The required return is what investors actually demand on the issue, and it will fluctuate through time. The coupon rate and required return are equal only if the bond sells for exactly at par. Some investors have obligations that are denominated in dollars; i. Their primary concern is that an investment provide the needed nominal dollar amounts.
Pension funds, for example, often must plan for pension payments many years in the future. If those payments are fixed in dollar terms, then it is the nominal return on an investment that is important. Companies pay to have their bonds rated simply because unrated bonds can be difficult to sell; many large investors are prohibited from investing in unrated issues. Treasury bonds have no credit risk since it is backed by the U.
The term structure is based on pure discount bonds. The yield curve is based on coupon-bearing issues. Bond ratings have a subjective factor to them. Split ratings reflect a difference of opinion among credit agencies. As a general constitutional principle, the federal government cannot tax the states without their consent if doing so would interfere with state government functions.
At one time, this principle was thought to provide for the tax-exempt status of municipal interest payments. However, modern court rulings make it clear that Congress can revoke the municipal exemption, so the only basis now appears to be historical precedent.
One measure of liquidity is the bid-ask spread. Liquid instruments have relatively small spreads. Looking at Figure 7. Generally, liquidity declines after a bond is issued. Some older bonds, including some of the callable issues, have spreads as wide as six ticks.
Companies charge that bond rating agencies are pressuring them to pay for bond ratings. When a company pays for a rating, it has the opportunity to make its case for a particular rating.
With an unsolicited rating, the company has no input. A year bond looks like a share of preferred stock. In particular, it is a loan with a life that almost certainly exceeds the life of the lender, assuming that the lender is an individual.
With a junk bond, the credit risk can be so high that the borrower is almost certain to default, meaning that the creditors are very likely to end up as part owners of the business. The yield to maturity is the required rate of return on a bond expressed as a nominal annual interest rate.
For noncallable bonds, the yield to maturity and required rate of return are interchangeable terms. Unlike YTM and required return, the coupon rate is not a return used as the interest rate in bond cash flow valuation, but is a fixed percentage of par over the life of the bond used to set the coupon payment amount.
For the example given, the coupon rate on the bond is still 10 percent, and the YTM is 8 percent. Price and yield move in opposite directions; if interest rates rise, the price of the bond will fall.
This is because the fixed coupon payments determined by the fixed coupon rate are not as valuable when interest rates rise—hence, the price of the bond decreases. We will use this par value in all problems unless a different par value is explicitly stated.
The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this problem assumes an annual coupon. We will use this shorthand notation in remainder of the solutions key. Here we need to find the YTM of a bond.
First, we know the YTM has to be higher than the coupon rate since the bond is a discount bond. That still leaves a lot of interest rates to check. Here we need to find the coupon rate of the bond. To find the price of this bond, we need to realize that the maturity of the bond is 10 years. The bond was issued one year ago, with 11 years to maturity, so there are 10 years left on the bond.
Also, the coupons are semiannual, so we need to use the semiannual interest rate and the number of semiannual periods. Here we are finding the YTM of a semiannual coupon bond. This is a bond since the maturity is greater than 10 years. The coupon rate, located in the first column of the quote is 6. Here we are finding the YTM of semiannual coupon bonds for various maturity lengths.
Also, notice that the price of each bond when no time is left to maturity is the par value, even though the purchaser would receive the par value plus the coupon payment immediately. This is because we calculate the clean price of the bond. Any bond that sells at par has a YTM equal to the coupon rate.
Both bonds sell at par, so the initial YTM on both bonds is the coupon rate, 8 percent. The company should set the coupon rate on its new bonds equal to the required return. The required return can be observed in the market by finding the YTM on outstanding bonds of the company. Accrued interest is the coupon payment for the period times the fraction of the period that has passed since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six months is one-half of the annual coupon payment.
There are five months until the next coupon payment, so one month has passed since the last coupon payment. There are three months until the next coupon payment, so three months have passed since the last coupon payment. To find the number of years to maturity for the bond, we need to find the price of the bond. Since we already have the coupon rate, we can use the bond price equation, and solve for the number of years to maturity.
Treasury bond with a similar maturity. The column lists the spread in basis points. One basis point is one-hundredth of one percent, so basis points equals one percent. The spread for this bond is basis points, or 4. The bond price is the present value of the cash flows from a bond. The YTM is the interest rate used in valuing the cash flows from a bond. If the coupon rate is higher than the required return on a bond, the bond will sell at a premium, since it provides periodic income in the form of coupon payments in excess of that required by investors on other similar bonds.
