Download Optimum Bank Equity Capital and Value at Risk

Optimum Bank Equity Capital and Value at Risk
Udo Broll1,* and Jack E. Wahl2
1
Saarland University
2
Dortmund University
Abstract. This paper studies the impact of the value at risk paradigm on required equity
capital of a banking firm under different risk attitudes. We show that the bank’s asset
liability management cannot be separated from the decision how much equity the bank
owner should invest. That is to say, banking decisions and equity policy have to be
addressed by bank managers simultaneously.
J.E.L. Classification. G21
Keywords. Equity capital, value at risk (VaR), banking, risk management, asset liability
management.
*
Correspondence: Udo Broll, Universität des Saarlandes, Lehrstuhl für Internationale
Wirtschaftsbeziehungen, D-66041 Saarbrücken, E-mail: [email protected], Fax:
0681-302-4390.
Bank Equity and VaR
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1. Introduction
In recent years, value at risk (VaR) has become a heavily used risk management tool in
the banking sector. Roughly speaking, the value at risk of a portfolio is the loss in market
value over a risk horizon that is exceeded with a small probability. Bank management
can apply the value at risk concept to set capital requirements because VaR models allow
for an estimate of capital loss due to market risk (see, e.g., Duffie/Pan 1997,
Jackson/Maude/Perraudin 1997, Jorion 1997, Saunders 1999, Friedmann/SanddorfKöhle 2000, Hartmann-Wendels/Pfingsten/Weber 2000, and Simons 2000). The aim of
our study is to answer the question what is the optimum amount of equity capital of a
banking firm under the value at risk concept?
Institutionally, the Basle Committee on Banking Supervision mandates that banks
using models of value at risk to set aside capital for market risk of their financial
operations use a risk horizon of two weeks and a confidence level of 99 percent (Basle
Committee on Banking Supervision 1999).
In our model of a banking firm, a risk averse bank management which has to decide
on the bank’s business policy regarding assets and liabilities acts in a competitive
financial market (For an excellent discussion of bank management see Greenbaum/Thakor 1995, and for modelling a banking firm see, e.g., Broll/Wahl 2000.). The
return on the bank’s portfolio of assets is uncertain. Hence, the banking firm is exposed
to market risk and may not be able to meet its debt obligations. In this case our bank goes
bankrupt. Instead of coping with the exposure of the banking firm to financial risk by
using hedging instruments such as futures and options or by using a variable rate of
deposits (Wahl/Broll 2001), in the present model we incorporate as a risk management
tool the value at risk approach in order to address bankruptcy risk.
As depicted in figure 1 the bank faces a loss distribution. Given a confidence level of
99 percent, to the equity holder there is a 1 percent chance of losing VaR or more in
Bank Equity and VaR
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value. Hence, if VaR determines optimum bank equity, then the confidence level gives
the probability that the bank will be able to meet its debt obligations.
Probability
1 percent
99 percent
0
VaR
Loss
Figure 1: Loss distribution, VaR and bank equity
The study proceeds as follows: Section 2 presents our model of a banking firm in a
competitive market environment under uncertainty. The value at risk concept is
formulated for our economic setting. In section 3 we investigate how optimum volume of
equity capital is affected by value at risk. We demonstrate that managerial and market
factors determine optimal asset liability and equity policy of the bank and that the
probability of bankruptcy has a complex impact on the decision making of bank
management. Section 4 discusses the case of risk neutrality and reports the relationship
between optimum equity and VaR. Final section 5 concludes the paper.
2. A Banking Firm
In this section we study how a risk averse bank management, acting in a competitive
financial market, can use value at risk to deal with market risks from financial assets.
Bank Equity and VaR
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What is the optimal equity capital requirement of a banking firm under the value at risk
concept?
2.1 The Model
The bank invests in risky assets, A . At the beginning of the period the return of the
bank’s portfolio, ~
r , is random. The bank’s portfolio is financed by deposits, D , and
A
equity capital, E . The intermediation costs of the bank compounded to the end of the
period are determined by C ( A + D ) . Hence intermediation costs are modelled to depend
upon the sum of the bank’s financial market activities. The cost function C (⋅) has
properties C ′(⋅) > 0 and C ′′(⋅) > 0 , i.e., marginal cost are positive and increasing.
