Methodology

The formulas behind every number

Factibilidad.io doesn't use AI for calculations — it uses deterministic engineering economics. These are the exact derivations of every result we show.

📖Based on Blank & Tarquin, Engineering Economy, 8th ed. — the international standard
01 — Net Present Value

NPV (Net Present Value)

How much money your project will generate in the future is worth today, in today's currency.

The central principle of engineering economics is that a dollar today is worth more than a dollar tomorrow. NPV discounts all future cash flows at the minimum rate you require from your investment (MARR) and sums them. If the result is positive, the project creates value above that requirement.

General formula

VPN = −I₀ + Σ [Fₜ / (1 + i)ᵗ] para t = 1 … n
  • I₀Initial investment at t = 0 (CAPEX + Working capital). Always negative.
  • FₜNet cash flow for period t (revenues − expenses − taxes).
  • iMARR: Minimum Attractive Rate of Return (your opportunity cost).
  • nEvaluation horizon in years.
  • (1+i)ᵗDiscount factor: converts future money into today's money.

Step-by-step derivation

1

Calculate the net cash flow for each year

Fₜ = Ingresos − Costos variables − Costos fijos − Impuestos
Taxes are calculated on EBIT (earnings before interest and taxes), which includes depreciation as a tax shield.
2

Add the year-0 cash flow

F₀ = −(CAPEX + Capital de trabajo)
At the end of the horizon, working capital and the asset's salvage value are recovered.
3

Discount each cash flow

Fₜ_descontado = Fₜ / (1 + i)ᵗ
A $100 cash flow in year 3 with i = 15% is worth today: 100 / (1.15)³ = $65.75
4

Sum all discounted cash flows

VPN = F₀ + F₁/(1+i)¹ + F₂/(1+i)² + … + Fₙ/(1+i)ⁿ

Decision rule

VPN > 0 → the project creates value above the MARR. Accept.
VPN = 0 → the project exactly equals the MARR. Indifferent.
VPN < 0 → the project destroys value at this rate. Reject or reformulate.

Numerical example

CAPEX = $100.000 · Flujos: $30.000/año · i = 15% · n = 5 años VPN = −100.000 + 30.000/(1.15)¹ = +26.087 + 30.000/(1.15)² = +22.684 + 30.000/(1.15)³ = +19.725 + 30.000/(1.15)⁴ = +17.152 + 30.000/(1.15)⁵ = +14.914 ───────── VPN = +$ 562 (viable, though barely)
02 — Internal Rate of Return

IRR (Internal Rate of Return)

The effective interest rate your project is paying on your investment.

IRR is the rate i* that makes NPV exactly zero. There is no closed-form formula to calculate it — it is solved numerically. Factibilidad.io uses Newton-Raphson with a bisection fallback when the cash flow series has multiple sign changes.

Definition

TIR = i* tal que VPN(i*) = 0 0 = −I₀ + Σ [Fₜ / (1 + i*)ᵗ] para t = 1 … n

Newton-Raphson algorithm

1

Starting point

Initial estimate: 10%. The geometric mean of positive cash flows is used as a starting heuristic.
2

Iteration

Each step applies the Newton correction. Where the analytical derivative is:
iₙ₊₁ = iₙ − VPN(iₙ) / VPN'(iₙ)

VPN'(i) = −Σ [t · Fₜ / (1 + i)^(t+1)]
3

Convergence

Iterates until the difference is less than 1×10⁻⁸. Newton-Raphson converges quadratically — typically in 5–10 iterations.
4

Fallback: bisection

If Newton-Raphson doesn't converge (multiple sign changes, non-convex curves), bisection is applied over [0%, 200%] with tolerance 1×10⁻⁷. Bisection always converges but more slowly (linear convergence).

Decision rule

TIR > TMAR → the project returns more than your opportunity cost. Accept.
TIR < TMAR → the project doesn't meet your minimum requirement. Reject.
TIR = TMAR → equivalent to NPV = 0. Indifferent.