If the coupon rate is lower than the required return on a bond, the bond will sell at a discount since it provides insufficient coupon payments compared to that required by investors on other similar bonds.
For premium bonds, the coupon rate exceeds the YTM; for discount bonds, the YTM exceeds the coupon rate, and for bonds selling at par, the YTM is equal to the coupon rate.
Current yield is defined as the annual coupon payment divided by the current bond price. For premium bonds, the current yield exceeds the YTM, for discount bonds the current yield is less than the YTM, and for bonds selling at par value, the current yield is equal to the YTM. In all cases, the current yield plus the expected one-period capital gains yield of the bond must be equal to the required return.
The price of a zero coupon bond is the PV of the par, so: a. Previous IRS regulations required a straight-line calculation of interest. The company will prefer straight-line methods when allowed because the valuable interest deductions occur earlier in the life of the bond.
The repayment of the coupon bond will be the par value plus the last coupon payment times the number of bonds issued. The total coupon payment for the coupon bonds will be the number bonds times the coupon payment. For the cash flow of the coupon bonds, we need to account for the tax deductibility of the interest payments.
To do this, we will multiply the total coupon payment times one minus the tax rate. For the zero coupon bonds, the first year interest payment is the difference in the price of the zero at the end of the year and the beginning of the year.
This is because of the tax deductibility of the imputed interest expense. That is, the company gets to write off the interest expense for the year even though the company did not have a cash flow for the interest expense.
During the life of the bond, the zero generates cash inflows to the firm in the form of the interest tax shield of debt. We should note an important point here: If you find the PV of the cash flows from the coupon bond and the zero coupon bond, they will be the same. This is because of the much larger repayment amount for the zeroes. We found the maturity of a bond in Problem However, in this case, the maturity is indeterminate.
A bond selling at par can have any length of maturity. In other words, when we solve the bond pricing equation as we did in Problem 22, the number of periods can be any positive number.
We first need to find the real interest rate on the savings. To find the capital gains yield and the current yield, we need to find the price of the bond. The rate of return you expect to earn if you purchase a bond and hold it until maturity is the YTM. To find our HPY, we need to find the price of the bond in two years. The price of any bond or financial instrument is the PV of the future cash flows. Even though Bond M makes different coupons payments, to find the price of the bond, we just find the PV of the cash flows.
To calculate this, we need to set up an equation with the callable bond equal to a weighted average of the noncallable bonds. We will invest X percent of our money in the first noncallable bond, which means our investment in Bond 3 the other noncallable bond will be 1 — X. This combination of bonds should have the same value as the callable bond, excluding the value of the call.
In general, this is not likely to happen, although it can and did. The reason this bond has a negative YTM is that it is a callable U. Treasury bond. Market participants know this. Given the high coupon rate of the bond, it is extremely likely to be called, which means the bondholder will not receive all the cash flows promised. A better measure of the return on a callable bond is the yield to call YTC.
The YTC calculation is the basically the same as the YTM calculation, but the number of periods is the number of periods until the call date.
If the YTC were calculated on this bond, it would be positive. To find the present value, we need to find the real weekly interest rate. To find the real return, we need to use the effective annual rates in the Fisher equation. The real cash flows are an ordinary annuity, discounted at the real interest rate. To answer this question, we need to find the monthly interest rate, which is the APR divided by We also must be careful to use the real interest rate.
The nominal monthly withdrawals will increase by the inflation rate each month. To find the nominal dollar amount of the last withdrawal, we can increase the real dollar withdrawal by the inflation rate. We can increase the real withdrawal by the effective annual inflation rate since we are only interested in the nominal amount of the last withdrawal.
Enter 16 7. Enter 20 4. Enter 29 3. If both bonds sell at par, the initial YTM on both bonds is the coupon rate, 8 percent. The company should set the coupon rate on its new bonds equal to the required return; the required return can be observed in the market by finding the YTM on outstanding bonds of the company.
The company will prefer straight-line method when allowed because the valuable interest deductions occur earlier in the life of the bond. Enter 8 5. The value of any investment depends on the present value of its cash flows; i. The cash flows from a share of stock are the dividends.
