Kaleb Konstruction, Inc., has the following mutually exclusive projects available. The company has historically used a three-year cutoff for projects. The required return is 11 percent.

Kaleb Konstruction, Inc., has the following mutually exclusive projects available. The company has historically used a three-year cutoff for projects. The required return is 11 percent.

Year	Project F		Project G
0	–$	139,000	–$	209,000
1		58,000		38,000
2		52,000		53,000
3		62,000		92,000
4		57,000		122,000
5		52,000		137,000

Required:

(a)	Calculate the payback period for both projects. (Do not round intermediate calculations. Round your answers to 2 decimal places (e.g., 32.16).)

	Payback period
Project F	years
Project G	years

(b)	Calculate the NPV for both projects. (Do not round intermediate calculations. Round your answers to 2 decimal places (e.g., 32.16).)

	Net present value
Project F	$
Project G	$

(c)	Which project should the company accept?
	Project G

Explanation:

(a)

The payback period for each project is:

F:     2 + ($29,000 / $62,000) = 2.47 years

G:    3 + ($26,000 / $122,000) = 3.21 years

The payback criterion implies accepting project F because it pays back sooner than project G. Project G does not meet the minimum payback of three years.

(b)

The NPV for each project is:

F:     NPV = – $139,000 + $58,000 / 1.11 + $52,000 / 1.112 + $62,000 / 1.113 + $57,000 / 1.114

+ $52,000 / 1.115

NPV = $69,197.62

G:    NPV = – $209,000 + $38,000 / 1.11 + $53,000 / 1.112 + $92,000 / 1.113

+ $122,000 / 1.114 + $137,000 / 1.115

NPV = $97,187.84

The NPV criterion implies we should accept project G because project G has a higher NPV than project F.

(c)

Even though Project G does not meet the payback period of three years, it does provide the largest increase in shareholder wealth, therefore, choose Project G. Payback period should generally be ignored in this situation.

    

Calculator Solution:

 

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

   

Project F		Project G
CFo	–$139,000	CFo	–$209,000
C01	$58,000	C01	$38,000
F01	1	F01	1
C02	$52,000	C02	$53,000
F02	1	F02	1
C03	$62,000	C03	$92,000
F03	1	F03	1
C04	$57,000	C04	$122,000
F04	1	F04	1
C05	$52,000	C05	$137,000
F05	1	F05	1
I = 11		I = 11
NPV CPT		NPV CPT
$69,197.62		$97,187.84

Problem 8-9 Calculating NPV [LO 4] Consider the following cash flows: Year Cash Flow 0 –$ 32,000 1 14,200 2 17,500 3 11,600

Problem 8-9 Calculating NPV [LO 4]

Consider the following cash flows:

Year	Cash Flow
0		–$	32,000
1			14,200
2			17,500
3			11,600

Requirement 1:

What is the NPV at a discount rate of zero percent? (Do not round intermediate calculations.) Net present value $   Requirement 2: What is the NPV at a discount rate of 10 percent? (Do not round intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)   Net present value $   Requirement 3: What is the NPV at a discount rate of 20 percent? (Do not round intermediate calculations. Negative amount should be indicated by a minus sign.Round your answer to 2 decimal places (e.g., 32.16).)   Net present value $   Requirement 4: What is the NPV at a discount rate of 30 percent? (Do not round intermediate calculations. Negative amount should be indicated by a minus sign. Round your answer to 2 decimal places (e.g., 32.16).)   Net present value $   Explanation: 1: The NPV of a project is the PV of the outflows plus the PV of the inflows. At a zero discount rate (and only at a zero discount rate), the cash flows can be added together across time. So, the NPV of the project at a zero percent required return is:   NPV = – $32,000 + 14,200 + 17,500 + 11,600 NPV = $11,300   2: The NPV at a 10 percent required return is:   NPV = – $32,000 + $14,200 / 1.10 + $17,500 / 1.102 + $11,600 / 1.103 NPV = $4,087.15   3: The NPV at a 20 percent required return is:   NPV = – $32,000 + $14,200 / 1.20 + $17,500 / 1.202 + $11,600 / 1.203 NPV = –$1,300.93   4: And the NPV at a 30 percent required return is:   NPV = – $32,000 + $14,200 / 1.30 + $17,500 / 1.302 + $11,600 / 1.303 NPV = – $5,441.97   Notice that as the required return increases, the NPV of the project decreases. 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.     Calculator solution:   Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.    CFo  –$32,000 CFo  –$32,000 C01  $14,200 C01  $14,200 F01  1 F01  1 C02  $17,500 C02  $17,500 F02  1 F02  1 C03  $11,600 C03  $11,600 F03  1 F03  1   I = 0%   I = 10%   NPV CPT   NPV CPT   $11,300.00   $4,087.15     CFo  –$32,000 CFo  –$32,000 C01  $14,200 C01  $14,200 F01  1 F01  1 C02  $17,500 C02  $17,500 F02  1 F02  1 C03  $11,600 C03  $11,600 F03  1 F03  1   I = 20%   I = 30%   NPV CPT   NPV CPT   –$1,300.93   –$5,441.97

