29/09/2013 1 Blind man's buff : ON THE SEARCH OF THE OPTIMAL CAPITAL STRUCTURE IGNACIO VÉLEZ P AREJA, FELIPE MEJÍA JAMES K OLARI AUGUST 30, 2012 Abstract Abstract Abstract Abstract: In these slides we discuss the practical and conceptual difficulty of finding an Optimal Capital Structure. We propose a normative approach we call Implicit Bankruptcy Costs Theory. We proceed to find the optimal capital structure and value when leverage is both constant and variable from one period to the next and the discount rate for tax shields is the cost of levered equity, Ke. Numerical procedures and a recursive closed-form non-circular expressions for the finite-period and perpetuity cases are presented, which facilitate implementation including Monte Carlo simulations. • Number of Pages in PDF File: 32 • Keywords: Optimal capital structure, valuation, non-circularity, finite cash flows, perpetuities tax shield, cost of equity • JEL Classification: M21, M40, M46, M41, G12, G31, J33 29/09/2013 EOC Vélez Pareja, Mejía y Kolari 2
Transcript
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1
Blind man's buff :
ON THE SEARCH OF THE OPTIMAL CAPITAL STRUCTURE
IGNACIO VÉLEZ PAREJA,
FELIPE MEJÍA
JAMES KOLARI
AUGUST 30, 2012
AbstractAbstractAbstractAbstract::::
In these slides we discuss the practical and conceptual difficulty of finding anOptimal Capital Structure. We propose a normative approach we call ImplicitBankruptcy Costs Theory. We proceed to find the optimal capital structureand value when leverage is both constant and variable from one period tothe next and the discount rate for tax shields is the cost of levered equity, Ke.Numerical procedures and a recursive closed-form non-circular expressionsfor the finite-period and perpetuity cases are presented, which facilitateimplementation including Monte Carlo simulations.
• Number of Pages in PDF File: 32
• Keywords: Optimal capital structure, valuation, non-circularity, finite cash flows, perpetuities tax shield, cost of equity
• Kolari, James W. and Ignacio Vélez-Pareja. 2010. Corporation Income Taxes and the Cost of Capital: A Revision. Available at SSRN: http://ssrn.com/abstract=1715044)
• Vélez-Pareja, Ignacio, Mejía Felipe and James Kolari. 2012. Optimal Capital Structure for Finite Cash Flows
• (Work in process)http://ssrn.com/abstract=1799605
• Salas Pérez, Rafael, Gutiérrez Ruiz, Juan and Ignacio Velez-Pareja. 2011. Value of Debt Tax Shields in Colombia: An Empirical Study. http://papers.ssrn.com/abstract=1919305
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What is this paper about?• We explore the possibility of identifying a level of debt that results in
an optimal value of the firm.
• We propose a method for finding a normative optimal capital structure for finite flows in two versions: (1) an Excel spreadsheet method for optimal structure, and (2) an analytical formulation. Our approach works when it is assumed that debt interest tax savings are discounted at the cost of equity, Ke.
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Problem not solved in financeProblem not solved in financeProblem not solved in financeProblem not solved in finance
• Optimal capital structure (OCS) has been a very elusive issue for many years and experts believe that is one of the unsolved problems of corporate finance.
• Everybody speaks of the OCS, but no one has seen it.
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Theories of OCS
• Tradeoff theory
• Pecking order theory or hierarchy of available resource utilization
• Cash flow theory of Myers (1993)
• Implicit bankruptcy costs theory (consistent with our proposal)
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Tradeoff theory
• Tradeoff theory assumes that the firm defines an OCS and tries to reach it in the future. It recognizes that there are benefits called tax savings on interest tax deductions on debt payments versus costs of financial distress (including potential bankruptcy costs). Both increase with indebtedness and eventually cancel each other out at the OCS.
• This is the most popular theory, which can be found in corporate finance textbooks.
