Regulation and Control in the Banking Sector:
How Might an Anti-Herding Measure Work?
Alistair Tucker with Robert MacKay and Nicholas Beale
Complexity Science DTC, University of Warwick.
The Problem
Banks’ technology for balancing individual risk and reward may not be
beyond criticism, but it is well developed compared to the regulator’s
technology for balancing systemic risk and reward.
Downloaded from rsif.royalsocietypublishing.org on February 19, 2012
Cost Function
with H the step function and s greater than 1 so as to make a function
convex in the number of bank failures. 2. THE BASIC MODEL
ecology, the nodes in basic models are simply
This formulation has the advantage thatIn
it
naturally
favours
‘species’
that are
linked ‘diverse
to other dispecies/nodes as
or mutualist (May 2001;
versification’, that is, some heterogeneity prey,
in thepredator,
choicecompetitor
of asset allocation
Dunne et al. 2002). In epidemiological networks, the
among banks. This is generally regardednodes
as a are
good
thing. infected/infectious or recovsusceptible,
ered/immune individuals (Anderson & May 1991;
For these illustrative purposes we have Newman
modelled
asset
by inde2002).
But losses
in a minimally
complicated banking according
network, theto
nodes,
banks, have a more
pendent random variables, each distributed
the individual
same alphacomplex structure. Following NYYA and GK, we
stable distribution.
define such a node/bank as illustrated schematically
in figure 1.
Note that, in this deliberately oversimplified scheme,
the activities of any given bank are partitioned among
four categories. Two of the four categories represent
and external assets (ei). The
assets:
(li) requirements
It is proposed that the regulator find the
setIBoflending
capital
other two correspond to liabilities: IB borrowing (bi)
∗
z (X) to impose by solving an optimisation
problem.
For the
example
it indicate the
subscripts
and deposits
(di). Here
specificofbank,
with i ¼ 1, 2, . . . , N, where N is the
might choose to minimise the weighted sum
two functions,
Regulator’s Action
z ∗(X)
finds
min
X
z + λC(z, X).
J. R. Soc.
i Interface (2010)
z
i
By keeping the first term low, we hope to avoid a large negative impact
on the willingness of banks to lend. By keeping the second term low we
hope to avoid significant risk of financial system failure.
In fact we choose to find z ∗ through the constrained optimisation
X
∗
z (X)
finds
min
zi
subject to
C(z, X) ≤ C ∗,
z
finds
max min
λ
j
(1– θi)ai
external assets,
ei
IB
borrowing,
bi
θi ai
Figure 6: z1∗(X)
As indicated in figure 1, gi is the ‘net worth’ or ‘capital
Figure
4: bank
Nashi.Equilibrium
and the
Systemic
Optimum.
Horizontally the parameter controlled by B1, vertically the
bi change
in such
a
buffer’ of
If ei, li, di and/or
then Equilibrium
liabilities exceed
way that gcontrolled
parameter
by B2. Nash
at the intersection of the two red contours, Systemic Optimum at the
i becomes negative,
assets and theofbank
i is assumed
to fail. The two are close but do not coincide.
intersection
the two
blue contours.
In the studies by NYYA, GK and others, these banks
1.0
are now interconnected in a random,
Erdos– Renyi
network in which any one of the N banks is linked to
any other one, as lender or as borrower (or possibly
both), with probability p. In such a network, a bank’s
average number of incoming/borrowing or outgoing/
lending connections, z, will obviously be
z
X
zi + (C(z, X) − C ∗) λ.
i
Figures 2 and 3 show that problematic multiple minima appear when
asset allocations are heavily correlated. An optimal correlation of 1 for
the Gaussian means that the scheme makes little sense in the absence
of fat tails.
X
S(X) =
zi∗(X).
i
k
(5)
Figure 7:
j
This amounts to (5) where
X
Aij = Aji =
αk Xik Xjk
z2∗(X)
0.5
and
b = 0.
k
ð2:2Þ
But allow that A and b may also include the effect of regulatory antiherding action. Now the systemic cost is given by
S(f ) = f Tζ(f ) = f TAf + f Tb
and its derivative by
!
X
X
∂ X
0
hi(f ) =
fj Ajk fk +
f j bj = 2
Aij fj + bi
∂fi
jk
0.0
0.5
1.0
h1
Correspondence to Nash ‘Flows’
ua banks become as highly leveraged as they do
If we can assume wthat
ð2:3Þ
¼ :
z
because it confers some advantage, it might seem reasonable to suppose
Here a represents the value of the total assets of the averthat
each will adjust its holdings so as to minimise its capital buffer
age bank. Note that zi(out) will vary, according to the
number of links a given bank has. The calculation of
requirement.
(out)
Take the (fixed, possibly infinite) set of allowable asset allocations P =
{p1, p2, . . .} independently of the particular entities that may adopt
them,
w) is found by a process
We propose that the regulator provide each bank with a description of
how its capital requirement depends on its own asset allocation (all other
things being equal).
Figures 5, 6 and 7 have been created for the two-bank, two-asset scenario. Each of the two has just one degree of freedom subject to the
constraint
X
Xij = 1 for all i.
j
Nash Equilibrium is found where, for each Bi,
∂zi∗(X) ∂zi∗(X)
≤
for all j, k with Xij > 0.
∂Xij
∂Xik
Systemic Optimum is found where, for each Bi,
∂S(X) ∂S(X)
≤
for all j, k with Xij > 0.
∂Xij
∂Xik
(1)