A simplified balance sheet of a bank is at each point in time:
Bank balance sheet
Assets A
Deposits D
Equity E
Equity is held by shareholders and necessarily E = A − D . Optimum decision making of
bank’s management has to satisfy the bank’s balance sheet constraint, A = E + D , where
in our model the debt/equity ratio of the banking firm is endogenous. Given that the
bank’s assets have risky returns, bankruptcy of the banking firm occurs if the firm cannot
meet its debt obligations. Value at risk is a risk management tool which allows to cope
with bankruptcy risk.
If the bank’s loss in market value does not exceed equity capital at some confidence
level 1 − α , then value at risk, VaRα , measures the maximum size of that loss in the next
period. That is to say, E ≥ VaRα implies that the firm cannot meet its debt obligations
with probability α . Therefore, in our context, α measures the probability of bankruptcy
of the banking firm.
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In the literature value at risk is discussed as an indicator for minimum capital
requirements regarding the solvency of banks (see, e.g., Jackson/Maude/Perraudin 1997).
Our paper derives optimally required equity capital and optimum banking under the
value at risk concept.
We consider a competitive bank which faces the risky return ~
rA on its assets and the
given nonrandom market rate of deposits, rD (Freixas/Rochet 1997, Broll/Wahl 2001).
~
The bank’s risky profit, Π , can be stated as follows:
~
Π=~
rA A − rD D − C ( A + D) .
(1)
Our bank management ranks probability distributions of returns by applying the
expected utility hypotheses. Bank management is risk averse and uses exponential utility
− e − R Π , where R > 0 measures constant absolute risk aversion. Furthermore, bank
management assumes that ~
rA is normally distributed with expected value µ A and
variance σ A . These assumptions on preferences and expectations imply the well-known
2
mean-variance objective function below.
Bank management has to choose the amount of equity which maximizes expected
utility of profits:
max
E
{ ε (Π~ ) − Rν (Π~ ) / 2 }},
(2)
subject to
where
ε
νν
and
E+D−A=0,
(3)
E − VaRα ≥ 0 ,
(4)
denote the expected value operator and the variance operator,
respectively.
The aim of our competitive bank is to establish the optimum amount of equity capital,
*
E , satisfying the balance sheet constraint (3) and the value at risk constraint (4). As
will be shown, optimum amount of equity E * is positive if and only if the bank’s
expected margin is positive. Optimum equity implies optimum bank policy D * , i.e. the
Bank Equity and VaR
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optimum amount of deposits, and A* , i.e. the optimum amount of assets. This
*
*
determines optimum bank size A * + D * and optimum debt/equity ratio D / E .
2.2 Value at Risk
Risk management of the bank has to take into account bankruptcy risk. The risk of
bankruptcy means that because of the uncertainty of the market value of equity at the end
of the period the bank may not be able to meet its financial obligations vis-à-vis its
depositors without further contributions by the owners. If the owners are not able or not
willing to contribute the bank goes bankrupt. Value at risk is a risk management tool
which allows to control for the probability of bankruptcy.
In our model of a banking firm bankruptcy risk can be defined as the probability of
bankruptcy: Prob (−~
r A > E ) > 0 where 1 + r~ = (1 + ~
r ) /(1 + r ) (see Appendix A1). This
A
D
means that with positive probability the decline in market value of the bank’s assets can
be greater than equity capital of the bank. To control for bankruptcy risk bank
management has to limit the probability of bankruptcy by choosing α . This leads to the
solvency condition
Prob (−~
r A ≤ E) = 1 − α .
(5)
Statistically speaking, the value loss that leads the firm to bankruptcy has an α percent
probability of occurring at the end of the period.
rA is normally distributed, the random
Given that the return on the bank’s assets ~
2
~
variable r has also a normal distribution with expected value µ and variance σ ,
where µ =
ε (~r ) = (1 + µ A ) /(1 + rD ) − 1 and σ 2 =ν (~r ) = (σ A /(1 + rD )) 2 . It follows that
the solvency condition (5) can be stated as (see Appendix A2):
rα A = E ,
where rα = −( µ + uα σ ) > 0 and uα
(6)
denotes the α -fractile of the unit normal
distribution.
rα represents value at risk of a risky investment of 1 Euro and is denoted as
percentage–VaR, pVaR. Note that pVaR is a decreasing function of the probability of
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bankruptcy α and, hence, an increasing function of the confidence level 1 − α . Value at
risk is then determined by multiplying the amount of risky assets by pVaR: VaRα = rα A .
