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Capital Budgeting

10 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.

Project Evaluation

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=∑t=0nCFt(1+k)tNPV = \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.

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

Modified Internal Rate of Return (MIRR)

Annualized rate linking the present value of negative cash flows discounted at a finance rate to the terminal value of positive cash flows accumulated at a reinvestment rate.

MIRR=(FV(positive CFs at ReinvestRate)∣PV(negative CFs at FinanceRate)∣)1/n−1MIRR = \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=PV(future cash flows)∣CF0∣=1+NPV∣CF0∣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.

Payback=t∗+∣CumCFt∗∣CFt∗+1\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.

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

Equivalent Annual Annuity (EAA)

Level end-of-year value whose present value equals the project NPV over n years at discount rate k.

EAA=NPV×k1−(1+k)−nEAA = \frac{NPV \times k}{1 - (1 + k)^{-n}}

Value Creation

Economic Value Added (EVA)

Operating profit after tax minus a charge for the capital used to earn it: NOPAT less WACC times invested capital. Positive EVA means the business earned more than its capital cost; this is also called economic profit or residual income.

EVA=NOPAT−WACC×Invested Capital\text{EVA} = \text{NOPAT} - \text{WACC} \times \text{Invested Capital}

ROIC − WACC Spread

Return on invested capital minus the weighted average cost of capital: the value created per dollar of capital employed. Positive means every dollar invested earns more than it costs; negative means growth destroys value.

Spread=NOPATInvested Capital−WACC\text{Spread} = \frac{\text{NOPAT}}{\text{Invested Capital}} - \text{WACC}

Return on Incremental Invested Capital (ROIIC)

The return earned on the NEW capital invested over a period: the change in NOPAT divided by the change in invested capital. Where ROIC describes the whole capital base, ROIIC describes the marginal dollar, which is what growth actually earns.

ROIIC=NOPAT1−NOPAT0IC1−IC0\text{ROIIC} = \frac{\text{NOPAT}_1 - \text{NOPAT}_0}{\text{IC}_1 - \text{IC}_0}