Risk Accumulation, Contagion and the Rules for
Bank Failure*
Yukinobu Kitamura
Keio University

Shuji Kobayakawa
Bank of Japan

Abstract
This paper examines how we view the cost associated with negative externality,
often referred to as “contagion effect,” in the financial market and how effectively
various types of ex ante disciplines can prevent contagion from spilling over toother participants in the market. The ultimate objective of the regulator is to keep
the market open, i.e. to attract investors’ participation in financial transactions.
For that purpose, we aim to identify how the regulator sets the rule which prevents the contagion effect, while at the same time making as many participants as
possible take part in the transactions. Our Monte-Carlo simulation suggests thatsuch regulatory rules as early closure and capital enforcement are effective devices for preventing the contagion effect, although contagion itself can never be
eliminated fully under the arc-sine feature of the random walk process. The
results further suggest that a more practical resolution method is to combine
several policy rules.
Almost all human life depends on probabilities.
Voltaire, Essays

1

Introduction

This paper examines how we view the cost associated with negative externality,
often referred to as “contagion effect,” in the financial market and how effectively
various types of ex ante disciplines can prevent the contagion from spilling overto other market participants.
*
The authors are grateful to Jeff Lacker, the discussant in the conference, and participants in theseminar at Osaka University. The views expressed herein are the authors’ and not necessarily thoseof the Bank of Japan.

106

Yukinobu Kitamura and Shuji Kobayakawa

Suppose that banks are engaged in risky financial transactions, and the overall
results of these transactions are summarised by winning and losing scores, which
add up to zero (i.e. zero-sum game). If, however, the losers of these transactions failto pay their losses due to insufficient capital holdings, then the winners might not beable to collect all their gains. Unless the losses are compensated by a third party, thewinners will have to start the next round of transactions with less wealth than theywould have obtained if the losers had paid their losses in full. If the winners continuously fail to receive their full winning position at the end of each round oftransactions, this may generate a higher possibility that they will eventually reach athreshold point of bankruptcy. This is one way of illustrating the contagion effecttransmitted from one party to others.1 Here the contagion cost is measured by theamount of funds which could have been obtained, had all the losses been paid infull or covered by a third party, minus the amount actually received.
In this paper, we are interested in how this type of contagion effect accumulates over the long run for financial transactions. We believe this is a uniqueinterpretation. One often associates the contagion with a situation where one
strategy, such as a deposit withdrawal, can spread from a finite set of players to
the whole population. Our interpretation is unrelated to the choice of strategies.
Instead, we examine how a given strategy (i.e. there is no choice of strategies—
players keep playing the game) changes financial positions over time by undertaking different methods of resolution against bank failures. This method establishes
our general approach for analysing how the contagion effect is amplified throughthe course of play.
In order to perform this analysis, we need to identify what resolution signifies.
Prior research has focused on two means of resolution mechanism: market discipline and regulatory discipline as mechanisms of preventing the contagious effect.
Market discipline has been regarded as discipline imposed on bank managers andstockholders through monitoring debtholders’ behaviour. Here, debtholders canbe either uninsured depositors or subordinated debtholders, who have a stronger
incentive to monitor banking activities than ordinary debtholders.
Empirical research on market discipline, however, shows somewhat mixed
results. Park and Peristiani (1998) examine two effects: one being the price effectwhere uninsured depositors demand risk premium, and the other being the quantity effect where smaller amounts of uninsured deposits are collected. They conclude that both effects are present in the U.S. thrift industry. Avery, Belton andGoldberg (1988) examine market discipline from the perspective of subordinateddebtholders, who have a stronger incentive to monitor banks than uninsured
depositors, who can withdraw their funds and are likely to receive de facto insurance in some cases of resolution. Their empirical evidence, however, suggests nosuch discipline is observed. Recent studies by Billett, Garfinkel and O’Neal
1
This may cause a chain reaction—if the liabilities to one institution cannot be covered, thisinstitution may fail to pay its liabilities to another institution, and a sequence of domino effects
follows. In a broader sense, a network of debtor-creditor relationships transmits the effects of a
debtor’s defaults to legally unrelated agents. Such negative externality is called “systemic risk.”

Risk accumulation, contagion and the rules for bank failure

107(1998) contend that given the complementary nature of market discipline andregulatory discipline, banks faced with more severe market discipline (e.g. downgrading by Moody’s), will shift their source of funding toward insured deposits
while maintaining their riskiness; hence, market discipline will not have any
effect on banks’ behaviour regarding moral hazard.
In addition, the existence of market discipline is often undermined because
regulatory discretion is often granted. Gorton and Santomero (1990) contend thatthe presence of market discipline is non-linear and non-monotonic, and liabilities
should be priced according to option theory. Nonetheless, liability pricing is basedon the assumption that the regulator’s closure policy is exogenous, well-defined
and well-understood. In reality regulatory discretion is prevalent, and may takeseveral forms, such as a straightforward injection of capital, or purchase andassumption. In any case, pricing tends to be complex. This leads to Gordon andSantomero’s conclusion that risk premium cannot be priced accurately, because
the regulator’s closure policy is not implemented discretion-free.
These potential problems in measuring market discipline have led us to analyse the role of regulatory discipline. We examine the ex ante rules on how toresolve the problematic banks. We scrutinise two rules: exit enforcement andcapital injection. In the former, banks are asked to terminate operations once they
hit a threshold point of exit (e.g. insolvency). We study different scenarios such as
implementation of the early closure rule (i.e. banks are asked to leave the financial system once they fail to fulfil minimum capital standards), the capital enforcement rule (i.e. banks hold higher levels of capital to strengthen their financial positions), and the netting operation rule (i.e. changing the frequency of
netting that takes place between transactions in order to finalise winning andlosing scores). The third scenario demonstrates, as further discussed in footnote 1,
that systemic risk is often transmitted through the netting operation or the payment system. In order to prevent systemic risk from spreading, it may be better toincrease settlement frequency.
The next rule is capital injection whereby a third party injects a necessaryamount of capital in order to prevent bankruptcy. Here the third party plays therole of “lender of last resort.” We compare how costly each rule and scenario isfor the participants, and try to obtain a clearer view in order to assess the least
costly resolution to prevent bank failures.
Finally, we would like to stress that the ultimate objective of market participants, policy makers and the clearing house is to keep the markets open; there is
no profit source if the markets were closed and no more transactions were to takeplace. In this sense, our view is consistent with the alleged short-run objective of
market participants, i.e. profit maximisation.2 This statement implies that participation is the first step to profit maximisation. Consider players initially choosing
whether to participate or not, and then choosing strategies to maximise their
2
What is implicit in this statement is that limited liabilities are adopted and the participation
constraints are satisfied.

