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Subtopic:

Net present value vs internal rate of return

 

NPV and IRR methods are closely related because:

i) both are time-adjusted measures of profitability, and
ii) their mathematical formulas are almost identical.

So, which method leads to an optimal decision: IRR or NPV?

a) NPV vs IRR: Independent projects

Independent project: Selecting one project does not preclude the choosing of the other.

With conventional cash flows (-|+|+) no conflict in decision arises; in this case both NPV and IRR lead to the same accept/reject decisions.

Figure 6.1 NPV vs IRR Independent projects

If cash flows are discounted at k1, NPV is positive and IRR > k1: accept project.

If cash flows are discounted at k2, NPV is negative and IRR < k2: reject the project.

Mathematical proof: for a project to be acceptable, the NPV must be positive, i.e.

Similarly for the same project to be acceptable:

where R is the IRR.

Since the numerators Ct are identical and positive in both instances:

implicitly/intuitively R must be greater than k (R > k);
If NPV = 0 then R = k: the company is indifferent to such a project;
Hence, IRR and NPV lead to the same decision in this case.

b) NPV vs IRR: Dependent projects

NPV clashes with IRR where mutually exclusive projects exist.

Example:

Agritex is considering building either a one-storey (Project A) or five-storey (Project B) block of offices on a prime site. The following information is available:


Initial Investment Outlay

Net Inflow at the Year End

Project A

-9,500

11,500

Project B

-15,000

18,000

Assume k = 10%, which project should Agritex undertake?

= $954.55

= $1,363.64

Both projects are of one-year duration:

IRRA:

$11,500 = $9,500 (1 +RA)

= 1.21-1

therefore IRRA = 21%

IRRB:

$18,000 = $15,000(1 + RB)

= 1.2-1

therefore IRRB = 20%

Decision:

Assuming that k = 10%, both projects are acceptable because:

NPVA and NPVB are both positive
IRRA > k AND IRRB > k

Which project is a \"better option\" for Agritex?

If we use the NPV method:

NPVB ($1,363.64) > NPVA ($954.55): Agritex should choose Project B.

If we use the IRR method:

IRRA (21%) > IRRB (20%): Agritex should choose Project A. See figure 6.2.

Figure 6.2 NPV vs IRR: Dependent projects

Up to a discount rate of ko: project B is superior to project A, therefore project B is preferred to project A.

Beyond the point ko: project A is superior to project B, therefore project A is preferred to project B

The two methods do not rank the projects the same.

Differences in the scale of investment

NPV and IRR may give conflicting decisions where projects differ in their scale of investment. Example:

Years

0

1

2

3

Project A

-2,500

1,500

1,500

1,500

Project B

-14,000

7,000

7,000

7,000

Assume k= 10%.

NPVA = $1,500 x PVFA at 10% for 3 years
= $1,500 x 2.487
= $3,730.50 - $2,500.00
= $1,230.50.

NPVB == $7,000 x PVFA at 10% for 3 years
= $7,000 x 2.487
= $17,409 - $14,000
= $3,409.00.

IRRA =

= 1.67.

Therefore IRRA = 36% (from the tables)

IRRB =

= 2.0

Therefore IRRB = 21%

Decision:

Conflicting, as:

NPV prefers B to A
IRR prefers A to B


NPV

IRR

Project A

$ 3,730.50

36%

Project B

$17,400.00

21%

See figure 6.3.

Figure 6.3 Scale of investments

To show why:

i) the NPV prefers B, the larger project, for a discount rate below 20%

ii) the NPV is superior to the IRR

a) Use the incremental cash flow approach, \"B minus A\" approach
b) Choosing project B is tantamount to choosing a hypothetical project \"B minus A\".


0

1

2

3

Project B

- 14,000

7,000

7,000

7,000

Project A

- 2,500

1,500

1,500

1,500

\"B minus A\"

- 11,500

5,500

5,500

5,500

IRR\"B Minus A\"

= 2.09

= 20%

c) Choosing B is equivalent to: A + (B - A) = B

d) Choosing the bigger project B means choosing the smaller project A plus an additional outlay of $11,500 of which $5,500 will be realised each year for the next 3 years.

e) The IRR\"B minus A\" on the incremental cash flow is 20%.

f) Given k of 10%, this is a profitable opportunity, therefore must be accepted.

g) But, if k were greater than the IRR (20%) on the incremental CF, then reject project.

h) At the point of intersection,

NPVA = NPVB or NPVA - NPVB = 0, i.e. indifferent to projects A and B.

i) If k = 20% (IRR of \"B - A\") the company should accept project A.

