Athabasca University



Slide #1: Lecture 5 – NPV Calculation

Hello everyone. Welcome to Lecture 5: Net Present Value (NPV) calculation.

Slide #2: Topics covered in this lecture

We will cover 6 topics in this lecture:

1. Identification of incremental cash-flows.

2. The main types of cash-flows we will see in NPV calculations

3. The basic NPV formula

4. The steps one must take when calculating NPV

5. An in-depth numerical example

6. A sample question at the end for you to practice

Slide #3: Incremental cash-flows

An incremental cash-flow is a cash-flow that accrues on a particular project only if it occurs as a direct consequence of taking the project. Therefore, in considering this type of cash-flow, we would exclude all cash-flows that occur whether or not we choose to take on a particular project.

So, the question to ask is: Will the cash-flow or cash-flow stream occur if we do not invest in this project?

- If the answer is ‘yes,’ then it is NOT incremental to the project and should NOT be included in the NPV analysis.

- If the answer is ‘no,’ then it is incremental to the project and SHOULD be included in the NPV analysis.

Slide #4: Main types of cash-flows

There are two main types of cash-flows: Cash Outflows, and, Cash Inflows.

A cash outflow is any negative cash-flow, i.e., the company has to pay out (or otherwise, lose) money.

The two main types of cash outflows that we will see in NPV analysis are the initial project or asset cost, which we will denote with C0, and the initial investment in Net Working Capital, which we will denote as NWC0.

A cash inflow is any positive cash flow, i.e., the company will receive (or otherwise gain or save) money.

The 4 main types of cash inflows that we will see in NPV analysis are:

- the after-tax operating cash-flows (denoted as ATOCF) or after-tax cost-savings (denoted as ATCS);

- salvage value (denoted as S);

- recovery of net working capital investment (denoted as NWCN); and,

- depreciation or CCA tax shields (denoted as CCATS).

Please take a bit of time now (pause this video if you wish) to familiarize yourself with the acronyms for these cash-flows. Your ability to memorize and recall these acronyms will help you later when we use them in the NPV calculations. Note that N is used to denote the length of the project life.

We will not talk much specifically about these cash-flows, as they are usually covered quite well in textbooks. However, we will talk a little bit about the initial investment in Net Working Capital and its recovery, as these are particular cash-flows that sometimes stump students.

Slide #5: Net Working Capital (NWC)

Let’s start from the basics. Net working capital is calculated as the Current Assets minus the Current Liabilities:

NWC = Current Assets – Current Liabilities

Current assets include things like “cash reserves,” “inventory,” and “accounts receivable.”

Current liabilities include items such as “accounts payable” and “notes payable.”

The net working capital will increase if there is an increase in current assets (cash and/or accounts receivable and/or inventory).

In direct contrast, the net working capital will also increase if there is a decrease in current liabilities (accounts payable and/or notes payable).

A special note on net working capital investment: usually, we assume that this net working capital investment will be recovered at the end of the project or asset life. This is assumed to be true unless otherwise and specifically stated.

Slide #6: Net Present Value (NPV)

Okay, now that we know how to identify relevant incremental cash flows, we move on to discuss the basic net present value (NPV) formula. This is a good thing to remember for future reference: The basic NPV formula is:

NPV = PV(All future cash inflows) – PV(All future cash outflows).

Based on the typical cash flows we itemized before, NPV is normally calculated based on these 6 items:

PV(ATOCF) = present value of after-tax operating cash flow or after-tax cost savings

PV(S) = present value of salvage

PV(NWCN) = present value of NWC recovered

PV(CCATS) = present value of CCA tax shields

C0 = Initial project or asset cost

NWC0 = Initial investment in Net Working Capital

The basic NPV rule is: Accept a project or buy an asset if its NPV is greater than 0.

By the way, the material on this page is of particular importance to capital budgeting and should therefore be memorized if possible, as these concepts will be used over and over again in capital budgeting. Make sure that you can recall the basic NPV formula and the basic NPV rule.