Problem 8-8 Calculating IRR [LO 3] Consider the following cash flows: Year Cash Flow 0 –$ 32,000 1 14,200 2 17,500 3 11,600

Problem 8-8 Calculating IRR [LO 3]

Consider the following cash flows:

Year	Cash Flow
0		–$	32,000
1			14,200
2			17,500
3			11,600

Required:

What is the IRR of the above set of cash flows? (Do not round intermediate calculations. Enter your answer as a percentage rounded to 2 decimal places (e.g., 32.16).)

Internal rate of return

%

Explanation:

The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is:

0 = – $32,000 + $14,200 / (1 + IRR) + $17,500 / (1 + IRR)2 + $11,600 / (1 + IRR)3

Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:

IRR = 17.32%

   

Calculator solution:

 

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

  

CFo	–$32,000
C01	$14,200
F01	1
C02	$17,500
F02	1
C03	$11,600
F03	1
IRR CPT
17.32%

Problem 8-7 Calculating NPV and IRR [LO 3, 4] A project that provides annual cash flows of $2,800 for nine years costs $9,200 today. Requirement 1: At a required return of 11 percent, what is the NPV of the project? (Do not round intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)

Problem 8-7 Calculating NPV and IRR [LO 3, 4]

A project that provides annual cash flows of $2,800 for nine years costs $9,200 today.

Requirement 1:

At a required return of 11 percent, what is the NPV of the project? (Do not round intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)

NPV

$

Requirement 2:

At a required return of 27 percent, what is the NPV of the project? (Do not round intermediate calculations. A negative amount should be indicated by a minus sign. Round your answer to 2 decimal places (e.g., 32.16).)

NPV

$

Requirement 3:

At what discount rate would you be indifferent between accepting the project and rejecting it? (Do not round intermediate calculations. Enter your answer as a percentage rounded to 2 decimal places (e.g., 32.16).)

Discount rate

%

Explanation:

1:

The NPV of a project is the PV of the outflows plus the PV of the inflows. Since the cash inflows are an annuity, the equation for the NPV of this project at an 11 percent required return is:

NPV = – $9,200 + $2,800(PVIFA11%, 9)

NPV = $6,303.73

At an 11 percent required return, the NPV is positive, so we would accept the project.

2:

The equation for the NPV of the project at a 27 percent required return is:

NPV = – $9,200 + $2,800(PVIFA27%, 9)

NPV = –$36.22

At a 27 percent required return, the NPV is negative, so we would reject the project.

3:

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. The IRR of the project is:

0 = – $9,200 + $2,800(PVIFAIRR, 9)

IRR = .2686, or 26.86%

   

Calculator Solution:

 

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation.

  

CFo	–$9,200	CFo	–$9,200	CFo	–$9,200
C01	$2,800	C01	$2,800	C01	$2,800
F01	9	F01	9	F01	9
I = 11%		I = 27%		IRR CPT
NPV CPT		NPV CPT		26.86%
$6,303.73		–$36.22

Garage, Inc., has identified the following two mutually exclusive projects:

Garage, Inc., has identified the following two mutually exclusive projects:    

Year	Cash Flow (A)				Cash Flow (B)
0	–$	29,900			–$	29,900
1		15,300				4,750
2		13,200				10,250
3		9,650				16,100
4		5,550				17,700

  

a-1	What is the IRR for each of these projects? (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16))

	IRR
Project A	%
Project B	%

a-2	Using the IRR decision rule, which project should the company accept?
	Project A

a-3	Is this decision necessarily correct?
	No

b-1	If the required return is 11 percent, what is the NPV for each of these projects? (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16))

  

	NPV
Project A	$
Project B	$

b-2	Which project will the company choose if it applies the NPV decision rule?
	Project B

  

c.	At what discount rate would the company be indifferent between these two projects? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16))

Discount rate

%

Explanation: a.