Bank management restricts the size of loss in the value of the bank by choosing the
bankruptcy probability and, with that, defines the maximum loss of value on the portfolio
of the bank’s assets over the planning period at a given confidence level.
2.3 The Bank’s Profit
From the solvency condition (6) and the definition of the value at risk measure, the value
at risk constraint (4) has to hold as an equation. Taking into account the balance sheet
constraint A = E + D and the value at risk equation rα A = E the profit equation (1)
results in the following definition of bank’s random profit:
1 − rα
1
~ æ
Π = çç ~
− rD
rA
rα
è rα
ö
æ 2 − rα ö
÷E − Cç
E ÷÷ .
÷
ç r
ø
è α
ø
(7)
Hence the bank’s random profit is a function of the bank’s equity capital which has to
be chosen by bank management in order to maximize expected utility of profits.
3. Optimal Capital Requirement
In the following we derive the implications of the value at risk concept to optimum
equity and optimum management of assets and liabilities of our banking firm.
3.1 Main Results
The amount of equity capital which maximizes expected utility of profits depends upon
the probability of bankruptcy set by bank management. In order to get a closed form
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solution we assume quadratic intermediation costs: C ( A + D ) = θ ( A + D ) 2 / 2 , where
θ > 0.
We claim the following propositions:
Proposition 1 (Required Equity Capital)
Risk management by value at risk
implies that optimum amount of equity depends upon
(i) managerial factors such as the confidence level, the degree of risk aversion and
expectations about the return on risky assets and
(ii) market factors such as the rate of deposits and the intermediation costs.
Proof.
Maximizing the mean-variance function (2) under the profit equation (7) with
respect to equity E leads to:
E * = rα
µ A − rD (1 − rα )
.
Rσ A2 + θ (2 − rα ) 2
(8)
This proves the claim.
Proposition 2 (Effect of Confidence Level)
If the banking firm satisfies the inter-
mediary function, i.e. D > 0 , then optimum equity increases if the confidence level, i.e.
*
1 − α , increases.
Proof.
From the balance sheet constraint (3), the solvency condition (6) and a
positive expected margin µ A > rD (1 − rα ) , we get that D * > 0 implies rα < 1 . Hence,
∂ E* / ∂α < 0 .
Proposition 3 (Asset Liability Management)
Optimum equity of the banking firm
under the value at risk concept determines optimum amount of risky assets and optimum
amount of deposits.
Proof.
The proof is a direct implication of (8), the value at risk constraint (4) and the
balance sheet constraint (3).
Proposition 2 demonstrates that a higher managerially given confidence level calls for
a higher amount of equity capital. Hence, if bank management sets a smaller probability
Bank Equity and VaR
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of bankruptcy the owners of the bank have to contribute additional equity in order to
enable bank managers to realize an optimal asset liability policy.
Proposition 3 shows that the required amount of equity capital cannot be derived
solely from value at risk but, instead, has to be determined simultaneously with the
bank’s business policy. Furthermore, equation (8) shows that even under our simplifying
assumptions the relationship between the confidence level and the optimum amount of
equity capital is not a straightforward one.
A quite simple result is stated in
Corollary 1
A higher level of constant absolute risk aversion leads to a lower
optimum amount of equity capital.
Proof.
See equation (8).
The intuition behind this result is as follows: Increasing risk aversion decreases the
optimum amount of assets. Since pVaR is unaffected, from the solvency condition (6) we
see that optimum equity also decreases. Another way to explain Corollary 1 is that the
optimum amount of assets declines faster than the optimum amount of deposits, when
risk aversion increases, since D * = (1 − rα ) A* . In order to meet the balance sheet
constraint optimum equity must decrease.
A further implication of our model is reported in
Corollary 2
If bank management sets a higher probability of bankruptcy, then
optimum debt/equity ratio increases.
Proof.
The claim follows from the fact that D * / E * = (1 − rα ) / rα and ∂ rα / ∂ α < 0 .
Hence, the value at risk concept implies that the optimum debt/equity ratio is
determined by pVaR, only. The optimum amount of equity and the optimum amount of
deposits are negatively affected by the probability of bankruptcy. However, the structure
of the bank’s liabilities will change in a way which increases the proportion of debt.
