responses

Chapter

Tool Kit		Chapter 10					10/27/15
Basics of Capital Budgeting: Evaluating Cash Flows
10-1 An Overview of Capital Budgeting
Capital budgeting is the process of analyzing projects and deciding which ones to accept.
10-2 The First Step in Project Analysis
The capital budgeting process begins with estimating a project's expected cash flows. We explain this in the next chapter.
The next step is to put the estimated cash flows and other inputs (primarily the project's cost of capital) on a time line, as shown below. The figure below also reports evaluation measures, which we explain in the next sections.
Figure 10-1
Cash Flows and Selected Evaluation Measures for Projects S and L (Millions of Dollars)
Panel A: Inputs for Project Cash Flows and Cost of Capital, r
INPUTS:
r =	10%
	Initial Cost and Expected Cash Flows
Year	0	1	2	3	4
Project S	−$10,000	$5,300	$4,300	$1,874	$1,500
Project L	−$10,000	$1,900	$2,700	$2,345	$7,800
Panel B: Summary of Selected Evaluation Measures
		Project S	Project L
	Net present value, NPV	$804.38	$1,048.02
	Internal rate of return, IRR	14.69%	13.79%
	Modified IRR, MIRR	12.15%	10.19%
	Profitability index, PI	1.08	1.10
	Payback	2.21	3.39
	Discounted payback	3.21	3.80
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
10-3 Net Present Value (NPV)
To calculate the NPV, we find the present value of the individual cash flows and then sum those discounted cash flows. The sum is the value the project adds to or subtracts from shareholder wealth.
Figure 10-2
Finding the NPV for Projects S and L (Millions of Dollars)
INPUTS:
r =	10%
	Initial Cost and Expected Cash Flows
Year	0	1	2	3	4
Project S	−$10,000	$5,300	$4,300	$1,874	$1,500
	4,818.18	← ← ⤶	↓	↓	↓
	3,553.72	← ← ← ← ← ←	← ← ⤶	↓	↓
	1,407.96	← ← ← ← ← ←	← ← ← ← ← ←	← ← ⤶	↓
	1,024.52	← ← ← ← ← ←	← ← ← ← ← ←	← ← ← ← ← ←	← ← ⤶
NPVS =	$804.38	Long way: Sum the PVs of the CFs to find NPV
	Initial Cost and Expected Cash Flows
Year	0	1	2	3	4
Project L	−$10,000	$1,900	$2,700	$2,345	$7,800
NPVL =	$1,048.02	Short way: Use Excel's NPV function =NPV(B47,C59:F59)+B59
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
10-4 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 IRRs for Project S and L are shown below, along with the data entry for Project S.
Figure 10-3
Finding the IRR for Projects S and L (Millions of Dollars)
INPUTS:
	Initial Cost and Expected Cash Flows
Year	0	1	2	3	4
Project S	−$10,000	$5,300	$4,300	$1,874	$1,500
	4,621.33	← ← ⤶	↓	↓	↓
	3,269.26	← ← ← ← ← ←	← ← ⤶	↓	↓
	1,242.34	← ← ← ← ← ←	← ← ← ← ←	← ← ⤶	↓
	867.07	← ← ← ← ← ←	← ← ← ← ←	← ← ← ← ←	← ← ⤶
NPVS =	$0.00	Long way: Try a value for r, sum the PVs of the CFs to find NPV. If NPV is not zero, try another value for r. Or use Goal Seek to find the value of r that makes the NPV = 0.
IRR = r =	14.69%	Value of r that makes NPV = 0.
	Initial Cost and Expected Cash Flows
Year	0	1	2	3	4
Project L	−$10,000	$1,900	$2,700	$2,345	$7,800
IRRL =	13.79%	Short way: Use Excel's IRR function =IRR(B88:F88)
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
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 greater IRR. In our example, each project has an IRR that exceeds the cost of capital (10%) so both projects should be accepted if they are independent. If, however, the projects are mutually exclusive, we would choose Project S because it has the higher IRR. Recall that this differs from our conclusion when using the NPV method. So, we have a conflict between the NPV and the IRR methods for ranking Projects S and L.
A Potential Problem with IRR: The Possibility of Multiple IRRs
Because of the mathematics involved, it is possible for some (but not all) projects that have more than one change of signs in the 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." The procedure for correcting the problem is to 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, several hundred percent).
Suppose Project M has the following cash flows:
INPUTS:
Year	0	1	2
Project M	-1.60	10	-10
