PRACTICE PROBLEMS

1.Project K has a cost of $52,125, its expected net cash inflows are $12,000 per year for 8 years, and its cost of capital is 12 percent. (Hint: Begin by constructing a time line.)

Year
0 1 2 3 4 5 6 7 8
(52,125) 12,000 ------------------------------------------------12,000

The arrow line means that from year 1 to 8, there will be the net cash inflows of $12,000.

A. What is the projected payback period (to the closest year)?

Payback period = Incremental investment/cash flow per year
= $52,125/$12,000
= 4.34 years

B. What is the project's discounted payback period?

First, we need to find the discount net cash inflow for each year by using the cost of capital of 12%.

Discount factor = 1/(1 + k)n where k is the cost of capital while n is the period

Year Discount factor Net cash inflow Discounted cash inflow

1 0.8929 12,000 10,714.8
2 0.7972 12,000 9,566.4
3 0.7118 12,000 8,541.6
4 0.6355 12,000 7,626.0
5 0.5674 12,000 6,808.8
6 0.5066 12,000 6,079.2
7 0.4523 12,000 5,427.6
8 0.4039 12,000 4,846.8

Total 59,611.2

Discounted payback period = 6 + (2,788.2/5,427.6)
= 6.51 years

C. What is the projected NPV?

Find NPV by finding the present value of each cash flow, including both inflows and outflows, discounted at the project's cost of capital.

NPV = sum of CFt where CF is the cash flow
(1 + k)n k is the cost of capital
n is the period.

NPV = - 52,125 + 59,611.2 =7,486.2

D. What is the projected IRR?

The internal rate of return is defined as that discount rate which equates the present value of a project's expected cash inflows to the present value of the project's costs:

PV(Inflows) = PV(Investment costs), or the rate which forces the NPV to equal zero.

IRR = 16%

E. What is the project's MIRR?

While the internal rate of return (IRR) assumes the cash flows from a project are reinvested at the IRR, the modified IRR assumes that all cash flows are reinvested at the firm's cost of capital. Then, we will find how much MIRR is

IRR = 16%

First, we need to find the future net cash inflow for each year by using the cost of capital of 12%.

Future discount factor = (1 + k)n where k is the cost of capital while n is the period

Year Discount factor Net cash inflow Discounted cash inflow

1 2.2107 12,000 26,528.4
2 1.9738 12,000 23,685.6
3 1.7623 12,000 21,147.6
4 1.5735 12,000 18,882.0
5 1.4049 12,000 16,858.8
6 1.2544 12,000 15,052.8
7 1.1200 12,000 13,440.0
8 1.0000 12,000 12,000.0

Total terminal value 147,595.2
The present value of terminal value must be equal to the cash outflow.

MIRR = 13.89%

2.Your division is considering two investment projects, each of which requires an up-front expenditure of $15 million. Your estimate that the investments will produce the following net cash flows:

Year Project A Project B

1 5,000,000 20,000,000

2 10,000,000 10,000,000

3 20,000,000 6,000,000

What are the two projects net present value, assuming the cost of capital is 10 percent? 5 percent? 15 percent?

At 10%

Project A

NPV = -15,000,000 + 5,000,000 + 10,000,000 + 20,000,000
(1.10)1 (1.10)2 (1.10)3

= -15,000,000 + 27,835,500

= 12,835,500

Project B

NPV = -15,000,000 + 20,000,000 + 10,000,000 + 6,000,000
(1.10)1 (1.10)2 (1.10)3

= -15,000,000 + 30,953,800

= 15,953,800

At 5%

Project A

NPV = -15,000,000 + 5,000,000 + 10,000,000 + 20,000,000
(1.05)1 (1.05)2 (1.05)3

= -15,000,000 + 31,108,000

= 16,108,000

Project B

NPV = -15,000,000 + 20,000,000 + 10,000,000 + 6,000,000
(1.05)1 (1.05)2 (1.05)3

= -15,000,000 + 33,300,800

= 18,300,800

At 15%

Project A

NPV = -15,000,000 + 5,000,000 + 10,000,000 + 20,000,000
(1.15)1 (1.15)2 (1.15)3

= -15,000,000 + 25,059,000

= 10,059,000

Project B

NPV = -15,000,000 + 20,000,000 + 10,000,000 + 6,000,000
(1.15)1 (1.15)2 (1.15)3

= -15,000,000 + 28,898,000

= 13,898,000

3.Edelman Engineering is considering including two pieces of equipment, a truck and an overhead pulley system, in this year's capital budget. The project are independent. The cash outlay for the truck is 17,000, and that for the pully system is 22,430. The firm's cost of capital is 14 percent. After-tax cash flows, including depreciation, are as follows:

Year Truck Pully
1 5,100 7500
2 5,100 7500
3 5100 7500
4 5100 7500
5 5100 7500

Calculate the IRR, the NPV, and the MIRR for the project, and indicate the correct accept/reject decision for each.

