Industry Survival Rate, Entrepreneur Historical Performance and
Personal Wealth: A Probabilistic Model for Optimizing SMEs’ Capital
Structure
Dr. Andrea Moro
Corresponding author
University of Leicester, School of Management
Ken Edwards Building – University Road
Leicester
LE1 7RH, United Kingdom
Tel. 0044 (0) 116 252 3376
Dr. Sandra Nolte
University of Leicester, School of Management
Ken Edwards Building – University Road
Leicester
LE1 7RH, United Kingdom
1
Industry Survival Rate, Entrepreneur Historical Performance and Personal Wealth: A
Probabilistic Model for Optimizing SMEs’ Capital Structure
Abstract
This work proposes a theoretical model for calculating the return on investment by looking at
the risk entrepreneurs incur from possible bankruptcy of the firm: the cluster risk (cluster
survival rate) and the firm specific risk (historical lack of success of the entrepreneur). They
are related through a Markov transition probability matrix. The risk is then weighted by the
personal wealth entrepreneurs invest in the venture as equity and as debt collateralized with
personal assets. The equilibrium between the cost of equity and the cost of debt collateralized
by personal assets and optimal capital structure is worked out and discussed.
JEL: G31, G32
Keywords: Capital Structure, Small and Medium Sized Enterprises, Return on Equity
1. Introduction
Two different branches of the literature on the capital structure of firms can be distinguished.
On the one hand, there is Modigliani and Miller (1958) seminal model, which is based on the
assumption of efficient markets (Fama 1970). This stream of research focuses mainly on
modeling theoretically the decisions on capital structure of large corporations. It concentrates
on investor expectations as the core determinant of the cost of capital (capital asset pricing
model, Sharpe (1964), and on the key role of portfolio diversification as a tool for dealing
2
with diversifiable risk incurred by investors (Markowitz 1952). On the other hand, there is the
empirical research on capital structure of SMEs that, based on the research on corporate
finance, tries to work out the expected return of the entrepreneur, the cost of capital and the
capital structure of the firm by looking at the opportunity cost, that is by evaluating the
potential benefit of investing in a specific venture with respect to the cost of missing the
opportunity to invest somewhere else (Müller 2011). In fact, this approach works well for a
traditional investor in shares of listed corporations but less well when it is applied to
entrepreneurs who have their own business idea and are interested in developing it (Timmons
1999). Actually, the predictors of capital structure in SMEs are found to be different from
large corporations. Aspects like management values and goals, an optimistic/pessimistic
vision of the firm’s future, access to finance and human capital, gender, the life stage of the
firm and its size, ownership structure, profitability, potential growth, and the provision of
collateral are found to affect SMEs’ capital structure. All in all, traditional financial models
developed for large listed corporations cannot properly be applied to SMEs (Chaganti et al.
1995).
This research, at variance with previous approaches, proposes a theoretical model for
calculating the expected return of the entrepreneur and then the cost of capital and its
structure by modeling the entrepreneur’s financial decision as that of a peculiar provider of
funds (in fact, a very peculiar kind of lender). Typically, lenders measure the risk they incur
by lending to the borrower based on the probability of default of the borrower and the
potential loss they might incur in case of default (i.e. the amount of money the borrower will
not be able to pay back). They then decide whether to lend and the interest rate premium to
charge the borrower. Similarly, entrepreneurs and owner/managers who invest their personal
3
wealth in a venture may evaluate the risk they incur by checking the probability that their
venture might collapse and the impact that such a potential adverse outcome would have on
their personal wealth. Then they may work out the minimum expected return they require in
order to compensate for the risk incurred. Clearly, the impact of an adverse outcome on
entrepreneurs’ personal wealth is expected to be particularly high: entrepreneurs are typically
very concentrated (Heaton and Lucas 2000) and capable of accepting high levels of risk
(Moskowitz and Vissing-Jørgensen 2002). In addition, the theoretical model emphasizes the
fact that entrepreneurs have to take into consideration not only the equity invested in the
venture but also the personal collateral that they provide to the firm in order to grant it access
to credit. Indeed, personal collateral increases the entrepreneur’s stake at risk.
The consequences of the proposed theoretical model are twofold. Firstly, the cost of
capital has to take into consideration not only equity and debt but also the specific cost
incurred by the firm when it is financed with debt collateralized by an entrepreneur’s personal
assets. Secondly, there could be scenarios in which the cost of debt that is collateralized by
personal assets is greater than the cost of equity invested in the venture.
The paper is structured as follows: Section I explains the model’s underpinnings
examining why previous solutions neither solve the problem of the return on equity for the
entrepreneur nor provide a useful model for building the capital structure of the
entrepreneurial firm. Section 3 presents a probabilistic approach to working out the expected
return, while Section 4 illustrates the consequences of the probabilistic approach to the return
for the entrepreneur on the weighted cost of capital. Section 5 comments on some
implications of the theoretical model and Section 6 draws conclusions.
4
Before moving on to introduce our model, an additional explanation is required: the
theoretical model presented is applicable to those who invest in a firm whose shares are not
traded in a liquid stock market. Thus the words ‘entrepreneurial firm’ and ‘SME’ are to be
considered interchangeable.
2. Theoretical Model Underpinnings
The capital structure of firms is often defined as a ‘puzzle’. The metaphor shows
effectively how difficult it is to identify the ideal financing structure for firms and projects.
Fundamental finance literature considers theoretical modeling of the optimal capital structure
for corporations and is based on Modigliani and Miller (1958) seminal work. Later research
investigates the role of taxes (Boyce and Kalotay 1979; Brick and Ravid 1985), the impact of
refinancing costs (Jun and Jen 2003) and the probability of bankruptcy (Philosophov and
Philosophov 2005). Further papers address agency costs (James S. Ang et al. 2000; Brau
2002), moral hazard risks (Stiglitz 1974) and the cost of financing the firm by using different
sources of finance such as in the pecking order theory (Myers and Majluf 1984). This branch
of research traditionally works out the expected return of the investor based on the capital
asset pricing model (Sharpe 1963, 1964).
In fact, research on SMEs’ capital structure points out that SMEs are peculiar since
they are characterized by a mix between personal and firm wealth (Avery et al. 1998), by a
short life expectancy, by the importance of personal relationships, by a great potential for
making mistakes, by inter-generational issues (James S. Ang 1992) and by lack of separation
between business and personal risk (James S. Ang et al. 1995). SMEs’ capital structure is
affected by management values and goals, by an optimistic/pessimistic vision of the firm’s
5
future, by access to finance and human capital, by gender (women typically face greater
problems in accessing credit), by entrepreneur preference for risk and related strategy
(Chaganti et al. 1995). The life stage of the firm also affects capital structure, since younger
firms face greater difficulties in accessing any form of finance (Frielinghaus et al. 2005) and
very often rely on informal sources of finance, defined as bootstrap finance, such as loans
from friends and relatives, use of entrepreneurs’ personal credit cards and entrepreneurs’
personal loans, etc. (Wingborg and Landström 2000; Ebben and Johnson 2006; Voordeckers
and Steijvers 2006). SMEs’ size, ownership structure, profitability, historical and prospective
growth and capability to provide collateral also influence SMEs’ capital structure (Cassar
2004; Cassar and Holmes 2003; James S. Ang et al. 2000; Brau 2002; Chaganti et al. 1995).
Capital structure is country specific (Hall et al. 2004) and linked to the industry the firm is
part of, suggesting that industry competition, agency conflicts, and heterogeneity in the
technology employed by the venture are also important drivers of capital structure (Degryse
et al. 2012). Furthermore, grants and earnings from a salaried job are among the most
important sources of funds: they affect entrepreneurs’ decisions to start up a firm and
implicitly affect the capital structure of the firm, at least in the early stages of its life. In fact,
wealth appears to have a positive impact on the probability of starting up a firm (Elston and
Audretsch 2011). Financing decisions in SMEs are also hampered, since raising arm’s length
finance is subject to constraints for small opaque firms that suffer from large information
asymmetries (Berger et al. 2001): only when firms become older, larger and more
informationally transparent can they access public equity funding as well as public debt
(Gregory et al. 2005).
6
In summary, entrepreneurs’ decisions to invest, their expected return on investments,
the cost of capital and the capital structure of small unlisted firms are affected by multiple
variables. However, irrespective of such complexity, it is essential for the entrepreneur to
work out the return on his/her investment and thus the overall cost of capital and capital
structure.
