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# Ranking conflicts Answer: a Diff: E

## 84. MIRR and missing cash flow Answer: d Diff: M

The up-front cost can be calculated using the payback:

$400 + ($500)(0.5) = $650.

The terminal value of the cash inflows are:

($400)(1.1)^{2} + ($500)(1.1) + $200 = $1,234.

Use your calculator to obtain the MIRR:

Enter N = 3; PV = -650; PMT = 0; FV = 1234; and then solve for MIRR = I = 23.82%.

85**.** **MIRR,
payback, and missing cash flow Answer: d Diff: M**

Step 1: Solve for the CF_{0} by knowing the payback is
exactly 2.0:

The CF_{0} for the project is $1 + $1.5 = $2.5 million.

Step 2: Find the FV of the cash inflows:

FV = $2.50 + ($2.00)(1.12)^{1} + ($1.50)(1.12)^{2} +
($1.00)(1.12)^{3}

= $2.50 + $2.24 + $1.88160 + $1.40493

= $8.026530 million.

Step 3: Solve for the MIRR:

Enter the following input data in the calculator:

N = 4; PV = -2.5; PMT = 0; FV = 8.026530; and then solve for

I =
MIRR = 33.85881%
33.86%.

86. MIRR and IRR Answer: e Diff: M

Time line:

Calculate MIRR_{T}:

Find TV of cash inflows:

N = 5; I = 8; PV = 0; PMT = 7000; and then solve for FV = TV = $41,066.21.

Find MIRR_{T} = 15.48%:

N = 5; PV = -20000; PMT = 0; FV = 41066.21; and then solve for I = MIRR = 15.48%.

Sum = 22.11% + 15.48% = 37.59%.

**87**

. Mutually exclusive projects Answer: b Diff: M

Financial calculator solution:

Project A: Inputs: CF_{0} = -100000; CF_{1} =
39500; N_{j} = 3.

Output: IRR_{A} = 8.992%
9.0%.

Project B: Inputs: CF_{0} = -100000; CF_{1} = 0;
N_{j} = 2; CF_{2} = 133000.

Output: IRR_{B} = 9.972%
10.0%.

88The
firm’s cost of capital is not given in the problem; so use the IRR
decision rule. Since IRR_{B} > IRR_{A}; Project
B is preferred.

. Before-tax cash flows Answer: b Diff: M

Financial calculator solution:

Inputs: N = 10; I = 15; PV = -10000; FV = 0. Output: PMT = $1,992.52.

Before-tax CF = $1,992.52/0.6 = $3,320.87 $3,321.

89. Crossover rate Answer: b Diff: M

Find the differences between the two projects’ respective cash flows as follows:

(CF_{A} - CF_{B}). CF_{0} = -5,000 -
(-5,000) = 0; CF_{1} = 200 - 3,000 = -2800; CF_{2} =
-2200; CF_{3} = 2200; CF_{4} = 4800. Enter these
CFs and find the IRR = 16.15%, which is the crossover rate.

**90**

. Crossover rate Answer: b Diff: M

First, find the differential CFs by subtracting Team A CFs from Team B CFs as follows:

CF_{0} = -5.5; CF_{1} = 0; CF_{2} = 0; CF_{3}
= 4; CF_{4} = 4; and then solve for IRR = 11.35%.

**91**

. Crossover rate Answer: b Diff: M

Subtract Project 2 cash flows from Project 1 cash flows:

CF_{0} = -100; CF_{1} = -600; CF_{2} = -200;
CF_{3} = 0; CF_{4} = 400; CF_{5} = 700.
Enter these in the cash flow register and then solve for IRR =
5.85%.

**92**

. Crossover rate Answer: d Diff: M

Find the differential cash flows by subtracting B’s cash flows from A’s cash flows for each year.

CF_{0} = -2000; CF_{1} = -6000; CF_{2} =
1000; CF_{3} = 5000; CF_{4} = 5000. Enter these
cash flows and then solve for IRR = crossover rate = 13.03%.

## 93. Crossover rate Answer: c Diff: M

The crossover rate is the point where the two projects will have the
same NPV. To find the crossover rate, subtract CF_{B }from
CF_{A}:

-$100,000 - (-$100,000) = 0.

$40,000 - $30,000 = $10,000.

$25,000 - $15,000 = $10,000.

$70,000 - $80,000 = -$10,000.

$40,000 - $55,000 = -$15,000.

Enter these into your CF register and then solve for IRR = 11.21%.

## 94. Crossover rate Answer: d Diff: M

Find the differential cash flows to compute the crossover rate. Subtracting Project A cash flows from Project B cash flows, we obtain the following differential cash flows:

CFs

__Year__ __B - A__

0 -$100

1 70

2 40

3 20

4 0

5 -20

Input the cash flows into your calculator’s cash flow register and solve for the IRR to obtain the crossover rate of 9.32 percent.

