NPV and IRR

The investment decision criteria Net Present Value (NPV) and Internal Rate of Return (IRR) are the two main criteria for evaluating investment projects. They help answer the question: does the project create value and is it worth investing in? Net Present Value (NPV) NPV is the sum of the present values of all project cash flows, including the initial investment. NPV = Σ CFt / (1+r)^t, where t ranges from 0 to n, CF₀ is usually negative (investment). NPV Rule: invest if NPV > 0. Positive NPV means that the project creates value beyond the required return. NPV = 0 – the project gives exactly the required return.

NPV

NPV as value creation: NPV shows in dollars how much value the project creates. NPV = $1M means a wealth increase of $1M in today's dollars.

NPV calculation example

Project: invest $100,000 today, receive $30,000 per year for 5 years. Required return is 12%.

NPV = -100,000 + 30,000/(1.12) + 30,000/(1.12)² + 30,000/(1.12)³ + 30,000/(1.12)⁴ + 30,000/(1.12)⁵ = -100,000 + 108,143 = $8,143.

NPV > 0, accept the project. The project creates $8,143 of value.

Internal Rate of Return (IRR)

IRR is the discount rate at which NPV = 0. This is the “internal” return of the project, implied by its cash flows.

IRR Rule: invest if IRR > required return (hurdle rate). If IRR exceeds cost of capital, the project creates value.

Calculation: IRR is found by solving NPV = 0 for r. No closed-form solution for uneven cash flows – iterative calculation (Excel: IRR function).

IRR calculation example

Same project: -$100,000 today, $30,000 per year for 5 years. Find r where NPV = 0.

Trial: at 15%, NPV = -100,000 + 30,000 × [(1-1.15^(-5))/0.15] = -100,000 + 100,565 = $565 (positive). At 16%, NPV ≈ -$1,000 (negative). IRR is between 15% and 16%, approximately 15.24%. IRR (15.24%) > hurdle (12%), accept the project. Consistent with NPV decision.

NPV vs IRR: when they agree

For independent, conventional projects (one initial outflow, then inflows), NPV and IRR provide the same accept/reject decision.

Conventional: if initial NPV is negative, NPV increases as the discount rate decreases, crossing zero at IRR. If IRR > hurdle, NPV > 0.

NPV vs IRR: when they diverge

Mutually exclusive projects: choosing between projects. NPV ranks correctly (choose the highest NPV), IRR may mislead.

Example:
Project A: invest $100, IRR 50%, NPV $20.
Project B: invest $1,000, IRR 30%, NPV $150.

IRR says A is better, NPV says B. NPV is correct – B creates more value.

Scale problem: IRR ignores scale. A small project with high IRR may be less valuable than a large project with moderate IRR.

Multiple IRRs

Non-conventional cash flows (sign changes more than once) may have multiple IRRs or none.

Example: -$100, +$230, -$132 (sign changes twice). May have IRR 10% and 20%. Which is “the” IRR?

Solution: use NPV, modified IRR (MIRR), or NPV profile analysis. IRR is problematic for non-conventional flows.

Modified IRR (MIRR)

MIRR addresses some IRR problems: assumes reinvestment at cost of capital (not IRR), produces a single solution.

Calculation: terminal value of inflows (compounded at cost of capital) / present value of outflows. Find the rate equating these.

More realistic reinvestment assumption, but adds complexity. NPV remains the gold standard.

Crossover rate

For mutually exclusive projects: crossover rate – the rate at which both projects have equal NPV. Below the crossover: one project is preferred. Above the crossover: the other project is preferred. Important for ranking decisions.

Calculate: find IRR of differential cash flows (A - B).

NPV profile

Graph NPV versus discount rate. Shows how NPV changes with rate. Visual representation of the project's sensitivity to the discount rate.

X-intercept = IRR (NPV = 0). For comparing projects, plot both profiles – crossover is visible.

Practical recommendations

Use NPV as the primary criterion – measures value creation in dollars, no ambiguity.

Report IRR for intuition – percentage return is intuitive for stakeholders.

Be cautious with IRR for non-conventional projects, mutually exclusive decisions, different scales.

NPV requires discount rate input – critical assumption. Sensitivity analysis on the rate is important.