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Then I borrow up to 100%
Discounting the tax savings (TS) at Ku or Kd yields the result that (ifnot adjusted for bankruptcy costs) the optimal capital structure is100% debt. However, this result is not reasonable, because as a firmborrows debt funds, there exists some contingent and/or hiddencosts associated with the fact that the firm may not be able to pay thedebt in the future and become insolvent. This means that there is anexpected value or cost of bankruptcy or financial difficulties thatreduce the value of the firm. The existence of these costs prevent, ingeneral, firms to borrow up to 100%.
This idea is known as the trade-off theory.
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Financial distress costs
When a firm starts to borrow, it increases the risk perceived by thirdparties, for example, the debt holders. A bank could charge more fornew loans. This higher cost is reflected in lower cash flow which inturn increases cash requirements and could increase indebtedness.
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Trade-Off Costs
Debt information spreads easily and providers can lose confidence and stop providing credit (at zero cost) and require payment in advance on outstanding debt. This reduces liquidity and increases the need for funding, which means a higher cost. Creditors can also reduce the amount of credit facilities such that the firm loses economies of scale and gross margin is reduced.Customers, who also learn of the situation, possibly will not buy the same amount because they prefer a secure supplier.
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Human Resources Costs
When the firm gets into financial difficulties, it is possible that qualityemployees resign from the company. Each new employee must betrained and the loss of intellectual capital is difficult to measure andreplace.
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A Vicious Circle
The firm can get into a vicious debt cycle that forces it to explore costlier financial sources. Financing costs above usury rates may not accepted by law to be deducted. That is, the tax savings or shields, TS, are lost. On reaching the extreme situation of near bankruptcy, advisers are required in different areas of the firm such as lawyers in particular.These added costs of financial difficulties can be considerable.
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EOC and firm value
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D/E = Debt/Equity Optimal leverage
Firm Value Vlevered = Vunlevered + PV(TS)
Firm Value
Present value of TS
Distress CostsMaximum value of levered firm Vlvd
Vunlevered
OptimalOptimalOptimalOptimal WACCWACCWACCWACC
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WACC
Mínimum WACC
D/E Optimal Capttal Structur
Κu (unlevered firm)
Kd(1-T) = cost of debt after taxes
WACC= KeE% + Kd(1-T)D%
Ke = cost of equity
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OCS and Other Theories
• The pecking order theory that proposes a hierarchy of use of available resources as well as cash flow theory do not identify an OCS.
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In Search of the OCS
• Many researchers in finance have been trying to discover how and why firms borrow and how they get (if at all) to the OCS.
• The trade-off theory is most popular approach to finding an OCS.
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Some Empirical Evidence
The different theories have to be compared with what we see in reality.Below there is some evidence from Colombia, the U.S., and LatinAmerica.
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Colombia: 25 traded firms 2001-2010
y = 0.3559x-0.16
R² = 0.0401
0
1
2
3
4
5
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Tota
l val
ue/
Bo
ok
valu
e
D%
Total value/Book value
(D+E)/Vass
Potencial ((D+E)/Vass)
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Regression LnTVt TSt and Dt
Dependent variable Tax shields Debt
Relation Does not affect Does not affect
Variable dependiente: Ahorros en Impuestost Deudat
Ln (VTt) 3,62E-07 (5,01E-07)
-
Ln (VTt) - 2,86E-07 (1,80E-07)
Relación No afecta No afecta
Regression LnTVt TSt-1 and Dt-1
Dependent variable Tax shields Debt
Relation Positive Does not affect
Variable dependiente: Ahorros en Impuestost-1 Deudat-1
Ln (VTt) 7,21e-06
(5,59e-07)***
Ln (VTt) - 1,61e-07 (1,71e-07)
Relación Positiva No afecta
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Regression TVt TSt and Dt
Dependent variable Tax shields Debt
Relation Negative Does not affect
Variable dependiente: Ahorros en Impuestos t Deuda t
-26,12253
(3,329189)***
1,266308
-1,401219
Relación Negativa No afecta
VTt -
VTt -
Regresión VTt TSt-1 y Dt-1
Dependent variable Tax shields Debt
Relation Does not affect Does not affect
Variable dependiente: Ahorros en Impuestost-1 Deudat-1
VTt 4.500237 (4,745917)
-
VTt - 1,271748 (1,448643)
Relación No afecta No afecta
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PV(TS at Kd)/TotAsst vs D%
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0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
VTS
_Kd
/To
tAss
t
D%_t-1
VTS_Kd/TotAsst
PV(TS at Ke)/TotAsst vs D%t-1
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0
0.05
0.1
0.15
0.2
0.25
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
VTS
_Ke
/To
tAss
t
D%_t-1
VTS_Ke/TotAsst
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Regression statistics
VTS(Ke) vs D%t-1
R2 0,46416
R2 adjusted 0,455902
Observations 188
Coeff. t p-value
D%t-12 -0,29762 -7,17238 1,69E-11
D%t-1 0,312791 10,92598 8,75E-22
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With Kd and Ku there is no OCS
• Clearly, as Kd and Ku are not dependent from D%, for the basic model of M & M, the greater D% the higher VTS is, a linear relationship. Kd can be modeled as a function of D%. But the increase in Kd is NOT the only cost of bankruptcy. Other costs are captured by Ke, as will be seen.
