NPV and IRR methods are closely related because:
i) both are timeadjusted 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 k_{1}, NPV is positive and IRR > k_{1}: accept project.
If cash flows are discounted at k_{2}, NPV is negative and IRR < k_{2}: 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 C_{t} 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 onestorey (Project A)
or fivestorey (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 oneyear duration:
IRR_{A}: _{}
$11,500 = $9,500 (1 +R_{A})
_{}
_{}
= 1.211
therefore IRR_{A} = 21%
IRR_{B}: _{}
$18,000 = $15,000(1 + R_{B})
_{}
= 1.21
therefore IRR_{B} = 20%
Decision:
Assuming that k = 10%, both projects are acceptable because:
NPV_{A} and NPV_{B} are both positive
IRR_{A} > k AND IRR_{B} > k
Which project is a \"better option\" for Agritex?
If we use the NPV method:
NPV_{B} ($1,363.64) > NPV_{A} ($954.55): Agritex should choose Project B.
If we use the IRR method:
IRR_{A} (21%) > IRR_{B} (20%): Agritex should choose Project A. See figure 6.2.
Figure 6.2 NPV vs IRR: Dependent projects
Up to a discount rate of k_{o}: project B is superior to project A, therefore project B is preferred to project A.
Beyond the point k_{o}: 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%.
NPV_{A} = $1,500 x PVFA at 10% for 3 years
= $1,500 x 2.487
= $3,730.50  $2,500.00
= $1,230.50.
NPV_{B} == $7,000 x PVFA at 10% for 3 years
= $7,000 x 2.487
= $17,409  $14,000
= $3,409.00.
IRR_{A} = _{}
_{}
= 1.67.
Therefore IRR_{A} = 36% (from the tables)
IRR_{B} = _{}
_{}
= 2.0
Therefore IRR_{B} = 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,
NPV_{A} = NPV_{B} or NPV_{A}  NPV_{B} = 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 I_{o} 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  I_{o} = 0
PV = I_{o}
Dividing both sides by I_{o} we get:
_{}
PI of 1.2 means that the project\'s profitability is 20%. Example:

PV of CF 
I_{o} 
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 shortterm 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 straightline 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.