The Analysis of the Influence of Debt Financing on the Efficiency of Investment Projects in the Perpetuity (Modigliani–Miller) Approximation

P. N. Brusov^{1, *}, T. V. Filatova², N. P. Orekhova^{3, 5}, I. K. Shevchenko⁴, A. Y. Arkhipov⁵, V. L. Kulik⁶

¹Applied Mathematics Department, Financial University under the Government of Russian Federation, Moscow, Russia

²Dean of GMM Faculty, Financial University under the Government of Russian Federation, Moscow, Russia

³Investment and Taxation Laboratory, Research Consortium of Universities of the South of Russia, Rostov-on-Don, Russia

⁴Research and Innovation Projects, Southern Federal University, Rostov-on-Don, Russia

⁵High School of Business, Southern Federal University, Rostov-on-Don, Russia

⁶Management Department, Financial University under the Government of Russian Federation, Moscow, Russia

Abstract

The analysis of effectiveness of investment projects within the perpetuity (Modigliani–Miller) approximation (Modigliani et al 1958, 1963, 1966) has been done. Based on the obtained in previous papers results for NPV (Brusov et al 2011a, b, c, d, e; 2012 a, b; 2013 a, b, c; 2014 a, b; Filatova et al 2008) we analyze the effectiveness of investment projects for three cases.

Keywords

Debt Financing, Long-Term Investment Projects, Effectiveness of Investment Project, Brusov-Filatova-Orekhova (BFO Theory)

Received: April 8, 2015
Accepted: April 26, 2015
Published online: June 14, 2015

@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/

1. Introduction

In this paper we conduct the analysis of effectiveness of investment projects within the perpetuity (Modigliani–Miller) approximation (Modigliani et al 1958, 1963, 1966). Based on the obtained in previous papers results for NPV (Brusov P, Filatova T, Orehova N, Eskindarov M 2015, Brusov et al 2011a,b,c,d,e; 2012 a, b; 2013 a, b, c; 2014 a, b; Filatova et al 2008) we analyze the effectiveness of investment projects for three cases:

1)  at a constant difference between equity cost (at ) and debt cost ;

2)  at a constant equity cost (at ) and varying debt cost ;

3)  at a constant debt cost  and varying equity cost (at ) .

The dependence of NPV on investment value and/or equity value will be also analyzed. The results are shown in the form of tables and graphs.

It should be noted that the obtained tables have played an important practical role in determining of the optimal, or acceptable debt level, at which the project remains effective. The optimal debt level there is for the situation, when in the dependence of NPV on leverage level L there is an optimum (leverage level value, at which NPV reaches a maximum value. An acceptable debt level there is for the situation, when NPV decreases with leverage. And, finally, it is possible that NPV is growing with leverage. In this case, an increase in borrowing leads to increased effectiveness of investment projects, and their limit is determined by financial sustainability of investing company.

2. The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only

2.1. With the Division of Credit and Investment Flows

At a constant value of the total invested capital (I = const)

      (1)

1) At the constant values of  NPV practically always decreases with leverage. At small L for many pairs of values  and  (for example,  (14%) and  (12%) ;  (18%) and  (16%) and many others) there is an optimum in the dependence of NPV(L) at small

For higher values of  (and, accordingly, ) curves NPV(L) lie below. With increase of NOI all curves NPV(L) is shifted in parallel upwards.

2) At the constant values of  NPV practically always decreases with leverage, passing through (most often), or not passing (more rarely) through optimum in the dependence of NPV(L) at small

All curves NPV(L) at the constant values of  are started (at L=0) from a single point, and with the increasing of  (and, respectively, a decrease of ) curves NPV(L) lie above. With increase of NOI all curve NPV(L) are shifted in parallel upwards practically.

3) At the constant values of  NPV practically always  decreases with leverage, optimum in the dependence of NPV(L) отсутствует.

All curves NPV(L) at the constant values of  are started (at L=0) from a single point. With the increasing of  (and, respectively, a increase of ) curves NPV(L) is shifted into region of higher NPV values. With increase of NOI all curves NPV(L) are shifted in parallel upwards practically.

Table 1. Dependence of NPV on leverage level NOI=1200 I=2000, k0-kd=const.

Table 1. (continue)

Fig. 1. Dependence of NPV on leverage level at fixed values of  and .

Fig. 2. Dependence of NPV on leverage level at fixed values of  and .