Investors believe the company will eventually start paying dividends or be sold to another company. In general, companies that need the cash will often forgo dividends since dividends are a cash expense. Young, growing companies with profitable investment opportunities are one example; another example is a company in financial distress.
This question is examined in depth in a later chapter. The general method for valuing a share of stock is to find the present value of all expected future dividends. The dividend growth model presented in the text is only valid i if dividends are expected to occur forever, that is, the stock provides dividends in perpetuity, and ii if a constant growth rate of dividends occurs forever.
A violation of the first assumption might be a company that is expected to cease operations and dissolve itself some finite number of years from now. The stock of such a company would be valued by applying the general method of valuation explained in this chapter.
This stock would also be valued by the general dividend valuation method explained in this chapter. The common stock probably has a higher price because the dividend can grow, whereas it is fixed on the preferred. However, the preferred is less risky because of the dividend and liquidation preference, so it is possible the preferred could be worth more, depending on the circumstances.
The two components are the dividend yield and the capital gains yield. For most companies, the capital gains yield is larger. This is easy to see for companies that pay no dividends. For companies that do pay dividends, the dividend yields are rarely over five percent and are often much less. If the dividend grows at a steady rate, so does the stock price. In other words, the dividend growth rate and the capital gains yield are the same.
In a corporate election, you can buy votes by buying shares , so money can be used to influence or even determine the outcome. Many would argue the same is true in political elections, but, in principle at least, no one has more than one vote.
Investors buy such stock because they want it, recognizing that the shares have no voting power. Presumably, investors pay a little less for such shares than they would otherwise. Presumably, the current stock value reflects the risk, timing and magnitude of all future cash flows, both short-term and long-term.
If this assumption is violated, the two-stage dividend growth model is not valid. In other words, the price calculated will not be correct. Depending on the stock, it may be more reasonable to assume that the dividends fall from the high growth rate to the low perpetual growth rate over a period of years, rather than in one year. So, if we know the stock price today, we can find the future value for any time in the future we want to calculate the stock price.
In this problem, we want to know the stock price in three years, and we have already calculated the stock price today. We need to find the required return of the stock. Using the constant growth model, we can solve the equation for R. The required return of a stock is made up of two parts: The dividend yield and the capital gains yield. The question asks for the dividend this year. The price of any financial instrument is the PV of the future cash flows.
The price a share of preferred stock is the dividend divided by the required return. This is the same equation as the constant growth model, with a dividend growth rate of zero percent. Remember, most preferred stock pays a fixed dividend, so the growth rate is zero.
This stock has a constant growth rate of dividends, but the required return changes twice. To find the value of the stock today, we will begin by finding the price of the stock at Year 6, when both the dividend growth rate and the required return are stable forever. The price of the stock in Year 6 will be the dividend in Year 7, divided by the required return minus the growth rate in dividends.
We need to find the price here since the required return changes at that time. Here we have a stock that pays no dividends for 10 years. Once the stock begins paying dividends, it will have a constant growth rate of dividends. We can use the constant growth model at that point. We simply discount the future stock price at the required return. The price of a stock is the PV of the future dividends.
This stock is paying four dividends, so the price of the stock is the PV of these dividends using the required return. With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the futures stock price, plus the PV of all dividends during the supernormal growth period.
Here we need to find the dividend next year for a stock experiencing supernormal growth. We know the stock price, the dividend growth rates, and the required return, but not the dividend. Now we need to find the equation for the stock price today. The constant growth model can be applied even if the dividends are declining by a constant percentage, just make sure to recognize the negative growth. We are given the stock price, the dividend growth rate, and the required return, and are asked to find the dividend.
The price of a share of preferred stock is the dividend payment divided by the required return. We know the dividend payment in Year 10, so we can find the price of the stock in Year 9, one year before the first dividend payment. We are asked to find the dividend yield and capital gains yield for each of the stocks. All of the stocks have a 15 percent required return, which is the sum of the dividend yield and the capital gains yield. To find the components of the total return, we need to find the stock price for each stock.
Using this stock price and the dividend, we can calculate the dividend yield. The capital gains yield for the stock will be the total return required return minus the dividend yield. High growth stocks have an appreciable capital gains component but a relatively small current income yield; conversely, mature, negative-growth stocks provide a high current income but also price depreciation over time.