2.1 Market Value and Portfolio Diversification
The Efficient Market Hypothesis suggests that the market provides the correct value
of firms (Fama 1970). Moreover, if markets are perfect, the current value of a firm’s shares
matches the expected return of the shareholders (Sharpe 1964). Investors can build up their
optimal portfolio by looking for the mix of shares that maximizes return by minimizing risk
thanks to portfolio diversification (Markowitz 1952). However, this logic does not properly
apply to entrepreneurs. Entrepreneurs hold a portfolio that is necessarily under-diversified
and, therefore, inefficient in terms of risk diversification. This is why Kerins et al. (2004)
state that entrepreneurs are exposed to a high idiosyncratic risk, definitely much higher than
that of a diversified investor (Müller 2011; Moskowitz and Vissing-Jørgensen 2002;
Hamilton 2000). Previous research tries to work out the premium that entrepreneurs gain
from such under-diversification in the portfolio by comparing entrepreneur returns to those of
the shareholders (investors) of an optimally diversified portfolio of shares (Heaton and Lucas
2000; Moskowitz and Vissing-Jørgensen 2002; Müller 2011; Kerins et al. 2004; Haney and
Holmes 2008). For instance, Kerins et al. (2004) find that the return for an under-diversified
entrepreneur/investor is two to four times higher than that of a well-diversified investor while
Müller (2011), by focusing on the fact that entrepreneurs suffer from high idiosyncratic risk
7
and under-diversification, calculates that for each 10% increase in concentration there is an
increase of about 15% in the return on invested funds. On the contrary, Moskowitz and
Vissing-Jørgensen (2002) do not find any significant difference in the return on investment
between investing in non-traded shares and traditional investment in a portfolio of traded
shares. Thus, the empirical findings are not conclusive, raising the question of whether such
an approach is the right one for calculating the expected return on equity for shareholders of
unlisted firms.
In fact, this branch of research considers that entrepreneurs’ decisions to invest
mirrors that of investors, i.e. entrepreneurs are simply interested in maximizing their
investments. Entrepreneurs are supposed to be in a position that may allow them to quite
easily sell the stake they hold in a venture when they are unsatisfied with their firm’s
performance. Thus, the return provided by the firm (dividends and capital gain) represents the
expected return for the entrepreneur. Nevertheless, at variance with the shares of firms listed
in stock markets that can be easily sold, entrepreneurial firms are very illiquid and
entrepreneurs can hardly ‘vote with their feet’, that is sell their shareholding if they are
unsatisfied with the firm’s performance. Even when firms are listed on regulated markets,
problems in selling shares can occur due to restrictions imposed by contracts on initial public
offerings (IPOs), due to the consequence of mergers and acquisitions as well as legal
restrictions (Kahl et al. 2003). When an SME is run by a family, shareholders can face
additional problems since, according to Romano et al. (2000), the desire of the owners and
entrepreneurs to maintain control of the firm is a very important determinant in avoiding the
sale of their shares. Indeed, LaPorta et al. (1999) by looking at large corporations find that
often the entrepreneurs/families hold a large majority of the shares and that their opportunity
8
to sell the shareholding is reduced. If this holds true for large listed family controlled firms, it
is even truer for small family dominated firms. Moreover, the existence of the entrepreneurial
firm is linked to the entrepreneur and it is often impossible to consider the firm without the
entrepreneur. In other words, the value of the firm is strongly related to the entrepreneurial
and managerial competences of the entrepreneur. Thus, the entrepreneur often cannot exit the
shareholding without compromising the value of the firm. All in all, illiquidity of shares,
legal and contractual restrictions, ownership structure as well as the peculiar role of the
entrepreneur can constrain the real value of shares as modeled by Kahl et al. (2003).
Consequently, relating the return on investment with the expectation of the investor is correct
when we consider an investor in liquid stock markets (where investors can ‘vote with their
feet’ by simply selling the shares if the return they are enjoying from the firm they have
invested in does not match the return they expect from such an investment). Linking the
return on investment with the expectation of the entrepreneur is not correct when we consider
an entrepreneur who has a stake in an illiquid firm (and no or very limited possibility of
‘voting with their feet’), so that
PROPOSITION 1: The expected return of an entrepreneur cannot be derived from the return
generated by the shares of a small firm.
2.2 Drivers of Entrepreneurs’ Investment Decisions
What drives entrepreneurs in their decisions to invest in the venture is mainly the
business idea the entrepreneurs have and the opinion that no one else is as capable as
themselves of exploiting it. They hope to maximize the pecuniary and non-pecuniary return
9
on that idea (irrespective of the amount of money and effort invested in the venture). Indeed,
entrepreneurs and SME shareholders involved in the management of the firm either directly
(as managers) or indirectly (as relatives or friends of the management), have a broad view of
the benefit provided by their investment. The literature on entrepreneurship stresses their
desire for independence (Delmar 2000), prestige, the desire to be free to make decisions
(Hamilton 2000) and the fact that entrepreneurs enjoy non-pecuniary benefits as high as 20%
of their investment (Moskowitz and Vissing-Jørgensen 2002). In addition, entrepreneurs are
less affected by moral hazard than traditional investors (James S. Ang et al. 2000) because of
the control they can exert over the firm and their higher risk tolerance, compared to that of
traditional investors in listed shares: they may accept higher return volatility, i.e. lower mean
return but higher maximum possible return on invested equity (Moskowitz and VissingJørgensen 2002). Interestingly, these psychological and social benefits are uncorrelated to the
amount of money invested in the venture and add an additional layer of complexity to
working out the expected financial return on equity for the entrepreneur. As a consequence, it
is no surprise that SME entrepreneurs and SME shareholders have difficulties in stating their
expected return on their investment.
2.3 Entrepreneurs’ different perspectives
Entrepreneurs invest a large part of their personal wealth in a venture and they are
legitimately entitled to expect remuneration for this investment. In fact, when entrepreneurs
act as providers of funds, they are mirroring the role of lenders even if they are very peculiar
lenders, in the sense that they are not only willing to accept very high levels of risk but also a
highly uncertain remuneration related to the profitability of the business.
10
Lenders decide whether to finance a venture and how to charge it according to the
riskiness of the venture. They work out the riskiness of the venture by using the probability of
default of the prospective borrower. The probability of default is then used firstly to decide
whether the borrower is sufficiently meritorious to be provided with funds, and secondly to
price the loan and attach conditions/covenants to it. Similarly, Cheung (1999) suggests the
use of the probability of bankruptcy of the industry in which the firm operates as a means of
measuring the firm’s riskiness from the entrepreneur’s point of view and then to calculate the
expected return accordingly. The argument is that entrepreneurs can evaluate their decision to
invest by looking at the probability that the venture in which they are investing their personal
wealth could go bankrupt. Cheung (1999) suggests a simple way of determining venture
riskiness by looking at the average survival rate of the cluster of firms the venture/project
belongs to (i.e. age, industry in which they operate, dimensions, etc.). Indeed, Cheung (1999)
on the one hand models the probability of bankruptcy simply as the average market survival
rate (thus he does not consider entrepreneurs’ specific skills, capabilities and success) while,
on the other hand, he does not address the problem of the loss in case of bankruptcy.
When lenders evaluate the risk they incur in lending to a specific borrower, they also
assess the loss at default they could potentially incur if the borrower were to be incapable of
paying back the loan. Similarly, entrepreneurs have to consider how their personal wealth
could be adversely affected if their venture enters the liquidation stage: all else being equal,
the greater the stake at risk in the venture, the higher the risk incurred in the venture.
Typically, entrepreneurs are more concentrated than a traditional portfolio of loans provided
by a lender (Müller 2011). Thus their loss in case of liquidation might impact heavily on their
11
personal wealth and therefore the risk related to loss of personal wealth in case of bankruptcy
is particularly important for them.
PROPOSITION 2: The risk entrepreneurs incur is the probability of bankruptcy of the
venture weighted for the impact that the loss at bankruptcy would have on the entrepreneur’s
personal wealth.
Lenders charge a premium on risk-free interest rates that is a function of the risk they
incur (that is the probability of default of the borrower and the loss in case of default). The
same logic may be applied in order to calculate the return for entrepreneurs: they need to
receive remuneration on the invested funds proportional to the probability that their venture
might go bankrupt weighted for the impact the venture’s liquidation would have on their
personal wealth. This remuneration is expected to compensate for the risk incurred by the
entrepreneurs, irrespective of their propensity towards risk. Thus this remuneration can be
defined as the ‘minimum’ return for the risk entrepreneurs incur when they fund the venture.
DEFINITION: The ‘minimum’ rate of return on the funds provided by an entrepreneur is the
remuneration that compensates for the risk of bankruptcy of the venture in which the
entrepreneur has invested, weighted for the impact the loss at liquidation would have on the
entrepreneur’s personal wealth.
Interestingly, the ‘minimum’ rate of return is independent of any personal valuation
made by the entrepreneur and is therefore unaffected by the overconfidence that characterizes
12
entrepreneurs (Landier and Thesmar 2009) as well as by any other benefit the entrepreneur
may enjoy (Hamilton 2000; Moskowitz and Vissing-Jørgensen 2002) or by the entrepreneur’s
propensity towards risk, that is whether they are risk adverse or risk takers. In other words, it
does not necessarily represent the expected return of the specific entrepreneur. Indeed, it
represents the minimum return entrepreneurs must require from their investment in order to
compensate for the risk they incur.
The minimum expected return for the entrepreneur is developed mathematically in the
next section while the implications at capital structure level are elaborated and discusses in
section 4.
