Capital Budgeting

7 formulas

Capital-budgeting formulas evaluate whether a project, investment, or capital outlay is worth pursuing. Net present value (NPV) discounts an arbitrary cash-flow stream at the cost of capital; IRR finds the rate that makes NPV zero; MIRR fixes the reinvestment-rate flaw in IRR; payback period and discounted payback measure time-to-recovery; profitability index and equivalent annual annuity make like-for-like comparisons across projects of different sizes and lifespans.

Net Present Value (NPV)

Present value of an arbitrary cash flow stream discounted at the required rate of return. Sums each cash flow CF_t divided by (1+k)^t. Positive NPV = project creates value; negative NPV = destroys value.

NPV = \sum_{t=0}^{n} \frac{CF_t}{(1 + k)^t}

Internal Rate of Return (IRR)

The discount rate that makes the NPV of a cash flow stream exactly zero. Solved numerically via Newton-Raphson with bisection fallback.

\text{Find IRR such that: } \sum_{t=0}^{n} \frac{CF_t}{(1 + IRR)^t} = 0

Modified Internal Rate of Return (MIRR)

A modified IRR that uses explicit finance and reinvestment rates rather than the IRR's implicit assumption that intermediate cash flows are reinvested at the IRR itself. Computes the future value of positive cash flows at the reinvestment rate, the present value of negative cash flows at the finance rate, then solves for the rate that links them over n years.

MIRR = \left( \frac{FV(\text{positive CFs at ReinvestRate})}{|PV(\text{negative CFs at FinanceRate})|} \right)^{1/n} - 1

Profitability Index (PI)

Ratio of the present value of future cash flows to the initial investment. PI > 1 means the project creates value; PI < 1 means it destroys value. Mathematically equivalent to NPV/|initial investment| + 1.

PI = \frac{PV(\text{future cash flows})}{|CF_0|} = 1 + \frac{NPV}{|CF_0|}

Payback Period

Time required to recover the initial investment from cumulative undiscounted cash flows. Linearly interpolates within the period when cumulative cash flow first turns non-negative.

\text{Payback} = t^* + \frac{|\text{CumCF}_{t^*}|}{CF_{t^*+1}}

Discounted Payback Period

Time required to recover the initial investment from cumulative discounted cash flows. The same logic as Payback Period but each cash flow is first discounted at the required rate before being summed.

\text{DiscountedPayback} = t^* + \frac{|\text{CumDiscCF}_{t^*}|}{CF_{t^*+1}/(1+k)^{t^*+1}}

Equivalent Annual Annuity (EAA)

Constant annual cash flow that, over the project's life and at the project's discount rate, would produce the same NPV. Standardizes NPV across projects of different lifespans for fair comparison.

EAA = \frac{NPV \times k}{1 - (1 + k)^{-n}}