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

Abstract

We consider a simple investment project with the following parameters: 0 I  : Ini-tial outlay which is amortizable in n years; n : Number of years the investment allows pro-duction with constant output per year; 0 A  : Annual amortization (/ A I n ); 0 Q  : Quan-tity of products sold per year; 0 v C  : Variable cost per unit; 0 p  : Price of the product with v pC  ; 0 f C  : Annual fixed costs; e t : Tax of earnings; r : Annual discount rate. We also assume inflation is negligible. We derive a closed expression of the financial break-even point f Q (i.e. the value of Q for which the net present value (NPV) of the investment project is zero) as a function of the pa-rameters I , n , v C , f C , e t , r , p . We study the behavior of f Q as a function of the discount rate r and we prove that: (i) For r negligible f Q equals the accounting break-even point c Q (i.e. the earnings before taxes (EBT) is null); (ii) When r is large the graph of the function   ff Q Q r  has an asymptotic straight line with positive slope. Moreover,   f Qr is an strict-ly increasing and convex function of the variable r ; (iii) From a sensitivity analysis we con-clude that, while the influence of p and v C on f Q is strong, the influence of f C on f Q is weak; (iv) Moreover, if we assume that the output grows at the annual rate g the previous re-sults still hold, and, of course, the graph of the function   , ff Q Q r g vs r  has, for all 0 g  , the same asymptotic straight line when r  as in the particular case with g=0.