Important limitation

If cash flows change sign more than once (e.g., additional investment mid-project), there may be more than one IRR or none. In that case NPV is the reliable criterion. The tool warns when this occurs.
03 — Sensitivity Analysis

Sensitivity Analysis

Which variable, if changed, moves NPV the most? The tornado chart ranks risks by real impact.

Sensitivity analysis evaluates the robustness of NPV to individual variations in each input parameter. It identifies the critical variable — the one that deserves the most attention when validating the market.

±10% procedure

1

Select variables

Evaluated: unit price, initial volume, unit variable cost, monthly fixed cost, CAPEX, growth rate and discount rate (MARR).
2

Perturb one variable at a time (ceteris paribus)

For each variable X, two scenarios are calculated. All other variables remain at their base value.
VPN⁻ = VPN(X × 0.90) // −10%
VPN⁺ = VPN(X × 1.10) // +10%
3

Calculate the impact range

Impacto(X) = |VPN⁺ − VPN⁻|
This is the total range of NPV variation for that perturbation.
4

Sort from highest to lowest (Tornado Chart)

Variables are plotted top to bottom by impact. Longest bar = highest risk. Red bar = unfavorable scenario; green = favorable.

Practical interpretation

If unit price has a much longer bar than volume, a 10% error in price destroys (or creates) much more value than the same error in volume. That is the variable where you should invest the most time validating before committing capital.

Note on sign

Para precio: VPN⁺ > VPN⁻ (más precio = más VPN — relación positiva) Para CAPEX: VPN⁺ < VPN⁻ (más CAPEX = menos VPN — relación negativa) Para costos: VPN⁺ < VPN⁻ (más costo = menos VPN — relación negativa)

The Tornado Chart shows the correct direction for each variable.

04 — Break-even Point

Break-even Point

The minimum sales volume where revenues exactly equal total costs.

Break-even analysis separates costs into two categories: fixed (do not change with volume) and variable (change proportionally). The intersection of the revenue line with the total cost line defines the operating break-even point.

Derivation from contribution margin

1

Unit contribution margin

MC = Precio − Costo variable unitario
Each unit sold contributes to covering fixed costs. MC is the difference between what comes in from each sale and what goes out as variable cost.
2

Contribution margin ratio (CMR)

RCM = MC / Precio
Expresses what fraction of each dollar of revenue is available to cover fixed costs. A CMR of 0.60 means $0.60 of every $1 in sales contributes to fixed costs.
3

Break-even in units

Equilibrium is reached when the sum of contribution margins exactly covers fixed costs.
PE_unidades = Costos fijos anuales / MC
4

Break-even in dollars

PE_pesos = Costos fijos anuales / RCM
PE_pesos = PE_unidades × Precio

Consolidated formulas

MC = P − Cv // Margen de contribución unitario RCM = MC / P // Razón de contribución marginal PE_u = CF / MC // Break-even en unidades PE_$ = CF / RCM // Break-even en pesos MS = (Q − PE_u) / Q // Margen de Seguridad (0 a 1) Donde: P = Precio unitario de venta Cv = Costo variable unitario CF = Costos fijos anuales (fixedCostMonthly × 12) Q = Volumen proyectado en año 1

Safety Margin

The Safety Margin (SM) answers: how much can volume fall before entering a loss?

MS < 20% → risk zone. Small demand variations generate losses.
MS 20%–40% → acceptable margin for most sectors.
MS > 40% → project is robust against demand drops.

Numerical example

P = $1.000 · Cv = $400 · CF anual = $360.000 · Q = 800 unidades/año MC = 1.000 − 400 = $600/unidad RCM = 600 / 1.000 = 60% PE_u = 360.000 / 600 = 600 unidades/año PE_$ = 360.000 / 0.60 = $600.000/año MS = (800 − 600) / 800 = 25% → acceptable

Apply these formulas to your real project

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Methodology based on Blank & Tarquin, Engineering Economy 8th edition