To sum up, equation (8) shows that optimum bank equity capital under VaR depends
upon several key factors:
Bank Equity and VaR
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E = E (α , µ A , σ A , rD , R , θ ) ,
*
2
where the comparative static impact is, in general, indetermined for the inputs µ A
(expected return on risky assets), σ A (variance of the return on risky assets) and rD
2
(market rate of deposits). The reason for this indetermination is that pVaR ( rα ) is also
affected by these inputs. The probability of bankruptcy (α ) , the level of risk aversion
(R ) and the unit variable intermediation costs (θ ) lead to a lower level of optimum
equity if these factors increase.
3.2 An Expository Case
For expository reasons let us consider a special case which leads to a plain relationship
between the probability of bankruptcy and optimum amount of equity capital. If there are
no intermediation costs (i.e. θ = 0 ) and if the bank’s expected margin is zero –before
adjustment for pVaR – (i.e. µ A = rD ), then optimally required equity reads:
E * = uα2
rD
R (1 + rD ) 2
.
Hence optimum equity is monotonically increasing (and convex) when the probability of
bankruptcy is decreasing or, in other words, when the required confidence level by bank
management is increasing (see figure 2).
Note that in our expository case optimum equity level does not depend upon the
variance of the random return on assets. Furthermore, in contrast to the general case,
optimum equity is strictly increasing when the market rate of deposits (or, respectively,
the expected return on risky assets) increases.
E*
0
=
Bank Equity and VaR
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Figure 2: Optimum equity and the probability of bankruptcy
4. Bank Equity Under Risk Neutrality
Now let us consider a risk neutral bank management (set R ≡ 0 in equation (2)). With
this risk attitude bank management has to choose the amount of equity capital which
maximizes expected profits:
max
E
ε ( Π~ ) ,
(9)
subject to
where, as before,
ε
E+D− A=0,
(3)
E − VaRα ≥ 0 ,
(4)
~
denotes the expectation operator and the bank’s risky profit, Π , is
determined by equation (1).
Since the bank’s random profit can still be represented by equation (7), optimum
amout of equity for the banking firm immediately follows from equation (8):
E * = rα
µ A − rD (1 − rα )
θ ( 2 − rα )
2
.
(10)
Bank Equity and VaR
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Therefore the following claim holds:
Proposition 4 (Optimum Equity)
The value at risk concept under risk neutrality
implies that optimum amount of equity depends upon
(i) managerial factors such as the confidence level and expectations about the return on
risky assets and
(ii) market factors such as the rate of deposits and the intermediation costs.
Although bank management is indifferent to return risk notwithstanding it takes into
account the probability of bankruptcy. Proposition 2 holds under risk neutrality and
required equity capital increases if the confidence level increases. Furthermore, the
variance of the return on risky assets still affects optimum equity. Therefore, optimum
amount of assets and deposits of the bank are also affected by the volatility of the assets’
return. The reason is that the value at risk on the equity at some confidence level is not
independent from the variance σ A of the risky returns.
2
We state the following observation:
Corollary 3
Let bank management exhibit risk neutral behavior. Then a mean
preserving spead in return on risky assets will increase optimum equity.
Proof.
when σ
The claim follows from equation (10) and the fact that pVaR rα increases
2
A
increases.
Hence, under risk neutrality the impact of the volatility of the assets’ risky return
upon optimum amount of equity capital becomes unambiguous.
5. Concluding Remarks
Bank Equity and VaR
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Models of value at risk have become a widespread risk management approach in many
different types of organizations. Our paper uses the value at risk concept to analyze
optimum equity capital requirement for a banking firm.
Our study shows that the optimum endowment of a bank in equity depends upon
managerial and market factors. Especially the confidence level set by bank management
for the assumed risk horizon has a nontrivial impact on the optimum amount of equity
capital even under restrictive assumptions of the bank model. Additionally, we
demonstrate that the asset liability policy under VaR cannot be separated from the
decision of optimally required equity capital.
Appendixes
A1
(Bankruptcy Risk):
In our model of a banking firm bankruptcy risk exists if and only if
Prob ( A(1 + r~ ) − D(1 + r ) < 0) > 0.
A
D
From the balance sheet constraint this condition is equivalent to
r ) − ( A − E )(1 + r ) < 0) > 0 ,
Prob ( A(1 + ~
A
D
which in turn becomes
r ) − A + E ) < 0) > 0
Prob ((1 + rD )( A(1 + ~
using 1 + r~ = (1 + ~
rA ) /(1 + rD ) . By simplifying we get
r A > E) > 0 .
