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Chapter 10. Tool Kit for The Basics of Capital Budgeting: Evaluating Cash Flows
In this file we use Excel to do most of the calculations explained in Chapter 10. First, we analyze Projects S and L, whose cash flows are shown immediately below in both tabular and a time line formats. Spreadsheet analyses can be set up vertically, in a table with columns, or horizontally, using time lines. For short problems, with just a few years, we generally use the time line format because rows can be added and we can set the problem up as a series of income statements. For long problems, it is often more convenient to use a tabular layout. Expected after-tax net cash flows (CFt) Project S Project L ($1,000) ($1,000) 500 100 400 300 300 400 100 600 Project S 0 (1,000) 1 500 2 400 Project L 0 (1,000) 1 100 2 300 3 400 4 600 3 300 4 100
Year (t) 0 1 2 3 4
Capital Budgeting Decision Criteria
Here are the five key methods used to evaluate projects: (1) payback period, (2) discounted payback period, (3) net present value, (4) internal rate of return, and (5) modified internal rate of return. Using these criteria, financial 'analysts seek to identify those projects that will lead to the maximization of the firm's stock price. Payback Period The payback period is defined as the expected number of years required to recover the investment, and it was the first formal method used to evaluate capital budgeting projects. First, we identify the year in which the cumulative cash inflows exceed the initial cash outflows. That is the payback year. Then we take the previous year and add to it unrecovered balance at the end of that year divided by the following year's cash flow. Generally speaking, the shorter the payback period, the better the investment. Project S Time period: Cash flow: Cumulative cash flow: 0 (1,000) (1,000) FALSE 0.00 2.33 2.33 1 500 (500) FALSE 0.00 2 400 (100) FALSE 0.00 3 300 200 TRUE 2.33 4 100 300 FALSE Use Logical "AND" to determine 0.00 the first positive cumulative CF. Use Logical IF to find the Payback. Use Statistical Max function to display payback.
Payback: Alternative calculation: Project L Time period: Cash flow: Cumulative cash flow: Payback: 0 (1,000) (1,000)
1 100 (900) 3.33
2 300 (600)
3 400 (200)
4 600 400
Uses IF statement.
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A B C D E F G H I J Discounted Payback Period Discounted payback period uses the project's cost of capital to discount the expected cash flows. The calculation of discounted payback period is identical to the calculation of regular payback period, except you must base the calculation on a new row of discounted cash flows. Note that both projects have a cost of capital of 10%. WACC = Project S Time period: Cash flow: Disc. cash flow: Disc. cum. cash flow: Discounted Payback: Project L Time period: Cash flow: Disc. cash flow: Disc. cum. cash flow: Discounted Payback: 0 (1,000) (1,000) (1,000) 3.88 1 100 91 (909) 2 300 248 (661) 3 400 301 (361) 4 600 410 49 0 (1,000) (1,000) (1,000) 2.95 1 500 455 (545) 2 400 331 (215) 3 300 225 11 4 100 68 79 10%
Cash Flows Discounted back at 10%.
Uses IF statement.
Uses IF statement.
The inherent problem with both paybacks is that they ignore cash flows that occur after the payback period mark. While the discounted method accounts for timing issues (to some extent), it still falls short of fully analyzing projects. However, all else equal, these two methods do provide some information about projects' liquidity and risk. Net Present Value (NPV) To calculate the NPV, we find the present value of the individual cash flows and find the sum of those discounted cash flows. This value represents the value the project add to shareholder wealth. WACC = Project S Time period: Cash flow: Disc. cash flow: NPV(S) = Project L Time period: Cash flow: Disc. cash flow: NPV(L) = $49.18 0 (1,000) (1,000) 1 100 91 2 300 248 $ 3 400 301 49.18 4 600 410 = Uses NPV function. $78.82 0 (1,000) (1,000) = Sum disc. CF's. 1 500 455 2 400 331 or 3 300 225 $78.82 4 100 68 5 10%
= Uses NPV function.
