Corporate Finance, Module 4: “Net Present Value vs Other Valuation Models”

Homework Assignments

(The attached PDF file has better formatting.)

Answer either one of the following two homework exercises.

Exercise 4.1: Project Duration

A firm has two potential projects, with the following expected cash flows:

    Cash Flows in Year X
    Yr 1    Yr 2    Yr 3    Yr 4
Project #1    (400)    220    310    –
Project #2    (400)    130    190    260


The firm’s opportunity cost of capital is 13% per annum.

A.    What is the net present value of Project #1?
B.    What is the net present value of Project #2?
C.    What project is better from an NPV perspective?
D.    What is the internal rate of return of Project #1? (You may use financial calculator to determine the IRR, or you may solve a quadratic equation)
E.    Is the internal rate of return of Project #2 higher or lower than the IRR for Project #1? (To see if the IRR for Project #2 is higher or lower than the IRR for Project #1, use the Project #1 IRR as the hurdle rate to compute the NPV of Project #2.)
F.    What project is better from an IRR perspective?
G.    Explain why the two performance measures give different answers. (See the practice problems for the explanation.)


Exercise 4.2: Project Size

A firm has two potential projects, with the following expected cash flows:

    Cash Flows in Year X
    Yr 1    Yr 2    Yr 3    Yr 4
Project #1    (800)    360    360    360
Project #2    (400)    130    190    260


The firm’s opportunity cost of capital is 13% per annum. (Project #2 is the same as in the previous homework assignment.)

A.    What is the net present value of Project #1?
B.    What is the net present value of Project #2?
C.    What project is better from an NPV perspective?
D.    What is the internal rate of return of Project #1?
E.    What is the internal rate of return of Project #2?
F.    What project is better from an IRR perspective?
G.    Explain why the two performance measures give different answers. (See the practice problems for the explanation.)

{Comments from a discussion forum thread:}

Question: What is the initial investment?

Answer: The initial investment is the negative cash flow at time 1.

Question: Does a project always have a negative cash flow followed by positive cash flows?

Answer: Not necessarily. A loan is a project. The borrower has a positive cash flow at time 0 and negative cash flows at each coupon payment date and at maturity.

Question: Are there other examples besides loans?

Answer: A state lottery collects money from consumers, and pays the winnings a few weeks later.

Question: Why do the examples generally have a negative initial cash flow followed by positive cash flows?

Answer: That is the most common sequence. Even for the state lottery, the state must spend money (an initial investment) preparing the structure: tickets, sales offices, advertisements, and so forth. This is an initial investment. The sequence of cash flows is a small initial investment at time 0, a large cash inflow at time 1, and a cash outflow at time 2.

Question: If a firm borrows money, is there a similar initial investment?

Answer: No one walks into a bank and says: “I’m setting up a firm which needs to borrow $100,000.” A person first sets up a firm, creates a business plan, creates a prototype of the product, and does all the work to show that the firm might succeed. This may cost $100,000, which is the initial investment. The firm then borrows $200,000 from the bank, to cover its production and operating costs.

Question: Does the initial investment have to be at time zero?

Answer: The time index is arbitrary. We can call it time 0, time 1, or time 2006.


Question: What should we focus on for the final exam?

Answer: The final exam may give a multiple choice question asking which project has the higher IRR and which has the higher NPV. Understand why a larger project or a longer project may have a higher NPV but a lower IRR. Know also two principles:

●    Both profit measures give the same accept/reject decision.
●    A change in the opportunity cost of capital changes the NPV but not the IRR.

Question: These two principles are not consistent. If a change in the opportunity cost of capital changes the NPV but not the IRR, why do the two measure give the same accept vs reject decision?

Answer: The hurdle rate for the IRR measure is the opportunity cost of capital. A higher opportunity cost of capital reduces the NPV (for investment projects, not loans) and increases the hurdle rate.