108

Yukinobu Kitamura and Shuji Kobayakawa

payoff. As long as they are secured by limited liabilities, they will initially choose
to participate in the transactions. In this paper, we do not concern the second stageof the game, i.e. the choice of strategies, but our objective is to compare andcontrast ourselves with the ways in which the financial market can continuously
operate without the severe contagion effect by altering the rules to identify thoseplayers who must leave.

2

Coin-tossing game reflecting banks’ investment activities

Let us describe how we analyse the contagion and study the menu of policy options
designed to prevent such contagion. We will treat bank investment activity as acoin-tossing game, the simplest random walk process, which will enable us to
examine a detailed mechanism of the contagion effect. We will then study where thedrawbacks lie in the classical random walk model as we interpret it as financialtransactions. We will review the main theorem of the random walk model. We willthen show a non-technical explanation of the arc-sine theorem and describe its
implication in our model. Lastly, we will explain a formal set-up of the game,especially the importance of the netting activities to settle the winning and losing
scores between banks. The simulation results are presented in sections 3 and 4.

2.1

Coin-tossing game revisited

Unlike most bank studies where collecting deposits and making loans are carefully modelled, we will focus on financial transactions that take place between
financial institutions, such as derivative transactions. A simple way to illustratesuch activities is to assume the outcome of the transactions is summarised in the
results of the random walk process, such as a coin-tossing game.3
Some may feel that a game so simple will not capture all essential features oftoday’s highly sophisticated financial transactions. Nonetheless, we believe that
one crucial feature of such financial transactions is their zero sum nature, namelythat the payoff of each participant will total zero, and this feature is easily incorporated in the coin-tossing game. To illustrate, think of a call option, where thebuyer of the option has the right to buy stock at a specified exercise price. Suppose further that the market price of the share is higher than the exercise price.
Then the buyer can exercise the right and earns profit equivalent to the difference
between the market price and the exercise price. On the other hand, the seller ofthe option suffers an equivalent loss. If we add up the buyer’s profit and theseller’s loss, we expect a total of zero, because the profit simply reflects the loss
borne by the counterpart. In this sense, we are not concerned with the nature of
the derivative transactions, but instead we wish to emphasise that buying and
3
The coin-tossing game has been developed extensively in probability theory. The origin ofprobability theory can be traced back to Pascal and Fermat who formulated the theory of chance inth
17 century France. Since then, the notion of chance has found its way into almost all branches ofknowledge.

Risk accumulation, contagion and the rules for bank failure

109

selling the derivatives will enable them to achieve higher efficiency in terms ofrisk management, whereby we expect social welfare to improve. The welfare
level is not zero sum after the derivative transactions. This is, however, not peculiar to financial derivatives. Any transaction which transfers risk from one agent
to another implies that the social welfare level will change as long as the transaction is carried out among agents with differing risk attitudes.4 For example, recall
Diamond and Dybvig (1983), where the bank is assumed to be risk neutral anddepositors are risk averse. The demand deposit contract, which insures depositors
against liquidity risk, will improve overall efficiency to a level that could nothave been achieved without the deposit contract. This implies that as long as thegame is measured in terms of Pareto efficiency, we cannot conclude that the gamepossesses the zero-sum feature any more. In the end, let us refer to Feller (1957),
who notes that “the (coin-tossing) model may serve as a first approximation to
many more complicated chance-dependent processes in physics, economics, andlearning theory.”5 We now review some stylised features of the coin-tossing gameand explain how we depart from the classical set-up of the game in order to bridge
the gap between the classical coin-tossing game and the nature of financial transactions.
Let us next explain how the gap is bridged by incorporating several factors
which involve further features of the financial practice. We focus on two established types of the random walk: the random walk with absorption barriers andthe random walk with reflection barriers. One example of the random walk withabsorption barriers is the Gambler’s Ruin problem. Let us call this problem the“classical ruin problem,” where two players play the game and their aggregate
endowments are fixed. As soon as one player’s position reaches zero (i.e. he loses
his entire endowment after playing the game), the game stops. The main point of
the classical ruin problem is to identify the probability of the gambler’s ultimate
ruin and to calculate the duration of the game.6 This set-up, nevertheless, has
several drawbacks when we try to interpret the problem as financial transactions.
4This issue is somewhat related to the St. Petersburg Paradox. According to the paradox, thegame runs as follows: suppose a coin is tossed repeatedly until the head appears. The gambler will
be paid 2n dollars if the head appears in the n-th toss. Then the question is, what price is the gambler
willing to pay for the privilege of gambling? The expected return from the gamble which thegambler wants to maximise is,2(1/2)+22 (1/2)2+…+2n (1/2)n=1+1+…+1=∞;

