If the Arrow–Pratt relative index of risk aversion −xu x/u x is decreasing respectively increasing, then the default probability of an unregulated bank is an increasing respectively de
Trang 1portfolio for a bank having an “adjusted net worth” equal to K By the
arguments above we know that
x(K) = σ (K)xM, where xMis the market portfolio, normalized in such a way that it has
a unit variance (i.e., xM = V −1 ρ/σ ) σ (K) is a nonnegative constant,
equal to the standard deviation of the argument maximum of (P) It is the maximum of σ → U(K + λσ , σ ) and, in particular, the mean return
Since λ > 0, and N increasing, the proof is completed.
Proposition 8.6 may seem a good justification of capital requirements
Independently of the choice of risk weights α1, , αN, but provided thatthe numerator of the ratio is adjusted to incorporate intermediationprofits on deposits, the capital ratio is an increasing function of thedefault risk The trouble is that as soon as the capital requirement isimposed, the banks’ behavior changes and proposition 8.6 ceases to betrue This will be the subject of the next section
As a conclusion to the present section, we examine the dependence
of the default risk of an unregulated bank on its (corrected) net worth
Is CR(K) a monotonic function of K? In other words, if no capital
regulation were imposed, would the more capitalized banks be more orless risky than the less capitalized ones? It turns out that the answer to
this question depends on the properties of the utility function u More
specifically, we have the following proposition
Proposition 8.7 If the Arrow–Pratt relative index of risk aversion
−xu (x)/u (x) is decreasing (respectively increasing), then the default probability of an unregulated bank is an increasing (respectively decreas- ing) function of its “adjusted” net worth K.
Trang 2Proof By proposition 8.6, the default probability of an unregulated bank
is an increasing function of
τ(K) = σ (K)
K , where τ(K) is the solution to maxτ Eu[K(1 + τ ˜ RM)] and ˜ RM is therandom return of the market portfolio By well-known results of Arrow(1974) and Pratt (1964), if−(xu (x))/(u (x)) is monotonic, then τ(K)
is monotonic in the other direction
As a consequence of proposition 8.7, the most frequently used ifications of VNM utility functions, namely exponential and isoelasticfunctions, lead to a default probability that is, respectively, a decreasing
spec-and a constant function of K.
8.6 Introducing Capital Requirements in the Portfolio Model
In order to concentrate on one distortion at a time, we will assume thatthe capital regulation requires the adjusted capital ratio to be less than
1 Or, equivalently, we neglect the intermediation margin K − K0 Thenew feasible set is now restricted to
A1(K) = {(σ , µ), ∃x ∈RN
+ , µ − K = x, ρ, x, V x = σ2, α, x, K}.
As before, let us denote by A11(K) the “efficient set under regulation,”
i.e., the upper contour of A1(K).
Proposition 8.8 In general the efficient set under regulation is
com-posed of a subset of the “market line” A +0(K) and a nondecreasing curve (a portion of hyperbola) In the particular case when the risk weights αi are proportional to the systematic risks βi or to the mean excess returns
ρi, this portion of hyperbola degenerates into a horizontal line.
Proof See the appendix (section 8.9).
Proposition 8.8 shows that the consequences of imposing a capitalrequirement are very different according to the value of the risk weights
αi If these weights are “market-based,” in the sense that the vector α is proportional to the vector β of systematic risks, then the new efficient
set is a strict subset of the market line All banks continue to choose
an efficient portfolio Those which are constrained by the regulationchoose a less risky portfolio than before As a consequence, their defaultprobability decreases
Trang 3Standard deviationσ
On the other hand, if the risk weights are not “market-based,” theadoption of the capital regulation has two consequences: for those bankswhich are constrained, the total “size” of the risky portfolio (as measured
by α, x) decreases, but the portfolio is reshuffled, by investing more
in those assets i for which ρi/αi is highest, and investing less in theother assets The total effect on the failure probability is ambiguous Asshown by the example given in the appendix (section 8.10), it may verywell increase in some cases, the reshuffling effect dominating the “size”
effect A similar result has been obtained before in Kim and Santomero(1988)
As a conclusion to this section, let us examine the following question:
which banks are going to be constrained by the capital regulation? Theanswer is again related to the monotonicity of the relative index of riskaversion
Proposition 8.9 If the relative index of risk aversion −(xu (x))/
(u (x)) is decreasing (increasing), the banks with the highest (lowest) net worth will be constrained by the capital regulation.
Trang 4Standard deviationσ
Proof A bank is constrained by the capital regulation if and only if the
portfolio chosen in the unregulated case has a variance greater than ¯σ2=
ρ, V −1 ρ /α, V −1 ρ 2 The result then follows from proposition 8.7
8.7 Introducing Limited Liability in the Portfolio Model
As remarked by Keeley and Furlong (1990), it is ironic that the verysource of the problem under study, namely the limited liability of banks,has been so far neglected in the portfolio model When it is taken intoaccount, the objective function of the bank becomes
,
where C 0 represents a (fixed) bankruptcy cost, and u has been
normalized in such a way that
u(0) = 0.
