have to find the risk premium adjusted for tax, rmt:
Figure 9.10 Cost of risky debt.
(ii) Using the interest rate paid
An easy and much-used proxy for the cost of debt is the average cost of debt of existing debt. One divides last period’s (year’s) interest costs on the average debt in the same period (year).
Example 9.5
Let us look at some data from the profit and loss statement, and a balance sheet for a company. Find the cost of debt for this company.
Figure 9.11 Profit and loss statement, and a balance sheet.
In the accompanying spreadsheet we observe that net debt for each year is: Long term debt + short term debt – cash (“=B5+B6–B4”). Net interest for 2010 is computed in cell B13 as interest costs – interest income. As balance sheet items are shown as discrete point estimates, while interest is for the whole year, we should use the average debt when computing the cost of debt.
Figure 9.12 Calculating the cost of capital from actual interest paid.
In cell B15 we have “=B13/AVERAGE(B12:C12)”, which is 6.85 per cent. This is pre-tax, so in order to find the after-tax cost of debt, we deduct the tax from interest costs (28 per cent):
6.85 per cent · (1 – 0.28) = 4.94 per cent
If a company has issued bonds that are listed and traded in the financial market we can use the current yield to maturity to find the cost of debt (see chapter 11 for more on yield to maturity). The yield at which these bonds are traded will then be the best estimate of its cost of debt. This requires some liquidity in the bond, i.e. some fairly recent transactions in the secondary market.
9.2 Valuing a project
Choosing the proper discount rate
In chapter 7 we addressed the concept of capital budgeting. However, we did not consider the financing of a project. We just took the discount rate as given. As we shall see, it may make a difference whether we finance a project with debt, equity or some mix of the two. According to one very famous theorem, Miller and Modigliani’s theorem, the choice between debt and equity financing is irrelevant, or, more precisely: a company’s leverage has no effect on its weighted average cost of capital. That is, the cost of equity capital is a linear function of the debt–equity ratio. Increased use of lower-cost debt will also increase risk to the equity and thus increase the cost of the equity, counterbalancing any benefit from taking on more of the cheaper debt. Therefore we can use the cost of equity as the cost of all capital (opportunity cost of capital will be the same regardless of debt level).
The existence of tax complicates the matter, however. The tax deductibility of interest creates an incentive to increase the amount of debt until potential lenders require too high interest to compensate for the increased financial risk from more leverage. There might therefore exist an optimal mix of debt and equity at which the total cost of capital is the lowest.
What about other financing instruments? The total cost of capital for a company is the average cost of capital of the different components of financing, which might constitute equity, different debt instruments and hybrid securities such as convertible bonds or preference shares. We just weigh the costs of the different instruments in order to find the average cost of capital.
We can consider a company’s cost of capital as its funding cost. The company has to ask itself: In order to finance a new project, what do we have to pay the capital providers (lenders and shareholders) in return? Investors and lenders will compare your investment proposal with projects of similar risk, so it is like issuing new shares (representing the cost of equity, rE) or issuing bonds (representing the cost of debt, rD).
To simplify, we look at only two components, debt (D) and equity (E), and their weights are found by dividing their value by the total capital (D + E), as shown in equation 9.1:
Cost of total capital, rTC, is also known as the weighted average cost of capital (WACC), which was introduced in equation 9.1 above. It is presented both before and after tax.
Imagine a company with 2 million in debt and 6 million in equity. After-tax return on equity is 31.8 per cent and after-tax borrowing cost is 5.04 per cent. What is the company’s return on capital?
WACC is the appropriate cost of capital when the riskiness of a cash-flow stream from a project is similar to riskiness of the cash flows received by shareholders and debtholders. In some instances, WACC can be used both in valuing projects (as in this chapter) and in valuing the entire company (as will be done in chapter 12).
Finding the cash flows
Any capital budgeting must start by estimating the expected cash flows. A proper discount rate is of little use if the cash flows are estimated wrongly. Of course, all future cash flows are subject to uncertainty, due to market risk (equity), solvency risk (debt), or tax and inflation risk (both). We can take this uncertainty into consideration by discounting very conservative estimates of cash flow at the risk-free rate, or we can discount our estimates for most probable cash flows by a cost of capital that is adjusted to the riskiness of the project. Only the latter method will be treated here.
Cash flow is not the same as earnings. All companies therefore report both a profit and loss statement (earnings) and a cash-flow statement. The major difference is the concept of expenditures and costs. When we buy a machine, the expenditure occurs at the time of the purchase, while the cost of the machine is distributed over the life expectancy of the asset (or by the number of years the tax rules allow). This cost is known as depreciation.
