Question

In: Finance

If a project requires $10 million investment at year 0, and creates a stream of annual...

If a project requires $10 million investment at year 0, and creates a stream of annual payoffs that grow at 2% per year forever: the first payoff of $1 million arrives in year 1, the payoff in year 2 is $1.02 million (that is, it grows by 2%), and so on. Assume that the cost of capital is 10% per year, and that you face no financial constraint.

a. (5 pt) What is the NPV of the project? Would you accept the project based on NPV rule?

b. (5 pt) Based on IRR rule, would you accept the project? You need to show your calculation.

c. (5 pt) If the required payback period is 5 years, would you accept the project based on the payback rule? Again, you need to show your work (calculations, explanations).

d. (5 pt) If the required payback period is 5 years, would you accept the project based on the discounted payback rule? Show your work (calculations, explanations).

Solutions

Expert Solution

(a) NPV refers to the Net Present Value of a project. It helps us to know what increment will be made to the wealth of shareholders when a particular project is accepted. It is calculated by finding the present value of cash inflows and reducing present value of cash outflows from it. The present value of cash flows is calculated by discounting the cash flows by their respective discount rates. Discount rates are based on the firm's minimum required rate of return, which is helpful to know if firm will be able to generate atleast minimum required return from the project or not. A project is accepted for a positive NPV and rejected for a negative NPV.

Here we will start by understanding what kind of cash flows the firm is getting forever. Then we will find out their present value to compare with initial investment in the project.

Firm is getting a series of growing cash flows year - to - year. So, we will find PV of these cash flows first.

Present value of infinite cash flows = Cash flow / (r - g)

A series of cash flows that will be available for an infinite period of time and will grow periodically at a constant rate is actually a growing perpetuity. Perpetuity refers to a series of infinite equal cash flows. Since cash flows are growing annually by 2%, this is a case of growing perpetuity.

Here, r = Discount rate

g = rate of growth of cash flows

So, PV of growing perpetuity = $1,000,000 / (0.10 - 0.02)

= $1,000,000 / (0.08)

= $12,500,000

Now, NPV = Present value of cash inflows - Present value of cash outflows

Given that, investment or outflows required presently for the project are $10,000,000, so

NPV =   $12,500,000 - $10,000,000

= $2,500,000

As NPV is positive, therefore we will accept this project.

  

(b) IRR is a technique of evaluating capital budgeting decisions by inding out the rate of return at which present value of cash inflows will be equal to present value of cash outflows and hence NPV will be 0. It should be noted that IRR is a rate which is internal to a project and it is different from minimum required rate of return which is an external variable. A project is accepted if it's IRR is higher than the minimum required rate of return and rejected when IRR is lower than the required rate of return.

The project IRR can be found by discounting the project cash inflows at a higher rate. This is because IRR is a rate at which NPV is 0. Here PV of inflows is higher than PV of outflows, so we have to discount the inflows at a higher rate to fnd out the IRR (higher rate discounting will reduce present value of cash inflows).

Let us start by finding the PV of cash inflows of our growing perpetuity at higher rate of 11% -

PV of growing perpetuity =  Cash flow / (r - g)

= $1,000,000 / (0.11 - 0.02)

=   $1,000,000 / (0.09)

= $11,111,111.11

Now let us take the rate at 12% as our PV of cash inflows is still higher than initial cash outflows.

PV of growing perpetuity =  Cash flow / (r - g)

= $1,000,000 / (0.12 - 0.02)

=   $1,000,000 / (0.10)

= $10,000,000

  

As PV of cash inflows is equal to PV of cash outflows ($10,000,000), our required IRR is 12%. Since this IRR is higher than the cost of capital of 10%, we will accept the project based on IRR evaluation.

(c) Payback period is a method of evaluating capital budgeting decisions based on the recovery of investment by the firm. The period of recovery of initial cash outflows of the project is compared with the target payback period to see if project should be taken up or not. If payback period is higher than the target payback period, project is rejected and if payback period is lower than the target payback period,the project is accepted.

Payback period requires us to calculate cumulative cash inflows to find out the time required to recover the initial investment. So,

Year Cash flows Cumulative
cash flows
1 1,000,000 1,000,000
2 1020000 2,020,000
3 1040400 3,060,400
4 1061208 4,121,608
5 1082432.16 5,204,040
6 1104080.803 6,308,121
7 1126162.419 7,434,283
8 1148685.668 8,582,969
9 1171659.381 9,754,628
10 1195092.569 10,949,721

As we can seee here that the cumulative cash inflows will be equal to initial cash outflow of $10,000,000 after 9 years, we cannot accept the project based on payback period. This is because the payback period required for the project to be acceptable is 5 years and payback period we are getting is much higher than that.

(d) Discounted payback period is also used to find out the time required for the firm to recover its cost of the project. So it is just like payback period except for one condition that the cashflows have to be discounted first to time period 0, to find their present values. These PV of cash flows is then used to find the payback period which can then be compared with target payback period to decide whether the project should be accepted or rejected.

We will calculate the cumulative discounted cash inflows to see where they will equal our initial investment.

Year Cash inflows PVF factor PV of cash
inflows
Cumulative
cash inflows
1 1,000,000 0.91 909,091 909,091
2 1020000 0.83 842,975 1,752,066
3 1040400 0.75 781,668 2,533,734
4 1061208 0.68 724,819 3,258,553
5 1082432.16 0.62 672,105 3,930,659
6 1104080.803 0.56 623,225 4,553,883
7 1126162.419 0.51 577,899 5,131,783
8 1148685.668 0.47 535,870 5,667,653
9 1171659.381 0.42 496,898 6,164,551
10 1195092.569 0.39 460,760 6,625,311
11 1218994.42 0.35 427,250 7,052,561
12 1243374.308 0.32 396,177 7,448,738
13 1268241.795 0.29 367,364 7,816,103
14 1293606.63 0.26 340,647 8,156,750
15 1319478.763 0.24 315,873 8,472,623
16 1345868.338 0.22 292,900 8,765,523
17 1372785.705 0.20 271,598 9,037,121
18 1400241.419 0.18 251,846 9,288,967
19 1428246.248 0.16 233,530 9,522,497
20 1456811.173 0.15 216,546 9,739,042
21 1485947.396 0.14 200,797 9,939,839
22 1515666.344 0.12 186,194 10,126,033

Here we see that our initial investment of $10,000,000 is recovered after 21 years, which is much higher than required payback of 5 years. So we will not accept this project.


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