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.

Abstract.We consider a simple investment project with the following parameters: 0 I  : Initial 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  : Quantity 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  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 conclude 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 results still hold, and, of course, the graph of the function   , ff Q Q r g vs r  has, for all 0 g  ,
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 ( 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 has, for all 0 g  , the same asymptotic straight line when r  as in the particular case with g=0.

Investment Project with the Independent Variable Q
We assume that in each year ( ) the net cash flow is given by: and then, by using the calculation of the sum of the first n terms of a geometric sequence, the NPV of the investment project is given by (Brealey and Myers, 1993 which represents a straight line of the variable where the real function () f f r  is defined by the following expression: Taking into account that the financial break-even that is, and therefore the financial break-even point f Q , as a function of the discount rate r , is given by the following expression: where the real coefficients a and b are given by Taking into account the formula (6) of the financial break-even point or equivalently by Thus we have obtained an expression of the NPV as a function of the variable Q , the discount rate r , and the financial break-even point () Qr.The previous result can be summarized as follows.

Theorem 1
The investment project has the following properties: (i) The NPV , as a function of the units sold per year Q , is given by ( 1) where the yintercept h and the slope m are expressed by ( 2) and ( 3) respectively where () f f r  is the real function defined by ( 4).(ii) The financial break-even point f Q , as a function of the discount rate r , is given by ( ) ( ) where the real function and the coefficients a and b are given by the expressions ( 8) and ( 9) respectively.(iii) The

 
, NPV Q r can also be calculated as a function of the f Q by the expression (11).(iv) The sign of the ( , ) NPV Q r , as a function of the financial break-even point ()  4) and ( 13) respectively which have the following properties.
Theorem 2  is a strictly decreasing and convex function of the discount rate r with the following properties: () df r dr where the real functions which have the following properties: , defined in (13), is a strictly increasing function and has at r   an asymptotic straight line given by the equation yr  (straight line with slope 1 and y-intercept 0) and the following properties:   Proof.
All properties of the real functions ,, f F G and H can be proved by using elementary math- ematical analysis (derivatives, l'Hopital rule, increasing and convexity of functions, asymptotic straight lines, etc.).▄ Taking into account that the earnings before taxes (EBT) is calculated by and defining the accounting break-even point c Q as the value Q for which the EBT is zero then c Q is given by the following expression

Theorem 3
The financial break-even point given by (12), is a strictly increasing function of the discount rate r and has the following properties: Moreover, the function () has at r   a straight line given by the equation which has a slope 0 b  and y-intercept a , defined in ( 9) and ( 8) respectively.

Proof
Taking into account the properties of the functions () , obtained in Theorem 2, we have the following results: which is the accounting break-even point c Q defined by (28).On the other words, we have the following properties:   Moreover, the function () y F r  for r   becomes asymptotic to the straight line whose equation is given by the equation yr  , and then the function ()

Remark 1
The limit (33) has an interesting accounting-financial property: the limit of the financial break-even point () when the discount rate goes to zero (i.e., discount rate is negligible) is equal to the accounting break-even point c Q .▄

Remark 2
The financial break-even point of the investnment project, as a function of the discount rate r , is given by a strictly increasing and convex function () Q (accounting break-even point) and for r   tends asymptot- ically to the straight line of equation y a br  where a and b are defined in ( 8) and (9)   respectively.On the other hand, the curve () b and b , less than the slope b of the asymptote for r   .▄ Taking into account the result ( 29) an interesting question is to determine the rate of convergence of the financial break-even point () to the accounting break-even point c Q when the discount rate r goes to zero.

Lemma 4
We have and the rate of convergence is of order one and it is given by

Proof
Taking into account formulas (7) or ( 12) for () Qr, and formula (28) for c Q , we obtain: that is (38).Now, taking into account the properties ( 24) and ( 25) for the function F(r) we get (39).▄ Now we will study the NPV as a function of the two independent variables: the discount rate r and the number Q of units sold per year.
Theorem 5 (i) The NPV , as a function of the two independent variables Q and r , is given by the fol- lowing expression where the function () f f r  was defined in ( 4).
(ii) The

 
, NPV Q r is a strictly increasing function of the variable Q and a strictly decreas- ing function of the variable r which has for the extreme values 0 and  of the variables Q and r respectively the following expressions: where c Q is the accounting break-even point defined in (28), and () h h r  is the real function defined in ( 2).