Here is the method for transforming earnings into cash flow:
After-tax profit |
Most common measure of profitability but includes non-cash element as depreciation. On the other hand it does not include asset purchases or large build-up of working capital that might demand much capital. |
+ Depreciation |
This is a non-cash expense that should be added back. |
– Capital expenditures |
Some of the cash flows must be reinvested in the company either i) to maintain the existing assets or ii) to maintain and expand the asset base. Thus, high growth must be accomplished with high capital expenditure. |
+ Change in working capital |
Some funds will be tied up in inventory, accounts receivable, etc. so we have to adjust for changes in current assets and liabilities related to the firm’s operations. |
+ After-tax interest payments |
Free cash flow should not include financing activities, so we should add back after-tax cost of debt since interest is tax deductible. |
= free cash flow |
The amount of cash from operations. “Free” denotes cash available to shareholders and debtholders (dividends, stock repurchases and interest). |
We have additional considerations when making a project calculation:
Sunk cost: If the project under consideration has incurred costs previously, i.e. before we decide to undertake it, we should not include these costs. If we can raise capital today in order to invest, we should do so without any consideration of the capital previously used. The same applies to a project that is unprofitable. You should not care about costs that have occurred. They are irrelevant for new investments.
Side-effects: You should, however, take account of alternative usage of previously acquired assets. If any usage of these assets will take capacity away from alternative usage, we have to include this opportunity cost in our investment analysis.
Spill-over effects: If the project under consideration is part of a bigger project, it might spur additional income on aggregate, or, if this project diminishes income from other parts of the company, this must be deducted from the cash flows.
Some examples
If a transaction takes place in November year 3, the income will be recognized in year 3 even if the cash flow will occur in April year 4. In a cash-flow statement, April year 4 is the correct time for recognizing the transaction.
If a company borrows, only interest is recognized as cost each year, but the cash flows associated with borrowing are the borrowed amount as investment in year zero (start of project) and amortization (interest plus principal) each year.
Investments are recognized differently in an income statement and in a cash-flow statement. An investment usually involves a big cash outlay at the start of a project and smaller cash inflows in subsequent years. In an income statement, only the annual depreciation is recognized. Since depreciation reduces the taxable profit, we have to find the implicit effect of the depreciation on the cash flows. If a company invests 400,000 in new machinery and the machine is depreciated linearly over eight years, we have an annual depreciation of 400,000/8 = 50,000. This is the annual reduction of the taxable profit, and thus increases the cash flow by the depreciation times the tax rate. If the tax rate is 28 per cent, annual tax is therefore reduced by 50,000 · 0.28 = 14,000. We have to add this amount to the cash flow each year.
Tax is recognized the year the profit is made, but usually paid the next year.
If a company has spent 500,000 on marketing research ahead of an investment decision, this will not change the future profitability of the project. The project is not dependent on previously used money, and should therefore not be included in the investment analysis.
Assume that a company decides to use some of its land for a new project. If this land can be rented to external tenants for 200,000 per year, this (opportunity) cost must be included as a negative cash flow in the investment analysis.
If a producer of microwave ovens starts producing a new model that cannibalizes sales of the old, one has to deduct the (net) lost sale as a negative cash flow for the new project.
A software developer should include future revenue from future upgrades of the same software as future positive cash flow.
Fixed costs such as rent and administrative expenses that occur regardless of any new projects should not be included. Only fixed costs that will come in addition because of the project should be taken into account.
Changes in the general price level, also known as inflation, might introduce some nuisance in an investment analysis. A common mistake is to mix nominal and real numbers. If inflation is included in the discount rate (nominal discount rate), then all cash flows must be adjusted accordingly (in line with annual expected inflation). If the discount rate is in real terms, there is no need to adjust future cash flows with the inflation rate. Whatever term you choose, do not mix apples and oranges!
Interest and returns are usually denominated in nominal terms, i.e. inflation is included. Let nominal interest rate be denoted by rnom and real interest rate by rreal. Inflation is denoted i. If you deposit 1 on a bank account at nominal rate of 12 per cent annually, you will one year ahead have:
If the inflation is 8 per cent annually, what is the real return? We have to divide the nominal return by the loss from inflation):
Real return is therefore 1.037–1 = 0.037 = 3.7 per cent.