p1 X11 X12 · · · f1
p2 X21 X22 · · · f2
..
..
.. . . . ..
instead of
The total dollar amount invested by institutions adopting pi we denote
by
fi
and the configuration of the system as a whole by
T
f = f1 f2 · · · .
Every dollar devoted to pi incurs some penalty, determined by the market
or the regulator or both, that we denote by
and for the total penalty incurred in configuration f we write
X
S(f ) =
fi ζi(f ) = f Tζ(f ).
i
(2)

B1 X11 X12 · · ·
B2 X21 X22 · · · .
..
..
.. . . .
ζi(f )
j
j
otherwise written
S 0(f ) = (2Af + b)T .
Under these assumptions then, where conditions implying Nash Equilibrium are satisfied by f feasible for total dollar amount r, we see that
conditions implying Systemic Optimum are satisfied by f /2 feasible for
r/2.
Banks’ Reaction
both ai and ei (ai ¼ ei þ zi
i.e. ζ(f ) = Af + b.
Such a situation might come about naturally through the market mechanism. For example suppose that the penalty to be incurred by a dollar
invested in asset k is proportional to the total number of dollars chasing
k. Then the penalty incurred through pi as a whole will be
!
X
X
fi
Xik αk
fj Xjk
with constant of proportionality α.
ð2:1Þ
Various assumptions about these parameters can
now be made. For a start, more or less by definition,
total assets equal total liabilities, as shown in figure 1
(Gai & Kapadia 2007; Nier et al.0.02007).
Assumptions about total assets (ai), net worth ( gi),
IB loans as a fraction of total assets (ui) and the average
size of any one given bank’s individual loans (wi) can
vary. For any given bank, the number of IB loans is
equal to the outgoing Erdos – Renyi links (zi(out)), and
-0.5
similarly for the number of incoming
links (zi(in)). Simulations by different authors make various assumptions
about these parameters. Both NYYA and GK assume
that gi is a fixed fraction of ai. Both also assume
random, Erdos –Renyi networks. NYYA assume that
all banks share a common value for the average loan, w.
-1.0
The total value of all assets in the system
is fixed, as
is
-1.0
-0.5
the overall system’s average value of the ratio of all
outgoing loans to all assets, u. From this, the average
value of w, which is assumed to be the value of each
and every individual loan, can be calculated as
(4)
∂
ζi(f ) = Aij
∂fj
total number of banks (table 1). For any individual
bank, solvency requires that assets exceed liabilities
z ¼ pðN $ 1Þ:
h0i(f ) ≤ h0j (f )
Assume that there exist symmetric, positive semi-definite A and nonnegative b such that the penalty incurred by a dollar invested in strategy
pi depends on the total invested in strategy pj according to
Figure 1. Schematic model for a ‘node’ in the IB network.
gi ; ðei þ li Þ $ ðdi þ bi Þ % 0:
(3)
Bounding the Price of Anarchy
i
IB
loans,
li
ζi(f ) ≤ ζj (f )
for all i, j with fi > 0
(i.e. all strategies adopted have a common, minimum cost).
Equally where we have defined
∂
0
hi(f ) =
S(f ),
∂fi
Systemic Optimum is characterised by the condition
Figure 3: α = 1.5 (Heavy tails).
Near the central diagonal (where banks have highly correlated asset
allocations) we again see the effects of multiple minima.
i
or equivalently,
z ∗(X)
deposits,
di
Nash Equilibrium is characterised by the condition
for all i, j with fi > 0.
Note the similarity between (3) and (4).
‘net worth’, γi
h2
models can make a useful contribution, just as they
demonstrably
have over
past few decades
Notable among its tools is the capital buffer
requirement
itthe
specifies
for in ecology
and epidemiology (Anderson & May 1991; Dunne et al.
individual banks. We interpret this as a2002).
constraint
on the leverage
ofbest rough carSuch oversimplified
models are at
icatures of reality. They have both the merits and the
the bank.
faults of caricature. They are also, of course, only part
of asystem
larger canvas:
more, but whose
no less.
We follow others who have modelled the
as anonetwork
The present paper builds on important earlier such
nodes are balance sheets (Figure 1). Bank
Bithe
willpropagation
fail if a offall
in the
work on
shocks
in model banking
Essentially,
all
this earlier
work is based on
value of external assets V causes a lossnetworks.
that
exceeds
its
capital
buffer
numerical simulations. One distinctive element of the
zi. That is, when
present paper is the use of analytic approximations
X
approximations’ for the network), which
Xij Vj − zi(‘mean-field
>
0.
can sharpen and generalize some insights.
The paper is organized as follows. In §2, following
j
previous authors, we define a class of deliberately overwith X the matrix of banks’ asset allocations.
simplified models in which individual banks are the