(iii) The real function () h h r
 has the following properties: and it is a strictly increasing (decreasing) function when At C t  .
In the particular case in with

Proof.
The properties follow from the previous results and elemenarty mathematical computations.
In particular, the partial derivatives of

 
, NPV Q r with respect to the variables Q and r are given by the following expressions: and then the properties (i) and (ii) hold.

Therefore, derivative of () h h r
 is given by: where   G G r  was defined in (20).The sign of () hr  depends of the sign of

 
1 f e e C t At    , which will be positive (i.e. h is a strictly increasing function of the variable r ) when At C t  and so on.In the particular case   At C t  we get that ( ) 0, 0 h r r     , and then () hr is a constant given by ( ) , 0 h r I r     .▄

Remark 3
In the Theorem 5 we showed that the increasing or decreasing behavior of the real function () h h r  depends on the sign of the expression which has a financial-accounting interpretation.▄

Remark 4:
Owing to the properties ( 46) and ( 47) of the

 
, NPV Q r , as a real function of the two inde- pendent variables Q and r , and the expression (11) of the NPV with parameters , , , , , , or equivalently by where   f Qr is the financial break-even point for the quantity Q as a function of the dis- count rate r , defined by ( 12), and , where c Q is the accounting break-even point given by (28).▄

Numerical Results for an Investment Project
We will consider the investment project with the following parameters (Brealey and Myers (1993)):  initial outlay:  explicit forecasted period of the project: n = 10;  annual amortization:  price per unit:  annual fixed costs:  tax of earnings: Taking into account the same parameters given before we can obtain the values of the financial break-even point   f Qr as a function of the discount rate r (see Table 1), and then we can get the graph of the real function   f Q r vs r (see Figure 1).Moreover, we can also plot the financial break-even point   f y Q r  and the asymptotic straight line y a b r  as a function of the discount rate r (see Figure 2).Now we will perform a sensitivity analysis of the investment project in a neighboord of a reference point 0.10 r  . We get the sensitivity analysis of the financial break-even point (see Figure 3).The sensitivity analysis of the financial break-even point   f Qr suggests that price per unit p lies on a more sensible point in comparison to the costs of any kind.We consider that the investment project defined in Section 2 has a constant annual growth rate 0 g  in each year.In this case, we obtain the results given in the next theorem.

Theorem 6
The investment project, defined previously, with a annual growth rate 0 g  has the following properties: (i) The NPV , as a function of the three variables , and Q r g , is given by: ( , , ) ( ) ( , ) where (ii) The financial break-even point f Q is given, as a function of the discount rate r and the growing rate g, by the following expression: where the real functions ( ) and ( ) , and the coefficients a and b are given by the expressions (4), ( 13), ( 8) and ( 9) respectively.which has a slope 0 b  and y-intercept a , defined by ( 9) and ( 8) respectively.The asymptot- ic straight line is independent of the growth rate 0 g  and coincides with the straight line for the particular case 0 g  .Proof.We follow the method developed in Section 2. ▄

Conclusions
For a simple investment project an explicit expression of the corresponding net present value ( NPV ) as a function of the independent variable Q in order to obtain a closed expression of the financial break-even point has, for all 0 g  , the same asymptotic straight line when r  as in the particular case with g=0.

( 14 )
In order to analyse the matematical behavior of the function   , NPV NPV Q r  and the financial break-even point ()  ff Q Q r we need to study the behavior of the real functions ()

Figure 2 Figure 3 C p 4 .
Figure 2 Graph of (55), is a strictly increasing function of the discount rate r and it has the following properties: g vs r  has at r   an asymptotic straight line given by the equation: y a b r  (59)

fQQQ
(i.e. the value of Q for which NPV is zero) as a function of the parameters I , as a function of the discount rate r is studied and it is proved 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; (iii)From a sensitivity analysis we conclude that, while the influence of p and 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   , ff Q Q r g vs r  1 n  : Numbers of years of the explicit forecasted period of the investment project which make the same activities per year with only one product; 0 Q  : Quantity of products sold per year; 0 v C  : Variable cost per unit; 0 p  : Price per unit with v pC  ; 0 f C  : Annual fixed costs; 0 t  : Time; 0 e t  : Tax of earnings (legal tax rate); 0 r  : Annual discount rate; 0 g  : Annual growth rate.

Table 1 Values of the financial break-even point
If we look into the costs, it is seen that annual fixed costs f Qr vs. r