Slide #7: Steps for calculating NPV

Here are the steps for calculating a project’s or asset’s NPV. FYI, these are not really steadfast rules; it is just my mental process to make the NPV analysis easier to handle.

Step 1 involves identifying incremental cash-flows.

Step 2 involves identifying the types of cash-flows.

Step 3 involves identifying the formulae needed to calculate the present values of cash-flows.

Step 4 involves the actual calculation of present values.

Step 5 calculates the NPV as the present value of all cash inflows minus the present value of all cash outflows.

We have previously talked about Step 1 and Step 5 on Slides #3, 4, 5, and 6.

Steps 2 to 4 are basically the tagging, matching, and chugging parts of the process.

In Step 2, you tag a cash-flow as either an annuity, lump sum, or perpetuity.

In Step 3, you then match the cash-flow type to the formula for an annuity, lump sum or perpetuity.

In Step 4, you then plug in the numbers to the formulae and chug out the resulting present values.

Here’s a memory aid for you: ITFCN (In The Funky Cat’s Nose)

I – Incremental cash-flows

T – Types of cash-flows

F – Formulas

C – Calculate

N – Net present value

I know, it’s pretty lame, so if you can think up a better one, please feel free to use it.

Slide #8: Numerical Example

Okay, enough theorizing and conceptualizing. Let’s play with some numbers!!

XYZ Ltd. is considering a project that will bring in before-tax operating cash-flows of $2 million per year for the next 15 years. This project will require $13 million in initial project costs, as well as initial investment of $300,000 in net working capital. At the end of the project, the assets used in this project can be sold at an estimated $500,000. These assets have a CCA rate of 30%, and it is assumed that the CCA asset-class will remain open at the end of the project. If XYZ’s marginal corporate tax rate is 40% and its required rate of return on this type of projects is 10%, should the company take on this project?

Here, we have a company, XYZ, that is considering a project that will bring in before-tax operating cash flows of $2 million per year for the next 15 years. This gives us before-tax OCF of $2,000,000, and N of 15.

This project has initial project costs of $13 million, as well as initial investment in net working capital of $300,000. This gives us C0 of $13,000,000 and NWC0 of $300,000.

At the end of the project, the assets used in this project can be sold at an estimated $500,000, so S is $500,000.

These assets have a CCA rate of 30% (let’s represent this with the symbol d), and it is assumed that the CCA asset-class will remain open at the end of the project (which means that we do not have to worry about terminal loss or recaptured CCA).

We ask again: XYZ’s marginal corporate tax rate is 40 percent and its required rate of return on this type of projects is 10 percent (so, T = 40 percent and r = 10 percent). Should the company take on this project?

To summarize, we have

Before-tax OCF = $2,000,000

N = 15

C0 = $13,000,000

NWC0 = $300,000

S = $500,000

d = 0.3

T = 0.4

r = 0.1

Slide #9: Numerical Example (cont.) – Step 1

So, now let’s follow the 5 steps I outlined earlier.

Step 1: Identify the incremental cash outflows and inflows:

The cash outflows consist of initial cost of 13 million, so

C0 = $13,000,000,

and initial investment in NWC of $300,000, so

NWC0 = $300,000.

The cash inflows consist of before tax operating cash flow of $2,000,000. We then calculate

after tax operating cash flow as:

ATOCF = $2,000,000 x (1 – 0.4) = $1,200,000.

Salvage is $500,000, and so

S = $500,000.

Net working capital recovery is $300,000, and so

NWCN = $300,000

And last, but not least, CCA Tax Shield is a declining balance process to infinity, which will be discussed in more depth later.

Slide #10: Numerical Example (cont.) – Step 2

Step 2: Identify Types of cash-flows:

The C0 and NWC0 are both lump sum cash-flows occurring at time 0.