hence, the gambler should be willing to pay any large sum of money in order to enter this gamble.
Several suggestions have been made to challenge this paradox, including one by Daniel Bernoulli,
who proposes that one maximise the expected non-linear utility rather than the expected return.
Taking a finite number for the expected utility, one can derive an upper bound for the price that thegambler is willing to pay. In order to understand why the agents play a zero-sum game (expected
return is zero), it is more natural to think that the agents use the expected utility (not the expected
return) for the basis of their decision making.
5
Feller (1957) p.71.
6
It is worth mentioning well-established results in the classical gambler’s ruin problem. Under afair game with the same probability of win and loss (i.e. 1/2), the probability that the gambler’s
wealth reaches N (combined capital) starting from i is i/N. The expected duration of the game is
D = i(N-i). See Feller (1957), Maitra and Sudderth (1996) or Ross (1996) for more details.

110

Yukinobu Kitamura and Shuji Kobayakawa

On this front, we intend to present the “new ruin problem.” In the following
paragraphs, we will describe the main departure from the classical ruin problem.
First, the game in practice normally consists of more than two players. Thisimplies that even if one player is forced to quit, there remains a possibility thatthe game continues among the remaining players. Our game is thus extended byincorporating more than two players in the game. It should be pointed out, however, that the study of the ruin problem deals with the case where there are morethan two players. But the study does not address how long each player can survive, nor does it address how each player’s position is influenced if the gamecontinues among the remaining players after other players are eliminated.
Second, we believe that the ruin decision essentially inherits a mechanism
underlying the closure rule. In other words, whether a player is forced to retire or
not depends on such ex ante rules as (1) the player is ruined if he fails to pay theliabilities fixed through the netting operation, or (2) the player is ruined if his
capital ratio falls below a certain level (prompt corrective action). By incorporating different rules of closure, we can derive some policy implications of the ruindecision. This is another extension we will put forth in the following section.
A related issue in this ruin decision is how to decide the bank’s closure. The
key lies in the payment operation, through which the settlement amount is calculated, allowing for assessment of the bank’s net wealth. In extreme cases, theoperation can take place every time the coin is tossed. This is equivalent to thephilosophy of the “real time” settlement. In the following section, we will see
how the amount of capital necessary for each player changes by alternating thefrequency of the settlement interval.
Turning to the random walk with reflection barriers, one normally assumes
that when one player reaches either the upper or lower barrier, the other playercompensates him for the same amount and lets him stay in the game. Again, thisis crucial if the game consists of only two players; the game cannot be played any
more if one player does not save the other. This operation does not have any
practical meaning, once we allow more than two players in the game. Imagine agambling house;7 it does not make sense why an almost ruined player’s position
has to be covered by the winners. By leaving the almost ruined player without any
compensation, winners may not be able to collect all of their winning positions,but they can still keep playing the game had they not saved the ruined player.
Let us summarise our new ruin problem. First, we allow more than two players in the game and we let the settlement operation take place a number of times.
Second, we specify the closure rule in various ways; we can let banks go bankrupt
if they become insolvent, or we can let them fail if their net wealth position
recedes from a certain positive level (i.e. early closure rule). Third, we explicitly
introduce, as a second rule, rescue operations by a third party. Note this operation

7
In practice, most legally approved gambling houses sell tokens or tickets in exchange for cash
before admitting people to play the game with these tokens or tickets, which allows for guaranteed
payments for winners.

Risk accumulation, contagion and the rules for bank failure111

is specifically intended to show the effect of capital injection. We are aware thatsuch a policy may generate moral hazard, but our intention is to present thebottom line of the cost associated with the contagion effect, i.e. the cost can stillbe generated even if we assume banks do not engage in any risk-taking behaviour.

