Properties of the Financial Break-Even Point in a Simple Investment Project As a Function of the Discount Rate

Domingo Alberto Tarzia

Abstract


We consider a simple investment project with the following parameters: I>0: Initial outlay 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; : 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) of the investment project 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  and we prove that: (i) For  negligible Qf equals the accounting break-even point Qc (i.e. the earnings before taxes (EBT) is null); (ii) When  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 ; (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; (iv) Moreover, if we assume that the output grows at the annual rate g the previous results still hold, and, of course, the graph of the function Qf = Qf (r,g) vs r has, for all g>0 the same asymptotic straight line when r trends to infinite as in the particular case with g=0. From our point of view, a result of this type is the first time which is obtained by a simple investment project being the cornerstone of our proof the explicit expression of the net present value and the corresponding financial break-even point value. A policy implication of our findings is that the results can be taken into account for investment projects, especially in countries with very small or very large discount rates.


Keywords


Financial break-even point; Investment project; Net present value; Discount rate; Accounting break-even point; Asymptotic behavior; Sensitivity analysis.

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DOI: http://dx.doi.org/10.18533/jefs.v4i02.226

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