• There is a contradiction between the behavior of VTS calculated with Ke and the behavior of TV (D + # stocks * Price). The data suggest that VTS = PV (TS at Ke) (which is what generates the optimum) behave nearly as an inverted U. But the market does not recognize it. Not so with VTS = PV (TS at Ku or Kd). The behavior is decreasing when D% grows.
TheTheTheThe Blind Man's Buff Blind Man's Buff Blind Man's Buff Blind Man's Buff • We want to find a D% that maximizes the value of the firm, but managers do not know
how or where it is located.
• If there is debt, the firm might earn tax benefits. It is known that the tax savings or tax shields are there, but shareholders do not directly see them because the dividend payments they receive do not clearly show these tax benefits.
• At the same time, with debt comes some financial costs. The hard part is that it is not clear what those future potential bankruptcy costs. Everyone knows how they arise – for example, commercial, financial, and legal costs – but they do not know how to measure them. And nobody says how.
• Although debt has some tax benefits, shareholders’ investment is risky due to being the last in the chain to receive their investment and profits.
• At the same time, many people misunderstand what was said by Modigliani and Miller in 1958 that the capital structure does not affect the value of the firm. This is true only if there are no taxes.
• In short, you have to guess the OCS. No one knows how to find it, or how to calculate it – it is a game of blind man's buff!
Facsimile of illustration of a class of 捉迷藏 in a book of Chinese elementary school in 1912.
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In Search of the EOCS
• There are works that try to find the optimal debt, others seek the speed with which the firm approaches the OCS, etc.
• It is a frantic search for the simple reason that no one knows where or how to calculate the OCS. Everyone talks about the EOC but no one has seen it.
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Implicit Bankruptcy Costs Theory (1)
• While the trade-off theory leaves things "alone" in order to reach an optimal, the proposed discounting of TS at the levered cost of equity, Ke, is normative and prescriptive.
• After discounting the TS with Ke, an optimum is obtained, and you can tell management that there is an optimum and can make decisions to reach that optimal level of leverage.
• It is not left to the "invisible hand" to handle the costs of financial distress and bankruptcy.
• The trade-off theory does not tell management what to do, but says that there are some theoretical costs that may appear and then get an optimum.
• Our proposal that tax savings or shields, TS, are discounted at the cost of equity, due to the fact that TS belongs to the shareholder (CFE = FCF + TS - CFD), there is a firm valuation effect associated with debt that does not occur with discount rates Kd and Ku are used (i.e., commonly proposed in corporate finance literature).
• The Ke formula when it is supposed that Ke is the discount rate for TS is:
• By having the debt involved in the formula for Ke, it captures the effect of debt, and an optimum is obtained, as seen in the chart below.