We can then use this interest rate to find the equivalent annual dividend. In other words, when we receive the quarterly dividend, we reinvest it at the required return on the stock. So, the effective quarterly rate is: Effective quarterly rate: 1. This would assume the dividends increased each quarter, not each year. Here we have a stock with supernormal growth, but the dividend growth changes every year for the first four years.
We can find the price of the stock in Year 3 since the dividend growth rate is constant after the third dividend. The price of the stock in Year 3 will be the dividend in Year 4, divided by the required return minus the constant dividend growth rate. Here we want to find the required return that makes the PV of the dividends equal to the current stock price. To find the value of the stock with two-stage dividend growth, consider that the present value of the first t dividends is the present value of a growing annuity.
Additionally, to find the price of the stock, we need to add the present value of the stock price at time t. The discounted payback includes the effect of the relevant discount rate. If a project has a positive NPV for a certain discount rate, then it will also have a positive NPV for a zero discount rate; thus, the payback period must be less than the project life. If NPV is positive, then the present value of future cash inflows is greater than the initial investment cost; thus PI must be greater than 1.
Payback period is simply the accounting break-even point of a series of cash flows. To actually compute the payback period, it is assumed that any cash flow occurring during a given period is realized continuously throughout the period, and not at a single point in time. The payback is then the point in time for the series of cash flows when the initial cash outlays are fully recovered.
Given some predetermined cutoff for the payback period, the decision rule is to accept projects that payback before this cutoff, and reject projects that take longer to payback. The worst problem associated with payback period is that it ignores the time value of money.
In addition, the selection of a hurdle point for payback period is an arbitrary exercise that lacks any steadfast rule or method. The payback period is biased towards short-term projects; it fully ignores any cash flows that occur after the cutoff point.
Despite its shortcomings, payback is often used because 1 the analysis is straightforward and simple and 2 accounting numbers and estimates are readily available. Materiality consider- ations often warrant a payback analysis as sufficient; maintenance projects are another example where the detailed analysis of other methods is often not needed.
Since payback is biased towards liquidity, it may be a useful and appropriate analysis method for short-term projects where cash management is most important.
The discounted payback is calculated the same as is regular payback, with the exception that each cash flow in the series is first converted to its present value. Given some predetermined cutoff for the discounted payback period, the decision rule is to accept projects whose discounted cash flows payback before this cutoff period, and to reject all other projects. The primary disadvantage to using the discounted payback method is that it ignores all cash flows that occur after the cutoff date, thus biasing this criterion towards short-term projects.
As a result, the method may reject projects that in fact have positive NPVs, or it may accept projects with large future cash outlays resulting in negative NPVs.
In addition, the selection of a cutoff point is again an arbitrary exercise. Discounted payback is an improvement on regular payback because it takes into account the time value of money. For conventional cash flows and strictly positive discount rates, the discounted payback will always be greater than the regular payback period.
The average accounting return is interpreted as an average measure of the accounting perfor- mance of a project over time, computed as some average profit measure attributable to the project divided by some average balance sheet value for the project. This text computes AAR as average net income with respect to average total book value. Given some predetermined cutoff for AAR, the decision rule is to accept projects with an AAR in excess of the target measure, and reject all other projects.
AAR is not a measure of cash flows and market value, but a measure of financial statement accounts that often bear little resemblance to the relevant value of a project. In addition, the selection of a cutoff is arbitrary, and the time value of money is ignored. For a financial manager, both the reliance on accounting numbers rather than relevant market data and the exclusion of time value of money considerations are troubling.
Despite these problems, AAR continues to be used in practice because 1 the accounting information is usually available, 2 analysts often use accounting ratios to analyze firm performance, and 3 managerial compensation is often tied to the attainment of certain target accounting ratio goals. NPV specifically measures, after considering the time value of money, the net increase or decrease in firm wealth due to the project. NPV is superior to the other methods of analysis presented in the text because it has no serious flaws.
The method unambiguously ranks mutually exclusive projects, and can differentiate between projects of different scale and time horizon. The only drawback to NPV is that it relies on cash flow and discount rate values that are often estimates and not certain, but this is a problem shared by the other performance criteria as well.
IRR can thus be interpreted as a financial break-even rate of return; at the IRR discount rate, the net value of the project is zero. NPV is preferred in all situations to IRR; IRR can lead to ambiguous results if there are non-conventional cash flows, and it also ambiguously ranks some mutually exclusive projects. The profitability index is the present value of cash inflows relative to the project cost. The profitability index decision rule is to accept projects with a PI greater than one, and to reject projects with a PI less than one.