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The NPV method of capital budgeting dictates that all independent projects that have positive NPV should accepted. The rationale behind that assertion arises from the idea that all such projects add wealth, and that should be the overall goal of the manager in all respects. If strictly using the NPV method to evaluate two mutually exclusive projects, you would want to accept the project that adds the most value (i.e. the project with the higher NPV). Hence, if considering the above two projects, you would accept both projects if they are independent, and you would only accept Project S if they are mutually exclusive.
Internal Rate of Return (IRR) The internal rate of return is defined as the discount rate that equates the present value of a project's cash inflows to its outflows. In other words, the internal rate of return is the interest rate that forces NPV to zero. The calculation for IRR can be tedious, but Excel provides an IRR function that merely requires you to access the function and enter the array of cash flows. The IRR's for Project S and L are shown below, along with the data entry for Project S. Expected after-tax net cash flows (CFt) Project S Project L ($1,000) ($1,000) 500 100 400 300 300 400 100 600
Year (t) 0 1 2 3 4
IRR S = IRR L =
14.49% 11.79%
The IRR function assumes payments occur at end of periods, so that function does not have to be adjusted.
Notice that for IRR you must specify all cash flows, including the time zero cash flow. This is in contrast to the NPV function, in which you specify only the future cash flows.
The IRR method of capital budgeting maintains that projects should be accepted if their IRR is greater than the cost of capital. Strict adherence to the IRR method would further dictate that mutually exclusive projects should be chosen on the basis of the greatest IRR. In this scenario, both projects have IRR's that exceed the cost of capital (10%) and both should be accepted, if they are independent. If, however, the projects are mutually exclusive, we would chose Project S. Recall, that this was our determination using the NPV method as well. The question that naturally arises is whether or not the NPV and IRR methods will always agree. When dealing with independent projects, the NPV and IRR methods will always yield the same accept/reject result. 'However, in the case of mutually exclusive projects, NPV and IRR can give conflicting results. One shortcoming of the internal rate of return is that it assumes that cash flows received are reinvested at the project's internal rate of return, which is not usually true. The nature of the congruence of the NPV and IRR methods is further detailed in a latter section of this model.
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Because of the mathematics involved, it is possible for some (but not all) projects that have more than one change of signs in the set of cash flows to have more than one IRR. If you attempted to find the IRR with such a project using a financial calculator, you would get an error message. The HP-10B says "Error - Soln", the HP-17B says '"Many/No Solutions, and the HP12C says Error 3; Key in Guess" when such a project is evaluated. The procedure for correcting the problem isto store in a guess for the IRR, and then the calculator will report the IRR that is closest to your guess. You can then use a different "guess" value, and you should be able to find the other IRR. However, the nature of the mathematics creates a scenario in which one IRR is quite extraordinary (often, a few hundred percent). Consider the case of Project M. Project M: 0 (1.6) 1 10 2 (10)
We will solve this IRR twice, the first time using the default guess of 10%, and the second time we will enter a guess of 300%. Notice, that the first IRR calculation is exactly as it was above.
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25.0%
400%
The two solutions to this problem tell us that this project will have a positive NPV for all costs of capital between '25% and 400%. We illustrate this point by creating a data table and a graph of the project NPVs. Project M: r = NPV = 25.0% 0.00 0 (1.6) 1 10 2 (10)
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B NPV $0.0 (1.60) 0.00 0.62 0.85 0.90 0.87 0.80 0.71 0.62 0.53 0.44 0.36 0.28 0.20 0.13 0.06 0.00 (0.06) (0.11) (0.16) (0.21) (0.26) (0.30)
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Multiple Rates of Return
$1.50 $1.00
Max.