Prob (−~
A2
(Solvency Condition):
Bank Equity and VaR
If ~
x is normally distributed with expected value µ x =
14
ε (x~)
and variance σ x2 =
ν ( ~x ) ,
then the N ( µ x , σ x2 ) -fractile of order α is defined by (see, e. g., Fisz 1977)
x ≥ xα ) = 1 − α ,
Prob ( ~
where xα = µ x + uα σ x and uα is the N(0, 1)-fractile of order α . Hence the solvency
r
condition (5) becomes for a normally distributed ~
Prob (r~ ≥ −( E / A)α ) = 1 − α .
It follows that
−( E / A)α = µ + uα σ ,
where µ =
ε (r~)
and σ 2 =
ν (~r ) . Defining rα = −(µ + uα σ ) we get equation (6).
References
Basle Committee on Banking Supervision, 1999, Credit risk modelling: Current practices and
applications, Basel.
Broll, U.; J. Wahl, 2001, Risikomanagement in Banken: Futures-Hedging, das wirtschafts-studium
(wisu), 30. Jg., 213-219.
Duffie, D.; J. Pan, 1997, An overview of value at risk, The Journal of Derivatives 4, 7-49.
Fisz, M., 1977, Wahrscheinlichkeitsrechnung und mathematische Statistik, 7. Aufl., Berlin.
Freixas, X.; J.-C. Rochet, 1997, Microeconomics of banking, Cambridge (MASS), London.
Friedmann, R.; W. G. Sanddorf-Köhle, 2000, Finanzmarktökonometrie am Beispiel des Value at
Risk, magazin forschung 2 (Universität des Saarlandes), 26-32.
Greenbaum, S. I.; A. V. Thakor, 1995, Contemporary financial intermediation, Forth Worth (TX)
et al.
Hartmann-Wendels, Th.; A. Pfingsten; M. Weber, 2000, Bankbetriebslehre, 2. Aufl., Berlin et al.
Jackson, P.; D. J. Maude; W. Perraudin, 1997, Bank capital and value at risk, The Journal of
Derivatives 4, 73-90.
Jorion, P., 1997, Value at risk: the new benchmark for controlling market risk, New York et al.
Bank Equity and VaR
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Saunders, A., 1999, Credit risk measurement, New York et al.
Simons, K., 2000, The use of value at risk by institutional investors, New England Economic
Review, Nov./Dec., 21-30.
Wahl, J. E.; U. Broll, 2000, Financial hedging and banks’ assets and liabilities management, in:
Frenkel, M.; U. Hommel; M. Rudolf (eds.); Risk management: Challenge and opportunity,
Berlin et al., 213-227.
Wahl, J.; U. Broll, 2001, Zur Vorteilhaftigkeit des Hedgings für Banken, Kredit und Kapital, 34.
Jg., Heft 4, 1-11.
Zusammenfassung
Optimale Eigenkapitalausstattung von Banken und Value at Risk
In internen Risikomodellen findet das Value at Risk-Konzept verstärkt Anwendung.
Dieses Konzept dient der Erfassung, Steuerung und Kontrolle von Markt- und
Kreditrisiken. Bei gegebener Eigenkapitalausstattung definiert der Value at Risk einen
maximalen Verlustbetrag der Bank in Bezug auf Marktwertminderungen der Bankaktiva.
Die Gefährung der Bankeinlagen ist mit einer hohen Wahrscheinlichkeit ausgeschlossen,
wenn im Aktivgeschäft der Bank der Verlustbetrag nicht erreicht wird. Solange das
Eigenkapital den Value at Risk der Bank deckt, sind die Gläubigeransprüche mit hoher
Wahrscheinlichkeit gesichert.
Unser Beitrag untersucht Marktrisiken und leitet den Zusammenhang zwischen optimaler
Eigenkapitalausstattung der Bank und der Wahrscheinlichkeit einer Bankinsolvenz auf
der Grundlage des Value at Risk-Konzepts ab. Es zeigt sich, dass die optimale
Eigenkapitalausstattung
einmal
von
managementbestimmten
Einflussgrößen
(Insolvenzwahrscheinlichkeit, Risikoverhalten, Renditeerwartungen) und zum anderen
von marktbestimmten Einflussgrößen (Einlagenzins, Kosten der Finanzintermediation)
abhängt. Des Weiteren können die optimalen Volumina von Aktiv- und Passivgeschäft
der Bank nicht isoliert bestimmt werden, sondern sind zusammen mit der optimalen
Eigenkapitalausstattung festzulegen.