The formula is
By rearranging equation 9.6 we get the expression for nominal rate:
In our example we used 8 per cent inflation rate and need a 12 per cent nominal interest rate in order to achieve a real interest rate of 3.7 per cent:
Why can’t we just add inflation to the real interest rate or subtract the inflation to get the real rate? The nominal rate, rnom, is calculated at the end of the period, so we have to adjust the real rate, rreal, to the price level at the end of the period (1 + i). We therefore add rreal(1 + i) to the inflation rate i. In this way we adjust both the principal and the interest.
We can use nominal or real numbers in a net present value analysis, but we have to be careful not to mix the two. This goes both for the cash flows and the discount rate.
Example 9.6
Rocky Marciano applies a cost of capital of 15 per cent. He computes the net present value to 1,689 for the following nominal cash flows (in thousands):
(–5,000, 1,500, 3,000, 3,000, 2,000)
Show that you will get the same net present value (NPV) using real numbers and an inflation rate of 9 per cent.
Using equation 9.6, the real cost of capital becomes
We use this in cell B5 in the spreadsheet in figure 9.13.
Figure 9.13 Transforming nominal cost of capital into real cost of capital.
In cells C10:F10 we have transformed the cash flows to real numbers in this way:
In cell G8 we have calculated the NPV = 1,689 by discounting the nominal cash flows (cells C8:F8) by the nominal discount rate (cell B3). In cell G10 we have calculated the same value for NPV by discounting the real cash flows by the real discount rate (cell B5).
In this example we would have got the same NPV regardless of which inflation rate we had used. Let us show a general case in which we get the same NPV for cash flows (–I, C1, C2, C3) whether we use nominal or real values. Remember: The net present value is the initial investment, I, minus all future cash flows (Ct) discounted at the appropriate discount rate.
If we are using real values we have to adjust by the inflation rate, i. C1 must be divided by (1 + i), etc. We also have to apply a real discount rate.
Note that inflation impacts cash flows in different ways. Prices of raw materials might increase more than the general price level, while the tax deductibility is constant and unrelated to the inflation rate. The NPV analysis should, however, give the same result regardless of nominal and real values, as long as they are not mixed.
Tax and investments
Depreciation is a cost, but not an expense and should therefore not be included in the cash flows. But depreciation reduces the taxable profit and hence increase the cash flow after tax. To complicate this even more, we have different depreciation methods, both among different assets and across different countries. When estimating the cash flows, we must use the depreciation rate that is given by the tax code in the individual country.
Straight-line (linear) depreciation
The most common European depreciation method is the straight-line depreciation method. Here the annual depreciation is fixed and equal to the investment divided by the useful life of the asset. The tax code tells us the life expectancy for the asset in question. If a machine is worth 20,000 at the time of investment and can be written down over five years, the annual depreciation will be 20,000/5 = 4,000.
Figure 9.14 Depreciation and book value.
In Excel, we can use the function SLN (cost, salvage, years).
Figure 9.15 Straight-line depreciation and book value using Excel formula.
The declining balance or reducing balance method involves a fixed depreciation rate each year, but a gradually declining charge. The method does not depreciate the asset fully by the end of its useful life, so we have to either convert to a straight-line depreciation at some point in time or just write off the remaining book value at the end of the asset’s useful life.
The tax authorities decide the depreciation rate for categories of various assets, depending on the useful life expectancy. Naturally, computers will typically have much shorter useful life than a building.
Figure 9.16 Depreciation using the declining balance method.
In Excel, we can use the function DDB (cost, salvage, life, period, factor) where:
cost is the initial cost of the asset,
salvage is the value at the end of the depreciation period,
life is the time period over which the asset is depreciated (also known as useful life),
period is the time for the depreciation to be calculated,
factor is the rate at which the balance declines.
Box 9.1 Norwegian depreciation charges. Declining balance method.
Modified Acceleration Cost Recovery System (MACRS)
In the USA a more progressive depreciation method is in use. Modified Acceleration Cost Recovery System (MACRS) allows a more rapid depreciation than the linear method. After some years of depreciations, however, the linear method will return a larger annual depreciation. MACRS then switches to linear depreciation. MACRS is thus a mix of declining balance and straight-line (linear) depreciation.
The assets are divided into different groups depending on their usage and life expectancy. Here are some examples:
When the straight-line method eventually returns a higher depreciation than the declining balance depreciation, a company can change to the straight-line method at the year that optimizes the depreciation deductions.
Figure 9.17 MACRS applicable percentage for various classes.