This justifies the use of NPV criterion.

Advantage of NPV:

It ensures that the firm reaches an optimal scale of investment.

Disadvantage of IRR:

It expresses the return in a percentage form rather than in terms of absolute dollar returns, e.g. the IRR will prefer 500% of $1 to 20% return on $100. However, most companies set their goals in absolute terms and not in % terms, e.g. target sales figure of $2.5 million.

The timing of the cash flow

The IRR may give conflicting decisions where the timing of cash flows varies between the 2 projects.

Note that initial outlay Io is the same.


0

1

2

Project A

- 100

20

125.00

Project B

- 100

100

31.25

\"A minus B\"

0

- 80

88.15

Assume k = 10%


NPV

IRR

Project A

17.3

20.0%

Project B

16.7

25.0%

\"A minus B\"

0.6

10.9%

IRR prefers B to A even though both projects have identical initial outlays. So, the decision is to accept A, that is B + (A - B) = A. See figure 6.4.

Figure 6.4 Timing of the cash flow

The horizon problem

NPV and IRR rankings are contradictory. Project A earns $120 at the end of the first year while project B earns $174 at the end of the fourth year.


0

1

2

3

4

Project A

-100

120

-

-

-

Project B

-100

-

-

-

174

Assume k = 10%


NPV

IRR

Project A

9

20%

Project B

19

15%

Decision:

NPV prefers B to A
IRR prefers A to B.

The profitability index - PI

This is a variant of the NPV method.

Decision rule:

PI > 1; accept the project
PI < 1; reject the project

If NPV = 0, we have:

NPV = PV - Io = 0
PV = Io

Dividing both sides by Io we get:

PI of 1.2 means that the project\'s profitability is 20%. Example:


PV of CF

Io

PI

Project A

100

50

2.0

Project B

1,500

1,000

1.5

Decision:

Choose option B because it maximises the firm\'s profitability by $1,500.

Disadvantage of PI:

Like IRR it is a percentage and therefore ignores the scale of investment.

The payback period (PP)

The CIMA defines payback as \'the time it takes the cash inflows from a capital investment project to equal the cash outflows, usually expressed in years\'. When deciding between two or more competing projects, the usual decision is to accept the one with the shortest payback.

Payback is often used as a \"first screening method\". By this, we mean that when a capital investment project is being considered, the first question to ask is: \'How long will it take to pay back its cost?\' The company might have a target payback, and so it would reject a capital project unless its payback period were less than a certain number of years.

Example 1:

Years

0

1

2

3

4

5

Project A

1,000,000

250,000

250,000

250,000

250,000

250,000

For a project with equal annual receipts:

= 4 years

Example 2:

Years

0

1

2

3

4

Project B

- 10,000

5,000

2,500

4,000

1,000

Payback period lies between year 2 and year 3. Sum of money recovered by the end of the second year

= $7,500, i.e. ($5,000 + $2,500)

Sum of money to be recovered by end of 3rd year

= $10,000 - $7,500

= $2,500

= 2.625 years

Disadvantages of the payback method:

It ignores the timing of cash flows within the payback period, the cash flows after the end of payback period and therefore the total project return.

It ignores the time value of money. This means that it does not take into account the fact that $1 today is worth more than $1 in one year\'s time. An investor who has $1 today can either consume it immediately or alternatively can invest it at the prevailing interest rate, say 30%, to get a return of $1.30 in a year\'s time.

It is unable to distinguish between projects with the same payback period.

It may lead to excessive investment in short-term projects.

Advantages of the payback method:

Payback can be important: long payback means capital tied up and high investment risk. The method also has the advantage that it involves a quick, simple calculation and an easily understood concept.

The accounting rate of return - (ARR)

The ARR method (also called the return on capital employed (ROCE) or the return on investment (ROI) method) of appraising a capital project is to estimate the accounting rate of return that the project should yield. If it exceeds a target rate of return, the project will be undertaken.

Note that net annual profit excludes depreciation.

Example:

A project has an initial outlay of $1 million and generates net receipts of $250,000 for 10 years.

Assuming straight-line depreciation of $100,000 per year:

= 15%

= 30%

Disadvantages:

It does not take account of the timing of the profits from an investment.

It implicitly assumes stable cash receipts over time.

It is based on accounting profits and not cash flows. Accounting profits are subject to a number of different accounting treatments.

It is a relative measure rather than an absolute measure and hence takes no account of the size of the investment.

It takes no account of the length of the project.

it ignores the time value of money.

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