**Properties Of The Financial Break-Even Point In A Simple Investment Project As A Function Of The Discount Rate**

**Domingo A. Tarzia**

Univ. Austral, FCE, Mathematics Department & CONICET

March 16, 2016

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*Journal of Economics and Financial Studies, Vol. 4 No. 2 (2016), pp 31-45*

**Abstract: **

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.

### Properties Of The Financial Break-Even Point In A Simple Investment Project As A Function Of The Discount Rate – Introduction

A rigorous evaluation of an investment project is crucial for the evaluation of its convenience, see the books (Alhabeeh, 2012; Bierman and Smidt, 1993; Bodmer, 2015; Brealey and Myers, 1993; De Pablo, Ferruz and Santamaria, 1990; Lopez Dumrauf, 2003; Suarez Suarez, 1991). Among the different methods to evaluate investment project we choose the net present value ( NPV ) (Baker and Fox, 2003; Beaves, 1988; Bric and Weaver, 1997; Chung and Lin, 1998; Grinyer and Walker, 1990; Hadjdasinski (1993, 1995, 1996, 1997); Hartman and Schafrick, 2004; Hazen, 2003; Kim and Chung, 1990; Lan, Chung, Chu and Kuo, 2003; Lohmann, 1994; Lohmann and Baksh, 1993; Moon and Yun, 1993; Pasin and Leblanc, 1996; Pierru and Feuillet Midrier, 2002; Prakash, Dandapani and Karels, 1988; Reichelstein, 2000; Roumi and Schnabel, 1990; Shull, 1992; Stanford, 1989; and Zhang (2005)). Assuming the absence of inflation we consider a simple investment project with the following parameters:

I > 0: Initial outlay which is amortizable in n years;

n >1: Numbers of years of the explicit forecasted period of the investment project which make the same activities per year with only one product;

Q > 0 : Quantity of products sold per year;

0 v C > : Variable cost per unit;

p > 0: Price per unit with v p > C ;

0 f C > : Annual fixed costs;

t > 0: Time;

0 e t > : Tax of earnings (legal tax rate);

r > 0 : Annual discount rate;

g > 0 : Annual growth rate.

Moreover, we also consider the annual amortization ( A = I / n > 0 ) which depends of two parameters I and n.

In Fernandez Blanco (1991) a first study for the NPV of an investment project was done, and we complete and improve it. There exist several papers on NPV but, from our point of view, we have not found in the literature a study of the mathematical-financial properties of the financial break-even point and this is the main objective of the present paper. We derive first an explicit expression of the NPV as a function of the independent variable Q in order to obtain a closed formula of the financial break-even point f Q (the value of Q for which the NPV of the investment project is zero) as a function of the parameters ( I, n, Cv, Cp, te, r, p).

Recent applications of the net present value and the break-even analysis can be found: break-even point between short-term and long-term capital gain (loss) strategies, Yang and Meziani, 2012; break-even procedure for a multi-period project, Kim and Kim, 1996; to mazimize the net present value of projects, Schwindt and Zimmermann, 2001; Vanhoucke, Demeulemeester and Herroelen, 2001a,b; the internal rate of return as a financial indicator, Hajdasinski, 2004; private competitiveness, production costs and break-even analysis of representative production units, Martinez Medina et al., 2015; the net present value as an optimal criterium of investing, Machain, 2002; break-even points for storage systems as a substitute to conventional grid reinforcements, Nykamp et al. 2014; to measure and analyse the operating risk, financial risk, financial break-even point and total risk of a selected public sector, Sarkar and Sarkar, 2013; risk diagnosis in the context of economic crisis, Suciu, 2010; marginal break-even between maintenance strategies alternatives, Gokiene, 2010; relationship among discount cash flow, free cash flow, economic value added and net present value, Hartman, 2010; Shieres and Wachowicz, 2001; running a profitable company, Paek, 2000;

The break-even analysis is a useful tool to study the relationship among fix costs, variable costs and returns. The break-even analysis computes the volume of production at a given price necessary to cover all costs. We study the behavior of Qf = Qf (r) with respect of the discount rate r and we prove the following results:

(i) When r is negligible (r goes to zero) then f Q tends to the accounting break-even point c Q (the value of Q for which the earnings before taxes (EBT) of the investment project is equals to zero);

(ii) When r is large ( r goes to infinity) the graph of the function Qf = Qf (r) has an asymptotic straight line. Its positive slope and the y-intercept point at r = 0 are determined explicitly. Moreover, Qf (r) r is an strictly increasing and convex function of the variable r ;

(iii) By a sensitivity analysis we obtain that p and Cv have an important influence on Qf , but Cf has a negligible influence on Qf ;

(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 as in the particular case with g=0.

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