Break-Even and Sensitivity Analysis - Theory & Concepts

Break-Even and Sensitivity Analysis

$Learning Objectives$

Derive and compute operating break-even quantity from fixed cost, unit variable cost, and unit revenue.
Interpret contribution margin and distinguish ordinary, infeasible, and degenerate break-even cases.
Perform one-way and two-way sensitivity analysis on important engineering-economic assumptions.
Distinguish sensitivity analysis from scenario analysis and probabilistic risk analysis.
Identify decision thresholds at which an economic recommendation changes.

$Break-Even Point$

A break-even point is the value of a decision variable at which two economic outcomes are equal, such as total revenue equaling total cost or two alternatives having equal present worth.

$Contribution Margin$

Contribution margin per unit is the selling price or unit benefit minus the unit variable cost. Each unit sold contributes this amount toward recovering fixed cost and, after break-even, toward profit.

Operating Break-Even Quantity

Quantity at which total revenue equals fixed plus variable cost for a linear cost-revenue model when contribution margin is positive.

Q_{\text{BE}}=\frac{F}{p-v}

Variables

Symbol	Description	Unit
$Q_{\text{BE}}$	Break-even quantity	-
$F$	Fixed cost over the relevant period	-
$p$	Revenue or selling price per unit	-
$v$	Variable cost per unit	-

$Break-Even Existence and Degenerate Cases$

For positive fixed cost $F>0$ , a nonnegative finite break-even quantity requires a positive contribution margin $p-v>0$ . If $p\le v$ , positive output cannot recover the fixed cost under the linear model.

When $F=0$ , special cases occur. If $p=v$ , revenue and cost are identical at every quantity, so every quantity is a break-even point. If $p<v$ , only $Q=0$ is break-even and every positive unit produces a loss. If $p>v$ , $Q_{\text{BE}}=0$ and every positive unit produces a positive operating contribution.

$Interactive Break-Even and Sensitivity Explorer$

Change fixed cost, unit cost, price, and the sensitivity band to see the exact crossing condition, degenerate zero-fixed-cost cases, and how the threshold shifts when uncertain assumptions move.

$Linear Operating Break-Even Conditions$

Fixed cost $F$	Contribution margin $p-v$	Break-even interpretation
$F>0$	$p-v>0$	Finite positive $Q_{\text{BE}}=F/(p-v)$
$F>0$	$p-v=0$	No finite quantity recovers fixed cost
$F>0$	$p-v<0$	Positive production increases the modeled loss; no nonnegative break-even quantity
$F=0$	$p-v>0$	Break-even at $Q=0$ ; positive quantities have positive contribution
$F=0$	$p-v=0$	Every quantity is a break-even point
$F=0$	$p-v<0$	Only $Q=0$ is break-even; positive quantities have negative contribution

Fixed costPHPVariable cost per unitPHP/unitSelling price per unitPHP/unit

Sensitivity change20.00%

Break-even quantity

2,174 units

At this quantity, modeled total revenue equals modeled total cost.

Contribution margin

₱230

Selling price minus variable cost per unit.

Loading chart...

One-way scenario	Break-even units
Price down	3,572 units
Base	2,174 units
Price up	1,563 units
Variable cost down	1,825 units
Variable cost up	2,689 units

Two-way sensitivity: break-even quantity when price and variable cost move together.
Variable cost \ Price	-20%	Base	+20%
-20%	2,718	1,825	1,374
Base	3,572	2,174	1,563
+20%	5,209	2,689	1,812

Sensitivity analysis does not assign probabilities. It exposes how strongly a decision measure responds to uncertain inputs and where the model becomes infeasible or degenerate because the contribution margin is non-positive.

MARR10.00%

Common study life8 years

Alternative A

First costPHPAnnual net cash flowPHP/yearSalvagePHP

Alternative B

First costPHPAnnual net cash flowPHP/yearSalvagePHP

PW — Alternative A

₱1,741,625

Meets the MARR screen.

PW — Alternative B

₱1,968,658

Meets the MARR screen.

Preferred on this common basis

Alternative B

PW(B) − PW(A) = ₱227,033.

For unequal lives, do not force a common-life comparison without stating the repeatability assumption. Annual worth or a fixed study period is often clearer.

$Sensitivity Analysis$

Sensitivity analysis measures how an output such as NPV, AW, B/C, or break-even quantity changes when one or more input assumptions change over specified ranges.

$One-Way Sensitivity Analysis$

One-way sensitivity changes one input at a time while holding the other modeled assumptions constant, revealing which variables have the strongest local effect on the decision metric.

$Two-Way Sensitivity Analysis$

Two-way sensitivity changes two inputs together and can reveal combinations of assumptions that form an accept/reject or alternative-selection boundary.

$Scenario Analysis$

Scenario analysis evaluates internally consistent sets of assumptions—such as base, adverse, and favorable cases—rather than changing one variable independently.

$Sensitivity Method and Interpretation$

Method	Inputs varied together	Learner question answered	Probability implied?
One-way	One input; others fixed	Which input changes the metric most over its range?	No
Two-way	Two inputs over a grid	Which combinations move the result across a threshold?	No
Scenario	A coherent set of inputs	What happens under an internally consistent case?	No
Probabilistic risk analysis	Distributions and dependencies	How are outcomes distributed under modeled uncertainty?	Yes, only when supported by data/model assumptions

$Decision Threshold$

A decision threshold is the input value at which the preferred alternative or accept/reject conclusion changes, often found by setting the relevant economic difference equal to zero.