Example 9.7
An asset has an initial cost of 800,000.
a) Calculate yearly depreciation when the asset is written down linearly (straight line) over five years.
b) Calculate the depreciation for the five first years when using the declining balance method with a 30 per cent depreciation rate.
c) Calculate yearly depreciation using five-year MACRS.
We have entered the numbers in the spreadsheet in figure 9.18.
a) Straight-line depreciation is calculated by using the function “=SNL($B$3;0; $B$4)” in cells B9:B13. It can also be found as initial cost divided by the number of years (“=$B$3/$B$4”).
b) Declining balance depreciation is calculated by using the DDB function in Excel.
c) Depreciation according to MACRS, class five years, is computed in cells E9:E14. Depreciation rate is multiplied by initial cost. In cell E10 you will for instance find the formula “=D10*$B$3” which returns 0.32 · 800,000 = 256,000.
Figure 9.18 Depreciation using different methods.
Tax on capital gains
When an asset is sold, most often there will be a difference between the sales price and the book value of the asset. A positive difference (sales price > book value) is a capital gain that will be taxed. A negative difference (sales price < book value) is a capital loss that can be deducted from the profit.
Consider a truck that has a book value of 200,000 and sold for 600,000. In some tax regimes, and for the sake of illustration here, we consider the whole total sales price as a gain and thus taxable at the prevailing tax rate of 28 per cent. This gives a tax charge of 600,000 · 0.28 = 168,000. However, we can keep the residual value (book value) of 200,000 as a basis for future depreciation and therefore lower taxes going forward. If the depreciation charge is 20 per cent with the declining balance method, we get the following lowering of taxes (“tax shield”) next year: 200,000 · 0.2 · 0.28 = 11,200.
The present value of the future tax shield is the sum of an infinite geometrical series with a growth factor of g = –0.2. If the cost of capital is 12 per cent, the net present value of the capital gains tax is:
This is the maximum tax on the capital gain, but obviously not optimal in relation to the tax code. If you pay tax on the capital gain only, the tax becomes (600,000 – 200,000) · 0.28 = 112,000. This is less than the net present value of the capital gains tax above (133,000), but it can still be unprofitable compared to what the tax code allows.
In the Norwegian tax code (see box above) there is an opening for writing off the entire sales value from the same asset category (within (a)–(i) in the box above).
In our example the total book value for the asset category is reduced by 600,000, of which 200,000 represents the truck. The book value of the remaining assets in this category is then 400,000 less than it would have been if the truck had been sold at book value. This implies some 400,000 less of future depreciation for the remaining assets in this category, and thus a smaller tax shield. It is this smaller tax shield that represents the capital gains tax. More specifically, the net present value of the tax shield is:
Consequently, this approach is the more profitable one.
We conclude this chapter by showing a complete investment analysis.
Example 9.8
The company Floyd Patterson Ltd is considering production and sales of canoes for the next five years. A consultant was hired to research the market outlook, a job that had a cost of 50,000. The research showed a huge market potential for canoes in the next five years.
New machinery will cost 8 million. This will be financed by a 2 million serial loan at a 7 per cent annual interest rate. The loan is amortized (principal and interest paid) at the end of each year of the five years the project runs.
The machinery is depreciated by 20 per cent annually using the declining balance method. At the end of year 5 the salvage value is 3 million. Any capital gain is taxed the same year.
The company has some vacant space for production at its facility. This can be let out to external tenants for 240,000 per year if the company decides not to use it for the project. Any rent is adjusted by 4 per cent annually as of year 2.
Projected sales in the first year are 3,000 canoes. From year 2 onwards, the sales volume is expected to level out at 4,000 per year. Price per canoe is set at 3,500 with variable cost of 1,700 per unit. Fixed costs, i.e. costs that are unrelated to the production volume, are estimated at 2,800,000. Necessary working capital is estimated to 100,000 in cash, 1,200,000 in accounts receivable and 800,000 in inventory as of and including year 1. Short-term debt is estimated at 700,000 in year 1. Sales price and costs are expected to follow the projected overall inflation rate of 4 per cent per year from year 2. Current assets and short-term debt are expected to increase by the same percentage. The tax rate is 28 per cent.
Use a cost of equity of 12 per cent to find the profitability of the canoe project.
Let us plug in all the data in a spreadsheet.
Figure 9.19 Input in spreadsheet.
We use the net present value method to evaluate the project. We therefore have to find the cash flows from the project. We start with the taxable net profit, as shown in figure 9.20. Here we also show how to compute interest, depreciation and working capital.