2.2

Arc-sine theorem

The purpose of this section is to direct the reader’s attention to a striking result
derived in the coin-tossing game. The theorem was established as part of the probability theory in the 1950s,8 but has been somewhat neglected mainly because of its
nature—it contradicts generally accepted views based on the law of large numbers.
More precisely, the random walk process has two seemingly contradictoryfeatures; a mean reversion (recurrent) process and the process summarised in thearc-sine theorem.9 To show these contradicting features of the policy, we derive acompletely different implication.
The mean reversion process implies that a player will return to his initial
position if the game is played ad infinitum. As a corollary, the wealth positions of
all players at any point in time form a normal distribution. This feature is the
keystone of modern finance theory.10 Recent discussion of the value-at-risk (VaR)
method of market risk management is also based on this feature.
The arc-sine theorem, on the other hand, implies that it is more likely forplayers to stay continuously on the winning side or the losing side than to frequently switch between the two sides.11 This neglected theorem has a remarkablepolicy implication for the recent financial market crisis. While the mean reversion
process tends to imply a wait-and-see policy or a no-intervention policy, namelythat no matter how badly banks perform, many of them will return to the mean;
hence, they can be left alone. However, the arc-sine theorem justifies an earlyclosure policy and is sceptical of the wait-and-see policy, namely that those who
perform poorly have most likely been performing poorly and unless an infinite
8
The arc-sine theorem was first discovered by Paul Levy in his study of Brownian motion, thendeveloped by Evolos and Kac as a limit theorem for sums of independent random variables withfinite second moments in line with Lindeberg conditions (see Loeve (1977) Vol. 1. p. 380). In ourdaily life, we often come across situations where an unfortunate event causes another unfortunate
event. It is such a rare event for one who has been suffering substantial losses in the casino to all ofa sudden recover the accumulated losses and start making gains. Similar observations can be seen in
the coin-tossing game.
9
Ross (1996, p.148) cunningly points out that the arc-sine theorem indicates the proportion oftime the symmetric random walk is positive does not converge to the constant value 1/2. On theother hand, it is clear by symmetry, the random walk process returns to an initial position (recurrent)
if the process continues infinitely. The reason why the above two seemingly contradicting
statements are in fact not a contradiction is that the expected cycle time is infinite. See Appendix 1
and 2 for more details. As Keynes wrote “in the long run we are all dead” (A Tract on Monetary
Reform, 1924), recurrence in infinity may not have a practical implication.
10
In the modern finance theory, most prices of financial assets are assumed to be non-zero, thus
to follow geometric Brownian motion (i.e. random walk with reflection barrier at zero). The mean
reversion process is reinforced by this assumption. See Dixit (1993) and Duffie (1996).
11
In terms of the fair coin-tossing game, extreme scores such as large number of heads (or large
number of tails) tend to appear quite frequently.

112

Yukinobu Kitamura and Shuji Kobayakawa

time interval is granted, it will probably take a substantially long time to recover;hence, given a finite time period (e.g. players’ performance has to be evaluatedevery accounting year), they should not be left alone but rather they can be expelled from the system. The most disturbing aspect of this theorem is that evenafter expelling poor performing players, this financial game will generate endogenously other poor performing players. In other words, this game will notconverge to the stable equilibrium where there are no more players who perform
poorly, but keep generating divergence between winners and losers. So the gameis inherently unstable and is constantly exposed to the contagion risk.
In the following, we will prove this rather counter intuitive theorem and see if
we can apply it to our Monte Carlo simulation. To give readers a more concrete feelfor the arc-sine theorem, we will borrow a numerical example from Feller (1957).
In this example, the game conducts many coin tosses at the rate of one per second,
day and night, for 365 days. The total number of tosses will be approximately 31.53
million. The probability that a gambler whose score is eventually negative, butduring the course of the game, has a positive score (i.e. more heads than tails at thatpoint) for a total time less than the listed time of lead, is summarised as follows.

Table 1: Time of lead in the coin-tossing game
Probability of
The Unfortunate
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.02
0.01

Time of Lead
153.95 days
126.10 days
99.65 days
75.23 days
53.45 days
34.85 days
19.89 days
8.93 days
2.24 days
13.5 hours
2.16 hours
32.4 minutes
Source: Feller (1957, p.83)

The first row of the table describes that 90% of the losers (i.e. unfortunate
gamblers with negative scores at the end of the game) have a positive scorethroughout the game for only 153.95 days. What is even more surprising is thatapproximately one in two losers enjoys a positive score for only 53.45 days of theyear. This example highlights that it is more likely for an unfortunate event to lastlong, even in a perfectly fair game such as coin-tossing. Let us now formallypresent the theorem.

113

Risk accumulation, contagion and the rules for bank failure

Arc-sine Theorem (Feller (1957, p.82))
The probability that in the time interval from 0 to 2n the position spends 2ktime units on the positive side (i.e. the position lies above the initial level) and
2n-2k time units on the negative side (i.e. the position lies below the initiallevel) equals R{2k ,2n} ≡ P ( 2k ,0) × P ( 2n − 2k ,0) , where n 
P ( n, r ) =  n + r  × 2 − n .


 2 
Hence,

 2k   2n − 2k  −2nR{2k ,2n} =   × 
× 2 .
 k   n−k For a detailed theoretical discussion and proof, see Appendix 3. The following
figure graphs the theorem. The x-axis denotes k/n, i.e. the probability that theposition stays positive, and the y-axis denotes R{2k,2n} times the number ofbanks, which is 100 in our simulation.

Figure 1: Graphical presentation of the arc-sine theorem
12

10

8

6

4

2
0.2

0.4

0.6

0.8

1

The figure explicitly shows the main feature of the arc-sine distribution
whereby the probability that the position is on the positive side half the time is theleast likely outcome, and the probability that the position stays either on the

114

Yukinobu Kitamura and Shuji Kobayakawa

positive or negative side most of the time is the most likely outcome.
The implications of this theorem are rather straightforward. Those who fall on
the negative side tend to stay on the same side and therefore have less chance torecover their losses, even if they continue to play the game for a substantiallylong period of time. Specifically, the only practical solution for those who havehuge losses is to leave the game as soon as possible, if feasible.
In the following section, we will see how this feature can be attained in ourMonte Carlo simulation, but first let us describe the basic set-up of the game.

