NPV и IRR

Критерии инвестиционных решений Net Present Value (NPV) и Internal Rate of Return (IRR) — два главных критерия оценки инвестиционных проектов. Они помогают ответить на вопрос: создаёт ли проект стоимость и стоит ли инвестировать? Net Present Value (NPV) NPV — сумма present values всех cash flows проекта, включая initial investment. NPV = Σ CFt / (1+r)^t, где t от 0 до n, CF₀ обычно negative (investment). NPV Rule: инвестировать если NPV > 0. Positive NPV означает, что проект создаёт стоимость сверх required return. NPV = 0 — проект даёт exactly required return. NPV NPV как value creation: NPV показывает в долларах, сколько стоимости создаёт проект. NPV = $1M означает wealth increase на $1M в today's dollars. NPV calculation example Project: invest $100,000 today, receive $30,000/year for 5 years. Required return 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 project. Проект создаёт $8,143 value. Internal Rate of Return (IRR) IRR — ставка дисконтирования, при которой NPV = 0. Это «внутренняя» доходность проекта, implied by its cash flows. IRR Rule: инвестировать если IRR > required return (hurdle rate). Если IRR превышает cost of capital, проект создаёт value. Calculation: IRR found by solving NPV = 0 for r. No closed-form solution для uneven cash flows — итерационный расчёт (Excel: IRR function). IRR calculation example Same project: -$100,000 today, $30,000/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 between 15% and 16%, approximately 15.24%. IRR (15.24%) > hurdle (12%), accept project. Consistent with NPV decision. NPV vs IRR: когда они согласуются Для independent, conventional projects (one initial outflow, then inflows), NPV и IRR дают same accept/reject decision. Conventional: если initial NPV negative, NPV increases as discount rate decreases, crossing zero at IRR. If IRR > hurdle, NPV > 0. NPV vs IRR: когда расходятся Mutually exclusive projects: выбор между проектами. NPV ranks correctly (choose highest NPV), IRR может mislead. Пример: Project A: invest $100, IRR 50%, NPV $20. Project B: invest $1,000, IRR 30%, NPV $150. IRR says A better, NPV says B. NPV correct — B creates more value. Scale problem: IRR игнорирует scale. Маленький проект с high IRR может быть менее valuable than large project с moderate IRR. Multiple IRRs Non-conventional cash flows (sign changes more than once) могут иметь multiple IRRs или none. Пример: -$100, +$230, -$132 (sign changes twice). Может иметь IRR 10% и 20%. Which is «the» IRR? Solution: use NPV, modified IRR (MIRR), или NPV profile analysis. IRR problematic для non-conventional flows. Modified IRR (MIRR) MIRR addresses some IRR problems: assumes reinvestment at cost of capital (not IRR), produces single solution. Calculation: terminal value of inflows (compounded at cost of capital) / PV of outflows. Find rate equating these. More realistic reinvestment assumption, но adds complexity. NPV remains gold standard. Crossover rate Для mutually exclusive projects: crossover rate — ставка, где оба проекта имеют equal NPV. Below crossover: one project preferred. Above crossover: other project preferred. Important для ranking decisions. Calculate: find IRR of differential cash flows (A - B). NPV profile Graph NPV vs discount rate. Shows how NPV changes with rate. Visual representation of project's sensitivity to discount rate. X-intercept = IRR (NPV = 0). For comparing projects, plot both profiles — crossover visible. Practical recommendations Use NPV как primary criterion — measures value creation in dollars, no ambiguity. Report IRR for intuition — percentage return is intuitive for stakeholders. Be cautious с IRR for non-conventional projects, mutually exclusive decisions, different scales. NPV requires discount rate input — critical assumption. Sensitivity analysis на rate important.