95**.** **Crossover
rate Answer: b Diff: M**

Step 1: Calculate the differential cash flows:

Project A Project B CFs

__Year__ __Cash Flow__ __Cash Flow__ __ B – A __

0 -$100,000 -$190,000 -$90,000

1 30,000 30,000 0

2 35,000 35,000 0

3 40,000 100,000 60,000

4 40,000 100,000 60,000

Step 2: Determine the crossover rate:

Enter the following inputs in the calculator:

CF_{0} = -90000; CF_{1} = 0; CF_{2} = 0; CF_{3}
= 60000; CF_{4} = 60000; and then solve for IRR = 8.5931%.

96**.** **Crossover
rate Answer: b Diff: M**

Step 1: Calculate the difference in the cash flows of the 2 projects:

Project A Project B CFs

__Year__ __ Cash Flow __ __ Cash Flow __ __ B –
A __

0 -$215 million -$270 million $55 million

1 20 million 70 million -50 million

2 70 million 100 million -30 million

3 90 million 110 million -20 million

4 70 million 30 million 40 million

Step 2: Calculate the IRR of the CFs:

Enter the following data (in millions) in the calculator:

CF_{0} = 55; CF_{1} = -50; CF_{2} = -30; CF_{3}
= -20; CF_{4} = 40; and then solve for IRR = 19.36%.

97**.** **Crossover
rate and missing cash flow Answer: e Diff: M**

Step 1: Determine the NPV of Project A at the crossover rate:

NPV_{A} = -$4 + $2/1.09 + $3/(1.09)^{2} + $5/(1.09)^{3}

_{ } = -$4 + $1.83486 + $2.52504 + $3.86092

_{ } = $4.22082 million.

Step 2: Determine the PV of cash inflows for Project B at the crossover rate:

NPV_{B} = CF_{0} + $1.7/1.09 + $3.2/(1.09)^{2}
+ $5.8/(1.09)^{3}

_{ } = CF_{0} + $1.55963 + $2.69338 + $4.47866

_{ } = CF_{0} + $8.73167 million.

Step 3: Determine the cash outflow at t = 0 for Project B:

At the crossover rate, NPV_{A} = NPV_{B}; NPV_{A}
- NPV_{B} = 0.

NPV_{A} = $4.22082 million; NPV_{B} = CF_{0}
+ $8.73167 million.

$4.22082 - CF_{0} - $8.73167 = 0

-CF_{0} = $8.73167 - $4.22082

CF_{0} = -$4.51085 million.

98. Multiple IRRs Answer: c Diff: T

Numerical solution:

This problem can be solved numerically but requires an iterative process of trial and error using the possible solutions provided in the problem.

Investigate first claim: Try k = IRR = 13% and k = 12.5%

NPV_{k = 13%} = -10,000 + 100,000/1.13 - 100,000/(1.13)^{2}
= 180.91.

NPV_{k = 12.5%} = -10,000 + 100,000/1.125 - 100,000/(1.125)^{2}
= -123.46.

The first claim appears to be correct. The IRR of the project appears to be between 12.5% and 13.0%.

Investigate second claim: Try k = 800% and k = 780%

NPV_{k = 800%} = -10,000 + 100,000/9 - 100,000/(1 + 8)^{2}

= -10,000 + 11,111.11 - 1,234.57 = -123.46.

NPV_{k = 780%} = -10,000 + 100,000/8.8 - 100,000/(1 + 7.8)^{2}

= -10,000 + 11,363.64 - 1,291.32 = 72.32.

The second claim also appears to be correct. The IRR of the project flows also appears to be above 780% but below 800%.

Below is a table of various discount rates and the corresponding NPVs.

__Discount rate (%)__ __ NPV __

12.0 ($ 433.67)

12.5 (123.46)

12.7 (1.02) IRR_{1}
12.7%

13.0 180.91

25.0 6,000.00

400.0 6,000.00

800.0 (123.46)

787.0 2.94 IRR_{2}
787%

780.0 72.32

By randomly selecting various costs of capital and calculating the project’s NPV at these rates, we find that there are two IRRs, one at about 787 percent and the other at about 12.7 percent, since the NPVs are approximately equal to zero at these values of k. Thus, there are multiple IRRs.

99. NPV Answer: c Diff: T

Step 1: Run a regression to find the corporate beta. It is 1.1633.

Step 2: Find the project’s estimated beta by adding 0.2 to the corporate beta. The project beta is thus 1.3633.

Step 3: Find the company’s cost of equity, which is its WACC because it uses no debt:

k_{s} = WACC = 7% + (12% - 7%)1.3633 = 13.8165%
13.82%.

Step 4: Now find NPV (in millions):

CF_{0} = -100; CF_{1-5} = 20; CF_{6-10} =
30; I = 13.82; and then solve for NPV = $23.11 million.

100. NPV Answer: c Diff: T

Step 1: Run a regression to find the corporate beta. Market returns are the X-input values, while Y’s returns are the Y-input values. Beta is 1.2102.

Step 2: Find the project’s estimated beta by subtracting 0.5 from the corporate beta. The project beta is thus 1.2102 - 0.5 = 0.7102.

Step 3: Find the project’s cost of equity, which is its WACC because it uses no debt:

k_{s} = WACC = 5% + (11% - 5%)0.7102 = 9.26%.