At a constant equity value (S = const)

            (2)

1) At the constant values of  NPV practically always decreases with leverage. The optimum in the dependence of NPV(L) has been found for one pair of  and ( (8%) and  (6%)) only.

Fig. 3. Dependence of NPV on leverage level at fixed values of  and .

All curves NPV(L) at the constant values of   are started (at L=0) from one point and with growth of  (and, accordingly, ) all curves NPV(L) lie below. With growth of density of curves NPV(L) increases.

2) At the constant values of   NPV practically always decreases with leverage. Optimum in the dependence of NPV(L) is absent.

All curves NPV(L) at the constant values of   are started (at L=0) from one point and with growth of  (and, respectively, a decrease of ) all curves NPV (L) are shifted upwards. With growth of density of curves NPV(L) increases.

3) At the constant values of   NPV practically always  decreases with leverage. The optimum in the dependence of NPV(L) has been found for one pair of  and ( (8%) and  (6%)) only.

Fig. 4. Dependence of NPV on leverage level at fixed values of  and .

All curves NPV(L) at the constant values of  are started (at L=0) from one point and with growth of  (and, respectively, an increase of ) curves NPV (L) are shifted into region of smaller NPV values. With growth of density of curves NPV(L) increases.

1) At the constant values of  NPV practically always decreases with leverage. The optimum in the dependence of NPV(L) has been found for one pair of  and ( (8%) and  (6%)) only. All curves NPV(L) at the constant values of  are started (at L=0) from one point and with growth of  (and, respectively, ) curves NPV (L) lie below.

With growth of density of curves NPV(L) increases.

2) At the constant values of   NPV practically always decreases with leverage. Optimum in the dependence of NPV(L) is absent.

All curves NPV(L) at the constant values of  are started (at L=0) from one point and with growth of  (and, respectively, a decrease of ) curves NPV (L) are shifted upwards. With growth of density of curves NPV(L) increases.

3) At the constant values of   NPV practically always  decreases with leverage. The optimum in the dependence of NPV(L) has been found for one pair of  and ( (8%) and  (6%)) only. All curves NPV(L) at the constant values of  are started (at L=0) from one point and with growth of  (and, respectively, an increase of ) curves NPV (L) are shifted into region of smaller NPV values. With growth of density of curves NPV(L) increases.

Table 2. Dependence of NPV on leverage level  S=1000 b=0.1, k0-kd=const.

Table 2. (Continue)

2.2. Without Flows Separation

At a constant investment value (I = const)

                                (3)

At the constant values of  NPV demonstrates a limited growth with leverage with output into saturation regime. The main increase in NPV occurs when . With growth of  (and ) the сurves NPV(L) are lowered. Optimum in the dependence of NPV(L) is absent.

With growth of NOI all curves NPV(L) are shifted practically parallel upwards.

2) At the constant values of  NPV demonstrates a limited growth with leverage with output into saturation regime. All curves NPV(L) at the constant values of and different values of  are started (at L=0) from one point, the higher values of  correspond to more low lying curves NPV(L). Optimum in dependence of NPV(L) is absent. With growth of NOI all curves NPV(L) are shifted practically parallel upwards.

3) At the constant values of  NPV grows with leverage  with output into saturation regime. All curves NPV(L) at the constant values of  and different values of  are started (at L=0) from one point, the higher values of  (and higher values of ) correspond to more low lying curves NPV(L). Optimum in dependence of NPV(L) is absent.

Table 3. Dependence of NPV on leverage level NOI=1200 I=2000, k0-kd=const.

Table 3. (continue)

At a constant equity value (S = const)

                                     (4)

At the constant values of  NPV shows as an unlimited growth with leverage and unlimited descending with leverage. It is interesting to note, that the credit rate value  turns out to be a border at all surveyed values of , equal to 2%, 4%, 6%,10% (it separates the growth of NPV with leverage from descending of NPV with leverage). In other words, with growth of  the transition from the growth of NPV with leverage to its descending with leverage takes place, and at the credit rate  NPV does not depends on the leverage at all surveyed values of .

Thus, we come to conclusion, that for perpetuity projects NPV grows with leverage at a credit rate  and NPV decreases with leverage at a credit rate  (project remains effective up to leverage levels , ). Optimum in the dependence of NPV(L) is absent.