nodes in a network; we also discuss possible choices
· structure of, and parameter choices within,
B1 X11 X12 for· · the
a network. Section 3 outlines how failure-causing
· · ·
B2 X21 X22 such
.. ‘shocks’
..
..
. . . can arise in the network, and how they may
be propagated within the interbank (IB) lending/borrowing network and/or by liquidity effects (from asset
classes shared among banks, or more generally). The
stage thus set, in §4, we apply our mean-field approximation to the kinds of models studied by Nier et al.
(2007) (henceforth NYYA) and by Gai & Kapadia
(GK) to contagion
get some simple
results that agree well
We do not consider here the possibility(2007)
of direct
between
with their simulations and, as a consequence of the anabanks, so together z and X determine the
theintuitive
system.
lyticconfiguration
approximations, of
some
understanding of
the overall dynamics of the system. Sections 5 and 6
Define the ‘systemic cost’ C of a configuration
as aeffects
function
the brush
risk way explored
add liquidity
in theof
broad
by previous we
authors.
§7, we define
of bank failures. In particular for our numerics
haveInadopted
themore complex,
and arguably somewhat more realistic, classes of
formula proposed in [2],
‘strong liquidity shocks’ and ‘weak liquidity shocks’
 (henceforth SLS and
WLS)
s and explore their dynamical

consequences in some detail both with simulations and
X
X
our mean-fieldapproximation.
Section 8 draws


X
V
−
z
H  in
C(z, X) = EV 
ij jsome tentative
i
together
conclusions which emerge
jfrom this work.
i
shock
(to ei or li)
(out)
assets
(in)
liabilities
z2∗(X).
Figure 5: S(X) =
+
Horizontally
the parameter controlled by B1. Into the paper
the parameter controlled by B2. Vertically S(X)
the sum of the two capital buffers.
Figure 2: α = 2.0 (Normal). Horizontally the
‘correlation’ between the two banks’ portfolios.
Into the paper the difference between the two
banks’ capital buffers. Vertically their sum.
Figure 1: Schematic model for a ‘node’ in the IB
network (from [1])
On the one hand the regulator wishes to avoid the financial trauma of
824 Systemic risk R. M. May and N. Arinaminpathy
multiple bank failures, on the other it needs banks to continue to lend
In all this, deliberately oversimplified mathematical
to home-buyers and small businesses.
z1∗(X)
Let f ∗ denote the Systemic Optimum feasible for r. Then by convexity
T
f
f f
1
f
f
3
∗
0
≥ S(f ) + 2A + b
= S(f ).
S(f ) ≥ S
+S
2
2 2
4
2
2
4
By tracing the arguments of [3] we have found the same bound on the
‘Price of Anarchy’:
S(f )
4
≤ .
∗
S(f )
3
References
[1] May R.M. and Arinaminpathy N., Systemic risk: the dynamics of model banking
systems, J. R. Soc. Interface (2010) 7, 823-838.
[2] Beale N. et al., Individual versus systemic risk and the Regulator’s Dilemma,
PNAS (2011) 108 (31), 12647-12652.
[3] Roughgarden T., Selfish Routing and the Price of Anarchy, MIT Press (2005).