Step 4: Now find the project’s NPV (inputs are in millions):

CF_{0} = -500; CF_{1-5} = 100; CF_{6-10} =
50; I = 9.26%; and then solve for NPV = $10.42 million.

101. NPV profiles Answer: b Diff: T

Project X: Inputs: CF_{0} = -100; CF_{1} = 50; CF_{2}
= 40; CF_{3} = 30; CF_{4} = 10.

Output: IRR = 14.489% 14.49%.

Project Z: Inputs: CF_{0} = -100; CF_{1} = 10; CF_{2}
= 30; CF_{3} = 40; CF_{4} = 60.

Output: IRR = 11.79%.

Calculate the NPVs of the projects at k = 0 discount rate.

NPV_{X,k = 0%} = -$100 + $50 + $40 + $30 + $10 = $30.

NPV_{Z,k = 0%} = -$100 + $10 + $30 + $40 + $60 = $40.

Calculate the IRR of the differential project, that is, Project_{X
- Z}

IRR_{X - Z} Inputs: CF_{0} = 0; CF_{1} =
40; CF_{2} = 10; CF_{3} = -10; CF_{4} = -50.

Output: IRR = 7.167 7.17%.

Solely using the calculator we can determine that there is a crossover point in the relevant part of an NPV profile graph. Project X has the higher IRR. Project Z has the higher NPV at k = 0. The crossover rate is 7.17% and occurs in the upper right quadrant of the graph.

**102**

. MIRR and NPV Answer: c Diff: T

Find the MIRR of the Projects.

Calculate NPV of Projects:

Project X: Inputs: CF_{0} = -2000; CF_{1} = 200;
CF_{2} = 600; CF_{3} = 800; CF_{4} = 1400; I
= 12.

Output: NPV_{X} = $116.04.

Project Y: Inputs: CF_{0} = -2000; CF_{1} = 2000;
CF_{2} = 200; CF_{3} = 100; CF_{4} = 75; I =
12.

Output: NPV_{Y} = $63.99.

Note that the better project is X because it has a higher NPV. Its corresponding MIRR = 13.59%. (Also note that since the 2 projects are of equal size that the project with the higher MIRR will also be the project with the higher NPV.)

103. MIRR and IRR Answer: a Diff: T

Step 1: Calculate IRR by inputting the following into a calculator:

CF_{0} = -10000000; CF_{1} = 5000000; CF_{2}
= -1000000; CF_{3-4} = 5000000; and then solve for IRR =
13.78%.

Step 2: Calculate MIRR:

a. Calculate PV of the outflows:

CF_{0} = -10000000; CF_{1} = 0; CF_{2} =
-1000000; I = 15; and then solve for NPV = -$10,756,143.67.

b. Calculate FV of the inflows:

CF_{0} = 0; CF_{1} = 5000000; CF_{2} = 0;
CF_{4} = 5000000; N_{j} = 2; I = 15; and then solve
for NPV = $10,494,173.48.

c. Calculate MIRR:

N = 4; PV = -10756143.67; PMT = 0; FV = 18354375; and then solve for I = MIRR = 14.29%.

Step 3: Calculate the difference between the project’s MIRR and its IRR:

MIRR - IRR = 14.29% - 13.78% = 0.51%.

104. MIRR and missing cash flow Answer: b Diff: T N

Step 1: Determine the PV of cash outflows and the FV of cash
inflows. The PV of all cash outflows is -$500 + -X/(1.10)^{2}.
The FV of all cash inflows is $500 + $300(1.1) + $200(1.1)^{3}
= $500 + $330 + $266.20 = $1,096.20.

Step 2: Find the PV of the future value of cash inflows using the MIRR. N = 4; I = 12; PMT = 0; FV = 1096.20; and then solve for PV = $696.65.

Step 3: Determine the value of the missing cash outflow.

-$696.65 = -$500 - X/(1.10)^{2}

-$196.65 = -X/1.21

-$237.95 = -X

$237.95 = X.

105**.** **MIRR
and missing cash flow Answer: b Diff: T**

Step 1: Determine the missing cash outflow:

The payback is 2 years so the project must have cash inflows through t = 2 that equal its cash outflow.

-CF_{0} = CF_{1} + CF_{2}; CF_{0} =
-($100,000 + $200,000); CF_{0} = -$300,000.

Step 2: Calculate the present value of the cash outflows:

Enter the following inputs in the calculator:

CF_{0} = -300000; CF_{1} = 0; CF_{2} = 0;
CF_{3} = 0; CF_{4} = -100000; I = 10; and then solve
for NPV = -$368,301.3455.

Step 3: Calculate the future value of the cash inflows:

Enter the following inputs in the calculator:

CF_{0} = 0; CF_{1} = 100000; CF_{2} =
200000; CF_{3} = 200000; CF_{4} = 0; I = 10; and
then solve for NPV = $406,461.3073.

Enter the following inputs in the calculator:

N = 4; I = 10; PV = -406461.3073; PMT = 0; and then solve for FV = $595,100.

Step 4: Calculate the MIRR:

Enter the following inputs in the calculator:

N = 4; PV = -368301.3455; PMT = 0; FV = 595100; and then solve for I = MIRR = 12.7448% 12.74%.