Break-even analysis

Break-even analysis finds the value of an input parameter at which an outcome metric reaches a threshold, most commonly PW = 0, the breakeven between accept and reject. Equivalent question: “How wrong would my estimate need to be before the answer flips?”

Say a manufacturing project’s profitability depends on the selling price $p$. Compute the price at which PW equals zero. That’s the break-even price. If your actual price is comfortably above it you have margin to spare; if it’s marginal, the project is risky.

For the simpler cost-volume-profit form (no time-value-of-money), break-even quantity is

$q^{\ast }=\frac{C_F}{p-v}$

where $C_F$ is fixed cost, $p$ is selling price per unit, and $v$ is variable cost per unit (Total cost). Above $q^{\ast }$ you make money; below, you lose.

For the time-value form: set PW (or AW, or any other metric) equal to the threshold, treat one input as the unknown, and solve. In an engineering project with first cost $P$, annual revenue $R$, annual cost $C$, salvage $S$ at year $N$, and MARR $i$:

$PW=-P+(R-C)\cdot (P/A,i,N)+S\cdot (P/F,i,N)=0.$

Solve for the parameter of interest. If solving for $R$: $R=C+\frac{P-S\cdot (P/F,i,N)}{(P/A,i,N)}$. This is the break-even revenue, the annual revenue that makes the project economically neutral.

Strengths. Answers concrete “what would have to be true” questions. Good for scenario planning: “the project pays off if we sell at least 10,000 units/year, is that realistic?” Easy to explain to non-technical stakeholders.

Weaknesses. Only one variable at a time (same as Sensitivity analysis); doesn’t capture variable interdependencies. Doesn’t tell you the probability of being at break-even, only the value where it sits. A planning tool, not a complete risk model.

For probabilistic risk analysis see decision tree and Expected value (engineering economics).