2.3

Set-up of the simulation

There are 100 banks and all of them are engaged in investment activities. Throughthe simulation, we derive a minimum level of wealth required to generate theoutcome specified in the scenario. Each scenario will be described in greater
detail later in the section. There is also a clearing house whose role is to presideover the netting activities among the banks. We introduce three notations; round,
position and score. We define “round” as the duration of investment activities
between the netting operations. We assume that there are several rounds of play,and each time a round finishes, the netting takes place until the game completely
ends. Next we define “position” as the net-wealth of each bank. The position ofeach bank before the first round is equivalent to an initial level of wealth, andchanges as the game proceeds, according to each bank’s performance. In theworst case scenario, if a bank performs very poorly in one round, it may not havea sufficient position to cover the losses incurred during the round, in which casethe bank may go bankrupt or seek capital injection from the government. Lastly,
we define “score” as the outcome of investment activities during the round.
The game in our simulation proceeds as follows. Each bank tosses the coinand adds + 1 when it obtains heads and – 1 when it obtains tails. It independently
tosses the coin and records the number of heads and tails to compute the score.
For example, if the bank has 8 heads and 2 tails after 10 tosses, its score is 6.
Then at the end of the first round of play, when scores must be settled, the clearing house will undertake the following procedure.12 First the banks are divided
into two categories by their scores. If the score is positive, they are regarded as“winners,” whereas if the score is negative, they are regarded as “losers.” Amongthe losers, if a bank’s position is sufficient to cover the score, it will give theclearing house an equivalent amount to the score. On the contrary, if its position
is not sufficient to cover the score, a problem arises. In the following, we examine
different scenarios by incorporating how a bank with insufficient funds to coverlosses is treated. In the first scenario, we allow banks to go bankrupt as soon as
12
The following interpretation of the settlement operation may potentially underestimate thecontagion effect. In our simulation, the net debit position of the players is pooled and surviving
players will share the loss if any (loss sharing rule). Hence, each player’s exposure to a potential
loss is limited. If on the other hand each player is fully exposed to the potential loss caused by abilateral transaction, the impact of the counterpart’s default is much more serious, and we may
observe a severer degree of contagion effect.

Risk accumulation, contagion and the rules for bank failure

115

they become insolvent. In the second scenario, we examine the early closure rule,
i.e. banks become bankrupt once they fail to fulfil a required standard, such as asolvency ratio. In the third scenario, we change the frequency of netting operation(i.e. the duration of each round) and see if there is any effect on the bank’s initial
wealth which prevents bankruptcy.

3

Simulation result: Exit enforcement

In our GAUSS programme, the coin toss is based on the uniformly distributed
random variable between zero and one. If the variable is greater than (or equal to)
0.5, we assume the coin comes up heads, and add 1 to the bank’s score. On theother hand, if the variable is less than 0.5, we assume that the coin comes up tails,
and subtract 1 from the score.13 Each bank is assigned a specific number as a“seed.” Each seed contains a stochastic process, known as a Markov chain, i.e. theconditional distribution of any future state xn+1, given the past states x0, x1, ..., xn-1
and the present state xn, is independent of the past states and depends only on thepresent state. Having the same seed implies a specific Markov chain is assigned toa specific bank.

3.1

Basic result

In this simulation, we assume that banks which failed to cover their losses in around of play will go bankrupt after paying off their position to the clearing house.
This means that the clearing house is granted authority to seize the bank’s assets(position) if the bank fails to cover its losses from the round. After the money iscollected from the losers, the clearing house will distribute it to the winners. If
there were no banks which failed to pay their losses in full, winners will receive
their winnings in full. If the clearing house fails to collect all the money as aresult of the bankruptcy of losers, winners will not fully receive their winnings,
but will instead be paid their winnings in proportion to others’ winnings.
After the money is distributed among the players, the losers who failed to pay
their liabilities in full will go bankrupt, and they will not be engaged in the investment activities starting with the next round. Other players, i.e. winners andlosers who covered their losses but still managed to sustain a positive position,
will proceed to the next round, where the same activities will continue: tossing
the coin a certain number of times, counting the score, and executing of theclearing house netting operations. According to these specifications, we obtain thefollowing results:

13
In this set-up, the scores of winners and losers uniformly move one unit (±1), as we ignore thecase where a player bets more than one unit in a game. However, the optimal strategy of attaining acertain level of wealth or of maximising expected playing time is the timid strategy that always bets
1 unit when the probability of win or loss is 1/2. See Ross (1983, p.76-83).

116

Yukinobu Kitamura and Shuji Kobayakawa

Table 2: Required level of wealth which leads to 10 banks defaulting at
various stages of coin-tossing
n=1
10
20
50
100
150
200

sim=1
0
6
7
13
15
20
22

10
5
13
22
31
49
64
68

20
7
22
28
50
73
98
111

50
11
27
45
71
112
140
168

100
11
43
60
111
151
204
252

150
16
63
79
143
212
273
298

200
18
70
103
154
242
350
383

Table 2 shows the minimum amount of wealth each bank needs to hold if thesystem allows 10 banks to default through the course of play.14 The row (n) in thetable represents the number of coin tosses taking place during a single round. The
column (sim) represents the number of rounds. Hence, each bank will toss thecoin a number of times equal to the product of row and column (n×sim). At theend of each round, the operation which assesses the winning and losing scores forevery bank will take place. This is equivalent to the net settlement. For example,
the table says that each bank will need to hold 383 units of wealth if the game isto last 40,000 tosses during which the netting operation will be carried out every
time 200 coin tosses have taken place. It is quite obvious as the number of cointosses increases the amount of initial wealth necessary to create the same numberof defaults will increase. For the cases to be discussed later, we shadowed 151
units of wealth as a benchmark.15
In Figure 2, we show a 3-dimensional presentation of the wealth per coin toss.
In other words, it plots the amount of wealth necessary for each coin toss. For
example, 383 units for 40,000 coin tosses in Table 2 is plotted as approximately
0.01. This figure signifies that the wealth per coin toss will generally decrease andstay around 0.01, as the number of tosses increases.