• Depreciation and book values are calculated in cells C21:G22. In cell D22 depreciation for year 2 is “=$E$6*D21” = 0.2 · 6,400,000 = 1,280,000. The taxable gain is shown in cell G23 as sales price minus book value = 3,000,000 – (3,276,800 – 655,360) = 378,560.
• In cells C25:G27 we have interest, amortization and loan balance. Amortization (principal) is fixed at 400,000 per year (2,000,000/5). Interest costs are 7 per cent of loan balance.
• In cell C29 we have net working capital at the end of year 1. Cash + accounts receivable + inventory – short-term debt = 100,000 + 1,200,000 + 800,000 – 700,000 = 1,400,000. As of year 2, net working capital is expected to be last year’s times the inflation rate (4 per cent). At the end of year 5, the project is terminated and all current assets used (current assets = 0).
• In the area C32:G38 we have the inputs for estimation of taxable profit. In the first year, income in cell C33 is the number of canoes sold times the price per unit “=C32*$C10” = 3,000 · 3,500 = 10,500,000. In the following year, income is adjusted by the inflation rate (4 per cent and a higher number of canoes sold, 4,000). Income in year 2 thus becomes 4,000 · 3,500 · (1 + 0.04) = 14,560,000. For the remaining years we just add the inflation rate to the sales. In cell E33 we have “=D33*(1+$E$4)” = 14,560,000 · (1 + 0.04) = 15,142,400.
• Variable costs are calculated in cell C34:G34 in the same way as income, with cost per unit in C11 times the number of units sold (C32:G32).
• Fixed costs for year 1 are in cell C35; 2,800,000. For the remaining four years we have to adjust for inflation. For year 3, we adjust year 2’s fixed costs by the inflation rate “=D35 * (1+$E$4)” = 2,912,000 · (1 + 0.04) = 3,028,480.
· In cell C36:G37 we have depreciation and interest computed in C22:G22 and C26:G26.
• Taxable profit is the net sum of income and costs and calculated in cells C38:G38 for every year.
Figure 9.20 Finding the taxable income.
Cash flows are different from profit. We therefore have to adjust the profits and loss projections above in order to compute the cash flows to find the net present value of the project, as shown in figure 9.21:
• The costs associated with the marketing survey (50,000) are sunk and should therefore not be included in the analysis.
• In cell B43 and G43 in the next spreadsheet, we have entered the cash from investing in (minus) and disposal of (positive) the machinery.
• Cash inflows in cells C45:G45 are the same as the revenues in cell C33:G33. In cells C46:G46 we have the total costs per year (variable and fixed).
• In cell B44 we have the loan (2,000,000) which will be a cash inflow in year 0. Interest and principal that were estimated in cells C26:G27, are transmitted to cells C47:G48.
• In cells C49:G49 we have the payable tax per year. This is calculated as 28 per cent of taxable income (cells C38:G38). In cell G50 we have the tax on the capital gain from the sale of the machinery (“= –G23*$E$11” = –378,560 · 0.28 = –105,997).
• An increase in current assets represents cash outflows. In cells C51:F51 these are shown for years 1–4. The first year current assets increase by 1,400,000 – 0 = 1,400,000. Next year they increase by 1,456,000 – 1,400,000 = 56,000 and so forth. See also cells C29:G29. In the final year the project closes and current assets are used. This implies that current assets are reduced by 1,574,810 during the final year (cell G51). This is when the working capital is returned to the company.
· We have to subtract a potential rent of 240,000 per year. This is the amount that the company could have received if it had rented out the vacant space that the production takes up. This is entered in cells C52:G52 as loss of revenue. Rent is regulated at 4 per cent annually as of year 2 in cells D52:G52.
• In cells B54:G54 we estimate net cash flow for each year by summarizing each column from row 43 to row 52.
Figure 9.21 Finding the cash flows and net present value of the project.
The cash flows in cells B54:G54 are after cost of debt and therefore belong to the equity. In order to calculate the net present value we need the cost of equity. We have been given a rate of 12 per cent and are then ready to compute the net present value. Using the NPV formula in Excel in cell B57, we get 4,800,143 (from “= NPV(B54:G54)”). Compared to investing the 6,000,000 at 12 per cent per annum, with similar risk, this project returns 4,800,143 more. It is thus very profitable.
