NPV и IRR: что использовать и когда

Net Present Value (NPV, чистая приведённая стоимость) и Internal Rate of Return (IRR, внутренняя норма доходности) — это два показателя, через которые проходит почти каждое решение в корпоративных финансах. Они оба основаны на одной и той же математике — дисконтированных денежных потоках, — но отвечают на немного разные вопросы. В большинстве случаев они дают одинаковую рекомендацию. В заметной меньшинстве случаев они расходятся, и только один из них оказывается прав.

Эта статья рассказывает, что такое NPV, что такое IRR, в каких случаях они совпадают, о четырёх случаях, когда их выводы расходятся (и какому показателю доверять), о корректировке в виде Modified IRR, и почему большинство учебников по финансам в итоге рекомендуют NPV, а не IRR.

What NPV is

The Net Present Value of an investment is the sum of all future cash flows discounted to present value, minus the initial investment.

$\text{NPV}={\sum }_{t=0}^N\frac{C{F}_t}{(1+r{)}^t}$

where $C{F}_0$ is typically negative (the initial outlay) and $C{F}_t$ for $t\ge 1$ are the future cash flows.

The discount rate $r$ is typically WACC for projects evaluated by the firm.

Decision rule: accept the investment if NPV > 0, reject if NPV < 0.

The intuition: NPV is the value added to the firm. A positive NPV project produces returns above the cost of capital — it creates wealth. A negative NPV project destroys wealth.

Example 1: A project requires $1Mtodayandproduces$400k for each of the next 3 years. Discount rate = 10%.

$\text{NPV}=-1,000,000+\frac{400,000}{1.10}+\frac{400,000}{1.10^2}+\frac{400,000}{1.10^3}$

$=-1,000,000+363,636+330,579+300,526=-5,259$

NPV is slightly negative, so reject the project. The cash flows are not quite enough to overcome the 10% cost of capital plus the initial $1M.

What IRR is

The Internal Rate of Return is the discount rate that makes NPV equal zero. It is the rate at which the project just breaks even.

$0={\sum }_{t=0}^N\frac{C{F}_t}{(1+\text{IRR}{)}^t}$

There is no closed-form solution in general; IRR is found numerically.

Decision rule: accept the project if IRR > hurdle rate (typically WACC), reject otherwise.

Example 2 (same numbers as Example 1): Find IRR for −$1M,$400k, $400k,$400k.

By trial and iteration: IRR is approximately 9.7%. Since 9.7% < 10% (the hurdle rate), reject. This is consistent with the negative NPV.

The intuition: IRR is the rate of return implicit in the project's cash flows. If IRR is higher than what you can earn elsewhere on capital of similar risk, the project adds value. If lower, it destroys value.

When NPV and IRR agree

For the standard case — a single up-front investment followed by a stream of positive cash flows — NPV and IRR give the same recommendation. NPV > 0 if and only if IRR > hurdle rate.

This covers maybe 70% of capital budgeting decisions: build a new factory, invest in equipment, launch a product, develop a property. Both NPV and IRR work.

In this case, IRR is sometimes preferred because:

• IRR is a rate, easily compared across projects of different sizes.
• IRR doesn't require choosing a discount rate to compute (the rate is the output).
• IRR is intuitive to non-finance stakeholders ("this project returns 18%").

But IRR has known problems in specific situations.

When NPV and IRR disagree

Four cases where the metrics disagree, and only NPV is reliable.

Case 1 — Mutually exclusive projects of different scales

You can choose Project A or Project B, but not both. They have different sizes.

Project A: −$10Mnow,+$15M in year 1. NPV at 10% = $13.6M.IRR=50\ast \ast ProjectB\ast \ast :-$100M now, +$120Minyear1.NPVat10$109.1M. IRR = 20%.

IRR says A is better (50% > 20%). NPV says B is better ($109.1M>$13.6M).

Which is right? If A and B are mutually exclusive and you have to choose one, NPV is right. You want B — it adds more value. Yes, B's return rate is lower, but you can't reinvest the unused capital at A's 50% rate (otherwise you'd be doing A and whatever else; they're not mutually exclusive). The IRR comparison implicitly assumes you can reinvest at IRR, which is false.

Rule: when choosing among mutually exclusive projects, use NPV.

Case 2 — Mutually exclusive projects with different time profiles

Project A: −$1000now,+$2000 in year 1. NPV at 10% = $818.IRR=100\ast \ast ProjectB\ast \ast :-$1000 now, +$5000inyear5.NPVat10$2103. IRR = 38%.

IRR says A. NPV says B.

Same fix: NPV is right. B creates more value in dollar terms.

Case 3 — Multiple IRRs

For non-conventional cash flow patterns (cash flow signs change more than once), IRR can have multiple solutions or no real solution.

Example: −$1000now,+$3000 in year 1, −$2200 in year 2. (Investment, big payoff, then significant cleanup cost.)

Setting NPV = 0 and solving: there are two real IRRs, at approximately 10% and 110%. Both make NPV = 0.

Which is "the" IRR? Neither, really. The IRR concept breaks down for cash flow patterns that change sign more than once. NPV remains well-defined and unambiguous.

Case 4 — Comparing projects vs evaluating absolute value

IRR is a rate. It's useful for ranking among similar opportunities, but it doesn't tell you the dollar value created. A 50% IRR on $1kislessvaluablethana20$1M — but IRR alone wouldn't show that.

NPV gives you the dollar value directly.

The Modified IRR (MIRR) fix

Some of IRR's problems can be addressed with the Modified IRR (MIRR). The MIRR assumes:

• Negative cash flows are discounted at the firm's financing rate.
• Positive cash flows are reinvested at the firm's reinvestment rate (usually WACC).

The MIRR is the rate that equates the present value of the negative cash flows to the future value of the positive cash flows.

The MIRR fixes:

• The multiple-IRR problem.
• The implicit "reinvest at IRR" assumption.

In practice, MIRR is rarely used. It's a partial fix. The full fix is just to use NPV.

Why most textbooks recommend NPV

A small consensus exists in academic finance: when in doubt, use NPV.

The case for NPV:

1. It directly measures value added.
2. It handles all cash flow patterns without ambiguity.
3. It uses the correct reinvestment assumption (cash flows reinvested at the discount rate, not the project's IRR).
4. It additively combines across projects: NPV of the firm is the sum of NPV of its projects.

The case for IRR:

1. It's intuitive as a rate of return.
2. It doesn't require specifying a discount rate at decision time.
3. It's the language private equity uses ("our fund targets 25% IRR").

The right practice: compute both. Use NPV for the decision. Report IRR for context.

In practice

In an investment committee context — at a corporation, a fund, or any decision-making body — you will usually see both NPV and IRR on a project summary. The committee may focus on IRR because it's a single comparable number. But the rigorous analyst checks NPV underneath and flags any case where the two diverge.

This is how mature finance organizations operate. The two metrics together provide more information than either alone, and the discipline of checking for divergences catches errors.

In private equity specifically, IRR is the dominant performance metric (LPs care about fund-level IRR), but at the deal level, both NPV and IRR are computed, and the cases where they diverge are scrutinized.