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( )
( )TS
1i1i
1-iiiii
1-i
Un
1i
1-iiiii
VE
DKdKuKu Ke
DV
DKdKuKu Ke
−−
−
−−+=
−−+=
OCS when Ke is the discount rate of TS
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9.9
10
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
VL
D/V
Firm Value, VL vs D% perpetuity
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The case of finite cash flows and constant D%
• The procedure maximizes firm value with constant D%. The optimizer model isMax VLsubject to0 ≤ D ≤ 1% (single cell)VL is the firm value and D% is the constant leverage.This procedure generates a circularity and you have toiteratively calculate value to reach a solution.
• There is consistency among methods: FCF, APV, CCF, CFE
• Equilibrium equations among cash flows and values are fulfilled.
• The proposed Ke reveals the OCS:
• Note that Ke does not imply a constant D% nor a constant WACC (WACC).
( )
( )TS1i1i
1-iiiii
1-i
Un
1i
1-iiiii
VE
DKdKuKu Ke
DV
DKdKuKu Ke
−−
−
−−+=
−−+=
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TheTheTheThe case of case of case of case of finitefinitefinitefinite cashcashcashcash----flow and variable D%flow and variable D%flow and variable D%flow and variable D%
• The procedure maximizes firm value with constant D%. The optimizer model isMax VLsubject to0 ≤ D ≤ 1% (several cells D% for each period)VL is the firm value and D% is the constant leverage.This procedure generates a circularity and iteratively calculate the solution.
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With variable D%
Year 0 1 2 3 4
D% 72.983% 75.600% 78.566% 82.324%
FCF 17.00 20.00 22.00 25.00
Debt D 45.04 39.79 30.90 18.32
Principal payment 5.25 8.90 12.57 18.32
Interest 4.95 4.38 3.40 2.02
Tax shields. TS 1.73 1.53 1.19 0.71
CFD 10.20 13.28 15.97 20.34
CFE = FCF - CFD + TS 8.53 8.26 7.22 5.37
PV(FCF at Ku) 58.66 50.46 38.03 21.74
Ke = Ku + (Ku-Kd)D/(VU- D) 28.22% 29.92% 32.31% 36.45%
• Although the formulation of Ke when the TS are discounted at the same rate Ke appears Kdt implying that Kd can vary, in Kd it is not included implicitly the variation we are interested in: that is, the effect of the value of leverage on Kd.
• Examining lending rates for Colombia between 1998 and 2010 and "risk free" rates (TES bonds), a risk premium for debt RPD can be estimated as the difference between the two. With that RPD a relationship between RPD and leverage D% was established. Analyses were performed for non-financial sectors using a total of 771 observations (sector/year). We used the accounting D% and the database of the Superintendence of Companies of Colombia. The use of book value of leverage is justified because it is the most evident measure of leverage perceived by the market.
RPD = risk premium for debt according to each type of loans; Kd = cost of debt; Rf = risk free rate.
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Algunos resultados
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Example
Year 1 2 3 4
T 35% 35% 35% 35%
Rf 7.00% 7.00% 7.00% 7.00%
Ku 15.00% 15.00% 15.00% 15.00%
FCF 17.00 20.00 22.00 25.00
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Definition of Definition of Definition of Definition of KdKdKdKd
• We assume that Kd is estimated based on the risk premium for debt, PRDKd = Rf + RPDRPDpref = 0,018028755 + 0,021257514 D%This estimate is obtained using Solver optimal.
Do Do Do Do interestinterestinterestinterest paymentspaymentspaymentspayments affectaffectaffectaffect OCS?OCS?OCS?OCS?
•Perhaps not. What debholders have to pay in taxes should not affect TS earned by the firm, which tbelong to shareholders.
• It is similar in logic to assuming that tax savings due to labor payments affect TS obtained by the firm due to the taxes employees pay.
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Do personal Do personal Do personal Do personal taxestaxestaxestaxes destroydestroydestroydestroy TS?TS?TS?TS?
• In principle, this occurs in countries with double taxation. That is theposition of those that asume that there is no such thing as OCS.