Im a little it unsure about the reason as to why the NPV and IRR give different results as to which project to choose...i checked the practice problems, but I am still a little fuzzy...anyboy have any ideas

[NEAS: Suppose a firm with a 10% opportunity cost of capital has $100,000 and two mutually exclusive potential projects:

~ Project Y requires $80,000 and return $11,200 after 1 years, for a 14% return (IRR).

~ Project Z requires $40,000 and return $6,000 after 1 years, for a 15% return (IRR).

Project Y has the higher NPV; Project Z has the higher IRR.

~ NPV implicitly assumes that after choosing a project, the rest of the firm’s cash earns the opportunity cost of capital.

~ IRR implicitly assumes that the rest of the firm’s cash earns the same return as the chosen project.]

""Im a little it unsure about the reason as to why the NPV and IRR give different results as to which project to choose...i checked the practice problems, but I am still a little fuzzy...anyboy have any ideas""

I think it's because a high IRR on a tiny investment won't make you much money (even though the interest rate is wonderful) and a smaller IRR on a large project might make you more. For example, 20% interest on a $10 project will make you $2, and 10% interest on a $100 project will make $10. By IRR, you'd take the first; but in reality you'd probably want the second because you actually make more money. NPV takes the scale of the project into account.

Another aspect to consider, based on the problem, is the time frame of the project. If you are using the NPV rule to compare a project with cash flows over 4 years to one with cash flows over 3 years (i.e. exercise 4.1), the rule assumes that the alternative in year 4 is to invest only at the opportunity cost of capital (which may or may not be the case). So in 4.1, project #1 has a higher IRR over the 3 year duration, but a lower NPV because #2 has cash flows over 4 years. If you calculated the IRR over 4 years for project #1 using the opportunity cost for year 4 (13%), the IRR would be lower than #2 (confirming the NPV findings).

[NEAS: See the thread on practice problems, which discusses the IRR vs NPV perspectives.]

I understand the reasoning for 4.1  G, but I'm having some trouble with 4.2  G.  I have read through the students and NEAS postings here and I think it has to do with the initial investment (800 in Project 1 and 400 in project 2) but I am not wrapping my mind around HOW this affects the NPV and IRR.  Can anyone help explain in different terms?

i am also having trouble with the part Gs of this homework. any help?

[NEAS: See the thread on practice problems, which discusses the IRR vs NPV perspectives.]

I also did the same thing and noted in my homework that I've assumed the cash flows occurred at the end of each year. Is this okay?

For the final exam, if the question does not state whether the cash flow occurs at the beginning or the end of the year, should I always assume it occurs at the end of the year?

[NEAS: Yes.  The exam problems should specify when the cash flow occurs.]

For questions 4.1 & 4.2, the initial investment takes place in Year 1. Is this the beginning of Year 1 or the end of Year 1?

I ask because I was discussing this problem with a coworker that is also taking the course, and he believes the question is saying the initial investment takes place at t=1 and that the other cash flows are at t=2, t=3, and t=4. Therefore, he thinks, in the case of 4.1, for example, that the NPV should be considered at t=0. So he says the NPV is -400/1.13 + 220/1.13^2 + 310/1.13^3

I initially thought the answer was simply -400 + 220/1.13 + 310/1.13^2, but now, after discussing this with him, I'm not so sure.

I assumed these transactions happened at the beginning of the year because it wasn't stated clearly in the homework, and I figured I'd do less work to understand the same concept. Also, Mr. Neas and Ms. Vee stated in the assignment that you could use a QUADRATIC equation to solve for IRR, implying that there are squares, but no cubes.

Conclusion: Do it however you want, but make sure you understand the concept.

[NEAS: Correct: An exam problem might use a two period IRR equation, not a three period equation.]

I was wondering about this as well. NEAS, can you verify this?

[NEAS: Assume cash flows occur at the beginning of each year, the end of each, or the middle of each year. Just be consistent with the same point in each year.]