$0.50 $0.00 -100% -$0.50 -$1.00 -$1.50 -$2.00 0% 100% 200% 300% 400% 500%
NPV Profiles NPV profiles graph the relationship between projects' NPVs and the cost of capital. To create NPV profiles for Projects S and L, we create data tables of NPV at different costs of capital. Net Cash Flows Project S Project L -$1,000 -$1,000 $500 $100 $400 $300 $300 $400 $100 $600
Year 0 1 2 3 4
WACC = NPV = IRR = Crossover =
10.0% Project S $78.82 14.49% 7.17%
Project L $49.18 11.79%
Project S's NPV Profile $400 $300 $200 NPV $100 $0
Both Projects' Profiles
$600
NPVs
Accept Reject
NPV
Conflict
No conflict
$400 NPVL $200 $0 -$200
-$100 IRR = 14.49% S -$200 0% 5% 10% WACC 15% 20% 25%
Crossover
= 7.17%
-$400 0% 5% 10% 15% 20% 25% WACC
Points about the graphs: 1. In Panel a, we see that if WACC < IRR, then NPV > 0, and vice versa. 2. Thus, for "normal and independent" projects, there can be no conflict between NPV and IRR rankings. 3. However, if we have mutually exclusive projects, conflicts can occur. In Panel b, we see that IRRS is always greater than IRRL, but if WACC < 11.56%, then IRRL > IRRS, in which case a conflict occurs. 4. Summary: a. For normal, independent projects, conflicts can never occur, so either method can be used.
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B C D E F G H b. For mutually exclusive projects, if WACC > Crossover, no conflict, but if WACC < Crossover, then there will be a conflict between NPV and IRR.
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Previously, we had discussed that in some instances the NPV and IRR methods can give conflicting results. First, we should attempt to define what we see in this graph. Notice, that the two project profiles (S and L) intersect the x-axis at costs of capital of 14% and 12%, respectively. Not coincidently, those are the IRR's of the projects. If we think about the definition of IRR, we remember that the internal rate of return is the cost of capital at which a project will have an NPV of zero. Looking at our graph, it is a logical conclusion that the IRR of a project is defined as the point at which its profile intersects the x-axis.
Looking further at the NPV profiles, we see that the two project profiles intersect at a point we shall call the crossover point. We observe that at costs of capital greater than the crossover point, the project with the greater IRR (Project S, in this case) also has the greater NPV. But at costs of capital less than the crossover point, the project with the lesser IRR has the greater NPV. This relationship is the source of discrepancy between the NPV and IRR methods. By looking at the graph, we see that the crossover appears to occur at approximately 7%. Luckily, there is a more precise way of determining crossover. To find crossover, we will find the difference between the two projects cash flows in each year, and then find the IRR of this series of differential cash flows. Expected after-tax net cash flows (CFt) Year (t) 0 1 2 3 4 Project S ($1,000) 500 400 300 100 Project L ($1,000) 100 300 400 600
Cash flow differential 0 400 100 (100) (500)
Alternative: Use Tools > Goal Seek to find WACC when NPV(S) = NPV(L). Set up a table to show the difference in NPV's, which we want to be zero. The following will do it, getting WACC = 7.17%. Look at B57 for the answer, then restore B57 to 10%. NPV S = $ 78.82 NPV L = $ 49.18 S-L= $ 29.64
IRR =
Crossover rate =
7.17%
G277
The intuition behind the relationship between the NPV profile and the crossover rate is as follows: (1) Distant cash flows are heavily penalized by high discount rates--the denominator is (1+r)t, and it increases geometrically, hence gets very large at high values of t. (2) Long-term projects like L have most of their cash flows coming in the later years, when the discount penalty is largest, hence they are most severely impacted by high capital costs. (3) 'Therefore, Project L's NPV profile is steeper than that of S. (4) Since the two profiles have different slopes, they cross one another. Modified Internal Rate of Return (MIRR) The modified internal rate of return is the discount rate that causes a project's cost (or cash outflows) to equal the 'present value of the project's terminal value. The terminal value is defined as the sum of the future values of the 'project's cash inflows, compounded at the project's cost of capital. To find MIRR, calculate the PV of the outflows 'and the FV of the inflows, and then find the rate that equates the two. Or, you can solve using the MIRR function.
