Thin capitalization is an observable concern in national law and in the OECD Model Tax Convention on Income and Capital. This paper attempts to establish a technical criterion to determine when a company is thinly capitalized and when it is not.

1. Corporate Tax Technical Note Spring 2010
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A FIRST INVESTIGATION ON THIN CAPITALIZATION
Ramon Serrallonga
ESADE Law School - Master in Business Law
Thin capitalization is an observable concern in national law and in the OECD Model Tax
Convention on Income and Capital. This paper attempts to establish a technical
criterion to determine when a company is thinly capitalized and when it is not.
1. Introduction
The first thing we should ask when studying thin capitalization is why we should care
about it. From the standpoint of maximizing a firm’s value, there is no conflict of
interests between debt and equity holders, like the risk-shifting incentives of leveraged
firms (Jensen & Meckling, 1976) or the debt overhang problem (Myers, 1977), when
they are the same entities. This is the case of related parties. From a credit risk
standpoint, we have to expect that rational creditors will take into account the
financial structure of the firm to determine the return they will demand for the risk
they are bearing. Under the arm’s length principle we should expect the same. From a
tax reuptake standpoint, however, there is a problem. If the authorities want to
intervene, they should know when a firm is thinly capitalized and by how much. The
following methodology is aimed at facilitating this work.
2. Financial management efficiency
Consider a firm characterized by equity (E), debt (D), tax rate (t), interest paid or
explicit cost of the investment (I), net income (B), leverage ratio (A), return on assets
(ROA), return on debt (ROD) and return on equity (ROE). We will assume (assumption
1) that the ROA is an exogenous variable. Then:
ROA = (B + I (A) + T) / (E + D)
ROD (A) = I (A) / D
ROE = B / E

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T = (B + T) · t
A = D / E > = 0
And
ROE (A) = (1 – t) · (ROA + A · (ROA – ROD (A)))
We will assume (assumption 2) that ROD (A) is a strictly increasing function on A with
ROD’ (A) > 0
And
ROD’’ (A) >= 0
In this case, the higher the A, the higher the credit risk and the higher the required
return. The shareholders want to find the optimal financial structure that maximizes
the ROE, so, differentiating with respect to A
ROE’ (A) = (ROA – ROD (A) – A · ROD’ (A)) · (1 – t)
Which is positive if
ROA > ROD (A) + A · ROD’ (A)
Taking the second derivative
ROE’’ (A) = (- 2 · ROD’ (A) – A · ROD’’ (A)) · (1 – t)
This is always negative under assumption 2. So, ROE (A) is a strictly concave function
with a global maximum at
A* = (ROA – ROD (A*)) / ROD’ (A*)
To understand this equation, let us rewrite it like this:
E · ROA = E · (ROD (A*) + A* · ROD’ (A*))
Maximizing the ROE with respect to A is equivalent to maximizing the net income of an
investment with respect to A. So A* is the leverage ratio that makes the marginal gross
income (ROA · E) equal to the marginal explicit cost of the investment (E · (ROD (A) + A
· ROD’ (A))) making the difference between gross income and the explicit cost of the
investment as well as the net income higher. It is very important to mention that the
optimum leverage ratio does not depend on t, thereby demonstrating that
opportunistic behavior with taxes does not play any role and it is all about financial
efficiency. The ROE (A) function is characterized by

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ROE (0) = ROA
ROE (A*) = ROA + (ROA – ROD (A*))2
/ ROD’ (A*)
ROE (Â) = ROA
Where
ROD (Â) = ROA
And
ROE (Ä) = 0
Where
Ä = ROA / (ROD (Ä) – ROA)
Again, we can observe that no critical issue depends on the tax rate.
If
ROD’’’ (A) = (- 3 · ROD’’ (A) – A x ROD’’’ (A)) · (1 – t) = 0
And so
ROD’’ (A) = ROD’’’ (A) = 0
ROE (A) would be a symmetric function with respect to the A* axis. It would have the
same effect on ROE (A) to be too much or too little indebted by the same proportion
with respect to the optimal one, in the interval [0 , Â] where now  will be twice the
amount of A* because of symmetry. Financial managers could have the target of
matching A* without caring about excessive leverage. Therefore, the following indexes
could be a good measure of the efficiency of financial management:
Optimal Leverage Position Index = A · 100 / A*
Financial Management Efficiency Index = (A* - IA* - AI) · 100 / A*
3. A linear model
A ROD (A) function linear on A has the good properties mentioned above. It is not
necessary to be linear in other factors like
ROD (A, x, y…) =  · A + f(x, y…)

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Assuming independence between factors. Consider the naïve model
ROD (A) = r +  · A
Where r is the risk-free interest rate and  is a parameter. Giving values to
ROA = 0.25
E = 1
r = 0.05
 = 0.1
t = 0
The critical points become
A* = 1
 = 2
Ä = 2.870…
ROE in function of A

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Gross income, explicit cost of the investment and net income in function of A
4. Conclusion
Under this model, to consider that a firm is thinly capitalized, it has to have a leverage
ratio substantially above A* which is steady over time. It would be inappropriate to
consider the case of an oscillating optimal leverage position index around A* or an
extraordinary high leverage ratio because of a shock in a variable. It would be
incomprehensible for an independent firm to be thinly capitalized because this goes
against the interests of the shareholders. But this does not have to hold true in the
case of related parties, like in a parent-subsidiary relationship, when equity holders are
also debt holders. In that situation, we should compare their financial structure with
the one they should have if they were independent, such as the one proposed in this
paper.