There are a number of reasons. Two of the most important have to do with transportation costs and exchange rates. Manufacturing in the U. It also reduces inventories because goods spend less time in transit. Higher labor costs tend to offset these savings to some degree, at least compared to other possible manufacturing locations.
Of great importance is the fact that manufacturing in the U. Since sales are in dollars, the net effect is to immunize profits to a large extent against fluctuations in exchange rates.
This issue is discussed in greater detail in the chapter on international finance. The single biggest difficulty, by far, is coming up with reliable cash flow estimates.
Determining an appropriate discount rate is also not a simple task. These issues are discussed in greater depth in the next several chapters. The payback approach is probably the simplest, followed by the AAR, but even these require revenue and cost projections. The discounted cash flow measures discounted payback, NPV, IRR, and profitability index are really only slightly more difficult in practice.
Yes, they are. Such entities generally need to allocate available capital efficiently, just as for-profits do. For example, charitable giving has real opportunity costs, but the benefits are generally hard to measure. To the extent that benefits are measurable, the question of an appropriate required return remains.
Payback rules are commonly used in such cases. The MIRR is calculated by finding the present value of all cash outflows, the future value of all cash inflows to the end of the project, and then calculating the IRR of the two cash flows. As a result, the cash flows have been discounted or compounded by one interest rate the required return , and then the interest rate between the two remaining cash flows is calculated.
As such, the MIRR is not a true interest rate. In contrast, consider the IRR. If you take the initial investment, and calculate the future value at the IRR, you can replicate the future cash flows of the project exactly.
The statement is incorrect. It is true that if you calculate the future value of all intermediate cash flows to the end of the project at the required return, then calculate the NPV of this future value and the initial investment, you will get the same NPV. However, NPV says nothing about reinvestment of intermediate cash flows. The NPV is the present value of the project cash flows. What is actually done with those cash flows once they are generated is not relevant.
Put differently, the value of a project depends on the cash flows generated by the project, not on the future value of those cash flows. One caveat: Our discussion here assumes that the cash flows are truly available once they are generated, meaning that it is up to firm management to decide what to do with the cash flows.
In certain cases, there may be a requirement that the cash flows be reinvested. It is true that if you calculate the future value of all intermediate cash flows to the end of the project at the IRR, then calculate the IRR of this future value and the initial investment, you will get the same IRR.
However, as in the previous question, what is done with the cash flows once they are generated does not affect the IRR. The IRR of the cash flows is 10 percent. Does the IRR change if the Year 1 cash flow is reinvested in the account, or if it is withdrawn and spent on pizza? Finally, consider the yield to maturity calculation on a bond. If you think about it, the YTM is the IRR on the bond, but no mention of a reinvestment assumption for the bond coupons is suggested.
The reason is that reinvestment is irrelevant to the YTM calculation; in the same way, reinvestment is irrelevant in the IRR calculation. Our caveat about blocked funds applies here as well. To calculate the payback period, we need to find the time that the project has recovered its initial investment. So, the payback period will be 2 years, plus what we still need to make divided by what we will make during the third year. The cash flows in this problem are an annuity, so the calculation is simpler.
Just divide the initial cost by the annual cash flow. This answer does not make sense since the cash flows stop after eight years, so again, we must conclude the payback period is never. When we use discounted payback, we need to find the value of all cash flows today. We know the payback period is between two and three years, so we subtract the discounted values of the Year 1 and Year 2 cash flows from the initial cost.
This is the numerator, which is the discounted amount we still need to make to recover our initial investment. We divide this amount by the discounted amount we will earn in Year 3 to get the fractional portion of the discounted payback. Our definition of AAR is the average net income divided by the average book value. We would be indifferent to the project if the required return was equal to the IRR of the project, since at that required return the NPV is zero. At a zero discount rate and only at a zero discount rate , the cash flows can be added together across time.
This will always be true for projects with conventional cash flows. Conventional cash flows are negative at the beginning of the project and positive throughout the rest of the project.
This may not be a correct decision; however, because the IRR criterion has a ranking problem for mutually exclusive projects. To find the crossover rate, we subtract the cash flows from one project from the cash flows of the other project. Here, we will subtract the cash flows for Project B from the cash flows of Project A. Once we find these differential cash flows, we find the IRR. We will subtract the cash flows from Project Y from the cash flows from Project X. It is irrelevant which cash flows we subtract from the other.