3. The Minimum Expected Return for The Entrepreneur
The previous section clearly indicates that the ‘minimum return’ mirrors the logic
behind the interest premium charged by lenders: lenders consider the probability of default of
a loan, entrepreneurs may consider the probability of bankruptcy. Lenders pay attention to the
loss at default of a loan, entrepreneurs may concentrate on the loss they would incur if the
venture were to face liquidation. What differentiates entrepreneurs from lenders is the fact
that, if the former consider the risk too high, there is no venture at all. When entrepreneurs
decide to launch a venture, they always accept (and cope with) high levels of risk. However,
entrepreneurs are also risk adverse, they are rational and their decisions to invest in a venture
are not a gamble. Indeed, they decide to invest in a venture if and only if the expected reward
to be gained in the event of a positive outcome (associated with a probability) at least
compensates for the loss that may be incurred in case of an adverse outcome (also associated
with a probability). In fact, entrepreneurs are willing to bear risk only if the potential
13
remuneration associated with favorable outcomes at least compensates for the number of
events in which a loss might occur: otherwise they would be worse off.
3.1 Assumptions
Let us assume that an entrepreneur’s investment may be diversified between
investment in the venture and other investments but that the correlation between investment
in the venture and other investments is negligible. Any possible outcome of a small or
medium venture is substantially uncorrelated to the performance of government or company
bonds, traded shares, investment trusts or mutual funds that the entrepreneur may hold as an
alternative investment. In fact, the correlation between investment in the venture and any
other investment could be modeled and included in the theoretical model we propose, but it
would increase its complexity without adding substantial value to it.
Let us assume that the overall risk (e.g. the probability of failure) that a venture faces
can be broken down into two components: a cluster risk and a firm specific risk. Different
cluster of firms (characterized by industry, age, market in which they operate, etc.) are
characterized by different survival rates and, thus, different probabilities of failure; different
entrepreneurs are characterized by different capabilities and competencies in running their
business and by different histories of success/failure. Thus, prospectively, different
entrepreneurs are distinguished by different probabilities of failure.
Let us also assume that firms/entrepreneurs have a previous history of running
projects. Some of them were successful, others unsuccessful. However, the unsuccessful ones
did not compromise the entrepreneurial activity of the entrepreneur. Unsuccessful projects
14
generated a loss but did not drive the entrepreneur into bankruptcy, i.e. the entrepreneur
survived all previous projects irrespective of their success.
A further assumption is that entrepreneurs may invest part of their personal wealth in
the firm in different forms namely, equity (𝐸), and that the firm’s debt that is collateralized
by personal assets (𝐷𝑐). In fact, very often entrepreneurs provide the firm with both equity
and personal collateral in order to allow the firm to access credit from banks.
Let us also assume that, in the worst case, entrepreneurs would lose their entire equity
invested in the venture. Thus, in case of liquidation, the firm would be able partly or totally
repay its debt but would not be able to repay the entrepreneur. This assumption is quite
reasonable, since the entrepreneur would recover the invested equity only after the firm had
repaid all other debts. Nevertheless, if the firm were to be unsuccessful and go bankrupt, it
would be highly unlikely that it would be able to reimburse the entrepreneurs the equity they
invested, since it would struggle to repay all the debts it owed to other creditors.
Let us define 𝑇𝑝𝑤 as the ratio of the entrepreneur’s overall personal wealth out of the
amount of wealth invested in the venture directly (as equity) and indirectly (as personal
collateral). Thus, 𝑇𝑝𝑤 =
𝐸+𝐷𝑐+𝑊
𝐸+𝐷𝑐
were W is the additional personal wealth of the entrepreneur
that is not invested in the venture. We assume that 𝑇𝑝𝑤 ≥ 𝐸 + 𝐷𝑐 since the entrepreneur
could at least allocate his/her entire personal wealth as equity and personal collateral and in
this case W is zero. In other words, the overall personal wealth invested by the entrepreneur is
at least equal to his/her investment in the venture. We are implicitly assuming that
entrepreneurs cannot provide guarantees for an amount of personal assets that they do not
have and that they cannot access additional personal debt to increase their investment in the
15
venture. In fact, from the entrepreneur’s point of view, any situation where the total invested
wealth is smaller than equity and the personal collateral provided, that is 𝑇𝑝𝑤 ≤ 𝐸 + 𝐷𝑐,
generates the same adverse impact as if Tpw = Dc + E. Thus,
𝑇𝑝𝑤 ≥ 1
(A1)
Let also assume that the value of the firm’s assets that can be used to repay the debt
collateralized by personal assets in the liquidation state, defined as 𝐴𝑏, is
0 ≤ 𝐴𝑏 ≤ 𝐷𝑐
(A2)
In fact, when the firm has assets worth more than the debt collateralized by personal assets,
the entrepreneur will not be adversely affected by a negative outcome of the venture due to
the personal guarantees provided to the bank. Thus, for any value of 𝐴𝑏 > 𝐷𝑐 the negative
impact on the entrepreneur equals what happens when 𝐴𝑏 = 𝐷𝑐, i.e. no negative impact at
all.
Finally, in order to simplify the notation let us assume that
1 = 𝐸 + 𝐷𝑐 + 𝐷
(A3)
where 𝐷𝑐 and 𝐸 are defined as above and 𝐷 is the uncollateralized debt.
16
3.2 Working out the minimum expected return
Let us start by specifying the cluster risk. In line with Cheung (1999), let us define
𝑡 = 1, … , 𝑇 as the 𝑡 -th time period, and 𝑖 = 1, … , 𝑁𝑡 as the 𝑖 -th firm belonging to a specific
cluster, where 𝑁𝑡 denotes the number of firms in the cluster at time 𝑡.1 This cluster is
𝑈
characterized by a specific level of risk, which represents the probability, 𝑝𝑡+1
, that the
average firm in the cluster is unsuccessful in a specific period 𝑡 + 1. Let 𝐏𝑡+1 be the vector
containing the probabilities that the average firm in the cluster is successful/unsuccessful in a
specific period 𝑡 + 1.
𝑝S
𝐏𝑡+1 = ( 𝑡+1
𝑈 ),
𝑝𝑡+1
(4)
where {𝑆, 𝑈} stand for successful and unsuccessful.
We can now turn our attention to the firm specific risk, namely the risk attached to
poor entrepreneurial performance. As stated in the assumptions, entrepreneurs have a
previous history of running projects: some of which were successful, others unsuccessful.
However, entrepreneurs survived all previous projects irrespective of their success. Clearly,
more capable entrepreneurs have a higher level of success (more projects that produced
profits) than less capable ones. Let us measure the entrepreneur’s 𝑖 performance through their
firm’s performance in period 𝑡, as PERFi,t , by using a vector of 𝑛 ratios. The choice of ratios
1
A cluster is characterized by age, the market in which the firm operates the product/service
it delivers, size, etc.
17
to include in the vector is beyond the topic covered by this paper and will therefore not be
discussed here. Nevertheless, one can question whether ratios that are based on accounting
figures are a good measure of firm performance. Indeed, previous research suggests that
analysis of official accounting information disclosed by firms provides some useful insight
into the firm’s current and prospective performance. For instance, Mramor and Valentincic
(2003) suggest that financial ratios and lagged liquidity indicators have predictive relevance
for identifying the firms that will not fail, even if they are not effective in identifying firms
that actually do fail subsequently. Pompe and Bilderbeek (2005) examine a large set of
bankrupt and non-bankrupt Belgian firms and discover that all the ratios they consider have
some predictive relevance, particularly for more mature businesses. Further, Beaver et al.
(2005) examine whether changes in accounting standards have affected the capability of
ratios to be used as predictive tools. Interestingly, they find that predictive capability is
unchanged, suggesting implicitly that ratios are robust tools for the analysis of firms’
performance. The above research suggests that, in general, ratios that use profit or cash flow
in the numerator in combination with total assets, turnover or total debt in the denominator
are significant in predicting the bankruptcy of a firm (Pompe and Bilderbeek 2005; Beaver
1966; Beaver et al. 2005). Thus, previous success can be measured, for instance and very
easily, by working out how many ratios out of the overall number of ratios selected to
measure the success of the firm/the risk of bankruptcy show a positive outcome. Clearly, it is
also possible to choose more sophisticated ways to calculate the level of previous success, for
instance by weighting different ratios according to their relevance or even by building more
complex algorithms.
18
Let 𝐏𝑖,𝑡+1be the vector containing the probabilities that the firm 𝑖 is
successful/unsuccessful in a specific period 𝑡 + 1.
𝐏𝑖,𝑡+1 = (
S
𝑝𝑖,𝑡+1
𝑈
𝑝𝑖,𝑡+1
),
(5)
where {𝑆, 𝑈} stand for successful and unsuccessful.
Assume that the cluster risk, 𝐏𝑡+1, and the entrepreneur’s 𝑖 risk, 𝐏𝑖,𝑡+1 are related
′
through the following Markov transition matrix Π𝑖,𝑡+1
′
𝐏𝑖,𝑡+1 = Π𝑖,𝑡+1
𝐏𝑡+1 ,
(6a)
where
′
Π𝑖,𝑡+1
= (
𝑆𝑆
𝑆𝑈
π𝑖,𝑡+1
π𝑖,𝑡+1
𝑈𝑈
π𝑈𝑆
𝑖,𝑡+1 π𝑖,𝑡+1
)
(6b)
where S and U denote the two states of the world - the successful, in which the entrepreneur
was successful in running the firm and the unsuccessful, in which the entrepreneur was not.