We can compare this to the cost of capital by computing the internal rate of return. In cell B58 we have used the function “=IRR(B54:G54)” to find the internal rate of return of 31.8 per cent. Since we have no change of sign from year 1 to year 5, we know this is the only solution. (When the sign changes more than once, the IRR may not be defined.)
We can make the calculation before debt and interest/principal payments. Then we would get the cash flow to the firm. In that case we have to use the firm’s cost of capital (WACC) to find the NPV.
Problems
9-1. X-treme Sports Corp. applies an 18 per cent nominal cost of capital. Management considers a project for the production and marketing of a new product that is expected to return the following net cash flows in today’s value: (–35, 8, 11, 13, 12, 6).
a. Compute the net present value when the inflation rate is 7 per cent.
b. How low must the inflation rate be in order for the project to be profitable?
9-2. Leon Spinks Inc. uses a nominal cost of capital of 15 per cent. The company considers a new project that is expected to produce the following cash flows (after tax) in today’s value (i.e. real numbers): (–500, 100, 100, 150, 150, 50). The investment will be depreciated linearly over the project period. The tax rate is 40 per cent and the inflation rate is 5 per cent. Estimate the net present value of the project.
9-3. An American company considers investing 2,000,000 in production machinery for a six-year project. Expected cash flows are 1,000,000 in the first year, 2,000,000 in the second year and 3,000,000 in the third, fourth and fifth year. Cost of raw materials is expected to represent 75 per cent of the income from sales, while other costs are expected to be 7 per cent the first year, then 5 per cent of sales. Working capital is 200,000 in the first year, increasing to 100,000 the next year, and staying there for the remaining years. The company uses the “modified accelerated cost recovery system”, MACRS, five-year class. At the end of the project (end of year 6), the machinery can be sold for 300,000. Assume that the book gain from sales is taxed the same year. Also assume that any loss can be netted against other income in the company so that the tax credit from this will occur the same year. Company tax rate is 36 per cent. Apply an after-tax cost of capital of 10 per cent to estimate the net present value of the project. Also estimate the net present value when using the declining balance method with a 30 per cent depreciation rate.
9-4. A company considers investing 70,000 in a five-year project for the production and marketing of a new product. The equipment is to be depreciated by 20 per cent annually using the declining balance method over the five-year period. At the end of year 5, it can be sold for 20,000 (nominal value) and the amount can be written off against the asset category. So far the company has spent 1,000 on a marketing survey. Estimated sales and costs per year are 50,000 and 25,000 respectively (in real prices). Working capital is expected to equal short-term debt. Inflation rate is expected to be 4 per cent in the next five years. The company applies a nominal after-tax cost of capital of 18 per cent. Use a tax rate of 28 per cent and find the net present value of the project.
9-5. A company invests 150,000 in new machinery for a project that is expected to last seven years. It will be partly financed by a loan of 50,000. The interest rate is 9 per cent annually and the principal is to be paid back as a fixed payment (serial loan) at the end of each year. The machinery is depreciated by 25 per cent annually using the declining balance method. At the end of year 7, the machinery is expected to be worthless. Assume that the depreciation will continue for ever. The tax rate is 28 per cent and the cash inflows and outflows from operations are shown below. Estimate return on equity (after tax) and return on total capital.
9-6. A company wants to produce tennis rackets. A machine costs 1,000,000 and is expected to last eight years and it is depreciated linearly (straight-line method). The machine is expected to produce 5,000 rackets per year. Price per racket is 50, and is expected to increase by 8 per cent per year. The materials cost 20 per racket, and material costs are expected to increase by 4 per cent per year. There is a need for three workers, each costing 25,000 per year. Their wages are expected to rise by 5 per cent per annum. The real discount rate is 4 per cent, expected inflation rate is 5 per cent, and the tax rate 28 per cent. Calculate the NPV using first nominal values and then real values.
9-7. A company wants to invest in a project for manufacture and sales of tennis balls. The project requires an investment of 20 million. After five years the project will terminate, and the machine will be sold for 2 million one year later. The book value of the machine will be depreciated linearly over the five years. Expenses are stable at 60 per cent of revenues. Working capital will increase with next year’s sales as inventory requires capital and accounts receivable to build up. The working capital ratio to revenues is 3 per cent (due to higher inventory and accounts receivable as sales increase). The company expects to sell 10 million tennis balls a year at a price of 1.5 per ball. Inflation will increase by 4 per cent and sales and expenses will increase by the same amount as the inflation rate. The tax rate is 28 per cent. The company uses a cost of capital of 10 per cent after tax. Find the net present value.