The after-tax OCFs are an annuity occurring from time 1 to time 15.

The salvage value and the NWC recovery are both lump sum cash-flows occurring at time 15.

And the CCA tax shield is a perpetuity based on a declining-balance calculation method.

Slide #11: Numerical Example (cont.) – Step 3

Step 3: Identify the present-value formula that goes with each type of cash-flows:

C0 and NWC0 are both lump sums occurring at time 0, and so there is no need to do any discounting to present value.

The after-tax OCFs (ATOCF), being an annuity, are discounted to present-value using the PV(annuity) formula:

PV(ATOCF) = ATOCF x [(1-(1/(1+r)N)) / r].

The salvage value (S) and NWC recovery (NWCN) are both discounted using the present value of lump sum formula:

PV(S) = S/(1+r)N

PV(NWCN) = NWCN / (1+r)N.

The present value of CCA tax shields is found by using the long formula for PV(CCATS):

PV(CCATS) = [(C0 d T/(r + d)) x ((1 + .5r)/(1 + r))] – [(S d T / (r + d)) / (1 + r)N].

The mechanics of using this PV(CCATS) formula are covered in another lecture: Lecture 7.

Slide #12: Numerical Example (cont.) – Step 4

Step 4: Calculate the present values of cash flows using the appropriately identified PV formulas:

For the cash outflows, we do not need to discount the initial investments C0 and NWC0, and so

PV(C0) = C0 = 13,000,000

PV(NWC0) = NWC0 = 300,000

Slide #13: Numerical Eample (cont.) – Step 4

The present value of the after-tax operating cash flows is calculated as:

PV(ATOCF) = 1,200,000 x (1 - (1/(1+0.1)15)) / 0.1 = 9,127,295.408

The present value of the salvage value is calculated as:

PV(S) = 500,000 / (1 + 0.1)15 = 119,696.0247

The present value of the NWC recovery is calculated as:

PV(NWCN) = 300,000 / (1 + 0.1)15 = 71,817.61481

The present value of the CCA tax shields is calculated as:

PV(CCATS) = ((13,000,000 x 0.3 x 0.4 / (0.1 + 0.3)) x ((1 + .5(0.1)) / (1 + 0.1))) – ((500,000 x 0.3 x 0.4 / (0.1 + 0.3)) x (1 / (1 + 0.1)15)) = 3,686,818.465

Slide #14: Numerical Example (cont.) – Step 5

We are now ready to do Step 5: Calculate the NPV.

The NPV is the present value of all cash inflows minus the present value of all cash outflows. This means that we can calculate the NPV as the sum of the present values of the ATOCF, S, NWC recovery, and CCATS, minus the initial cost and NWC investment.

This gives us

NPV = PV(All Cash Inflows) – PV(All Cash Outflows)

= PV(ATOCF) + PV(S) + PV(NWCN) + PV(CCATS) – C0 – NWC0

Plugging in the numbers we calculated before, we get

NPV

= 9,127,295.408 + 119,696.0247 + 71,817.61481 + 3,686,818.465 – 13,000,000 – 300,000 \

= -294,372.4876.

Therefore, with a negative NPV, the NPV rule says that the company should reject this project.

Slide #14: No Practice, No Gain

And now, to our favourite activity: practice, practice, practice. No practice, no gain! Try this question and see if you can get the project NPV.

ABC Inc is considering purchasing new equipment that will cost $1 million and will generate $500,000 in cost savings in each of the next 3 years. The equipment has a CCA rate of 30% and will have a salvage value of $100,000 at the end of the third year. If the new equipment is purchased, inventory will increase by $15,000 and accounts payable will increase by $10,000. ABC’s marginal corporate tax rate is 35% and its required rate of return on the equipment is 12%. Should ABC purchase this new equipment?

This project has an NPV of 69,144.72. The conclusion is to buy this piece of equipment.

And that, my friends, is the end of this lecture on NPV calculation.

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