The table shown below calculates Project M's NPV at the rates (i.e., cost of capital) shown in the left column. Notice that NPV = 0 at both 25% and 400%. Since the definition of the IRR is the rate at which the NPV = 0, there are two IRRs.
	r	NPV
	0%	-$1.600
	10%	-$0.774
	15%	-$0.466
	20%	-$0.211
	25%	$0.000	= IRR #1 = 25%
	30%	$0.175
	40%	$0.441
	50%	$0.622
	60%	$0.744
	70%	$0.822
	80%	$0.869
	90%	$0.893
	95%	$0.898
	97%	$0.899
	98%	$0.900
	99%	$0.900
	100%	$0.900
	101%	$0.900
	102%	$0.900
	103%	$0.899
	102%	$0.900
	105%	$0.899
	110%	$0.894
	120%	$0.879
	160%	$0.767
	200%	$0.622
	300%	$0.275
	400%	$0.000	= IRR #2 = 400%
	500%	-$0.211
Figure 10-4
Graph for Multiple IRRs: Project M (Millions of Dollars)
A Potential Problems When Using the IRR to Evaluate Mutually Exclusive Projects
The IRR and NPV can leading to conflicting decisions when choosing among mutually exclusive projects.
	NPV	IRR
Project S	$804.38	14.69%
Project L	$1,048.02	13.79%
An NPV profile shows how a project's NPV declines as r (the cost of capital used to calculate the NPV) increases. The crossover rate is the rate at which the NPV of Project S is equal to the NPV of Project L. The easiest way to find the crossover rate is to subtract one project's cash flows from the others and find the IRR of this differential cash flow stream.
Year	0	1	2	3	4
Project S	-$10,000	$5,300	$4,300	$1,874	$1,500
Project L	-10,000	1,900	2,700	2,345	7,800
Δ = CFS − CFL	$0	$3,400	$1,600	-$471	-$6,300
IRR Δ =	12.274%
NPV Profiles for Projects S and L
	Cost of Capital	NPVS	NPVL
	0%	$2,974.00	$4,745.00
	5%	1,800.73	2,701.28
	10%	804.38	1,048.02
Crossover =	12.3%	399.91	399.91	NPVS = NPVL
IRRL =	13.8%	145.90	0.00	NPVL = 0
IRRS =	14.7%	0.00	-227.16	NPVS = 0
	20%	-$789.35	-$1,423.03
Figure 10-5
NPV Profiles for Projects S and L (Millions of Dollars)
Crossovers are caused by timing differences and size differences as shown below.
Project Sooner has a higher proportion of its future cash flows occurring sooner than the proportion for Project Later. Notice that Sooner has a higher IRR but a lower NPV.
r =	10%
	Year
Project	0	1	2	NPV	IRR
Sooner	−$1,000	$1,020	$120	$26	12.7%
Later	−$1,000	$120	$1,120	$35	12.0%
Δ = CFS − CFL	$0	$900	−$1,000		11.1%
Both projects have half of their future cash flows occurring each year, but Project Smaller has a smaller scale that Project Larger. Notice that Smaller has a higher IRR but a lower NPV.
r =	10%
	Year
Project	0	1	2	NPV	IRR
Smaller	−$90	$12	$112	$13	18.4%
Larger	−$1,000	$120	$1,120	$35	12.0%
Δ = CFS − CFL	$910	−$108	−$1,008		11.3%
10-5 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. Alternatively, you can solve using Excel's MIRR function.
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 MIRR where reinvestment is likely to occur at a different rate than WACC.
If negative cash flows occur in years beyond Year 1, those cash flows should 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 a given year, the negative flows should be discounted, and the positive ones compounded, rather than just dealing with the net cash flow. This can make a difference.
Figure 10-6
Finding the MIRR for Projects S and L
INPUTS:
r =	10%
	Initial Cost and Expected Cash Flows
Year	0	1	2	3	4
Project S	−$10,000	$5,300	$4,300	$1,874	$1,500
	↓	↓	↓	⤷ → →	$2,061
	↓	↓	⤷ → →	→ → → → →	$5,203
Present Value of Negative CF (PV) =	↓	⤷ → →	→ → → → →	→ → → → →	$7,054
	−$10,000			Terminal Value of Positive CF (TV) =	$15,819
Calculator: N = 4, PV = −10000, PMT = 0, FV = 15819. Press I/YR to get:					MIRRS =	12.15%
Excel Rate function–Easier:		=RATE(F296,0,B301,F301)			MIRRS =	12.15%
Excel MIRR function–Easiest:		=MIRR(B297:F297,B294,B294)			MIRRS =	12.15%
Year	0	1	2	3	4
Project L	−$10,000	$1,900	$2,700	$2,345	$7,800
For Project L, using the MIRR function:				=MIRR(B308:F308,B294,B294)	MIRRL =	12.78%
Notes:
1. The terminal value (TV) is the future value of all positive cash flows. The present value (PV) is the present value of all negative cash flows.
2. Find the discount rate that forces the TV positive cash flows to equal the PV of negative cash flows. That discount rate is defined as the MIRR. PV of negative cash flows =(TV of positive cash flows)/(1+MIRR)N $10,000 = $15,819/(1+MIRR)4 We can find the MIRR with a calculator or Excel.