′
The individual transition matrix, Π𝑖,𝑡+1
, defined above, relates cluster risk to individual firm
𝑆𝑆
risk. π𝑖,𝑡+1
, resp. π𝑈𝑈
𝑖,𝑡+1 , is the transition probability that the project of 𝑖’s company is
successful, resp. unsuccessful, in 𝑡 + 1 given that the cluster to which the company belongs
19
to is a successful, resp. unsuccessful, one in 𝑡 + 1. We are going to call these transition
probabilities ‘no transition category’ later on, since there is no change in the state of
entrepreneurs: their projects are successful and the cluster to which they belong is successful
too. The transition probabilities on the off-diagonal indicate that the project of the 𝑖’s
company is successful, resp. unsuccessful, in 𝑡 + 1 given that the cluster to which the firm
belongs is unsuccessful, resp. successful is 𝑡 + 1.2 Consequently, using the performance
measure PERFi,t introduced earlier, the transition probabilities can be rewritten as
𝑆𝑆
𝑆
π𝑖,𝑡+1
= Pr[PERF𝑖,𝑡+1 > 0│𝑝𝑡+1
]
(7a)
𝑆𝑈
𝑈
π𝑖,𝑡+1
= Pr[PERF𝑖,𝑡+1 > 0│𝑝𝑡+1
]
(7b)
𝑆
π𝑈𝑆
𝑖,𝑡+1 = Pr[PERF𝑖,𝑡+1 < 0│𝑝𝑡+1 ]
(7c)
𝑈
π𝑈𝑈
𝑖,𝑡+1 = Pr[PERF𝑖,𝑡+1 < 0│𝑝𝑡+1 ]
(7d)
This means that the probability that firm 𝑖 is successful/unsuccessful in 𝑡 + 1 can be
written as
𝑆
𝑆𝑆
𝑆𝑈
𝑆
𝑈
𝑝𝑖,𝑡+1
= π𝑖,𝑡+1
× 𝑝𝑡+1
+ π𝑖,𝑡+1
× 𝑝𝑡+1
𝑈
𝑈𝑈
𝑆
𝑈
𝑝𝑖,𝑡+1
= π𝑈𝑆
𝑖,𝑡+1 × 𝑝𝑡+1 + π𝑖,𝑡+1 × 𝑝𝑡+1
2
(8a)
(8b)
Note that the sum of a column is equal to one by definition.
20
Given that the sum of the transition probabilities in a column are equal to one, we
only need to model and estimate two probabilities out of four. In order to model the transition
probabilities, we use a logit model given by3
𝑆𝑈
π𝑖,𝑡+1
=
π𝑈𝑆
𝑖,𝑡+1
=
exp(Λ𝑆𝑈
𝑖,𝑡+1 )
(9a)
𝑈𝑈
exp(Λ𝑆𝑈
𝑖,𝑡+1 ) + exp(Λ 𝑖,𝑡+1 )
exp(Λ𝑈𝑆
𝑖,𝑡+1 )
(9b)
𝑆𝑆
exp(Λ𝑈𝑆
𝑖,𝑡+1 ) + exp(Λ 𝑖,𝑡+1 )
𝑈𝑆
𝑆𝑆
𝑈𝑈
where the log-odds ratio Λ𝑆𝑈
𝑖,𝑡+1 , Λ 𝑖,𝑡+1 , Λ 𝑖,𝑡+1 , Λ 𝑖,𝑡+1 will be specified below. As a
normalization constraint, we use as the reference category the ‘no transition category’, i.e.
𝑈𝑈
Λ𝑆𝑆
𝑖,𝑡+1 = 0 and Λ 𝑖,𝑡+1 = 0, such that the odds ratios are defined as the quotient of a given
transition probability to the probability of no transition. Note that we only need one equation
to describe a variable with two categories, so the choice of the reference category does not
matter. It can always be obtained from the other category using a back transformation. For
example, in our case, using Λ𝑆𝑆
𝑖,𝑡+1 as the reference category leads to
π𝑈𝑆
𝑖,𝑡+1
3
=
exp(Λ𝑆𝑈
𝑖,𝑡+1 )
1 + exp(Λ𝑈𝑆
𝑖,𝑡+1 )
(10a)
Note that given the specification of our model, it can be easily extended to one containing a
transition matrix with more states of the world. In this case, the transition probabilities can be
modelled with a multinomial logit model and estimated in the same way with maximum
likelihood.
21
𝑆𝑆
π𝑖,𝑡+1
=
1
1 + exp(Λ𝑈𝑆
𝑖,𝑡+1 )
(10b)
The log-odds ratios given by
π𝑈𝑆
𝑖,𝑡+1
Λ𝑆𝑈
𝑖,𝑡+1 = ln [π𝑈𝑈 ]
(11a)
𝑖,𝑡+1
π𝑈𝑆
𝑖,𝑡+1
Λ𝑆𝑈
𝑖,𝑡+1 = ln [π𝑆𝑆 ]
(11a)
𝑖,𝑡+1
are then specified through a multivariate linear form. This specification will allow us to
introduce explanatory variables which are firm specific, and represent the previous success of
the entrepreneur for example, and explanatory variables that are cluster specific.
The transition probabilities can be expressed as the sum of the transition matrix and
the probabilities at cluster level, and can easily be estimated by maximum likelihood. This
leads to a log likelihood function, ln ℒ, of the form
𝑇
ln ℒ = ∑
𝑁
∑
𝑡=1
𝐿
∑
𝑖=1
𝑙=1
𝟙{ PERF𝑖,𝑡+1 |ℱ𝑖𝑡} ln𝐏𝑖𝑡+1,𝑙
(12)
𝑘𝑙
where 𝐏𝑖𝑡+1,𝑙 =∑𝐾
𝑘=1 π𝑖𝑡+1 ∙ 𝐏𝑡+1,𝑘 , and 𝑘, 𝑙 ∈ {𝑆, 𝑈}.
For simplicity of notation from now on, let us define the overall probability of
bankruptcy as worked out above as p.
22
As illustrated in the assumptions section, entrepreneurs’ investment in the firm has
different forms. In fact, very often, entrepreneurs provide the firm with both equity and
personal collateral in order to gain credit from banks. When entrepreneurs provide a bank
with personal collateral, they implicitly increase their stake in the venture since they increase
the quota of personal wealth invested in the venture. In this case, the bank is actually
transforming entrepreneurs’ assets that are not liquid (such as properties) or that the
entrepreneurs do not want to sell into liquidity, preventing the entrepreneurs from selling
them to finance the firm. An adverse outcome would wipe out the equity invested in the
venture, E, and the personal wealth that has been used as collateral, Dc, less the amount of
the firm’s assets that can be used to repay the debt collateralized by personal assets, Ab. The
impact on personal wealth of an adverse outcome is given by
𝐿𝐵 =
𝐸 + 𝐷𝑐 − 𝐴𝑏
,
𝑇𝑝𝑤
(13)
Since 𝑇𝑝𝑤 is always greater than 1 (assumption A1), and that 0 ≤ 𝐴𝑏 ≤ 𝐷𝑐 (assumption A2),
it follows that
𝐸
𝑇𝑝𝑤
≤
𝐸+𝐷𝑐−𝐴𝑏
𝑇𝑝𝑤
𝐸
≤ 1. Indeed, the wealth at risk is constrained between 𝑇
𝑝𝑤
when 𝐴𝑏 ≥ 𝐷𝑐 (i.e. when the firm’s assets can be used to repay the entire collateralized bank
debt), and 1 (when entrepreneurs lose their entire invested wealth during liquidation).
Consequently, given the overall probability of failure, p, worked out as explained
above (Eq 12), the overall minimum average expected premium required by the entrepreneur,
weighted by the amount of personal wealth invested in the venture, can be worked out as
23
𝑅𝑒𝜋 =
𝑝
𝐸 + 𝐷𝑐 − 𝐴𝑏
∙
,
1−𝑝
𝑇𝑝𝑤
(14)
and the minimum weighted return expected by the entrepreneur as
𝑅𝑒 = 𝑅𝑒𝑓𝑟𝑒𝑒 +
𝑝
𝐸 + 𝐷𝑐 − 𝐴𝑏
∙
,
1−𝑝
𝑇𝑝𝑤
(15)
where 𝑅𝑒𝑓𝑟𝑒𝑒 denotes the risk free rate.
Given that 0 ≤ 𝑝 ≤ 1 and 𝐸 ≤ 𝐸 + 𝐷𝑐 − 𝐴𝑏 ≤ 𝑇𝑝𝑤 equation (15) is always positive.
In addition, the curve represented by equation (15) has domain {0, 1} and an asymptote when
the probability of failure equals 1 (𝑅𝑒 = ∞). In fact, the curve is truncated somewhere on the
right earlier than the asymptote, since the entrepreneur would not invest in a very high-risk
venture mainly because a high-risk venture would not provide returns that would compensate
for the risk incurred. Evidently, the truncation depends on the remuneration capability of the
project.