The IRR is the interest rate that makes the NPV of the project equal to zero. The equation for the IRR of Project A is:

0 = –$29,900 + $15,300 / (1 + IRR) + $13,200 / (1 + IRR)2 + $9,650 / (1 + IRR)3 + $5,550 / (1 + IRR)4

Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:

IRR = 20.57%

The equation for the IRR of Project B is

0 = –$29,900 + $4,750 / (1 + IRR) + $10,250 / (1 + IRR)2 + $16,100 / (1 + IRR)3 + $17,700 / (1 + IRR)4

Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:

IRR = 18.58%

Examining the IRRs of the projects, we see that the IRRA is greater than the IRRB, so IRR decision rule implies accepting project A. This may not be a correct decision; however, because the IRR criterion has a ranking problem for mutually exclusive projects. To see if the IRR decision rule is correct or not, we need to evaluate the project NPVs.

b.

The NPV of Project A is:

NPVA = –$29,900 + $15,300 / 1.11 + $13,200 / 1.112 + $9,650 / 1.113 + $5,550 / 1.114

NPVA = $5,309.15

And the NPV of Project B is:

NPVB = –$29,900 + $4,750 / 1.11 + $10,250 / 1.112 + $16,100 / 1.113 + $17,700 / 1.114

NPVB = $6,130.13

The NPVB is greater than the NPVA, so we should accept Project B.

c.

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. The equation for the crossover rate is:

Crossover rate: 0 = $10,550 / (1 + R) + $2,950 / (1 + R)2 – $6,450 / (1 + R)3 – $12,150 / (1 + R)4

Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that:

R = 14.09%

At discount rates above 14.09 percent choose project A; for discount rates below 14.09 percent choose project B; indifferent between A and B at a discount rate of 14.09 percent.

  

Calculator Solution:

Note: Intermediate answers are shown below as rounded, but the full answer was used to complete  the calculation.

     

	Project A
	CFo	–$29,900	CFo	–$29,900
	C01	$15,300	C01	$15,300
	F01	1	F01	1
	C02	$13,200	C02	$13,200
	F02	1	F02	1
	C03	$9,650	C03	$9,650
	F03	1	F03	1
	C04	$5,550	C04	$5,550
	F04	1	F04	1
	IRR CPT		I = 11%
	20.57%		NPV CPT
			$5,309.15

  

	Project B
	CFo	–$29,900	CFo	–$29,900
	C01	$4,750	C01	$4,750
	F01	1	F01	1
	C02	$10,250	C02	$10,250
	F02	1	F02	1
	C03	$16,100	C03	$16,100
	F03	1	F03	1
	C04	$17,700	C04	$17,700
	F04	1	F04	1
	IRR CPT		I = 11%
	18.58%		NPV CPT
			$6,130.13

     

	Crossover rate
	CFo	$0
	C01	$10,550
	F01	1
	C02	$2,950
	F02	1
	C03	–$6,450
	F03	1
	CO4	–$12,150
	FO4	1
	IRR CPT
	14.09%

You are considering acquiring a firm that you believe can generate expected cash flows of

You are considering acquiring a firm that you believe can generate expected cash flows of $28,000 a year forever. However, you recognize that those cash flows are uncertain.

a.	Suppose you believe that the beta of the firm is 2.2. How much is the firm worth if the risk-free rate is 5% and the expected rate of return on the market portfolio is 8%? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Value of the firm

$

b.	How much is the overvalue of the firm if its beta is actually 2.5? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Overvalue

$

Explanation:

Some values below may show as rounded for display purposes, though unrounded numbers should be used for the actual calculations.

a.

The expected cash flows from the firm are in the form of a perpetuity. The discount rate is:

rf + β(rm − rf) = 5% + [2.2 × (8% − 5%)] = 11.6%

Therefore, the value of the firm would be:

P0 =	Cash flow	=	$28,000	= $241,379.31
	r		0.116

b.

If the true beta is actually 2.5, the discount rate should be:

rf + β(rm − rf) = 5% + [2.5 × (8% − 5%)] = 12.5%

Therefore, the value of the firm is:

P0 =	Cash flow	=	$28,000	= $224,000.00
	r		0.125

By underestimating beta, you would overvalue the firm by:

$241,379.31 − $224,000.00 = $17,379.31