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
10-6 Profitability Index (PI)
The profitability index is the present value of all future cash flows divided by the initial cost. It measures the PV per dollar of investment.
Figure 10-7
Profitability Index (PI)
Project S:	PIS =	PV of future cash flows ÷		Initial cost
	PIS =	$10,804	÷	$10,000
	PIS =	1.0804
Project L:	PIL =	PV of future cash flows ÷		Initial cost
	PIL =	$11,048	÷	$10,000
	PIL =	1.1048
Notes:
1. If Projects L and S are independent, both should be accepted as both have PI greater than 1.0. However, if they are mutually exclusive, Project L should be chosen as it has the higher PI.
2. PI and NPV rankings will be consistent if the projects have the same cost, as is true for S and L. However, if they differ in size, conflicts can occur. In the event of a conflict, the NPV ranking should be used.
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
10-7 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 the 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.
It's easy to calculate the payback manually--calculate cumulative cash flows and look to see when the cumulative CF turns positive, and recognize that the payback year is the prior year plus a fraction equal to the shortfall divided by the CF in the next year. However, it would be useful to have an automated procedure if you were calculating many paybacks or if you wanted to do sensitivity analysis for a given project, but this is more complicated. You can see the formula below, and the procedure is explained in detail in our Excel Tutorial. We use the formula only if we must do a number of payback calculations--for just one or two, the manual approach is much easier. If cash flows are normal (i.e., there is only one sign change) and begin with a negative cash flow, then there is an Excel function, PERCENTRANK, that will calculate payback. See the explanation of PERCENTRANK to the right.
									PERCENTRANK Function
									If cash flows are normal (i.e., there is only one sign change) and begin with a negative cash flow, then there is an Excel function, PERCENTRANK, that will calculate payback. If this is not the case, don't use the PERCENTRANK function to calculate payback! The PERCENTRANK function is easy to use but hard to explain! It looks at the range specified by the first argument, takes the second argument, finds where the second argument lies in the range, interpolates between the two values surrounding it, and coverts it to a percent. When multiplied by the number of years in the range, it gives the payback. (The last argument just tells Excel how many significant digits to use. ) See the formula in cell D380.
Figure 10-8
Payback Period
Project S	Year	0	1	2	3	4
	Cash flow	−$10,000	$5,300	$4,300	$1,874	$1,500
	Cumulative cash flow	−$10,000	−$4,700	−$400	$1,474	$2,974
	Intermediate calculation		—	—	2.21	—
					↑
		Payback S = 2 + $400/$1,874 =	2.21		↑
				Intermediate calculation:
		Excel calculation of Payback S =	2.21	=IF(AND(E367<=0,F367>=0),E365+ABS(E367/F366),"—")					For example, for Project L, the range used in the PERCENTRANK function is the row for cumulative cash flow, C376:G376. The second argument in the PERCENTRANK function is 0, the target value for the cumulative cash flow (i.e., the project is "paid back" when the cumulative cash flows is 0). This range has five columns (Years 0, 1, 2, 3, and 4) and a value (i.e., the cumulative payback) for each column.
Project L	Year	0	1	2	3	4
	Cash flow	−$10,000	$1,900	$2,700	$2,345	$7,800
	Cumulative cash flow	−$10,000	−$8,100	−$5,400	−$3,055	$4,745
		Payback L = 3 + $3,055/$7,800 =	3.39		Payback switches from negative to positive cash flow.
