Net present value (NPV) of **investment appraisal** and Internal Rate of Return (IRR) of **capital budgeting** are the two methods of using the discounted cash flow (DCF) to evaluate capital **investment**. This article is a continuation of two previous investment appraisal articles **introduction to investment appraisal **and** investment appraisal- ROCE AND PAYBACK**

**WHAT IS DISCOUNTED CASH FLOW (DCF)?**

DCF is an investment appraisal technique which takes into account both the timings of **cash flows** and total profitability over a project’s life.

**IMPORTANT POINTS ABOUT DCF**

- DCF is concerned with the cash flows of a project, not the accounting profits. The reason for this is that cash flows show the costs and benefits of a project when actually executed and ignore notional costs like
**depreciation**. - Timing of cash flows- which is very important- is fully recognized by DCF by discounting the cash flows. The effect of discounting is to give a bigger value per $1 for each flow that occurs earlier – time vale of money ($1 earned today is much better than $1 earned at the end of one year.)

For other terms used in investment appraisal, read up **terms used in investment appraisal techniques**.

**TIMING OF CASH FLOWS: CONVENTIONS USED IN DCF**

For investment appraisals evaluation conducted using the NPV and IRR methods of capital budgeting to be meaningful, cash flows must be rightly and correctly timed. As a general rule, the following guidelines may be applied – especially if you are writing **accounting professional exams** that require you to time initial investment, working capital, and tax cash flows, **ACCA and CIMA** are good examples.

**Threes ways to time cash flows:**

- Any cash outflow to be incurred at the beginning of an investment project (‘now’) occurs in year 0. Note that the present value (PV) of $1 now, in year 0, is $1 irrespective of the value r. r= compound rate of return per time period, expressed as a proportion.
- Any cash outflow, saving, and inflow that occurs during the course of a time frame (one year for instance) is assumed to have occurred all at once at the end of the time period. For example, receipt of $12,500 during year 1 is taken to have occurred at the end of year 1.
- Any cash outlay or inflow that occurs at the beginning of a time period (at the beginning of year one for instance) is assumed to have occurred at the end of the year before i.e., previous year. So, a cash outlay of $1,200 at the beginning of year 2 is seen as occurring at the end of year 1.

Please make sure you are at home with the timing conventions of these cash flows for good understanding of NPV and IRR!

Net present value (NPV)

NPV is the value we get by discounting all cash outflows and inflows of a capital investment project by a chosen cost of capital or target rate of return. This is to say that the NPV method of investment appraisal compares present value (PV) of all cash inflows from an investment with the PV of all the cash outlays from an investment. Mathematically, NPV = PV of inflows – PV of outflows.

**Example 1: example of NPV calculation**

A company is considering capital investment, where the estimated cash flows for this project over a period of four years are as follows:

Years Cash Flows

$

0 (i.e. now) (100,000)

1 160,000

2 90,000

3 20,000

4 30,000

The company’s cost of capital is 15%. You are required to calculate the NPV of the project and assess whether it should be undertaken or not.

**Solution**

Years Cash flows Discount factor 15% Present value

$ $

0 (100,000) 1.000 (100,000)

1 160,000 1/ (1.15)^{1} = 0.870 139,200

2 90,000 1/ (1.15)^{2} = 0.756 68,040

3 20,000 1/ (1.15)^{3 }= 0.658 13,160

4 30,000 1/ (1.15)^{4 }= 0.572 17,160

NPV= 137,560

** **

Note: remember that the discount factor for any cash flow in year 0 (now) is always = 1, regardless of what the cost of capital is.

**NPV INVESTMENT APPRAISAL / CAPITAL BUDGETING DECISION CRITERION**

There are three things that manager can do after calculating the NPV of a project. They are:

- Accept the project when the NPV is positive. This is a situation where returns from investment’s cash inflow are in excess of the cost of capital.
- Reject the project when the NPV is negative. You refuse to undertake a project when the return from investment’s cash inflow is below cost of capital.
- Become indifferent from financial perspective. Managers consider other
**non-financial factors**before deciding on whether to undertake a project or not when the return from investment’s cash inflow is the same as cost of capital.

Based on the above investment decision criterion, we can accept the project as it is a good investment that has a positive NPV of $137,560.

Internal rate of return (IRR)

Unlike the NPV method of investment appraisal that is calculated by discounting streams of cash flows with a target rate of return or cost of capital, and the difference taken as the NPV, the IRR method of capital appraisal is to calculate the exact DCF rate of return which a project or investment opportunity is expected to achieve; in other words, the rate at which the NPV is zero. Investment decision under the IRR rule is to accept any project whose IRR or DCF yield exceeds a target rate or return. Without a computer, IRR is calculated using a trial and error, crude method called interpolation method.

**HOW TO CALCULATE INTERNAL RATE OF RETURN (IRR)**

Three steps are involved in calculating IRR, they are:

Step 1, calculate the NPV of the project using a company’s cost of capital

Step 2, calculate the NPV of the project using another discount rate,

- If the NPV is positive, use a second rate that is greater than the first rate
- If the NPV is negative, use a second rate that is less than the first rate.