Equation (15) specifies the minimum average weighted return an entrepreneur expects
from a venture in order to avoid being worse off. In fact, any return below the value of 𝑅𝑒
does not compensate for the risk of bankruptcy and consequent damage to personal wealth
that the entrepreneur incurs by investing in the venture. In other words, the reward gained
does not compensate for the probability of facing a negative outcome. Any average return
above the value of 𝑅𝑒 gives an extra reward with respect to the minimal reward that
compensates for the probability of an adverse outcome.
24
Interestingly, the minimum expected return worked out as above, has implications for
the capital structure of the firm. The implications are illustrated in the following section.
4. The Weighted Average Cost of Capital in SMEs
The minimum average expected return as worked out above contains two
components: entrepreneurs’ remuneration for equity and entrepreneurs’ remuneration for debt
collateralized by personal assets. It is easy to split the above formula into the two
components.
The remuneration for invested equity will be the remuneration for the risk of an
adverse outcome weighted for the impact that the loss of equity would have on the
entrepreneurs’ total wealth. Thus, it will be
𝑅𝑒 = 𝑅𝑒𝑓𝑟𝑒𝑒 +
𝑝
𝐸
∙
1 − 𝑝 𝑇𝑝𝑤
(16)
In this case, the probability of bankruptcy has to be weighted only for the impact the
loss of invested equity would have on the entrepreneur’s personal wealth.
When entrepreneurs provide private collateral for granting credit access to a firm,
they need to receive remuneration on the additional stake they are investing in the venture.
Also, in this case, remuneration for invested personal wealth will be remuneration for the risk
of adverse outcomes weighted for the impact the loss of personal wealth would have on the
entrepreneurs’ total wealth. However, unlike equity, here a quota of the debt collateralized by
personal assets might be repaid by the firm by selling its assets, implicitly reducing adverse
25
impact on personal wealth. The cost of debt collateralized by personal assets will be the
remuneration to entrepreneurs and the traditional interest rate charged by the bank.
𝑅𝑖𝑐 = 𝑅𝑒𝑓𝑟𝑒𝑒 +
𝑝
𝐷𝑐 − 𝐴𝑏
∙
+ 𝑅𝑖 ,
1−𝑝
𝑇𝑝𝑤
(17)
where 𝑅𝑖 is the interest rate charged by the bank.
Finally, the cost of debt that is not collateralized by personal assets will be, as usual,
simply the interest rate charged by the bank, that is
𝑅𝑖 .
(18)
Consequently, by putting together equations (16), (17) and (18) and keeping in mind
assumption A3, that is 𝐸 + 𝐷𝑐 + 𝐷 = 1, it is possible to work out the weighted average cost
of capital, WACC, as
𝑊𝐴𝐶𝐶 = [𝑅𝑒𝑓𝑟𝑒𝑒 +
+ [𝑅𝑒𝑓𝑟𝑒𝑒 +
𝑝
𝐸
∙
] ∙𝐸
1 − 𝑝 𝑇𝑝𝑤
𝑝
𝐷𝑐 − 𝐴𝑏
∙
+ 𝑅𝑖 ∙ (1 − 𝑡)] ∙ 𝐷𝑐
1−𝑝
𝑇𝑝𝑤
(19)
+𝑅𝑖 ∙ (1 − 𝑡) ∙ 𝐷
where 𝑡 is the tax rate. The first line represents the cost of equity, the second line the cost of
debt collateralized by personal assets and the third line the cost of uncollateralized debt.
26
The 𝑊𝐴𝐶𝐶 presented in equation (19) has two major implications. Firstly it leads to
situations where the cost of debt collateralized by personal assets is more expensive than the
cost of equity. This is a consequence of the fact that debt collateralized by personal assets
bears both the cost of debt and the cost of equity, even if the latter is ‘smoothed’ by the
relative impact that debt collateralized by personal assets has on personal wealth (that is by
the amount of debt collateralized by personal assets and by the value of the firm’s assets that
can be used to repay the debt collateralized by personal assets). Thus, when the interest rate
charged is sufficiently high or the ‘smoothing’ effect does not smooth too much of the cost of
debt collateralized by personal assets, it should be possible to have debt collateralized by
personal assets that is more expensive than equity. Secondly, it allows for the optimization of
the SME’s capital structure. Capital structure can be optimized because of the traditional
impact of taxes that generate a benefit in leveraging debt. Nevertheless, optimization is also a
result of the fact that the cost of the debt collateralized by personal assets on the one hand is
affected by the remuneration entrepreneurs demand for the additional risk they incur and, on
the other, by the smoothing effect generated by assets that the firm can use in order to repay
its debts.
Both implications are examined in the next two subsections.
4.1 Cost of debt collateralized by personal assets vs. cost of equity
The point of equilibrium between the overall cost of equity and the overall cost of
debt collateralized by personal assets is given for the following quantity of 𝐸 (and thus for a
quantity of 𝐷𝑐 = 1 − 𝐸 − 𝐷)4
4
The derivation of equation (20b) is given in Appendix A.
27
𝑝
𝑝
𝑝
2
(1 − 𝐴𝑏)
𝐷(𝐴𝑏
−
2)
1−𝑝
1−𝑝
1−𝑝𝐷
(1 − 𝐷)(𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡)𝑅𝑖) +
+
+ 𝑇
𝑇𝑝𝑤
𝑇𝑝𝑤
𝑝𝑤
𝐸=
(20a)
𝑝
𝑝
𝑝
2 1 − 𝑝 2 1 − 𝑝 𝐷 1 − 𝑝 𝐴𝑏
2𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
− 𝑇
− 𝑇
+ (1 − 𝑡)𝑅𝑖
𝑝𝑤
𝑝𝑤
𝑝𝑤
or, to simplify the notation
𝑝
1−𝑝
(1 − 𝐷) ∙ (𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖) +
[1
(𝐴𝑏 − 2) + 𝐷2 ]
𝑇𝑝𝑤 ∙ − 𝐴𝑏 + 𝐷 ∙
𝐸=
𝑝
1−𝑝
2 ∙ 𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
∙ (2 − 2𝐷 − 𝐴𝑏) + (1 − 𝑡) ∙ 𝑅𝑖
𝑝𝑤
(20b)
The equilibrium presented in equation (20b) has solutions for values of 0 ≤ 𝐸 < 1,
and, thus, it implies that there are scenarios when the cost of debt collateralized by personal
assets is higher than the cost of equity. This finding has interesting implications.
LEMMA 1: The proportion of equity the cost of which equates to the cost of the debt
collateralized by personal assets decreases (and respectively the amount of debt
collateralized by personal assets increases) with the increase of the risk of the venture.
Proof of Lemma 1
In fact
28
𝑝
1−𝑝
(1 − 𝐷) ∙ (𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖) +
[1
(𝐴𝑏 − 2) + 𝐷2 ]
𝑇𝑝𝑤 ∙ − 𝐴𝑏 + 𝐷 ∙
lim
𝑝
𝑝→1
1−𝑝
2 ∙ 𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
∙ (2 − 2𝐷 − 𝐴𝑏) + (1 − 𝑡) ∙ 𝑅𝑖
𝑝𝑤
(21)
depends mainly on the value of 𝐷.
Typically, the value of 𝐷 in SMEs is limited. Moreover, the reduced role of 𝐷 holds
particularly true when the project is very risky (𝑝 → 1). Indeed, in this context, lenders rarely
provide uncollateralized finance. Thus this scenario is of particular interest here. When
uncollateralized debt is negligible, then 𝐷 = 0 the limit simplifies into
𝑝
1−𝑝
(𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖) + 𝑇
∙ [1 − 𝐴𝑏]
𝑝𝑤
lim
𝑝
𝑝→1
1−𝑝
2 ∙ 𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
∙ (2 − 𝐴𝑏) + (1 − 𝑡) ∙ 𝑅𝑖
𝑝𝑤
(22)
and, in this case, a key role is played by the value of assets that a firm can use to repay debt.
When 𝐴𝑏 is very small (i.e. it tends to zero), the limit is
29
𝑝
𝑝
1−𝑝
1−𝑝
(𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖) + 𝑇
∙ [1 − 𝐴𝑏]
(𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖) + 𝑇
𝑝𝑤
𝑝𝑤
lim
=
𝑝
𝑝
𝑝→1
2∙1−𝑝
1−𝑝
2𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
∙ (2 − 𝐴𝑏) + (1 − 𝑡) ∙ 𝑅𝑖
2𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
+ (1 − 𝑡)𝑅𝑖
𝑝𝑤
𝑝𝑤
𝑝
1−𝑝
(𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖) + 𝑇
1
1
𝑝𝑤
= ∙
=
𝑝
2
2
1
1−𝑝
𝑅𝑒𝑓𝑟𝑒𝑒 + 2 ∙ (1 − 𝑡) ∙ 𝑅𝑖 + 𝑇
𝑝𝑤
(23)
Indeed, when the firm has no assets, the impact incurred by entrepreneurs when they
use debt collateralized by personal assets is high, since there is no hedge (firm assets) on the
personal collateral they have provided. In this case, remuneration to the entrepreneur
becomes a major component of the cost of debt collateralized by personal assets and thus it
approaches the cost of equity. Nevertheless, this happens in addition to the interest rate paid
to the bank. Thus, when the remuneration requested by the entrepreneur is high due to the
high risk incurred by the venture, the cost of debt collateralized by personal assets equates to
the cost of equity for a smaller proportion of equity (up to 50% of the mix).