Alternative Excel calculation of Payback L =									The PERCENTRANK function takes the target 0, and identifies where it would be within the range of cumulative cash flows, which go from -$10,000 at Year 0 to $4,745 at Year 4. In this example, it would be between the cumulative cash flow of -$3,055 at Year 3 and the cumulative cash flow of $4,745 at Year 4. But instead of looking at Year number (which isn't included in the list of arguments), PERCENTRANK looks at the columns and determines what "percent" of the total range is covered by each column. For example, the 2nd column (Year 1) is 25% of the way from the first column (Year 0) to the last column (Year 4); the3rd column (Year 2) is 50% of the way from the first column (Year 0) to the last column (Year 4); the 4th column (Year 3) is 75% of the way from the first column (Year 0) to the last column (Year 4); and the 5th column (Year 4) is 100% of the way from the first column (Year 0) to the last column (Year 4). Because the 4th column (Year 3) has the last cumulative cash flow less than the target of 0 and because the 4th column is 75% of the way from the first column to the last column, the column immediately prior to payback is 75% of the way from Year 0 to Year 4.
			3.39	=PERCENTRANK(C376:G376,0,6)*G374
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
The regular payback has two major flaws. First, it does not take account of any cash flows that occur past the payback year, no matter how large those flows might be. Second, the payback does not take account of the time value of money. This second problem is addressed with the discounted payback as discussed below, but the failure to consider beyond-payback cash flows is a problem for both payback methods.
Figure 10-9
Discounted Payback
Project r =	10%
Project S	Year	0	1	2	3	4
	Cash flow	−$10,000	$5,300	$4,300	$1,874	$1,500
	Discounted cash flow	−$10,000	$4,818	$3,554	$1,408	$1,025
	Cumulative discounted CF	−$10,000	−$5,182	−$1,628	−$220	$804
									PERCENTRANK then identifies the distance from $4,745 (the first cumulative cash flow greater than 0) and -$3,055 (the last cumulative cash flow less than 0); this is $4,745 - (-$3,055) = $7,800. This is just the cash flow for Year 4. PERCENTRANK then identifies the percent of the distance between the two columns needed to reach the target (which is the same step taken in the manual approach): -(-$3,055)/$7,800 = 0.39166667. This means that the target payback value of 0 occurs 0.3916667 of the way between column 4 (Year 3) and column 5 (Year 4). Column 4 (Year 3) is 75% of the way from the first column, and column 5 (Year 4) is 100% of the way, so 25% of the total distance occurs between Year 3 and year 4: 100% - 75% = 25%. Therefore, the portion of the year as a percentage of the total number of years is 0.391667(25%) = 9.791667% of the total distance.
		Discounted Payback S = 3 + $220.14/$1,024.52 =	3.21		Switches from negative to positive cash flow.
		Excel calculation of Discounted Payback S =
		=PERCENTRANK(C397:G397,0,6)*G394	3.21
Project L	Year	0	1	2	3	4
	Cash flow	−$10,000	$1,900	$2,700	$2,345	$7,800
	Discounted cash flow	−$10,000	$1,727	$2,231	$1,762	$5,328
	Cumulative discounted CF	−$10,000	−$8,273	−$6,041	−$4,279	$1,048
		Discounted Payback L = 3 + $4,279.49/$5,327.50 =	3.80		Switches from negative to positive cash flow.
		Excel calculation of Discounted Payback L =
		=PERCENTRANK(C406:G406,0,6)*G403	3.80
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
10-8 How to Use the Different Capital Budgeting Methods									PERCENTRANK then takes the percentage of the total to get to column 4 (Year 3), which we found to be 75%, and the percentage of the total that is used in the column to get to the target of 0, which we found to be 9.791667%, and adds them together to get the overall total distance to get to the payback target of 0: 75% + 9.791667% = 84.791667%. This is the result given by the PERCENTRANK function. To convert this percentage to the number of years needed to reach payback, we multiply it by the total number of years: (84.791667)(4) = 3.39. This is the same payback for Project L that we calculated previously. Like we said, the function is easy to use but hard to explain!