Let us remark that our normality assumption implies that the bank’s
utility still depends on (µ, σ ) and not on the truncated moments of
Trang 5Standard deviationσ
U = constant
µ
Figure 8.7. The shape of the bank’s indifference
curves under limited liability.
˜
K However, the properties of W will differ markedly from the utility
function under full liability, given by
In fact, our first result asserts that if the absolute index of risk aversion
is bounded above, then
• for small µ and large σ , W is increasing in σ (the bank exhibits
locally a risk-loving behavior!);
• W is not everywhere quasiconcave.
Proposition 8.10 We assume that
Trang 6As a consequence, for σ large enough, W is increasing in σ Moreover,
W is not everywhere quasiconcave.
Proof See the appendix (section 8.9).
The shape of the indifference curves in the (µ, σ )-plane is given by
figure 8.7
We are now in a position to study the portfolio choice of a limited
liability bank For simplicity, from now on we are going to take C = 0.
Let us begin with the unregulated case Since W is increasing in µ, we can limit ourselves to A +0(K) = {(µ, σ )/µ = K + λσ } In order to find the maximum of W on A +0(K), we have to study the auxiliary function
ω(σ ) = W (K + λσ , σ ).
Proposition 8.11 We assume that −(u (x))/(u (x)) a for all x and that C = 0 For all K < 1/a, ω(σ ) is increasing with σ Therefore its supremum is attained for σ = +∞.
Proof See the appendix (section 8.9).
Of course, proposition 8.11 does not mean that a bank with smallenough own funds would choose an “infinitely risky portfolio.” Indeed,
we have neglected nonnegativity constraints on asset choices Therefore,even in the unregulated case, only a portion of the market line is infact attainable When nonnegativity constraints start to be binding, theefficient set becomes a hyperbola similar to the one we found in theregulated case The correct interpretation of proposition 8.11 is that, for
K < 1/a, the bank will choose a very “extreme” portfolio with (at least
partial) specialization on some assets The convexity of preferences due
to limited liability eventually dominates risk aversion
Although we do not provide a full characterization of the behavior
of a limited liability bank, propositions 8.10 and 8.11 together have aninteresting consequence Even with correct risk weights, a capital ratiomay not be enough to induce an efficient portfolio choice of the bank
This is explained by figure 8.8
When K is small enough, W is increasing on the efficient market line but it may happen that W (¯ µ, σ ) becomes larger than W (¯ µ, ¯ σ ) for σ large
enough Consequently, and in contradistinction with the full liability
case, the bank would not choose (¯ µ, ¯ σ ) As a consequence it may be
necessary to impose an additional regulation in the form of a minimalcapital level ¯K as suggested by figure 8.9.
Trang 7Standard deviationσ
Figure 8.8. Portfolio choice with limited liability and
“correctly weighted” solvency ratio but no minimum capital.
8.8 Conclusion
Of course one should not take too literally all the conclusions of thevery abstract and reducing model presented in the paper However, wehave clarified several elements of the polemic between value maximizingmodels and utility-maximizing models
If we accept the assumption of complete contingent markets (the onlycorrect way to justify value-maximizing behavior), then it is true thatunder fixed-rate deposit insurance, absence of capital regulations wouldlead to a very risky behavior of commercial banks However, capitalregulations (at least of the usual type) are a very poor instrument forcontrolling the risk of banks: they give incentives for choosing “extreme”
asset allocations, and are relatively inefficient for reducing the risk
of bank failures The correct instrument consists in using “actuarial”
pricing of deposit insurance, which implies computing risk-related miums A pricing formula incorporating interest rate risk is obtained in
pre-a comppre-anion ppre-aper (Kerfriden pre-and Rochet 1991)
On the other hand, if we take into account incompleteness of financialmarkets and adopt the portfolio model (utility-maximizing banks), thecorrect choice of risk weights in the solvency ratio becomes crucial Ifthese risk weights are related to credit risk alone (as is the case forthe Cooke ratio and its twin brother, the EEC ratio), this may againinduce very inefficient asset allocations by banks We suggest insteadthe adoption of “market-based” risk weights, i.e., weights proportional
Trang 8Standard deviationσ
Figure 8.9. Portfolio choice with limited liability,
“correctly weighted” solvency ratio and minimum capital R.