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PV:
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Terminal Value:
4 600 440.0 363.0 133.1 1,536.1
The advantage of using the MIRR, relative to the IRR, is that the MIRR assumes that cash flows received are reinvested at the cost of capital, not the IRR. Since reinvestment at the cost of capital is more likely, the MIRR is a 'better indicator of a project's profitability. Moreover, it solves the multiple IRR problem, as a set of cash flows can have but one MIRR .
Note that if negative cash flows occur in years beyond Year 1, those cash flows would be discounted at the cost of capital and added to the Year 0 cost to find the total PV of costs. If both positive and negative flows occurred in some year, the negative flow should be discounted, and the positive one compounded, rather than just dealing with the net cash flow. This makes a difference. Also note that Excel's MIRR function allows for discounting and reinvestment to occur at different rates. Generally, MIRR is defined as reinvestment at the WACC, though Excel allows the calculation of a special MIRR where reinvestment occurs at a different rate than WACC.
Finally, it is stated in the text, when the IRR versus the NPV is discussed, that the NPV is superior because (1) the NPV assumes that cash flows are reinvested at the cost of capital whereas the IRR assumes reinvestment at the IRR, and (2) it is more likely, in a competitive world, that the actual reinvestment rate is more likely to be the cost of capital than the IRR, especially if the IRR is quite high. The MIRR setup can be used to prove that NPV indeed does assume reinvestment at the WACC, and IRR at the IRR.
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Reinvestment at WACC = 10%
PV outflows PV of TV NPV
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Terminal Value:
$
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Thus, we see that the NPV is consistent with reinvestment at WACC.
Now repeat the process using the IRR, which is G118 as the discount rate. Project S IRR = 14.49% 0 (1,000) 1 500 2 400 3 300 4 100 343.5 524.3 750.3 1,718.1
Reinvestment at IRR = 14.49%
PV outflows PV of TV NPV
-$1,000.00 $1,000.00
Terminal Value:
$0.00 Thus, if compounding is at the IRR, NPV is zero. Since the definition of IRR is the rate at which NPV = 0, this demonstrates that the IRR assumes reinvestment at the IRR.
Profitability Index (PI) The profitability index is the present value of all future cash flows divided by the intial cost. It measures the PV per dollar of investment. For project S: PI(S) = PV of future cash flows PI(S) = $ 150,216.88 PI(S) = 1.138006667 For project L: PI(L) = PV of future cash flows PI(L) = $ 1,049.18 PI(L) = 1.049
÷ ÷
Initial cost
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PROJECTS WITH UNEQUAL LIVES If two mutually exclusive projects have different lives, and if the projects can be repeated, then it is necessary to deal explicitly with those unequal lives. We use the replacement chain (or common life) approach. This procedure compares projects of unequal lives by equalizing their lives by assuming that each project can be repeated as many times as necessary to reach a common life span. The NPVs over this life span are then compared, and the project with the higher common life NPV is chosen. To illustrate, suppose a firm is considering two mutually exclusive projects, either a conveyor system (Project C) or a fleet of forklift trucks (Project F) for moving materials. The firm's cost of capital is 12%. The cash flow timelines are shown below, 'along with the NPV and IRR for each project. Project C WACC: 11.5% End of Period: 0 1 2 3 4 5 6