NPV is the single best criterion because it provides a direct measure of the value a project adds to shareholder wealth. However, all methods provide helpful information.
NPV and IRR are the most widely used methods.
10-9 Other Issues in Capital Budgeting
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 first 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, a conveyor system (Project C) with a 6-year life for moving materials or a fleet of forklift trucks (Project F) with a 3-year life. The common life is 6 years, the firm's cost of capital is 12%, and the cash flow are as shown on the time lines below. The project's NPVs and IRRs are also provided.
Figure 10-10
Analysis of Projects C and F (r = 12%)
Project r =	12.0%
Data on Project C, Conveyor System:
Year	0	1	2	3	4	5	6
Cash flows for C	−$40,000	$8,000	$14,000	$13,000	$12,000	$11,000	$10,000
	NPVC =	$6,491		IRRC =	17.5%
Data on Project F, Forklifts:
Year	0	1	2	3
Cash flows for F	−$20,000	$7,000	$13,000	$12,000
	NPVF =	$5,155		IRRF =	25.2%
Common Life Approach with F Repeated (Project FF):
Year	0	1	2	3	4	5	6
CFt for 1st F	−$20,000	$7,000	$13,000	$12,000
CFt for 2nd F				−$20,000	$7,000	$13,000	$12,000
All CFs for FF	−$20,000	$7,000	$13,000	−$8,000	$7,000	$13,000	$12,000
	NPVFF =	$8,824		IRRFF =	25.2%
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
On the basis of this extended analysis, it is clear that Project FF is better, as its NPV and IRR both exceed those of C.
You could also think about the problem this way: F has a NPV of $5,155, but the firm can repeat the project at Year 3 and get another project with the same NPV. Discount the second NPV back to the present at 12%, add the two NPVs, and the result agrees with the one calculated above.
	NPV1	$5,155
	PV of NPV2	$3,669
	Total NPV for FF	$8,824
Equivalent Annual Annuity (EAA) Approach
Here are the steps in the EAA approach:
1. Find the NPV of each project over its initial life (we did this in our previous analysis).
NPVC =	$6,491
NPVF =	$5,155
2. Find the annual annuity payment whose PV is equal to the project's NPV, based on the project's life. Use a calculator or Excel's PMT function to find the EAA.
EAAC =	$1,579	=-PMT(B457,H459,B493)		N = 6, I/YR = 12, PV = -6,491, FV = 0, PMT = 1,579.
EAAF =	$2,146	=-PMT(B457,E464,B494)		N = 3, I/YR = 12, PV = -5,155, FV = 0, PMT = 2,146.
Project F has the higher EAA, so it is the better project.
ECONOMIC LIFE VS. PHYSICAL LIFE
Sometimes an asset has a physical life that is greater than its economic life. Trucks and machinery, where maintenance costs become quite high if they are used too long, are examples. Consider the following asset, which has a physical life of 3 years. During its life, the asset will generate the indicated end-of-year operating cash flows. However, the project could be terminated and the asset sold for its salvage value at the end of any year. The following table shows the operating cash flows and the salvage value for each year--all values are on an after-tax basis. The cost of capital is 10%.
Figure 10-11
Economic Life Versus Physical Life
r =	10%			PVs of the Cash Flows
	Year	Operating Cash Flow	Salvage Value	Operating Cash Flow	Salvage Value
	0	−$4,800
	1	2,000	$3,000	$1,818.18	$2,727.27
	2	2,000	1,650	1,652.89	1,363.64
	3	1,750	0	1,314.80	0.00
NPV at Different Operating Lives:		Initial Cost	+	PV of Operating Cash Flows	+	PV of Salvage Value
Operate for 3 Years:
NPV3:	−$14.12	−$4,800	+	$4,785.88	+	$0.00
Operate for 2 Years:
NPV2:	$34.71	−$4,800	+	$3,471.07	+	$1,363.64
Operate for 1 Year:
NPV1:	−$254.55	−$4,800	+	$1,818.18	+	$2,727.27
Note: Numbers in the figure are shown as rounded values for clarity in reporting. However unrounded values are used for all calculations.
Figure 10-12
IOS and MCC Schedules
Investment Opportunity Schedule (IOS)			Marginal Cost of Capital (MCC)
		Highest to Lowest IRR	Cumulative Cost	Lowest to Highest WACC
Projects	Cost
A	$100	14.0%	$100	9.0%
B	$100	13.0%	$200	9.0%
C	$100	11.5%	$300	9.0%
D	$100	10.0%	$400	10.0%
E	$50	9.5%	$450	11.0%
F	$50	9.0%	$500	12.0%
G	$100	8.5%	$600	15.0%
Note:
Use WACC = 10% as the base rate for finding risk-adjusted project WACCs.