If we are evaluating whether or not to accept this project, we would not want to use the IRR to make our decision. The profitability index is defined as the PV of the cash inflows divided by the PV of the cash outflows. We would reject the project if the required return were 22 percent since the PI is less than one. The profitability index is the PV of the future cash flows divided by the initial investment.
Using the profitability index to compare mutually exclusive projects can be ambiguous when the magnitude of the cash flows for the two projects are of different scale.
In this problem, project I is roughly 3 times as large as project II and produces a larger NPV, yet the profit- ability index criterion implies that project II is more acceptable. In this instance, the NPV criteria implies that you should accept project A, while profitability index, payback period, discounted payback and IRR imply that you should accept project B.
The final decision should be based on the NPV since it does not have the ranking problem associated with the other capital budgeting techniques. Therefore, you should accept project A. The MIRR for the project with all three approaches is: Discounting approach: In the discounting approach, we find the value of all cash outflows to time 0, while any cash inflows remain at the time at which they occur. With different discounting and reinvestment rates, we need to make sure to use the appropriate interest rate.
The MIRR for the project with all three approaches is: Discounting approach: In the discounting approach, we find the value of all cash outflows to time 0 at the discount rate, while any cash inflows remain at the time at which they occur. Given the seven year payback, the worst case is that the payback occurs at the end of the seventh year.
Thus, the best-case NPV is infinite. Even with most computer spreadsheets, we have to do some trial and error. We would accept the project when the NPV is greater than zero. Here the cash inflows of the project go on forever, which is a perpetuity. Unlike ordinary perpetuity cash flows, the cash flows here grow at a constant rate forever, which is a growing perpetuity. If you remember back to the chapter on stock valuation, we presented a formula for valuing a stock with constant growth in dividends.
This formula is actually the formula for a growing perpetuity, so we can use it here. Here we want to know the minimum growth rate in cash flows necessary to accept the project.
The minimum growth rate is the growth rate at which we would have a zero NPV. Since the initial cash flow is positive and the remaining cash flows are negative, the decision rule for IRR in invalid in this case.
The NPV profile is upward sloping, indicating that the project is more valuable when the interest rate increases. The reason is that there is no real IRR for this set of cash flows. If you examine the IRR equation, what we are really doing is solving for the roots of the equation. Going back to high school algebra, in this problem we are solving a quadratic equation. First, we need to find the future value of the cash flows for the one year in which they are blocked by the government.
Since the cash inflows are blocked by the government, they are not available to the company for a period of one year. Thus, all we are doing is calculating the IRR based on when the cash flows actually occur for the company.
Calculator Solutions 7. Using trial and error, or a root solving calculator, the other IRR is — In this instance, the NPV criteria implies that you should accept project A, while payback period, discounted payback, profitability index, and IRR imply that you should accept project B. In this context, an opportunity cost refers to the value of an asset or other input that will be used in a project.
The relevant cost is what the asset or input is actually worth today, not, for example, what it cost to acquire. For tax purposes, a firm would choose MACRS because it provides for larger depreciation deductions earlier. These larger deductions reduce taxes, but have no other cash consequences. Notice that the choice between MACRS and straight-line is purely a time value issue; the total depreciation is the same, only the timing differs.
Current liabilities will all be paid, presumably. The cash portion of current assets will be retrieved. These effects tend to offset one another. The EAC approach is appropriate when comparing mutually exclusive projects with different lives that will be replaced when they wear out. This type of analysis is necessary so that the projects have a common life span over which they can be compared; in effect, each project is assumed to exist over an infinite horizon of N-year repeating projects.
Assuming that this type of analysis is valid implies that the project cash flows remain the same forever, thus ignoring the possible effects of, among other things: 1 inflation, 2 changing economic conditions, 3 the increasing unreliability of cash flow estimates that occur far into the future, and 4 the possible effects of future technology improvement that could alter the project cash flows.
Depreciation is a non-cash expense, but it is tax-deductible on the income statement. Thus depreciation causes taxes paid, an actual cash outflow, to be reduced by an amount equal to the depreciation tax shield tcD. A reduction in taxes that would otherwise be paid is the same thing as a cash inflow, so the effects of the depreciation tax shield must be added in to get the total incremental aftertax cash flows.
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