14
Admittedly the simulation is not sophisticated enough to allow banks to start with different
levels of wealth. Our simulation thus far treats all 100 banks homogeneously at the beginning. It is
an important extension, because the initial wealth represents the strength of the financial position(i.e. the larger amount of wealth, the less likely that the bank goes insolvent), and the financial
system is better represented by players with differing wealth positions. Nevertheless, our simulation
allows wealth positions of banks to vary from the second round onwards.
15
After 10,000 coin tosses, the marginal increase in required levels of wealth becomes
substantially small. We believe that the marginal gain of increasing the number of coin tosses
becomes relatively small and that small sample bias may disappear after 10,000 times. See Figure 2.

Risk accumulation, contagion and the rules for bank failure

117

Figure 2: Level of initial wealth per coin toss
1.5
1.0
Wealth

Trial

sim=1

20

100

200

n=1
100
Round

0.5
0.0

(Contagion effect)

We now proceed to show how the existence of the contagion effect affects thefinancial state of the other players in the game. The best way to see the effect is tomeasure wealth distribution at the end of the game. To be more precise, we firsttake the case where the contagion effect exists, namely that banks which failed to
pay their liabilities in full go bankrupt and other banks without receiving their full
winnings keep playing the game. We choose as a benchmark the situation wherebanks hold 151 units of initial endowments and play 100 rounds of the game,where each round consists of 100 coin tosses; hence the coin is tossed 10,000
times. This figure is taken from Table 2 which results in 10 bankruptcies. Second,
we assume that banks suffering financial difficulties are rescued by having their
liability positions covered by third parties.16 They are rescued by means of capital
injection, and the winners in each round of the game receive their full winnings.
After they pay the liabilities in full, they can keep playing the game. This coincides with the case where there is no contagion effect and the risk derived fromthe inability of losers to pay liabilities in full will not take place.
The next table and figure show the summary statistics for the distribution of
wealth with and without the contagion effect after 10,000 coin tosses.

16This is equivalent to the capital injection case studied in the next section. For a detailed
analysis, see section 4.

118

Yukinobu Kitamura and Shuji Kobayakawa

Table 3: Summary statistics of wealth distribution for the surviving
players
Contagion
No Contagion
Mean
167.8
156.3
Standard Deviation
90.5
96.2
Minimum Value
5.0
0.0
Maximum Value
422.4
424.9
# of Players
90
100
Table 4: Summary statistics of wealth distribution for the winningplayers
Contagion
No Contagion
Mean
224.2
226.4
Standard Deviation
64.2
64.1
Minimum Value
151.1
152.9
Maximum Value
422.4
424.9
# of Players
55
55

In the table, “Contagion” refers to the situation where the players toss the coin
10,000 times (n=100 and sim=100) with an initial wealth of 151. As seen from
Table 2, this is equivalent to the case where 10 banks go bankrupt; hence, we
expect to observe the contagion. On the other hand, “No Contagion” refers to thesituation where the players again toss the coin 10,000 times (n=100 and sim=100)
with an initial wealth of 151. This time, however, the players who fail to pay theirlosses have their debt positions covered by a third party; hence, we do not expect
to observe the contagion effect.
We have summarised two tables: the first table shows the statistics of wealthdistribution for all surviving players, and the second table shows the statistics ofwealth distribution for the winners. From Table 3, it is not apparent whether or
not “Contagion” really has a contagion effect, because the mean distribution ismuch higher for “Contagion.” This is, nonetheless, due to the fact that in cases of“No Contagion,” banks which played very badly receive injected capital, but theytend to perform at the very bottom of the wealth distribution.17 This pulls down
the mean wealth.
Table 4 shows the very weak contagion effect.18 By focusing on the wealthdistribution of the winning players, we see that the “Contagion” mean is margi17
Here we assume that the third party covers only debt positions; hence, the wealth of theinjected banks right after the injection is zero. We will further develop the analysis where theinjection policy is more accommodating, allowing for situations such as injections raising a bank’s
wealth position to 8% of the initial level.
18This is basically because the contagion effect is pooled among all winners, so that a marginal
impact per player is limited.

119

Risk accumulation, contagion and the rules for bank failure

nally lower than the “No Contagion” mean, with the same number of winningbanks. Both the minimum and maximum wealth positions in “No Contagion” aremarginally higher than those in “Contagion.” Although its effect seems rather
weak, it seems to be consistent with our proposition that contagion weakens thefinancial positions of sound players.
(Arc-sine feature)
Next, we present a result that exemplifies the arc-sine nature. We have seen abenchmark case where 100 banks hold 151 units of initial wealth and play 100
rounds of the game with each round consisting of 100 coin tosses. Figure 3 shows
the histogram of the first round, where the x-axis represents the probability thatthe accumulated score of coin tosses lies above zero, and the y-axis represents thenumber of banks. For example, if the accumulated record of 10 coin tosses follows a path, {1, 2, 1, 0, –1, –2, –3, –2, –1, 0}, then the record is 4, because theintervals (0 →1), (1 →2), (2 →1) and (1→0 ) remain above zero, i.e. on the positive side. The line in Figure 3 represents the number of banks theoretically derived from the arc-sine theorem. This is a pure random walk process. Here thehigher the probability is, the higher the number of times the score remains afloat.

Figure 3: Graphical presentation of arc-sine after the first round in thebenchmark case
20

15

10

5

0
0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0.44 0.48 0.52 0.56 0.6 0.64 0.68 0.72 0.76 0.8 0.84 0.88 0.92 0.96

1

The figure inherits the nature of the arc-sine distribution, where probabilities
staying on either the positive side or negative side are more likely to occur thanother probabilities.
The next figure presents the wealth distribution at the end of 100 coin tosses.