2) At the constant values of  NPV shows an unlimited growth with leverage as well as unlimited descending with leverage. NPV grows with leverage at a credit rate  and NPV decreases with leverage at a credit rate  (project remains effective up to leverage levels , ). All curves NPV(L) at the constant values of  and different values of  are started (at L=0) from one point, the higher values of  (and higher values of  ) correspond to more low lying curves NPV(L). Optimum in dependence of NPV(L) is absent.

3) At the constant values of  NPV as well as in case of constant values of  shows an unlimited growth with leverage  as well as unlimited descending with leverage. An analysis of the data leads to the same conclusion, that and, in 1): NPV grows with leverage at a credit rate  and NPV decreases with leverage at a credit rate  (project remains effective up to leverage levels , ).

Fig. 5. Dependence of NPV on leverage level at fixed values of  and .

It should be noted that this pattern should be taken into account by the Regulator which should regulate the normative base in such a way that credit rates of banks, that depends on basic rate of Central bank, not exceed, say, .

 All curves NPV(L) at the constant values of  and different values of  are started (at L=0) from one point, the higher values of   (and lower values of ) correspond to more low lying curves NPV(L). Optimum in dependence of NPV(L) is absent.

Table 4. Dependence of NPV on leverage level  S=1000 b=0.1, k0-kd=const.

Table 4. (continue)

3. The Effectiveness of the Investment Project from the Perspective of the Equity and Debt Owners

3.1. With the Division of Credit and Investment Flows

At a constant investment value(I = const)

       (5)

1) At the constant values of  NPV practically always decreases with leverage. At small L values for many pairs of values  and  (for example,  (24%) and  (22%) ;  (30%) and  (28%) and many others there is an optimum in the dependence of NPV(L) at small

For higher values of   (and, respectively, ) all curves NPV(L) lie below. With growth of NOI all curves NPV(L) are shifted parallel upwards.

2) At the constant values of  NPV practically always decreases with leverage, passing through (most often), or not passing (more rarely) through optimum in the dependence of NPV(L) at small

All curves NPV(L) at the constant values of  and different values of  are started (at L=0) from one point, the higher values of  (and lower values of  ) correspond to higher lying curves NPV(L). With growth of NOI all curves NPV(L) are shifted practically parallel upwards.

3) At the constant values of  NPV practically always decreases with leverage, optimum in the dependence of NPV(L) is absent. All curves NPV(L) at the constant values of   are started (at L=0) from one point, the higher values of  (and higher values of ) correspond to lower lying curves NPV(L). With growth of NOI all curves NPV(L) are shifted practically parallel upwards.

Table 5. Dependence of NPV on leverage level  NOI=1200 I=2000, k0-kd=const.

Table 5. (continue)

Fig. 6. Dependence of NPV on leverage level at fixed values of  and .

Fig. 7. Dependence of NPV on leverage level at fixed values of  and .

Fig. 8. Dependence of NPV on leverage level at fixed values of  and .

Fig. 9. Dependence of NPV on leverage level at fixed values of  and .

Fig. 10. Dependence of NPV on leverage level at fixed values of  and .

At a constant equity value(S = const)

            (6)

1) At the constant values of   NPV practically always decreases with leverage, optimum in the dependence of NPV(L) is absent. All curves NPV(L) at the constant values of   are started (at L=0) from one point, the higher values of  (and, respectively, ) correspond to lower lying curves NPV(L). With growth of the density of curves NPV(L) increases.

2) At the constant values of  NPV decreases with leverage, optimum in the dependence of NPV(L) is absent. All curves NPV(L) at the constant values of  are started (at L=0) from one point, the higher values of  (and, respectively, the lower values of ) correspond to higher lying curves NPV(L). With growth of the density of curves NPV(L) increases.

3) At the constant values of  NPV decreases with leverage, optimum in the dependence of NPV(L) is absent. All curves NPV(L) at the constant values of   are started (at L=0) from one point, the higher values of  (and, respectively, the higher values of ) correspond to lower lying curves NPV(L). With decrease of NOI the density of curves NPV(L) increases and they are shifted down.

Table 6. Dependence of NPV on leverage level NOI=1200 I=2000, k0-kd=const.

Table 6. (continue)

3.2. Without Flows Separation

At a constant investment value(I = const)

      (7)

At the constant values of  NPV demonstrates a limited growth with leverage with output into saturation regime. The main increase in NPV occurs when . With growth of  (and ) the сurves NPV(L) are lowered. Optimum in the dependence of NPV(L) is absent.