Find NPV by finding the present value of each cash flow, including both inflows and outflows, discounted at the project's cost of capital.

For Truck

NPV = sum of CFt where CF is the cash flow
(1 + k)t k is the cost of capital
t is the period.

NPV = -17,100 + 5,100 + 5,100 + 5,100 + 5,100 + 5,100
(1.14)1 (1.14)2 (1.14)3 (1.14)4 (1.14)5

= -17,100 + 17,508.81
= 408.81

For Pulley

NPV = -22,430 + 7,500 + 7,500 + 7,500 + 7,500 + 7,500
(1.14)1 (1.14)2 (1.14)3 (1.14)4 (1.14)5

= -22,430 + 25,748.25
= 3,318.25

Select Pulley because of higher NPV.

Calculate the IRR

IRR is defined as that discount rate which equates the present value of a project's expected cash inflows to the present value of the project's costs.

NPV = sum of CFt where CF is the cash flow
(1 + IRR)t IRR is the internal rate of return
t is the period.

For Truck

NPV = -17,100 + 5,100 + 5,100 + 5,100 + 5,100 + 5,100 = 0
(1+IRR)1 (1+IRR)2 (1+IRR)3 (1+IRR)4 (1+IRR)5

IRR = 15%

For Pulley

NPV = -22,430 + 7,500 + 7,500 + 7,500 + 7,500 + 7,500 = 0
(1+IRR)1 (1+IRR)2 (1+IRR)3 (1+IRR)4 (1+IRR)5

IRR = 20%

Select Pulley because of higher IRR.

Calculate the MIRR

MIRR is the modified IRR, which make it a better indicator of relative profitability.

First, we will find the terminal value, which is the future of the cash inflows. Then, we will discount to make the terminal value equal to the initial outlay to make NPV equal to 0 with MIRR.

TV = sum of CFt(1 + k)n - t where CF is the cash flow
MIRR is the internal rate of return
t is the period.
n is the total number of period.
For Truck

TV = 5,100(1.14)4 + 5,100(1.14)3 + 5,100(1.14)2 + 5,100(1.14)1 + 5,100(1.14)0
= 33,711.51

PV = FV where PV is the present value
(1 + MIRR)n FV is the TV
MIRR is the interest rate
n is the period

17,100 = 33,711.51
(1 + MIRR)5

MIRR = 14.54%

For Pulley

TV = 7,500(1.14)4 + 7,500 (1.14)3 + 7,500 (1.14)2 + 7,500 (1.14)1 + 7,500 (1.14)0
= 49,575.75

PV = FV where PV is the present value
(1 + MIRR)n FV is the TV
MIRR is the interest rate
n is the period

22,430 = 49,575.75
(1 + MIRR)5

MIRR = 17.19%

Select Pulley because of higher MIRR.

4.Your company is considering two mutually exclusive projects. X and Y, whose costs and cash flows are shown below:

Year X Y
0 ($1,000) ($1,000)
1 100 1,000
2 300 100
3 400 50
4 700 50

The projects are equally risky, and their cost of capital is 12 percent. You must make a recommendation, and you must base it on the modified IRR (MIRR). What is the MIRR of the better project?

First, we will find the terminal value, which is the future of the cash inflows. Then, we will discount to make the terminal value equal to the initial outlay to make NPV equal to 0 with MIRR.

TV = sum of CFt(1 + k)n - t where CF is the cash flow
MIRR is the internal rate of return
t is the period.
n is the total number of period.
For Project X

TV = 100(1.12)3 + 300(1.12)2 + 400(1.12)1 + 700(1.12)0
= 1,664.81

PV = FV where PV is the present value
(1 + MIRR)n FV is the TV
MIRR is the interest rate
n is the period

-1,000 = 1,664.81
(1 + MIRR)4

MIRR = 13.59%

For Project Y

TV = 1,000(1.12)3 + 100(1.12)2 + 50(1.12)1 + 50(1.12)0
= 1,636.34

PV = FV where PV is the present value
(1 + MIRR)n FV is the TV
MIRR is the interest rate
n is the period

-1,000 = 1,636.34
(1 + MIRR)4

MIRR = 13.10%

Project X is a better project with the MIRR of 13.59%.