0 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.97 0.98 0.99 1 1.01 1.02 1.03 1.02 1.05 1.1000000000000001 1.2 1.6 2 3 4 5 -1.6 -0.77355371900826375 -0.46578449905482167 -0.21111111111111169 0 0.17514792899408316 0.4408163265306122 0.62222222222222179 0.74374999999999991 0.8221453287197229 0.86913580246913558 0.89307479224376696 0.89835634451019031 0.89942023757375855 0.89974492398734851 0.89993687028105329 0.89999999999999991 0.89993812034355569 0.89975492598764806 0.89945400276638576 0.89975492598764806 0.89851279000594841 0.89433106575963706 0.8793388429752067 0.7668639053254438 0.62222222222222179 0.27499999999999991 0 -0.21111111111111103

Cost of Capital, r (%)

Project S 0 0.05 0.1 0.12274246316970694 0.13786030093241841 0.14685717425849906 0.2 2974 1800.731176824469 804.38494638344127 399.91415635804879 145.89652905413459 3.5981204564450309E-4 -789.35185185185219 Project L 0 0.05 0.1 0.12274246316970694 0.13786030093241841 0.14685717425849906 0.2 4745 2701.2818732935339 1048.0158459121612 399.9141563584908 0 -227.15802800329402 -1423.0324074074069

Cost of Capital

NPV ($)

MCC and IOS Schedules

MCC = WACC 100 200 300 400 450 500 600 0.09 0.09 0.09 0.1 0.11 0.12 0.15 IOS = IRR 100 200 300 400 450 500 600 0.14000000000000001 0.13 0.115 0.1 9.5000000000000001E-2 0.09 8.5000000000000006E-2

Dollars Raised and Invested

MCC and IOS

NPV = −$1.6 + $10/(1+r) + (−$10)/(1+r)2

IRR #1 = 25%

IRR #2 = 400%

Crossover: Conflict if WACC is to left of crossover; no conflict if WACC is to right. Since WACC = 10%, which is left of the crossover rate, there IS a conflict: NPVL > NPVS, but IRRS > IRRL.

L

S

IRRL

IRRS

At 10% WACC: NPVL

At 10% WACC: NPVS

MCC

Web10A

							10/27/15
Web Extension 10A: The Accounting Rate of Return
ACCOUNTING RATE OF RETURN (ARR)
The ARR is the second oldest criterion, after the regular payback. We have seen several versions of the ARR, but the one we've seen most often is shown below. In the example we assume that the projects both have $2,500 of depreciation per year.
ARR =	Average annual cash inflows – Average annual depreciation
	Average Investment
		Project S		Project L
Average CF		$3,244		$3,686.25
Average Depreciation		$2,500		$2,500
Years		4		4
Investment		$10,000		$10,000
Avg. Investment		$5,000		$5,000
ARRS =	$744	=	14.87%
	$5,000
ARRL =	$1,186.25	=	23.73%
	$5,000
We don't like the ARR, primarily because it ignores the time value of money and also because the IRR and the MIRR provide much more reasonable rate of return estimates. We include it here strictly for completeness.