120

Yukinobu Kitamura and Shuji Kobayakawa

All the banks start with 151 units of initial wealth.

Figure 4: Wealth distribution after 100 coin tosses
20

16

12

8

4

0
135

138

141

144

147

150

153

156

159

162

165

168

171

174

177

By construction of the game, we find the contagion effect is rather weak,although the results still support our proposition that the contagion will weaken
the financial positions of the sound players. On the other hand, the arc-sine theorem shows that those banks that are destined to perform poorly continue to perform poorly. From this result, it is quite costly to leave the poor performingbanks; a wait-and-see policy will allow these banks to perform even worse andthe contagion effect will become more prevalent. Thus it is better not to wait until
they recover in the unforeseeable future.
What we are more interested in is to see whether we still observe the differences as suggested above by depicting the observations at the end of each round.
Of course, the path which consists solely of the end-of-round observations does
not represent the random walk in a genuine sense—a sequence of the outcome
after the aggregation of each round’s 100 coin tosses, followed by the netting
operation in order to make the game zero sum, does not possess a random walkfeature, i.e. +1 with probability 0.5 and –1 with probability 0.5. Figure 5 is ahistogram that records the number of times that the banks’ position stays above
the initial wealth. There seems to be less observations under extreme probabilities,
but the figure still inherits the arc-sine feature—a tendency which suggests there
are more banks which either play poorly (i.e. the probability is close to 0) or
nicely (i.e. the probability is close to 1).

121

Risk accumulation, contagion and the rules for bank failure

Figure 5: Graphical presentation of arc-sine at the end of each round in
the benchmark case
20

15

10

5

0
0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36 0.4 0.44 0.48 0.52 0.56 0.6 0.64 0.68 0.72 0.76 0.8 0.84 0.88 0.92 0.96

1

The results in Figures 3 and 5 suggest a general trend tracing an arc-sine. In
one sense, the result in Figure 5 is quite striking; the path we follow in the figure
is not a purely random walk—the change in wealth position from round to roundis not uniform and the settlement process at the end of each round disturbs thesmoothness of the arc-sine.
What is more striking is that once we divide banks into two groups: thosebanks that are ultimately winners (i.e. banks whose position ends above the initial
wealth) and those that are ultimately losers (i.e. banks whose position ends below
the initial wealth), we find the winners’ history of end-of-round wealth tends tobe generated by good performance whereas the losers’ history of wealth is generated by poor performance. This observation is graphed in Figure 6.

Figure 6: Histogram of winners and losers at the end of each round
Losers
10
8
6
4
2
0
0.04

0.08 0.12

0.16

0.2

0.24 0.28

0.32

0.36

0.4

0.44

0.48

0.52

0.56

0.6

0.64

0.68

0.72

0.76

0.8

0.84

0.88

0.92 0.96

1

122

Yukinobu Kitamura and Shuji Kobayakawa

Winners
10
8
6
4
2
0
0.04

0.08 0.12

0.16

0.2

0.24 0.28

0.32

0.36

0.4

0.44

0.48

0.52

0.56

0.6

0.64

0.68 0.72

0.76

0.8

0.84 0.88

0.92 0.96

1

We see that the lower end of the arc-sine graph generally comes from losing
banks. On the contrary, the second figure is the histogram based on the number ofwinners. Here we understand that the upper end of the arc-sine graph consists ofthese winners. In sum, Figure 6 summarises our conjecture that the upper end of
the arc-sine histogram comes from the winners and the lower end of the histogram comes from the losers.
One implication we derive from this analysis is that those banks that aredestined to perform poorly (i.e. to stay on the negative side) will continue toperform poorly. From this point of view, it will be quite costly to leave the poor
performing banks; a wait-and-see policy will allow these banks to perform evenworse and the contagion effect will become more prevalent. Thus it is better not
to wait until they recover in the unforeseeable future. Owing to this observation,
the next section concerns several policy options and evaluates which policy seems
appropriate.
We examine three scenarios. First we study what if banks are asked to close
operation before they become insolvent. We study the case where the thresholdlevel of closing its operation is 8% of the initial wealth. We compare the performance of players and see how effective this early closure policy is to isolate thecontagion. Second we examine what if they start with higher levels of initial
wealth by raising additional capital. We identify four cases that generate nobankruptcy, 1 bankruptcy, 5 bankruptcies and 10 bankruptcies, and the minimum
levels of initial wealth to generate an assigned number of bankruptcies are computed. Third we examine what if the clearing house changes the frequency of the
netting operation. In an extreme case, the operation takes place every time banks
toss the coin. We calculate how many banks will go bankrupt if we change thefrequency of the operation while maintaining both the total number of coin tosses
and the initial wealth constant.
Lastly Figure 7 presents the wealth distribution at the end of the game (i.e.
100 rounds).