• They apply this formula: (1-Tdebt) = (1-Tcorp)(1-Tpers) and they say this makes TS ≈ 0.
• As we are concerned with the maximization of shareholders’ wealth, we should not worry about debtholders wealth. They receive contractual interest and principal payments. What we have to look for is if Int*Tcorp –Int*Tdebt – Div*Tpers = 0 personal taxes destroy TS (T is tax rate). Why this? Because firms do not always distribute 100% of net income. This meansthat TS=0 if Div/Int = (Tcorp- Tdebt) Tpers.
• The condition of full distribution could be tested with information of listed firms.
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WhyWhyWhyWhy personal personal personal personal taxestaxestaxestaxes doesdoesdoesdoes affectaffectaffectaffect TS? TS? TS? TS?
• Personal taxes on dividends affect TS because TS belong toshareholders. This occurs where there is doublé taxation.
• Taxes paid by debtholders cannot affect TS that belong to shareholders.
• This is, net TS after the effect of personal taxes on receiveddividends by the shareholders.
• TS is tax shields, Tcorp is the corporate tax rate, Div is the dividends paid by the firm to shareholders, and Tpers is the tax rate paid bythe shareholder.
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Relationship between dividends and interest payments
• If we define that dividends are some proportion of interest, we have
Div = b*Int (2)
• If we define that the personal tax rate, Tpers, is a portion of the corporate tax rate Tcorp, we have
Tpers =a*Tcorp (3)
• Then TS could be defined as
TS = Int*(Tcorp - b*Tpers) (4)
TS = Int*Tcorp *(1 - b*a) (5)
TS = Int* Tcorp - Div* Tpers � TS will be 0 if Div/Int = Tcorp/ Tpers
• This can be tested with actual data and is valid if we asume shareholders are individuals.
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What is the relationship when there is corporate shareholders?
• When equity is shared by personal and institutional (corporate) shareholders, the
previous relationship can be rewritten as
TS Net = TS – Dividends*ca*Tcorp – Dividends*pa* Tpers
where pa + ca = 1, with pa = % of equity owned by individuals and ca = % of equity
owned by corporate investors.
• What we are interested in is the case when net TS is zero. In that case,
Tcorp*Interest = Dividends*(ca*Tcorp +pa* Tpers)
Tcorp/(ca*Tcorp + pa* Tpers) = Dividends/Interest
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• Then we can examine the relationship between dividends and interest payments compared with the LHS of last equation. With this, we can estimate if it is true that personal taxes (on dividends) when there is doublé taxation destroys TS and, hence, there is NO OCS.
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Value of Div/Int for TSnet = 0
ca Tcorp/(ca*Tcorp + pa*Tpers) 0.0% 3.30
12.5% 2.56
25.0% 2.10
37.5% 1.77
50.0% 1.53
62.5% 1.35
75.0% 1.21
87.5% 1.10
100.0% 1.00
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What does the previous table tell us?
• Assuming Tcorp=33%, and Tpers = 10%, we obtain the previous table for different percent of institutional (corporate) shareholders. This assumesdouble taxation.
• The table says that, if shareholders are 100% corporations, then dividends and interest payments are identical, and TS vanishes. This means that there is no OCS; if the right column value is greater than 1, then net TS would be negative and OCS should be 100% debt. If the proportion in the first column is 50%, when Div/Int is at least 1.53, TS = 0. In the case there is 100% individuals, Div/Int must be at least 3.3 for TS = 0.
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DivDivDivDiv////IntIntIntInt in in in in somesomesomesome firmsfirmsfirmsfirms tradedtradedtradedtraded in in in in Colombia’sColombia’sColombia’sColombia’s Stock ExchangeStock ExchangeStock ExchangeStock Exchange
b = Div/Int
(times)Frecuency Cummulated%
0.0 26 28.89%
0.5 9 38.89%
1.0 3 42.22%
1.5 5 47.78%
2.0 3 51.11%
2.5 5 56.67%
3.0 2 58.89%
43.21 29 91.11%
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b = Div/Int (times) Frecuency Cummulated%
83.42 1 92.22%
123.63 1 93.33%
163.84 1 94.44%
204.06 2 96.67%
244.27 0 96.67%
284.48 1 97.78%
405.11 1 98.89%
>405.11 1 100.00%
In a sample of 11 firms traded on the stock Exchange of Colombia between 2001
and 2009, we obtained the following
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• We show some evidence that contradicts what has been studied in the literature on capital structure. Using the trade-off theory for finding an optimal structure, our graphic evidence implies a preference for low leverage.
