Financial Leverage for Multi-Period Levered Investments

Return on Equity (ROE) is one of the most popular performance metric related to equity involved in one-period investment. According to the Modigliani and Miller leverage formula, that applies to one-period levered investments, if the rate of Return on Investment (ROI) is not less than the Rate of Debt (ROD) then external financing increases ROE. The aim of this paper is to extend the Modigliani and Miller leverage formula to multi-period appraisals. First, we define the multi-period ROE as the Equity cash-flow Internal Rate of Return. Then, we achieve sufficient and necessary conditions to guaranteeing that external financing has beneficial effects on ROE. If projects are financed by up-front funds, as it is typical in longterm project financing, the Modigliani and Miller leverage formula can be extended. 1398 Simone Farinelli et al.


Introduction
The effects of debt on financial indices have long been a central question in numerous disciplines, including corporate finance, engineering economy and, in general, financial risk management (see [3] and [6] and the literature hereafter).In this paper we focus on the effects of financial leverage on one of the most popular measure of profitability of internal capital, the so called Return on Equity (ROE).Seminally [7] argue that if the Return on Investment (ROI) exceeds the Rate (or cost) of the Debt (ROD) then financial leverage has beneficial impact on ROE (see also [2]).Since typically long-term investments cover multi-period financing, we wonder whether the leverage rule still applies for multi-period appraisals.First, we extend the notion of ROE of multi-period appraisals as the Internal Rate of Return (IRR) of the Equity Cash-Flow.Sufficient and necessary conditions for leverage to increase ROE are obtained.In contrast to single-period projects, to detect the leverage impact on ROE, more complex conditions are to be checked.The augmented complexity stems from the presence of possible multiple ROEs at the same leverage level (for a seminal discussion on the IRR non-uniqueness, see [8]; and [5]).To guaranteeing the existence of a single ROE at any leverage level, we restrict our analysis to levered projects characterized by up-front financings.Typically that occurs in project financing when internal and external capitals are invested in one lump sum.In these circumstances the Modigliani and Miller leverage formula can be extended to multi-period valuations.Due the relevance of leverage in trade-off theory (see [10] and literature hereafter for trade-off theory) our results shed lights to further applications in the theory of capital structure.
The remainder of the paper is organized as follows.In Section 2 we introduce the basic notation.In Section 3 we derive sufficient and necessary conditions under which ROE increases with leverage.Section 4 concludes.

Notation setup
The impact of debt on ROE has been analysed extensively in the corporate finance literature.The leverage formula can be traced back to [7]    ROI is an IRR of the project cash-flow A, so it is a solution of the equation  ROD is an IRR of the unitary debt cash-flow D, so it is a solution of the equation  ROE  is an IRR of the equity cash-flow generated by the 100%

From Single vs. Multi-Period Project Appraisals
To get insights into the formula (1), we first analyze single-period projects.By definition ROE  is the solution of the equation (1), that takes for Plain remarks follow.ROE  exists and is positive if:   , i.e. the project is equity financed in positive percentage   , with 01   . (3) We are ready now to provide sound foundations to the classical single-period leverage formula (see [7]).Let Financial leverage for multi-period levered investments 1401 where x is a solution of the equation Proof (see the Appendix).
In contrast to single-period context, the impact of leverage on ROE no longer depends on the ROI and ROD spread only.The augmented complexity is imputable to the definition of IRR as a solution of an equation of n-th order.IRR drawbacks in this respect have been seminally pinpointed by [8] and long debated in academia.
Recently, [5] sums up "eighteen fallacies" of IRR; the most relevant in our context concerns the presence of multiple IRR (for a critical analysis for a practitioner audience, see [9])

Norstrøm (1972)'s conditions
To eliminate possible IRR inconsistencies and guarantee the existence of a single ,, ROI ROE ROD  at any leverage level  , we confine our analysis to levered projects that apply [8] conditions.So, we restrict our analysis to levered projects for which the sequence of non-discounted cash flow changes sign only once.To state it differently, we consider levered projects such that outflows come before inflows, i.e.In addition, to deal with meaningful financial indices, we concern unlevered projects with 0 ROI  and financings with 0 ROD  .The single-period leverage rule described in Theorem 1 is plainly extended to multiperiod appraisals.

Conclusion
In this short note we extend the Modigliani and Miller financial leverage rule to multi-period appraisals.Sufficient and necessary conditions guaranteeing that financial leverage increases ROE, referred as equity IRR, are stated.To avoid the undesirable occurrence of multiple ROEs, we restrict our focus on levered projects characterized by one lump internal and external capitals as it is typical to occur in long-term project financing.In these circumstances, the Modigliani and Miller single-period leverage ruleif ROI is not less than ROD, leverage arises ROEis extended to multi-period appraisals.

Proof of Theorem 2
The sketch of the proof follows the seminal path suggested in [4].By definition, . Since g has continuous partial derivatives, Dini's Implicit Function Theorem provides the derivative of is increasing with respect to  .Q.E.D.

Proof of Theorem 3
Let ROE  is nonnegative and increasing with respect to  .
By Theorem 2, it results ROE  is nonnegative and increasing at any leverage level  , with 01   .Now, let ROE  be nonnegative and increasing at any leverage level  , with 01   .Then 0 dROE d   and consequently, ROI ROD  .Q.E.D.
to Norstrøm (1972)'s conditions, if a cash-flow displays one change in sign, the equation ( ) 0 DCF x  admits a unique solution 1x  .So, for any 01   there exists a unique ROE  .As a consequence,  10 ROE   in(5).Since also the denominator of (5) is positive, it follows that The function ROE  results nonnegative and increasing with respect to  .Q.E.D.
[1] us now generalize previous results to multi-period levered projects.Multiperiod ROE is referred to as an Internal Rate of Return of equity cash-flow (see[1]).Sufficient and necessary conditions for leverage to increase ROE are given.