6.2 Net Present Value & Internal Rate of Return
Key Takeaways
- NPV = sum of discounted cash inflows − initial investment; accept the project if NPV > 0.
- IRR is the discount rate that makes NPV exactly zero; accept if IRR exceeds the hurdle rate (often WACC).
- For mutually exclusive projects, NPV is preferred because it measures dollar value added and assumes reinvestment at the cost of capital.
- IRR's flaw is the assumption that interim cash flows are reinvested at the IRR itself, overstating return for high-IRR projects.
- MIRR fixes the reinvestment problem by compounding inflows forward at the cost of capital.
Time Value of Money
A dollar today is worth more than a dollar tomorrow because it can be invested. Capital budgeting therefore discounts each future cash flow back to present value using a required rate of return (the hurdle rate), usually the firm's WACC (weighted average cost of capital).
The present value of a cash flow received in year t is:
PV = Cash flow_t / (1 + r)^t
Net Present Value (NPV)
NPV is the sum of the present values of all cash flows minus the initial investment:
NPV = Σ [ CF_t / (1 + r)^t ] − Initial investment
Decision Rule
- NPV > 0: accept — the project earns more than the cost of capital and adds shareholder value.
- NPV < 0: reject.
- NPV = 0: indifferent (earns exactly the required return).
NPV is the theoretically preferred metric because it is expressed in dollars of value added and uses a realistic reinvestment assumption (cash flows reinvested at the cost of capital).
Annuity Shortcut
When cash inflows are equal each year, multiply the constant cash flow by the present value of an annuity factor instead of discounting year by year. The annuity factor for n years at rate r equals the sum of the individual PV factors, which speeds up exam calculations. For unequal cash flows you must discount each year separately.
Worked NPV Example
A project costs $50,000 today and returns $20,000 at the end of each of years 1, 2, and 3. The required rate is 10%.
| Year | Cash flow | PV factor @10% | Present value |
|---|---|---|---|
| 1 | $20,000 | 0.9091 | $18,182 |
| 2 | $20,000 | 0.8264 | $16,528 |
| 3 | $20,000 | 0.7513 | $15,026 |
- PV of inflows = $18,182 + $16,528 + $15,026 = $49,736.
- NPV = $49,736 − $50,000 = −$264.
The NPV is slightly negative, so the project is rejected: it does not quite earn the 10% required return. (Using the 3-year annuity factor 2.4869: $20,000 × 2.4869 = $49,738, the small difference being rounding.)
Internal Rate of Return (IRR)
The IRR is the discount rate that makes NPV equal zero — the project's own compounded rate of return.
At IRR: Σ [ CF_t / (1 + IRR)^t ] − Initial investment = 0
Decision Rule
- Accept if IRR > hurdle rate (cost of capital); reject if IRR < hurdle rate.
In the example above, the IRR is just under 10% (because NPV at 10% is negative), confirming the reject decision. IRR is found by trial and error or interpolation between two discount rates that bracket a zero NPV.
Interpolating IRR
If NPV is +$1,200 at 12% and −$800 at 14%, the IRR lies between them. Interpolate: IRR ≈ 12% + [$1,200 / ($1,200 + $800)] × (14% − 12%) = 12% + 0.6 × 2% = 13.2%. Because the NPV profile is curved, interpolation gives an approximation, but it is accurate enough for exam purposes when the bracketing rates are close together.
NPV–IRR Conflict and the Reinvestment Assumption
For independent projects, NPV and IRR always give the same accept/reject answer. For mutually exclusive projects (you can pick only one), they can rank projects differently because of two effects:
- Scale: a small project can have a high IRR but a small NPV; a large project may have a lower IRR but far greater dollar value added.
- Timing: projects with different cash-flow patterns cross over at a specific discount rate.
Why NPV Wins
The core difference is the reinvestment-rate assumption. NPV assumes interim cash flows are reinvested at the cost of capital — realistic. IRR assumes reinvestment at the IRR itself, which overstates return for unusually high-IRR projects. When the two conflict, follow NPV because it measures actual dollar value created.
IRR has two further weaknesses: non-conventional cash flows (sign changes) can produce multiple IRRs, and IRR cannot rank projects of different scale reliably.
Modified Internal Rate of Return (MIRR)
MIRR repairs the reinvestment flaw. All cash inflows are compounded forward to the terminal year at the cost of capital (the reinvestment rate), giving a terminal value; MIRR is then the rate that equates the PV of outflows to that terminal value. MIRR yields a single, more realistic rate and resolves the multiple-IRR problem, while still expressing the answer as a percentage that managers find intuitive.
Practical Decision Guidance
Keep three rules straight for the exam:
- For a single, independent project, NPV and IRR agree — use either.
- For mutually exclusive projects or capital rationing, default to NPV (or PI), never the higher IRR.
- Use MIRR when an examiner stresses the unrealistic reinvestment assumption or a project has non-conventional cash flows.
A further subtlety is the crossover rate — the discount rate at which two projects' NPVs are equal. Below the crossover rate the project with the steeper NPV profile (larger, later cash flows) has the higher NPV; above it, the ranking flips. This explains why low-rate environments favor large, long-horizon projects.
Because NPV directly measures the wealth added to shareholders, it is the metric most consistent with the firm's goal of value maximization. That is why the CMA blueprint treats NPV as the benchmark against which the other techniques are judged.
A project costs $30,000 and returns a single cash inflow of $36,300 at the end of year 2. At a required rate of 10%, what is the NPV (PV factor for year 2 at 10% = 0.8264)?
Two mutually exclusive projects give conflicting rankings: Project A has the higher IRR while Project B has the higher NPV. Which should be chosen and why?