$Sensitivity Does Not Assign Probability$

A sensitivity chart shows response to assumed changes. It does not by itself say how likely those changes are. Probability distributions, expected value, simulation, or other risk methods require additional information.

$Direction Is Not the Same as Importance$

A sensitivity result should be interpreted using both direction and magnitude. A variable can have a predictable effect yet remain economically unimportant over its plausible range, while another variable can cross a decision threshold with only a small change.

Break-Even First Cost from Net Present Worth

Maximum first cost that yields zero NPV for a known future net-benefit stream.

P_{\max}=PW(\text{future net benefits at MARR})

Variables

Symbol	Description	Unit
$P_{\max}$	Break-even first cost at time zero	-

$Linear Break-Even Model Has a Relevant Range$

The formula $Q_{\text{BE}}=F/(p-v)$ assumes fixed cost remains fixed and unit price and variable cost remain constant over the quantity range. Capacity steps, volume discounts, overtime, nonlinear demand, taxes, and financing can require a piecewise or full cash-flow model.

$Sensitivity Analysis Workflow$

STEP-BY-STEP

Compute and document the base-case economic metric.
Identify uncertain inputs that could materially affect the decision.
Set defensible ranges and units for each input.
Change one input at a time for one-way sensitivity, or two together for interaction analysis.
Record the metric and the decision at each case.
Locate thresholds at which the recommendation changes.
Use scenarios for jointly plausible assumption sets.
Check whether a tested combination creates an infeasible or degenerate model state rather than forcing a numerical answer.
Do not attach probabilities unless a separate risk model justifies them.

Break-Even and Sensitivity Decision

Select the relevant break-even model, test its domain, and examine how plausible input changes affect the economic recommendation.

For Q_BE = F/(p−v), a positive fixed cost requires p−v > 0 for a finite nonnegative break-even quantity. Sensitivity ranges do not assign probabilities; probability requires a separate supported risk model.

Start → Operating quantity or alternative choice?; Operating quantity or alternative choice? — Quantity → State linear-model range and inputs F, p, v; Operating quantity or alternative choice? — Alternatives → Set alternative cash-flow difference to zero; State linear-model range and inputs F, p, v → F > 0 and p − v > 0?; F > 0 and p − v > 0? — Yes → Compute finite Q_BE = F/(p − v); F > 0 and p − v > 0? — No → Report no finite or degenerate case; Set alternative cash-flow difference to zero → Compute and document base-case metric; Compute finite Q_BE = F/(p − v) → Compute and document base-case metric; Report no finite or degenerate case → Compute and document base-case metric; Compute and document base-case metric → Test one input, two inputs, or coherent scenario?; Test one input, two inputs, or coherent scenario? — One input → Run one-way sensitivity; Test one input, two inputs, or coherent scenario? — Two inputs → Map two-way threshold boundary; Test one input, two inputs, or coherent scenario? — Scenario → Evaluate internally consistent scenarios; Run one-way sensitivity → Plausible range crosses decision threshold?; Map two-way threshold boundary → Plausible range crosses decision threshold?; Evaluate internally consistent scenarios → Plausible range crosses decision threshold?; Plausible range crosses decision threshold? — Yes → Flag recommendation as range-sensitive; Plausible range crosses decision threshold? — No → Report stable recommendation over tested range; Flag recommendation as range-sensitive → Need probability model?; Report stable recommendation over tested range → Need probability model?; Need probability model? — Yes → Use a supported probabilistic risk model; Need probability model? — No → Report threshold and model limits; Use a supported probabilistic risk model → Report threshold and model limits

Start: terminator
Operating quantity or alternative choice?: decision
State linear-model range and inputs F, p, v: process
F > 0 and p − v > 0?: decision
Compute finite Q_BE = F/(p − v): process
Report no finite or degenerate case: process
Set alternative cash-flow difference to zero: process
Compute and document base-case metric: process
Test one input, two inputs, or coherent scenario?: decision
Run one-way sensitivity: process
Map two-way threshold boundary: process
Evaluate internally consistent scenarios: process
Plausible range crosses decision threshold?: decision
Flag recommendation as range-sensitive: process
Report stable recommendation over tested range: process
Need probability model?: decision
Use a supported probabilistic risk model: process
Report threshold and model limits: terminator

Key Takeaways

Operating break-even occurs where total revenue equals total cost.
For positive fixed cost, a positive contribution margin $p-v$ is required for a nonnegative finite linear break-even quantity.
With zero fixed cost, $p=v$ makes every quantity break even, while $p<v$ leaves only zero activity at break-even.
Sensitivity analysis asks how outputs respond to input changes; it does not assign likelihood.
Two-way sensitivity can expose interaction effects and infeasible combinations that one-way checks miss.
Scenario analysis changes coherent sets of assumptions together.
Decision thresholds identify where an economic recommendation changes.
Linear break-even formulas should be restricted to the range in which their assumptions hold.

$References$

Engineering Economy, 18th edition — William G. Sullivan, Elin M. Wicks, and Joseph H. Wilck, Pearson, 2024; chapter 11 covers break-even and sensitivity analysis.

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