10-3

SECTION 10-3
SOLUTIONS TO SELF-TEST
Projects SS and LL have the cash flows shown below. If a 10% cost of capital is appropriate for both of them, what are their NPVs?
WACC = r =	10%
	0	1	2	3
SS	-700	500	300	100
LL	-700	100	300	600
	NPV
SS	$77.61
LL	$89.63
What project or set of projects would be in your capital budget if SS and LL were (a) independent or (b) mutually exclusive?
If the projects are independent, accept both. If the projects are mutually exclusive, accept Project LL.

10-4

SECTION 10-4
SOLUTIONS TO SELF-TEST
The cash flows for Projects SS and LL are as follows:
WACC = r =	10%
	0	1	2	3
SS	-700	500	300	100
LL	-700	100	300	600
What are the two projects’ IRRs, and which one would the IRR method select if the firm has a 10% cost of capital and the projects are (a) independent or (b) mutually exclusive?
	IRR
SS	18.0%
LL	15.6%
If the two projects are independent, accept both. If the two projects are mutually exclusive,
accept Project LL.
Project MM has the following cash flows:
r =	10%
	0	1	2	3
	-$1,000	$2,000	$2,000	-$3,350
Calculate MM’s NPV at discount rates of 0%, 10%, 12.2258%, 25%,
122.1470%, and 150%. What are MM's IRRs? If the cost of capital were
10%, should the project be accepted or rejected?
NPV =	-$45.83
WACC:	-$45.83
0%	-$350.00
10%	-$45.83
12.2258%	-$0.00
25%	$164.80
122.1470%	$0.00
150%	-$94.40

0 0.1 0.12225800000000001 0.25 1.2214700000000001 1.5 -350 -45.83020285499606 -2.9089235135870695E-4 164.7 9999999999995 1.1703486995884305E-4 -94.399999999999977

10-5

SECTION 10-5
SOLUTIONS TO SELF-TEST
Projects A and B have the cash flows shown below. Their cost of capital is 10%. What are the projects’ IRRs, MIRRs, and NPVs? Which project would each method select?
	0	1	2
A	-$1,000	$1,150	$100
B	-$1,000	$100	$1,300
WACC =	10%
	IRR	MIRR	NPV
A	23.1%	16.8%	$128.10
B	19.1%	18.7%	$165.29
We used Excel functions to calculate these values. See the Tutorial for instructions on
the NPV, IRR, and MIRR functions.

Multiple IRRs: Project MM

1 1

Cost of capital

NPV ($)

10-6

SECTION 10-6
SOLUTIONS TO SELF-TEST
A project has the following expected cash flows: CF0 = -$500, CF1 = $200, CF2 = $200, and CF3 = $400. If the project's cost of capital is 9%, what is the PI?
WACC = r =	9%
	0	1	2	3
	-$500	$200	$200	$400
PI =	PV of future cash flows			¸	Initial cost
	$660.70			¸	$500
PI =	1.3213912584

10-7

SECTION 10-7
SOLUTIONS TO SELF-TEST
Project P has a cost of $1,000 and cash flows of $300 per year for 3 years plus another $1,000 in Year 4. The project’s cost of capital is 15%. What are P’s regular and discounted paybacks?
Regular payback
		Years	0	1	2	3	4
			|	|	|	|	|
		Cash Flow	-1,000	300	300	300	1,000
		Cumulative Cash Flow	-1,000	-700	-400	-100	900
Regular payback =			3.10
Discounted payback
WACC	15%
		Years	0	1	2	3	4
			|	|	|	|	|
		Cash Flow	-1,000	300	300	300	1,000
		Discounted Cash Flow	-1,000	261	227	197	572
		Cumulative Discounted CF	-1,000	-739	-512	-315	257
Discounted payback =			3.55
If the company requires a payback of 3 years or less, would the project be accepted?
The payback rule of 3 years leads to a reject decision.
Would this be a good accept/reject decision, considering the NPV and/or the IRR?
NPV =	$256.72
IRR =	24.78%
The payback rule conflicts with both the NPV and IRR criteria, which would suggest accepting the project.