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Initially, it would appear that Project C is the better investment, based upon its higher NPV. However, if the firm chooses Project F, it would have the opportunity to make the same investment three years from now. Therefore, we must reevaluate Project F 'using extended common life of 6 years. The time lines are shown below. Note that only F's is changed. Common Life Approach Project C End of Period: 0 ($40,000) NPV IRR Project F 0 ($20,000) ($20,000) NPV IRR 1 $7,000 $7,000 $9,281 25.2% 2 $13,000 $13,000 3 $12,000 ($20,000) ($8,000) 4 $7,000 $7,000 5 $13,000 $13,000 6 $12,000 $12,000 1 $8,000 $7,165 17.5% 2 $14,000 3 $13,000 4 $12,000 5 $11,000 6 $10,000
On the basis of this extended analysis, it is clear that Project F is the better of the two investments (with both the NPV and IRR methods). Equivalent Annual Annuity (EAA) Approach (See the Chapter 10 Web Extension for details.) Here are the steps in the EAA approach. 1. Find the NPV of each project over its initial life (we already did this in our previous analysis). NPVC= 7,165 NPVF= 5,391 2. Convert the NPV into an annuity payment with a life equal to the life of the project. EEAC= 1,718 Note: we used the Function Wizard for the PMT function. EEAF= 2,225 Project F has a higher EEA, so it is a better project. ECONOMIC LIFE VS. PHYSICAL LIFE
Sometimes an asset has a physical life that is greater than its economic life. Consider the following asset which has a physical life of three years. During its life, the asset will generate operating cash flows. However, the project could be terminated and the asset sold at the end of any year. The following table shows the operating cash flows and the salvage value for each year-- all values are shown on an after-tax basis.
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Salvage Operating Cash Flow Value Year 455 ($4,800) $4,800 0 456 1 $2,000 $3,000 457 2 $2,000 $1,650 458 3 $1,750 $0 459 460 461 The cost of capital is 10%. If the asset is operated for the entire three years of its life, its NPV is: 462 PV of PV of Operating Salvage + 3-Year NPV = Intial Cost + Cash Flow Value 463 464 = ($4,800.00) + $4,785.88 + $0.00 3-Year NPV = ($14.12) 465 466 467 The asset has a negative NPV if it is kept for three years. But even though the asset will last three years, it might be 468 better to operate the asset for either one or two years, and then salvage it. 469 PV of PV of Operating Salvage 2-Year NPV = Intial Cost + + Cash Flow Value 470 = ($4,800.00) + $3,471.07 + $1,363.64 471 2-Year NPV = $34.71 472 473 PV of PV of Operating Salvage + 1-Year NPV = Intial Cost + Value Cash Flow 474 475 = ($4,800.00) + $1,818.18 + $2,727.27 1-Year NPV = ($254.55) 476
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Click fx > Logical > AND > OK to get dialog box. Then specify you want TRUE if cumulative CF > 0 but the previous CF < 0. There will be one TRUE. Click fx > Logical > IF > OK. Specify that if true, the payback is the previous year plus a fraction, if false, then 0. Click fx > Statistical > MAX > OK > and specify range to find Payback.
K L M N O P Q 52 53 54 55 56 57 58 59 60 61 ash Flows Discounted back at 10%. 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Notice that the NPV function isn't really a Net present value. Instead, 88 it is the present value of future cash flows. Thus, you specify only the 89 future cash flows in the NPV function. To find the true NPV, you 90 must add the time zero cash flow to the result of the NPV function. 91 92 93 94 95 96 97 98
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K 150 151 ge of signs in the set of cash 152 tor, you would get an error 153 Key in Guess" when such a 154 lculator will report the IRR that 155 RR. However, the nature of the 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188
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K L M 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 Data Table used to make graph: 227 Project NPVs 228 S L 229 WACC $78.82 $49.18 230 $300.00 $400.00 0% 231 5% $180.42 $206.50 232 7.17% $134.40 $134.40 233 10% $78.82 $49.18 234 11.79% $46.10 $0.00 235 14.49% $0.00 -$68.02 236 15.0% -$8.33 -$80.14 237 20% -$83.72 -$187.50 238 25% -$149.44 -$277.44 239 240 241 242 243 244 245 246 247 248 249 250
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