When 𝐴𝑏 is large, i.e. it tends to 𝐷𝑐, the limit is
𝑝
1−𝑝
(𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖) + 𝑇
∙ (1 − 𝐴𝑏)
𝑝𝑤
lim
=0
𝑝
𝑝→1
1−𝑝
2 ∙ 𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
∙ (2 − 𝐴𝑏) + (1 − 𝑡) ∙ 𝑅𝑖
𝑝𝑤
(24)
30
In fact, when the firm has assets that cover the debt collateralized by personal assets,
the risk incurred by an entrepreneur is negligible, irrespective of the fact that the venture
could be very risky and remuneration requested by the entrepreneur could be very high. Thus,
the cost of debt collateralized by personal assets decreases with respect to the cost of equity,
since the component of the cost of debt collateralized by personal assets that has to
remunerate the high risk incurred by the entrepreneur is close to zero. In other words, the cost
of debt collateralized by personal assets approaches the cost of uncollateralized debt,
implying that debt collateralized by personal assets tends to be cheaper than equity in the
great majority of cases (and, in the extreme case, always).
As explained above, cases of SMEs that are largely financed by uncollateralized debt
are rare. When a firm implements a very risky project, on the one hand lenders will provide a
reduced amount of finance and, on the other, they will demand more personal collateral in
order to finance the firm. Nevertheless, from a theoretical point of view, it is interesting to
examine what happens when the value of uncollateralized debt is large, that is 𝐷 tends to be
close to one and thus 𝐷𝑐 and, because of assumption A2, 𝐴𝑏 are close to zero. In this case the
numerator 1 − 𝐴𝑏 + 𝐷(𝐴𝑏 − 2) + 𝐷2 =0 and the denominator (2 − 2𝐷 − 𝐴𝑏)=0.
Thus
𝑝
1−𝑝
(1 − 1) ∙ (Refree + (1 − t) ∙ Ri) +
𝑇𝑝𝑤 ∙ 0
lim
=0
𝑝
p→1
1−𝑝
2 ∙ Refree + (1 − t) ∙ Ri + 𝑇
∙0
𝑝𝑤
(25)
31
In fact when 𝐷 tends to be close to one, the role played by equity and debt
collateralized by personal assets is highly reduced. In addition, debt collateralized by personal
assets benefits intensively from even a limited ‘smoothing’ effect of 𝐴𝑏, simply because it is
a small amount. Thus, due to the reduced role played by debt collateralized by personal assets
on the capital structure of the firm, the high remuneration demanded by an entrepreneur to
compensate for the high risk incurred by the venture is irrelevant. In summary, when the firm
is mainly financed with uncollateralized debt, the amount of equity that equates to the cost of
debt collateralized by personal assets is 0, implying that debt collateralized by personal assets
tends to be cheaper than equity in the great majority of mixes.
LEMMA 2: The proportion of equity, the cost of which equates to the cost of the debt
collateralized by personal assets, increases (and respectively the proportion of debt
collateralized by personal assets decreases) with the amount of personal wealth.
Proof of Lemma 2:
𝑝
1−𝑝
(1 − 𝐷) ∙ (𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖) +
[1
(𝐴𝑏 − 2) + 𝐷2 ]
𝑇𝑝𝑤 ∙ − 𝐴𝑏 + 𝐷 ∙
lim
𝑝
𝑇𝑝𝑤 →∞
1−𝑝
2 ∙ 𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
∙ (2 − 2𝐷 − 𝐴𝑏) + (1 − 𝑡) ∙ 𝑅𝑖
𝑝𝑤
=
(1 − 𝐷) ∙ (𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡)𝑅𝑖)
2 ∙ 𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡) ∙ 𝑅𝑖
(26)
32
since
𝑝
1−𝑝
𝑇𝑝𝑤
speaking,
, in both the numerator and the denominator tend to zero. In fact, mathematically
(𝑅𝑒𝑓𝑟𝑒𝑒 +(1−𝑡)∙𝑅𝑖)
2𝑅𝑒𝑓𝑟𝑒𝑒 +(1−𝑡)∙𝑅𝑖
tends to .5 but, since 𝑅𝑒𝑓𝑟𝑒𝑒 is typically quite small, then 𝑅𝑒𝑓𝑟𝑒𝑒 +
(1 − 𝑡)𝑅𝑖 ≅ 2𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡)𝑅𝑖. Here, a key role is played by 𝐷. When 𝐷 is large (a
scenario quite unlikely for SMEs), that is when the firm is financed mainly through
uncollateralized debt, the limit tends to 0. Alternatively, when the role of uncollateralized
debt is negligible (a typical situation for entrepreneurial firms), the limit tends to 1. In the
latter case, the greater the premium charged by the bank, the less the taxation, and the smaller
the amount of uncollateralized debt, the closer the value to 1.
In fact, an increase in personal wealth implies a reduction of the impact the venture
has on an entrepreneur’s personal wealth. When a venture has limited impact on an
entrepreneur’s personal wealth, the differential between the cost of equity and the cost of debt
collateralized by personal assets is small. Symmetrically, if the investment has a huge impact
on the entrepreneur’s personal wealth, the differential between the weighted cost of equity
and the weighted cost of debt collateralized by personal assets will be larger and,
consequently, a smaller amount of debt collateralized by personal assets is needed in order to
generate an overall cost on debt collateralized by personal assets that equates to the overall
cost of equity.
33
4.2 Capital structure optimization
Given assumptions A1, A2 and A3 and that
𝐸
𝑇𝑝𝑤
≤
𝐸+𝐷𝑐−𝐴𝑏
𝑇𝑝𝑤
≤ 1, a venture can
optimize its capital structure by working out the first derivative of WACC with respect to E
and equaling it to zero5
𝜕𝑊𝐴𝐶𝐶
𝑝
𝑝
𝑝
𝑝
= 4∙
∙𝐸−2∙
−2∙
∙𝐷+
∙ 𝐴𝑏 − 𝑅𝑖 ∙ (1 − 𝑡) ∙ 𝑇𝑝𝑤 = 0
𝜕𝐸
1−𝑝
1−𝑝
1−𝑝
1−𝑝
(27)
By solving for E, the proportion of E that optimizes capital structure is given by
𝐸=
1 𝐷 𝐴𝑏 𝑅𝑖 ∙ (1 − 𝑡) ∙ 𝑇𝑝𝑤
− −
+
𝑝
2 2
4
4∙
1−𝑝
In fact, the derivative in equation (28) is a minimum since
𝜕2 𝑊𝐴𝐶𝐶
𝜕2 𝐸
(28)
𝑝
1−𝑝
4
=
𝑇𝑝𝑤
>0.
The optimization of capital structure has interesting implications. First and foremost,
since the first derivative of the weighted average cost of capital always has real solutions, this
implies that it is possible to optimize capital structure for SMEs by minimizing the overall
cost of equity, debt collateralized by personal assets and uncollateralized debt given the
assumptions we set out at the beginning. In addition, it is possible to derive two lemmas.
5
The derivation of this formula can be found in Appendix B.
34
LEMMA 3: The lower the impact the venture has on the entrepreneur personal wealth,
the more the venture has to be financed with equity.
Proof of Lemma 3:
1 𝐷 𝐴𝑏 𝑅𝑖 ∙ (1 − 𝑡) ∙ 𝑇𝑝𝑤
− −
+
=∞
𝑝
𝑇𝑝𝑤 →∞ 2
2
4
41−𝑝
lim
(29)
But, given that 𝐸 + 𝐷 + 𝐷𝑐 = 1, the amount of invested equity cannot be larger than one,
meaning that the limit above has to be constrained to one even if mathematically it should be
tending to infinity. This leads to the fact that
1 𝐷 𝐴𝑏 𝑅𝑖 ∙ (1 − 𝑡) ∙ 𝑇𝑝𝑤
− −
+
=1
𝑝
𝑇𝑝𝑤 →∞ 2
2
4
41−𝑝
lim
(30)
Thus, the higher the proportion of personal wealth not invested in a venture (and thus, the
lower the impact of the venture on an entrepreneur’s wealth), the higher the amount of equity
invested in the venture. This occurs because higher personal wealth implies reduced impact
of an adverse outcome on an entrepreneur’s personal wealth. Such a reduced effect affects the
cost of equity more than the cost of debt collateralized by personal assets, since the latter is
characterized by the smoothing effect of the firm’s assets but also bears the cost of debt (the
interest rate). Thus, since the cost of equity benefits more than the cost of debt collateralized
by personal assets from any increase in the value of personal wealth, an entrepreneur is better
35
off by decreasing debt collateralized by personal assets (that also bears the interest rate
charged by the bank) and increasing the amount of equity invested in a venture.
