Properties of the financial break-even point in a simple investment project as a function of the discount rate

We consider a simple investment project with the following parameters: I>0: Initial investment which is amortizable in n years; n: Number of years the investment allows production with constant output per year; A>0: Annual amortization (A=I/n); Q>0: Quantity of products sold per year; Cv>0: Variable cost per unit; p>0: Price of the product with p>Cv; Cf>0: Annual fixed costs; te: Tax of earnings; r: Annual discount rate. We also assume inflation is negligible. We derive a closed expression of the financial break-even point Qf (i.e. the value of Q for which the net present value (NPV) is zero) as a function of the parameters I, n, Cv, Cf, te, r, p. We study the behavior of Qf as a function of the discount rate r and we prove that: (i) For r negligible Qf equals the accounting break-even point Qc (i.e. the earnings before taxes (EBT) is null) ; (ii) When r is large the graph of the function Qf=Qf(r) has an asymptotic straight line with positive slope. Moreover, Qf(r) is an strictly increasing and convex function of the variable r; (iii) From a sensitivity analysis we conclude that, while the influence of p and Cv on Qf is strong, the influence of Cf on Qf is weak.