to the systematic risks of these assets, measured by their market betas
However, contrarily to previous papers using the portfolio model, we donot neglect the limited liability of the banks under study We show that
it implies that insufficiently capitalized banks may exhibit risk-lovingbehaviors As a consequence it may be necessary to impose a minimumcapital level as an additional regulation
It may seem unrealistic to suggest the adoption of “market-based” riskweights for bank loans, which constitute a large proportion of banks’
assets and are a priori nonmarketable However, the success of thesecuritization activity in the United States has shifted the border betweenmarketable and nonmarketable assets Moreover, once nonsystematicrisk has been diversified there is not much of a difference between a pool
of loans and a government bond However, further research is needed tocorrectly account for the asymmetric information aspects of the bankingactivity
8.9 Appendix
8.9.1 Proof of lemma 8.1
By the projection theorem, the function x, V x (which equals Bx2,
where B denotes the “square-root” of V , i.e., the unique symmetric
positive definite matrix such that tBB = V) has a unique minimum x ∗
Trang 9on the convex set
minx, V x for x ∈RN
+ , such that ρ, x = µ − K, α, x K (P )When the second constraint is not binding, we are back to our initialproblem The solution of (P) is
In particular, if α, V −1 ρ 0, condition (8.13) is always satisfied for
µ K and the capital requirement is ineffective This case is completely
uninteresting Therefore, we may assume thatα, V −1 ρ > 0.
When the second constraint is binding, we have to distinguish betweentwo cases
Case 8.1 (∃h > 0, α = hρ) Then the feasible set of problem (P ) isnonempty only when condition (2) is satisfied, that is, when
When α is positive, A1(K) is in fact a triangle.
Trang 10Case 8.2 (α and ρ are linearly independent) The Lagrangian of ( P )can be written as
L = x, Vx − 2ν1ρ, x − 2ν2α, x,
and the first-order condition gives
V x = ν1ρ + ν2α, where ν1, ν2are determined by
M −1 = r s s t
!
is such that ∆= r t − s2> 0 and s < 0
(becauseα, V −1 ρ > 0).
Since M −1is positive definitive, (8.14) is the equation of a hyperbola
x is indeed the solution to ( P ) if and only if ν2 0, which is equivalentto
Trang 11eax u (x) is an increasing function of x, which implies
∀y yu (µ + yσ ) yu (µ)e −ayσ
We need a technical lemma
Lemma 8.2. ∀t A(t) −1/t + 1/t3; moreover, when t → +∞, these two functions are equivalent.
Using this lemma, we obtain
∂W
∂σ f
µ σ
Denoting by ψ(a) the term between brackets we have
Trang 12Since ∂W /∂σ is negative for σ small enough, W cannot be quasiconcave with respect to σ
1 0
ds
√
t2− 2 ln s . Let B(t) = t2(tA(t) + 1) and
(1 − (2 ln s)/t2) +1− (2 ln s)/t2, B(t) =
1 0
Trang 13Using the fact that x → e ax u (x) is increasing, we have
∀y (λ + y)u (µ + σ y) (λ + y)u (µ − σ λ)e −aσ (y+λ)
Thus
w (σ ) u (µ − σ λ)e +a(µ−σ λ)
+∞
−µ/σ (λ + y)e −a(µ+σ y) dN(y).
After straightforward computations the right-hand side can be pressed as follows:
ex-w (σ ) u (µ − σ λ)e +a(µ−σ λ) f
λ + K σ
We are going to prove that the graph of H crosses the horizontal axis at
most once, by showing that
Trang 14= 1− aK
a2 Thus, when K < 1/a, lim infσ →+∞ w (σ ) 0 and w is increasing.
8.10 An Example of an Increase in the Default Probability Consecutive to the Adoption of the Capital Requirement
We take an exponential specification: u(x) = (1/a)(1 − e −ax ), where
When the capital requirement is introduced, and the bank is indeed
constrained, the portfolio it chooses has a mean return µ1 that mizes
a − Ks
and σ12= tK2+ 1
r
1
a2 − K2s2
.
Trang 15µ0/σ0, and the failure probability of the bank increases when the capitalrequirement is adopted.
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Trang 16“pil-it seems strange to insist so much on the importance of supervisoryreview3 and market discipline as necessary complements to capitalrequirements while remaining silent on the precise ways4 this comple-mentarity can work in practice One possible reason for this imbalance is
a gap in the theoretical literature As far as I know, there is no tractablemodel that allows a simultaneous analysis of the impact of solvencyregulations, supervisory action, and market discipline on the behavior
3 For example, the Basel Committee (BIS 2003) insists on the need to “enable early supervisory intervention if capital does not provide a sufficient buffer against risk.”
4 In particular, despite the existence of very precise proposals by U.S economists (Calomiris 1998; Evanoff and Wall 2000; see also the discussion in Bliss 2001) for mandatory subordinated debt, these proposals are not discussed in the Basel II project.
...Journal of Finance 43:12 19? ??33.
Klein, M A 197 1 A theory of the banking firm Journal of Money, Credit and Banking 3:205–18.
Koehn, H., and A... reexamination of mean–variance analysis
of bank capital regulation Journal of Banking and Finance 14: 69? ??84.
Kerfriden, C., and J.-C Rochet 199 1 Measuring...
North-Holland.
Dothan, U., and J Williams 198 0 Banks, bankruptcy and public regulation.
Journal of Banking and Finance 4:65–88.