According to assumption A3, when the firm is financed solely with equity, 𝐷𝑐 = 0
and 𝐴𝑏 = 0. Thus, equation (28) becomes
1 𝑅𝑖 ∙ (1 − 𝑡) ∙ 𝑇𝑝𝑤
−
=1
𝑝
2
41 − 𝑝
(31a)
and by solving for 𝑇𝑝𝑤
𝑇𝑝𝑤
In other words, when 𝑇𝑝𝑤 ≥ 8 ∙
𝑝
1−𝑝
= 8∙
𝑅𝑖 ∙ (1 − 𝑡)
𝑝
1−𝑝
𝑅𝑖∙(1−𝑡)
(31b)
, the firm must be financed only with equity.
LEMMA 4: The higher the risk of a venture, the less it must be financed with equity.
Proof of Lemma 4:
1 𝐷 𝐴𝑏 𝑅𝑖 ∙ (1 − 𝑡) ∙ 𝑇𝑝𝑤 1 𝐷 𝐴𝑏
lim − −
+
= − −
𝑝
𝑝→1 2
2
4
2 2
4
41 − 𝑝
(32)
36
In fact, the increase in risk of the project and, therefore, the increase of the expected
return on invested equity decreases the amount of equity that minimizes the overall cost of
finance used in the project. This happens because the effect exerted by the expected return on
equity on the debt collateralized by personal assets is smoothed by the value of the assets that
can be used to repay the debt. Thus, because of such a smoothing effect the debt
collateralized by personal assets becomes progressively relatively cheaper than equity and an
entrepreneur is better off increasing the proportion of debt collateralized by personal assets at
the expense of equity.
5. Implications
The proposed theoretical model provides a threshold for making decisions about
prospective projects: running a project that remunerates entrepreneurs less than the minimum
average return is not sensible since the return will not compensate for the risk incurred by the
entrepreneurs. Thus, any additional monetary and non-monetary remuneration has to be over
and above the suggested minimum average return. The theoretical model has some interesting
characteristics and implications.
The first relevant implication of the theoretical model is about the pecking order of
financing sources. The model, by adding a category to the possible sources of finance,
namely debt collateralized by personal assets, suggests that the pecking order theory as
proposed by Myers and Majluf (1984) is too simplistic and not ideally suited to building the
capital structure of SMEs. Indeed, the theoretical model suggests that equity is not
necessarily the most expensive source of finance. This happens simply because the debt
37
collateralized by personal assets has to remunerate both the bank and the entrepreneur. In
fact, the additional stake at risk linked to guaranteeing the bank with personal collateral may
be very relevant for the entrepreneur. This depends on two elements: firstly the value of firm
assets since the lower the value of the firm’s assets that can be used to repay the debt
collateralized by personal assets, the bigger the additional stake at risk. All other things being
equal, in case of an adverse outcome, the entrepreneur will have to use a higher amount of
personal wealth to repay the firm’s debt if the value of firm’s assets is very low and cannot be
used to repay the debt collateralized by personal assets. Secondly, the amount of personal
wealth is a determining element, since the greater the percentage of personal wealth indirectly
invested in the venture via personal collateral, the greater the negative impact the
entrepreneur will suffer in the case of an adverse outcome. In fact, the entrepreneur should be
compensated for the additional negative impact that their personal wealth might suffer (that is
for the additional concentration of their investment). Such compensation is transferred as a
higher cost of funding to the venture. Thus, when the value of the firm’s assets that can be
used to repay debt collateralized by personal assets is very low and the personal wealth
invested in the venture is very high, the cost of debt collateralized by personal assets may be
higher than the cost of equity invested in the venture.
Secondly, the theoretical model has implications in terms of the financial structure of
the venture since it provides a possible additional explanation for the intense use of trade
credit by SMEs. Previous research finds that trade credit is widely used by SMEs, since it is
easily accessible and is also considered to be a signaling device about the firm, its products
and future prospects (Paul and Wilson 2006). Cuñat (2006) stresses that trade credit can be a
two or one part contract. In the former case, customers are entitled to receive a discount if
38
they pay immediately; in the latter case, customers do not receive any discount if they pay
cash. For the two part contract, the cost of trade credit is defined as the discount received by
customers when they pay cash. Previous research provides strong evidence that, in the case of
two part contracts, the cost of trade credit is very high (Cuñat 2006; Huyghebaert et al. 2007;
Petersen and Rajan 1994). However, irrespective of the cost, firms are said to rely on trade
credit because it is the easiest source of finance - definitely easier to obtain than bank finance
(Petersen and Rajan 1997). Suppliers are typically more supportive of customers when they
need extended credit than banks are (Howorth and Reber 2003) by supplying additional
services/goods (Cuñat 2006). Such additional extended trade credit is cost-free since
suppliers do not charge extra fees to the customers. In fact, suppliers are in a better position
than banks to evaluate the credit quality of customers, they have more tools to enforce proper
behavior of customers and therefore have greater control over credit provided (Cuñat 2006).
Thus Cuñat (2006) argues that the high price of trade credit incorporates an insurance
premium that customers pay in order to be sure of obtaining (non-bank) credit when other
sources of finance (typically banks) dry up. However, according to our proposed model, an
additional possible explanation can be provided for why firms rely on trade credit when it
appears to be so expensive. In fact, in the case of quite small and quite newly established
firms where an entrepreneur is required to provide personal collateral in order to obtain credit
(that is, the firm does not access uncollateralized credit or accesses only a very small amount
of it) and where the stake at risk can be very high since the entrepreneur has invested in the
venture the greater part of his/her personal wealth, the cost of debt collateralized by personal
assets can be very high. It can easily exceed the cost of trade credit. Thus, in this scenario, it
39
is perfectly rational for entrepreneurs to opt for trade credit rather than bank credit (or to at
least switch part of funding to trade credit).
The third relevant implication of the theoretical model is linked to capital structure
that minimizes the overall cost of capital for SMEs. Typically, firms face some problems in
implementing an optimal capital structure based on their (traditional) calculations, since
banks are very conservative in providing debt (and particularly uncollateralized debt). The
proposed theoretical model improves the calculation not only by providing a minimum
expected return that is specifically worked out for the entrepreneur, but also by considering
the peculiarities of a firm’s debt collateralized by personal wealth. In summary, the
theoretical model is close to what SMEs face on a daily basis: indeed, they have some
negotiation power between exchanging equity and debt collateralized by personal assets
while they have reduced room for maneuver in the case of uncollateralized debt. Thus, the
proposed theoretical model helps SMEs to optimize the mix of equity and debt collateralized
by personal assets: it gives them a target capital structure that is easier to pursue than the
traditional capital structure worked out without considering the entrepreneur-specific
minimum expected return and the differences in cost between debt collateralized by personal
assets and uncollateralized debt.
The last point emphasizes an additional intriguing implication of the model: it is easy
to apply. Whoever would like to use it needs just two inputs: the survival rate of the cluster
the venture is part of and the history of the entrepreneur’s previous success measured simply
as the performance of the firm/entrepreneur in previous periods. The former input is in the
public domain and can be easily accessed in the local chamber of commerce; the latter input
can be easily accessed by entrepreneurs by looking at the historical performance of their
40
venture or previous projects. It is worth noting that all the additional information needed in
order to work out the WACC is also accessible and known to the entrepreneur: the amount of
personal wealth, the amount of equity, collateralized and uncollateralized debt that will be
invested in the project, as well as tax and interest rates charged by the bank. Interestingly, the
theoretical model can also be used by banks in order to decide whether it is sensible for the
bank to finance the project the entrepreneur is interested in pursuing. In fact, banks too can
easily access the survival rate of the cluster a venture belongs to. Moreover, they can quite
easily access information regarding previous performance of the entrepreneur, in particular
when they have an established relationship with the entrepreneur. It is worth noting that all
the additional information needed in order to work out the WACC is also accessible and
broadly known to the bank: the amount of personal wealth, the amount of equity,
collateralized and uncollateralized debt that will be invested in the project, as well as tax and
interest rates charged by the bank. Indeed, this information is available to the bank when it
has an established relationship with the entrepreneur.
5. Conclusion
This paper presents a theoretical model for the minimum expected return on equity
invested in a venture by entrepreneurs. The existing finance literature is incomplete with
respect to the capital structure of SMEs and more specifically to the expected return of
entrepreneurs. In fact, previous literature tends to model the behavior of entrepreneurs
similarly to that of investors. This paper suggests viewing entrepreneurs mainly as providers
of funds who require a return for the risk they incur by investing in a venture, i.e. the risk that
the venture may go bankrupt and the impact that such an adverse event would have on
41
entrepreneurs’ personal wealth. Thus, it is suggested that entrepreneurs should demand a
‘minimum’ return worked out as the return that compensates for the probability of
bankruptcy of the venture weighted by the loss at liquidation that entrepreneurs might incur.
Such a return represents the border between a return which can be accepted by the
entrepreneur and one which cannot.
The suggested theoretical model is considered a step ahead in supporting analysis of
investment for SMEs. Even if it does not provide the ‘real’ expected return for an
entrepreneur, it provides a ‘minimum’ one that can be useful when firms need to evaluate a
project. This work goes further by examining when the overall cost of debt collateralized by
personal assets equals the overall cost of equity. This occurs because debt collateralized by
personal assets bears the cost of equity (that is ‘smoothed’ by the value of the assets the firm
can sell in order to repay bank debt and the entrepreneur’s personal wealth) and the cost of
debt (interest rate). Thus, there are scenarios when equity is cheaper than debt collateralized
by personal assets. Finally, the paper calculates the optimal capital structure for a firm that
relies on equity and debt collateralized by personal assets and uncollateralized debt.
