[savio13]

The internal rate of return (IRR)

• The IRR is the discount rate at which the NPV for a reject equals zero. This rate means that the present value of the cash inflows for the project would equal the present value of its outflows.
• The IRR is the break-even discount rate.
• The IRR is found by trial and error.
where r = IRR

IRR of an annuity:

where:
Q (n,r) is the discount factor
Io is the initial outlay
C is the uniform annual receipt (C1 = C2 =....= Cn).

Economic rationale for IRR:

If IRR exceeds cost of capital, project is worthwhile, i.e. it is profitable to undertake.
Net present value vs. internal rate of return
Independent vs dependent projects
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.
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:

ABC Ltd 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 ABC Ltd undertake?
NPVA = 11500/1.1 -9500
= Rs.954.55
NPVB = 18000/1.1 -15000
= Rs.1,363.64
Both projects are of one-year duration:
IRRA:
=11500/RA =9500
Rs.11,500 = Rs.9,500 (1 +RA)
RA = 11500/9500-1
= 1.21-1
therefore IRRA = 21%
IRRB:
Rs.18,000 = Rs.15,000(1 + RB)
RB = 18000/15000 -1
=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 ABC Ltd?
If we use the NPV method:
NPVB (Rs.1,363.64) > NPVA (Rs.954.55): ABC Ltd should choose Project B.
If we use the IRR method:
IRRA (21%) > IRRB (20%): ABC Ltd should choose Project A.
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 = Rs.1,500 x PVFA at 10% for 3 years
= Rs.1,500 x 2.487
= Rs.3,730.50 - Rs.2,500.00
= Rs.1,230.50.
NPVB == Rs.7,000 x PVFA at 10% for 3 years
= Rs.7,000 x 2.487
= Rs.17,409 - Rs.14,000
= Rs.3,409.00.
IRRA = 36%
IRRB = 21%
Decision:
Conflicting, as:
• NPV prefers B to A
• IRR prefers A to B
NPV IRR
Project A Rs. 3,730.50 36%
Project B Rs.17,400.00 21%

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