Step 3, use the two NPV values to estimate the IRR. The formula to apply is thus:

IRR= a + {[NPVa/ NPVa – NPVb] (b-a) } %

Where; a= the lower of the two rates of return used

b= the higher of the two rates of return used

NPVa = the NPV obtained using rate a

NPVb = the NPV obtained using rate b

**Example 2: example of IRR calculation**

A company ants to decide whether to buy a machine for $100,000 which will save costs of $25,000 per annum for 5 years and which will have a residual value of $15,000 at the end of year 5. If it is the company’s policy to undertake projects only if they are expected to yield a DCF return of 10% or more, decide whether this project should be undertaken or not.

**Solution:**

**Step 1 calculate the NPV using the company’s cost of capital of 10%**

Year Cash flow PV factor 10% PV of cash flow

$ $

0 (100,000) 1.000 (100,000)

1-5 25,000 3.791 94,775

5 15,000 0.621 9,315

NPV= 4090

Step 2, calculate the second NPV, using a rate that is greater than the first rate, as the first rate gave a positive answer.

Let us try 12%.

Year Cash flow PV factor 12% PV of cash flow

$ $

0 (100,000) 1.000 (100,000)

1-5 25,000 3.605 90,125

5 15,000 0.567 8,505

NPV= (1,370)

This figure is close to zero and is negative, but we haven’t got what we are looking for which is somewhere in-between.

Step 3, use the two values to interpolate.

Substituting the above values for our formula we have:

IRR= 10+ {[4,090/4,090+1,370] x (12-10)} %

= 10+ 4090/5460 x 2 /100

IRR= 10.10%

This project should be accepted since its IRR is marginally above the cost of capital.

I would like to bring to your attention that IRR and NPV as methods of investment appraisal are not without some disadvantages and limitations. I would have loved to discuss all those here but you will agree with me that this article is already long, and will there be too long if I include that. Watch out for another article on that.

THE FIGURE THAT YOU HAVE OBTAINED IS WRONG IT SUPPOSE TOBE

IRR= 10.01498…….

Aziz, thanks for pointing that error out. you are right. The correct answer is 10.015%

I have a finance homework question for you…..if you have a project that has a cost of 15,000, cost of capital is 14%, and the IRR is 17%, how would you figure out the capital budget?

Hi gabe,

could you please rephrase your question? Thanks for asking.

This is helpfull. However here is my question.

I invest $2000 in a project which is expected to achieve cost saving as follows

year 1 to year 3 equals $400 per year, year 4 and year 5 equals $500, year 6,7 and 8 equals $450, year 9 and 10 equals $400.

project will last for 10years, at the end of which it will have zero residual value. The company will require a target ARR of 25 percent and a disired payback period of 4 years.

The company also requires a return on capital of 15 percent after tax.

Calculate the payback period, NPV and IRR.Also advise the management on which criteria to use.

Thank you very much!

Welcome

thanks 4 ur solving

this is some good work done, thax alot

You are welcome, Rhodah.

This is quality work, it has really helped me, keep up with good work.

Thanks Mimiana. I am happy to hear that you did enjoy this article on capital budgeting and investment appraisal.

Two firms are identical in all respect except for their capital structure. Their NOI is as follows:

• -$ 500 with a probability of ¼

• $ 1 000 with a probability of ½

• $ 2 000 with a probability of ¼

The unlevered firm issues 10000 shares at $1 per share. The levered firm issues $ 5000 debt at 3% and 5 000 shares. You observe that the levered firm’s stock price jumps to $1.50 .

a. Suppose you hold 1000 shares for the levered firm. What is the distribution of the rate of return on your investment.?

b. What arbitrage transaction would you perform? Show the distribution of income after the transaction.

c. Suppose you invest the money saved in the arbitrage transaction of at an interest rate of 3%.what is the distribution of when you include the return on your savings.

d. How would your answer change if you had to pay a 0.5% transaction cost on the volume of stock purchase?

assist me

Two firms are identical in all respect except for their capital structure. Their NOI is as follows:

• -$ 500 with a probability of ¼

• $ 1 000 with a probability of ½

• $ 2 000 with a probability of ¼

The unlevered firm issues 10000 shares at $1 per share. The levered firm issues $ 5000 debt at 3% and 5 000 shares. You observe that the levered firm’s stock price jumps to $1.50 .

a. Suppose you hold 1000 shares for the levered firm. What is the distribution of the rate of return on your investment.?

b. What arbitrage transaction would you perform? Show the distribution of income after the transaction.

c. Suppose you invest the money saved in the arbitrage transaction of at an interest rate of 3%.what is the distribution of when you include the return on your savings.

d. How would your answer change if you had to pay a 0.5% transaction cost on the volume of stock purchase?

Two firms are identical in all respect except for their capital structure. Their NOI is as follows:

• -$ 500 with a probability of ¼

• $ 1 000 with a probability of ½

• $ 2 000 with a probability of ¼

The unlevered firm issues 10000 shares at $1 per share. The levered firm issues $ 5000 debt at 3% and 5 000 shares. You observe that the levered firm’s stock price jumps to $1.50 .

a. Suppose you hold 1000 shares for the levered firm. What is the distribution of the rate of return on your investment.?

b. What arbitrage transaction would you perform? Show the distribution of income after the transaction.

c. Suppose you invest the money saved in the arbitrage transaction of at an interest rate of 3%.what is the distribution of when you include the return on your savings.

d. How would your answer change if you had to pay a 0.5% transaction cost on the volume of stock purchase?

please assist me

dear,

it is nice