Investment Decision Rules: NPV, IRR, Payback

The four rules in one table

Rule	Decision	Strength	Failure mode
NPV	Accept if NPV > 0; choose highest NPV among mutually exclusive	Always correct	Requires a discount rate
IRR	Accept if IRR > cost of capital	Easy to communicate, useful diagnostic	Multiple IRRs for non-conventional CFs; misranks by scale and timing
Payback	Accept if recouped within cutoff	Quick screen for liquidity	Ignores TVM; ignores cash flows after cutoff
ARR	Accept if accounting return > hurdle	Aligns with financial reporting	Uses accounting income (not CFs); ignores timing

You can apply all four to most projects, but the rankings can disagree. When they do, NPV wins.

NPV in one minute

You already saw this in Unit 1:

$$\text{NPV} = C_0 + \sum_{t=1}^{T} \frac{C_t}{(1+r)^t}.$$

Positive: take it. Among mutually exclusive choices: take the highest.

Initial outflow CF₀: $-1000Base annual CF: $250Growth g: 3.0%Discount rate r: 8.0%Horizon T (years): 8Reset

NPV = $578, IRR ≈ 21.2%

Cashflows by year

NPV sensitivity to discount rate

Current parameters: CF₀ = $-1000, first annual cash flow = $250, growth = 3.0%, r = 8.0%, and T = 8 years. Cash flows grow from the year-1 base and are discounted at the same rate shown in the sensitivity chart.

IRR — when it fails

Failure 1: non-conventional cash flows. Suppose a project requires an initial outlay, generates positive cash flows for several years, then needs a large cleanup expenditure at the end (a mine that must restore the land, an oil well requiring abandonment cost). The cash flow sign changes more than once. Multiple IRRs exist. The IRR “rule” gives no unique answer.

Example. Cash flows: $-100$, $+230$, $-132$. The NPV equation has two real roots — IRR = 10% and IRR = 20%. Both make NPV = 0. Which is “the” IRR? Neither. The NPV at a 15% cost of capital is $+0.38$; the project marginally creates value. NPV gives a clean answer; IRR doesn’t.

Failure 2: scale. Two mutually exclusive projects, cost of capital = 10%:

• A: $-1$, $+2$ ⇒ IRR = 100%, NPV = $0.82
• B: $-1$ million, $+1.2$ million ⇒ IRR = 20%, NPV ≈ $90,!900

A has higher IRR; B has $90,900 in NPV vs A’s $0.82. Which do you want? B. IRR favors small projects with high percentage returns; NPV correctly rewards dollar value created.

Failure 3: timing. Mutually exclusive projects with different cash-flow patterns. Project Short has all cash flows in years 1-3; Project Long has them in years 3-10. At low cost of capital, Long might have higher NPV; at high cost of capital, Short might. IRR gives a single number that doesn’t capture this — you need NPV at the actual cost of capital.

Modified IRR (MIRR) — partial fix

MIRR addresses failure 1 by:

1. Discounting all negative cash flows back to time 0 at a finance rate.
2. Compounding all positive cash flows to the horizon $T$ at a reinvestment rate.
3. Solving for the single growth rate that turns PV of negatives into FV of positives.

This eliminates multiple IRRs and uses realistic reinvestment assumptions. But it still doesn’t solve failures 2 and 3. NPV still wins.

Payback — when to use it (and when not to)

Payback = the number of years to recoup the initial investment. Discounted payback = same but cash flows discounted.

Where it works. As a liquidity screen, especially for early-stage startups with limited cash runway. “If this doesn’t pay back in 18 months, we’re dead anyway, so don’t bother with NPV.” That’s defensible.

Where it fails.

• A 30-year solar farm with positive NPV of $50M but payback of 8 years fails a 5-year payback cutoff. You reject a hugely value- creating project.
• Two projects with identical 3-year paybacks but very different 10-year totals look identical to payback. NPV sees the difference.

Use payback as a sanity check, never as the primary rule.

ARR — used in practice, used wrongly

Accounting rate of return = average annual net income / average book value of assets. It’s used because it aligns with how managers are evaluated (financial-statement metrics). It’s wrong for project selection because:

• It uses accruals, not cash flows. Two projects with identical cash flows but different depreciation policies get different ARRs.
• It ignores timing.
• The denominator depends on book value, which depends on accounting choices unrelated to economic value.

Useful for performance review of existing units; useless for forward-looking project decisions.

When NPV and IRR agree

For a single conventional project (one negative cash flow then all positives), NPV and IRR agree on accept/reject. Specifically, NPV > 0 if and only if IRR > cost of capital. So if you’re choosing between accepting a single project and doing nothing, IRR is fine — but you’ve already had to specify the cost of capital to compare against.

The disagreements arise in ranking mutually exclusive choices and in non-conventional CF profiles. Real-world capital budgeting hits those cases constantly. Use NPV.

Profitability index: choosing under a constraint

NPV says take every positive-NPV project. Reality often imposes a fixed capital budget. When capital is the single binding resource, rank projects by the profitability index:

$$\text{PI} = \frac{\text{PV of future cash inflows}}{\text{Initial investment}} = 1 + \frac{\text{NPV}}{\text{Initial investment}}.$$

For divisible projects with capital as the only binding resource, take projects in descending PI order until the budget runs out. For all-or-nothing projects, use PI as a screening heuristic and compare the feasible combinations.

Worked example. A $100M capital budget and three independent projects:

Project	Investment	NPV	PI
A	$100M	$40M	1.40
B	$50M	$30M	1.60
C	$50M	$25M	1.50

Ranking by raw NPV grabs A first ($40M NPV), which exhausts the budget for a total of $40M. Ranking by PI takes B (1.60) then C (1.50), which also fits in $100M but creates $30M + $25M = $55M. In this example, the PI ranking is feasible and value-maximizing.

The rule has limits. It assumes a single binding constraint. With two constraints at once, or lumpy all-or-nothing projects, PI ranking can mislead and you need explicit optimization.

Practical decision template

When facing a capital-budgeting decision:

1. Forecast incremental free cash flows (next lesson).
2. Discount at the appropriate cost of capital (Units 4-5).
3. Compute NPV. If $> 0$, project is value-creating.
4. Compute IRR as a diagnostic — useful for communicating with non-technical colleagues, but defer to NPV when they disagree.
5. Skip payback unless a true liquidity constraint binds.