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04 // Interactive Formula Compendium // FINANCENPV = Σ_{t=1..T} [ CF_t / (1 + r)ᵗ ] − C₀

Net Present Value (NPV) & Internal Rate of Return (IRR) Formula Sheet

Net Present Value NPV = Σ_{t=1..T} CF_t / (1 + r)ᵗ − C₀ is the foundational capital budgeting rule: accept investments with NPV > 0 because they earn more than the opportunity cost of capital r.

Formula Reference
Main Formula
NPV = Σ_{t=1..T} [ CF_t / (1 + r)ᵗ ] − C₀
Rearranged Forms
IRR: Discount rate r* such that NPV(r*) = 0
PI = PV(Future Cash Flows) / C₀ = 1 + (NPV / C₀) (Profitability Index)
NPV_annuity = CF · [ 1 − (1 + r)⁻ᵀ ] / r − C₀ (Level Cash Flow Case)
Variables & Units
Variables & Units — Net Present Value (NPV) & Internal Rate of Return (IRR) Formula Sheet
SymbolNameUnit
NPVNet Present Value (NPV)USD
C0Initial Upfront Investment (C₀)USD
CFAnnual Level Net Cash Flow (CF)USD/yr
rDiscount / Hurdle Rate (decimal)1/yr
TProject Lifespan (T years)yr
Inputs & Parameters
[USD]
[USD/yr]
[1/yr]
[yr]
Calculated Result (Net Present Value (NPV))
NPV =13,723.6USD

Formula Notes & Background

  • By the time value of money, a cash flow CF_t received t years in the future has present value PV_t = CF_t / (1 + r)ᵗ.
  • Summing the present values of all future cash inflows over t = 1..T gives total gross present value.
  • Subtracting the immediate t = 0 outlay C₀ yields the net wealth created today: NPV.

Step-by-Step Examples

#1: Evaluate a $100,000 equipment investment that generates $30,000/year for 5 years at a 10% hurdle rate.
C₀ = $100,000CF = $30,000/yrr = 0.10T = 5 yrs
NPV = 30000 × [1 − (1.10)⁻⁵] / 0.10 − 100000 = 113,723.60 − 100,000
→ NPV = +$13,723.60 (Accept project; IRR = 15.24% > 10%)
#2: Re-evaluate the same project if the hurdle rate rises to r = 18%.
C₀ = $100,000CF = $30,000/yrr = 0.18T = 5 yrs
NPV = 30000 × [1 − (1.18)⁻⁵] / 0.18 − 100000 = 93,815.13 − 100,000
→ NPV = −$6,184.87 (Reject project)
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Frequently Asked Questions

Why is NPV superior to IRR when ranking mutually exclusive projects?+

IRR measures percentage return independent of scale and assumes interim cash flows are reinvested at the IRR itself, whereas NPV measures absolute dollar wealth added assuming reinvestment at the realistic hurdle rate r.

What does NPV = $0 mean?+

NPV = $0 means the project earns the exact required hurdle rate r (i.e., IRR = r), compensating investors fairly for risk without generating excess economic rent.

How is inflation handled in NPV calculations?+

Either discount nominal cash flows using a nominal discount rate, or discount real (inflation-adjusted) cash flows using a real discount rate via the Fisher equation (1 + r_nom) = (1 + r_real)(1 + π).