This paper opens into at least two streams of research. The first branch could consider
the development and improvement of measurement of the probability of failure that is
suggested in the present theoretical model (in particular developing a suitable vector of ratios
or an algorithm that can be used to work out firm specific risk). The second could test the
theoretical model empirically.
However, the proposed theoretical model indicates that the return of equity for
entrepreneurs and, more specifically, for a specific venture in a specific context can be
42
worked out. Thus, this research extends the scope for and effective application of present
value techniques in SME project evaluation.
43
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Appendix A. Derivation of the Equilibrium between the Cost of Equity and the Cost of
Collateralized Debt
According to equation (19), the point of equilibrium between the cost of equity and the cost
of collateralized debt is worked out as follows. In order to simplify the notation, let us assume
𝑝
that 𝛾 = 1−𝑝 and that 𝛿 = 𝑅𝑖 ∙ (1 − 𝑡)
[𝑅𝑒𝑓𝑟𝑒𝑒 + 𝛾
(1 − 𝐸 − 𝐷) − 𝐴𝑏
𝐸
] ∙ 𝐸 = [𝑅𝑒𝑓𝑟𝑒𝑒 + 𝛾
+ 𝛿] ∙ (1 − 𝐸 − 𝐷)
𝑇𝑝𝑤
𝑇𝑝𝑤
𝑅𝑒𝑓𝑟𝑒𝑒 𝐸 + 𝛾
𝑅𝑒𝑓𝑟𝑒𝑒 𝐸 + 𝛾
2𝑅𝑒𝑓𝑟𝑒𝑒 𝐸 +
𝐸2
𝑇𝑝𝑤
𝛾
𝛾𝐸
𝛾𝐷 𝛾𝐴𝑏
−
−
−
+𝛿
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝛾𝐸 𝛾𝐸 2 𝛾𝐸𝐷 𝛾𝐸𝐴𝑏
− 𝑅𝑒𝑓𝑟𝑒𝑒 𝐸 −
+
+
+
− 𝛿𝐸
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
𝛾𝐷 𝛾𝐸𝐷 𝛾𝐷2 𝛾𝐷𝐴𝑏
− 𝑅𝑒𝑓𝑟𝑒𝑒 𝐷 −
+
+
+
− 𝛿𝐷
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
= 𝑅𝑒𝑓𝑟𝑒𝑒 +
𝐸
+ 𝑅𝑒𝑓𝑟𝑒𝑒 𝐸
𝑇𝑝𝑤
𝛾𝐸 𝛾𝐸𝐷 𝛾𝐸𝐴𝑏
𝛾𝐸𝐷
−
−
+ 𝛿𝐸 −
𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
𝑇𝑝𝑤
𝛾
𝛾𝐷 𝛾𝐴𝑏
= 𝑅𝑒𝑓𝑟𝑒𝑒 +
−
−
+𝛿
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝛾𝐷 𝛾𝐷2 𝛾𝐷𝐴𝑏
− 𝑅𝑒𝑓𝑟𝑒𝑒 𝐷 −
+
+
− 𝛿𝐷
𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
+
2𝛾𝐸 2𝛾𝐸𝐷 𝛾𝐸𝐴𝑏
−
−
+ 𝛿𝐸
𝑇𝑝𝑤
𝑇𝑝𝑤
𝑇𝑝𝑤
𝛾
2𝛾𝐷 𝛾𝐴𝑏
−
−
+𝛿
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝛾𝐷2 𝛾𝐷𝐴𝑏
− 𝑅𝑒𝑓𝑟𝑒𝑒 𝐷 +
+
− 𝛿𝐷
𝑇𝑝𝑤
𝑇𝑝𝑤
= 𝑅𝑒𝑓𝑟𝑒𝑒 +
49
𝐸 (2𝑅𝑒𝑓𝑟𝑒𝑒 +
2𝛾 2𝛾𝐷 𝛾𝐴𝑏
𝛾(1 − 𝐴𝑏)
−
−
+ 𝛿) = (1 − 𝐷)(𝑅𝑒𝑓𝑟𝑒𝑒 + 𝛿) +
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
2
𝛾𝐷(𝐴𝑏 − 2) 𝛾𝐷
+
+
𝑇𝑝𝑤
𝑇𝑝𝑤
𝛾(1 − 𝐴𝑏) 𝛾𝐷(𝐴𝑏 − 2) 𝛾𝐷 2
+
+
𝑇𝑝𝑤
𝑇𝑝𝑤
𝑇𝑝𝑤
2𝛾 2𝛾𝐷 𝛾𝐴𝑏
2𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇 − 𝑇 − 𝑇 + 𝛿
𝑝𝑤
𝑝𝑤
𝑝𝑤
(1 − 𝐷)(𝑅𝑒𝑓𝑟𝑒𝑒 + 𝛿) +
𝐸=
𝑝
And by substituting back 𝛾 = 1−𝑝 and 𝛿 = 𝑅𝑖 ∙ (1 − 𝑡)
𝑝
𝑝
𝑝
2
(1
1 − 𝑝 − 𝐴𝑏) 1 − 𝑝 𝐷(𝐴𝑏 − 2) 1 − 𝑝 𝐷
(1 − 𝐷)(𝑅𝑒𝑓𝑟𝑒𝑒 + (1 − 𝑡)𝑅𝑖) +
+
+ 𝑇
𝑇𝑝𝑤
𝑇𝑝𝑤
𝑝𝑤
𝐸=
𝑝
𝑝
𝑝
2 1 − 𝑝 2 1 − 𝑝 𝐷 1 − 𝑝 𝐴𝑏
2𝑅𝑒𝑓𝑟𝑒𝑒 + 𝑇
− 𝑇
− 𝑇
+ (1 − 𝑡)𝑅𝑖
𝑝𝑤
𝑝𝑤
𝑝𝑤
50
Appendix B. Optimization of Capital Structure
𝑝
In order to simplify notation, assume 𝛾 = 1−𝑝 and that 𝛿 = 𝑅𝑖 ∙ (1 − 𝑡). According to the
model presented in equation (19) a venture can optimize its capital structure by
𝑊𝐴𝐶𝐶 = [𝑅𝑒𝑓𝑟𝑒𝑒 + 𝛾
(1 − 𝐸 − 𝐷) − 𝐴𝑏
𝐸
] 𝐸 + [𝑅𝑒𝑓𝑟𝑒𝑒 + 𝛾
+ 𝛿] ∙ (1 − 𝐸 − 𝐷) + 𝛿𝐷
𝑇𝑝𝑤
𝑇𝑝𝑤
𝐸2
𝛾
𝛾𝐸
𝛾𝐷 𝛾𝐴𝑏
𝛾𝐸
= 𝑅𝑒𝑓𝑟𝑒𝑒 𝐸 + 𝛾
+ 𝑅𝑒𝑓𝑟𝑒𝑒 +
−
−
−
+ 𝛿 − 𝑅𝑒𝑓𝑟𝑒𝑒 𝐸 −
𝑇𝑝𝑤
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
+
𝛾𝐸 2 𝛾𝐸𝐷 𝛾𝐸𝐴𝑏
𝛾𝐷 𝛾𝐸𝐷 𝛾𝐷2 𝛾𝐷𝐴𝑏
+
+
− 𝛿𝐸 − 𝑅𝑒𝑓𝑟𝑒𝑒 𝐷 −
+
+
+
− 𝛿𝐷 + 𝛿𝐷
𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
𝜕𝑊𝐴𝐶𝐶
2𝛾𝐸
𝛾
𝛾
2𝛾𝐸 𝛾𝐷 𝛾𝐴𝑏
𝛾𝐷
= 𝑅𝑒𝑓𝑟𝑒𝑒 +
−
− 𝑅𝑒𝑓𝑟𝑒𝑒 −
+
+
+
−𝛿+
𝜕𝐸
𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝑇𝑝𝑤
=
4𝛾𝐸 2𝛾 2𝛾𝐷 𝛾𝐴𝑏
−
+
+
−𝛿
𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤 𝑇𝑝𝑤
𝜕𝑊𝐴𝐶𝐶
= 0 ⟺ 4𝛾𝐸 − 2𝛾 + 2𝛾𝐷 + 𝛾𝐴𝑏 − 𝛿𝑇𝑝𝑤 = 0
𝜕𝐸
𝐸=
2𝛾 − 2𝛾𝐷 − 𝛾𝐴𝑏 + 𝛿𝑇𝑝𝑤 1 𝐷 𝐴𝑏 𝛿𝑇𝑝𝑤
= − −
+
4𝛾
2 2
4
4𝛾
Computation of the second order derivative with respect to E
51
𝜕 2 𝑊𝐴𝐶𝐶
4𝛾
=
>0
𝜕𝐸 2
𝑇𝑝𝑤
The curve has a minimum.
52