With growth of NOI all curves NPV(L) are shifted practically parallel upwards.

2) At the constant values of  NPV demonstrates a limited growth with leverage with output into saturation regime. All curves NPV(L) at the constant values of  and different values of  are started (at L=0) from one point, the higher values of  (and, respectively the lower values of ) correspond to higher lying curves NPV(L). Optimum in the dependence of NPV(L) is absent. With growth of NOI all curves NPV(L) are shifted practically parallel upwards.

Fig. 11. Dependence of NPV on leverage level at fixed values of  and .

3) At the constant values of  NPV shows a limited growth with leverage with output into saturation regime. All curves NPV(L) at the constant values of  and different values of  are started (at L=0) from one point, the higher values of  (and, respectively, the higher values of ) correspond to lower lying curves NPV(L). Optimum in the dependence of NPV(L) is absent.

Table 7. Dependence of NPV on leverage level  NOI=1200 I=2000, k0-kd=const.

Table 7. (continue)

At a constant equity value (S = const)

,                                                                  (8)

.                                                      (9)

1.  At the constant values of  NPV shows as an unlimited growth with leverage and unlimited descending with leverage. It is interesting to note, that the credit rate value  turns out to be a border at all surveyed values of , equal to 2%, 4%, 6%, 10% (it separates the growth of NPV with leverage from descending of NPV with leverage). In other words, with growth of  the transition from the growth of NPV with leverage to its descending with leverage takes place, and at the credit rate  NPV does not depends on the leverage level at all surveyed values of .

Thus, we come to conclusion, that for perpetuity projects NPV grows with leverage at a credit rate  and NPV decreases with leverage at a credit rate  (project remains effective up to leverage levels , ). Optimum in the dependence of NPV(L) is absent.

Fig. 12. Dependence of NPV on leverage level at fixed values of  and .

Fig. 13. Dependence of NPV on leverage level at fixed values of  and .

2.  At the constant values of  NPV shows an unlimited growth with leverage as well as unlimited descending with leverage. NPV grows with leverage at a credit rate  and NPV decreases with leverage at a credit rate  (project remains effective up to leverage levels , ). All curves NPV(L) at the constant values of  and different values of  are started (at L=0) from one point, the higher values of  (and higher values of  ) correspond to more low lying curves NPV(L). Optimum in dependence of NPV(L) is absent.

Fig. 14. Dependence of NPV on leverage level at fixed values of  and .

3.  At the constant values of  NPV as well as in case of constant values of  shows mainly an unlimited growth with leverage. Unlimited descending with leverage was shown for the pair  only. All curves NPV(L) at the constant values of  and different values of  are started (at L=0) from one point, the higher values of   (and lower values of ) correspond to more high lying curves NPV(L). Optimum in dependence of NPV(L) is absent.

Table 8. Dependence of NPV on leverage level S=1000 b=0.1, k0-kd=const.

Table 8. (continue)

4. Conclusions

We conduct the analysis of effectiveness of investment projects within the perpetuity (Modigliani–Miller) approximation (Modigliani et al 1958, 1963, 1966). Based on the obtained in previous papers results for NPV  (Brusov et al 2011a,b,c,d,e; 2012 a, b; 2013 a, b, c; 2014 a, b; Filatova et al 2008) we have analyzed the effectiveness of investment projects for three cases:

1)  at a constant difference between equity cost (at ) and debt cost ;

2)  at a constant equity cost (at ) and varying debt cost ;

3)  at a constant debt cost  and varying equity cost (at )

The dependence of NPV on investment value and/or equity value will be also analyzed. The results are shown in the form of tables and graphs.

The obtained tables have played an important practical role in determining of the optimal, or acceptable debt level, at which the project remains effective. The optimal debt level there is for the situation, when in the dependence of NPV on leverage level L there is an optimum (leverage level value, at which NPV reaches a maximum value. An acceptable debt level there is for the situation, when NPV decreases with leverage. And, finally, it is possible that NPV is growing with leverage. In this case, an increase in borrowing leads to increased effectiveness of investment projects, and their limit is determined by financial sustainability of investing company.

Fig. 15. Dependence of NPV on leverage level at fixed values of  and .

Fig. 16. Dependence of NPV on leverage level at fixed values of  and .

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