• An explanation for this behavior is that, given the ignorance of what the OCS is, managers and owners prefer to be on the safe side of the curve. That is, they tend to use low leverage. It is as if there were an horror debiti similar to horror vacuiifound in nature.
• We presented a normative model that allows management to identify in advance their firm’s optimal capital structure and focus efforts toward that goal.
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Conclusions (1)
• There is some indication of the behavior of VTS as optimizer when calculated with Ke although the market does not recognize it.
• The optimal capital structure is found by numerical methods using Excel Solver and analytical solution using data such as Ku, K, T, FCF and TS.
• Management must model the cashflow behavior that not only reflects the expected leverage but includes the effects of leverage on some variables of the mode. For example, it is possible to model the variable cost of the debt as a function of leverage. It could be done with other variables as this is not the only cost of bankruptcy.
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Conclusions (2)
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• There remains the problem that occurs when Ku - Kd is very small. This affects the optimum, Debt and TS when D and D% are very large. That is, when there are low levels of Ku – Kd, it might collapse when the leverage is high. We have to study this issue.
• Last but not least, we have to study what is the composition of ownership (individual and corporation) in order to measure the effect of doublé taxation on the existence of an OCS.
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Conclusions (3)
Summary
ψψψψ
TasaKd Ku Ke
WACC
FCF
WACC
CCFKu
(no circularity)
Ke(no circularity)
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( )1i
TS
V
VKd -Ku - Ku
1-i
−
( )1iV
TS1-iV
1iDUn
1iV
1iDKd -KuKu
−−−−
−+
( )1-tV
TS
1tVKdKu-
1tV
tTKu
−−
−
−S
1tV
tTKu
−
−S
( )1iV
TS1-iV
1iDUn
1iV
1iDKd -Ku
1iV
iTS- Ku
−−−−
−+
−
( )
1-tE
VTS
1tV
1tE
1-tD
KdKuKu
−
−
−
−+ ( )1tE
1-tDKdKu
Ku
−
−+
( )1-iD
Un
1iV
1-iDKdKuKu
−−
−+
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Thank you!
☺
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Ahorros en impuestos
Son un subsidio que el gobierno da a la firma por cada gasto deduciblede impuestos de renta. Esto se llama una externalidad. El value de estesubsidio es de TKdDt-1, donde T es la tasa de impuestos, Kd es el costo dela Debt y D es la Debt.
Así las cosas, el value de la firma se incrementa por el value presente delos ahorros en impuestos o escudo fiscal (tax shield). Es decir, una firmacon Debt vale más como un todo que una firma sin Debt.
VCD = VSD + VTS =PVatrimonio + VDebt
Estos valuees tienen asociados respectivamente los siguientes flujos decaja .
FCF + TS = CFE + CFD
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Recordar: El TS depende de la UO
1. Si UO+OI ≥ Gastos financieros (GF) entonces
TS = T × GF
1. Si 0 ≤ UO +OI< GF entonces
TS = T × (UO+OI)
1. Si UO+OI < 0 entonces
TS = 0
Esto significa que TS es
T × Máximo((Mínimo(UO+OI, GF), 0).
Si se puede amortizar pérdidas, los TS no ganados en un período se pueden recuperar en el futuro
El CPPC tradicional, CPPC = Kd × D%×(1−T) + Ke×P% aplica para el caso 1 si se pagan los impuestos en el mismo período en que se causan y los Interest son la únicafuente de TS. Es un caso particular deCPPCFCF_t = Kut – TSt/Vt-1