Risk accumulation, contagion and the rules for bank failure

123

Figure 7: Wealth distribution at the end of the game
16

12

8

4

0
25

3.2

50

75

100

125

150

175

200

225

250

275

300

325

350

375

400

425

Policy: Early closure rule

In the last section, we showed a possibility that the contagion effect results in aweakening of the winners’ positions. In this section, we extend the analysis to
study the scenario where the banks are forced into bankruptcy even if their net
wealth position remains positive.
The last section assumed that banks fail if they cannot cover their debt positions with their wealth. In other words, if the summation of the wealth and thescore of the game in each round is below zero, the banks are asked to leave thegame. Here we assume that they are forced to shut down their operation if thesummation of the wealth and the score fails to reach a certain level above zero,say 8% of the initial wealth position. In essence, this is in line with the earlyclosure rule, often referred to as “prompt corrective action.”
Prompt corrective action provisions are proposed as a possible approach toovercome time-inconsistency and eliminate any forbearance in financial regulation. For example the U.S. Federal Deposit Insurance Corporation Improvement
Act (FDICIA 1991) mandates that any bank whose tier 1 capital ratio falls below
2% is subject to mandatory provisions such as the appointment of a receiver/conservator within 90 days, suspension of payments on subordinated debt,
and restrictions on certain activities.
In the context of our game, the early closure maintains the following feature—due to the game’s losers being asked to leave before their net-wealth position reaches zero (strictly speaking, they are not yet insolvent), the winners areable to collect larger sums; hence, the severity of the contagion effect is limited.
Although we may still observe some degree of the contagion effect unless full

124

Yukinobu Kitamura and Shuji Kobayakawa

losses can be covered, we expect the effect to lie somewhere between the “Contagion” and “No Contagion” cases in the last section. The statistical results of anumerical example which enables us to run a comparative study with the benchmark cases is shown below.

Table 5: Summary statistics of the wealth distribution for the surviving
players
Reference
Early Closure
Contagion
No Contagion
Mean
177.6
167.8
156.3
Standard Deviation
85.2
90.5
96.2
Minimum Value
34.8
5.0
0.0
Maximum Value
424.1
422.4
424.9
# of Players
85
90
100
Table 6: Summary statistics of the wealth distribution for the winningplayers
Reference
Early Closure
Contagion
No Contagion
Mean
225.3
224.2
226.4
Standard Deviation
64.2
64.2
64.1
Minimum Value
152.5
151.1
152.9
Maximum Value
424.1
422.4
424.9
# of Players
55
55
55

From Table 5, we see that “Early Closure” results in more bank closures (15banks), but the average wealth position of the surviving banks further improves.
Table 6 reveals, as expected, that the average wealth position of the winners in“Early Closure” is higher than the position of the winners in “Contagion.” This
seems to suggest that the early closure policy is effective in the sense that thecontagion effect is restricted; hence, the policy would be of benefit to the financial system as a whole. The argument states that because losing banks tend to doworse, it is best to adopt a non-discretionary policy and let these banks failthrough adopting harsher standards, such as the early closure rule.

3.3

Policy: Capital enforcement rule

One way to avoid the risk from contagion is to implement a tighter capital en-Risk accumulation, contagion and the rules for bank failure

125

forcement rule.19 We know from Table 1 the minimum level of initial endowments
which would allow 10% of the total banks to go bankrupt. If the regulator asks
each bank to hold more endowments before the start of the game, how much
would each bank have to increase its level? This study is conducted and summarised in the following figure. The figure represents the initial level of wealthnecessary for no default (Case 1, i.e. none of 100 banks goes bankrupt), one
bank’s default (Case 2), five banks’ default (Case 3) and ten banks’ default (Case4) which is the same result obtained in the previous sub-section.

Figure 8: Minimum level of initial wealth needed to generate a specified
number of defaults
Case 2: Default = 1

Case 1: Default = 0

Case 3: Default = 5

Trial

200

500
400
300
Wealth
200
100
0

Trial

Case 4: Default = 10

500
400
300
Wealth
200
100
0

n=1

200

Round

100

200
20

200

100

n=1

20

Round

sim=1

150

20

n=1

100

Round

Trial

sim=1

200

sim=1

200

20

100

n=1
Round

sim=1

150

500
400
300
Wealth
200
100
0

500
400
300
Wealth
200
100
0

Trial

The general trend is as banks hold more wealth, they are less likely to face athreat of default. In order to achieve a “risk free system,” banks have to hold asubstantial amount of wealth, perhaps by means of raising additional capital, as
shown in the figure as Case 1: Default=0. None of the banks having sufficient
19
Note in this paper what we mean by “capital enforcement” is to raise additional capital at the
beginning of the game. Hence the degree of capital enforcement is not related to the performance ofbanks. As we will examine in the next section, we distinguish it from “capital injection,” which isanother resolution mechanism whereby additional capital is injected if banks perform poorly.

126

Yukinobu Kitamura and Shuji Kobayakawa

wealth will go bankrupt, and the rescue operation as specified in the last section
will not need to be implemented.
An interesting observation is to see how the capital enforcement rule and theearly closure policy will interact with each other in our model. Their fundamental
philosophy concerning how to resolve the systemic risk is, however, different.
In the capital enforcement rule, the policy guide is to raise additional capital,
so that the players have sufficient wealth to avoid bankruptcy beforehand. Thus
the more capital the banks raise, the less banks face the threat of going bankrupt.
The policy aims to prevent market risk by strengthening the financial position of
the players.
In the closure policy, the policy guide is how to pinpoint the poor performing
players in the early stage and to get rid of them from the financial system before
systemic risk becomes more prevalent. Thus the tighter the policy is, the greater
the number of banks that will face bankruptcy. Here the policy does not aim tostrengthen the financial position, but rather it takes the initial position as given
and implements the mechanism according to which contagion does not spill overto other players.
In this sense, we claim that the former is ultimately contagion-free, whereas
the latter more realistically lets the contagion happen but concerns how to minimise its effect.
In the following, we show a table where each cell identifies the improvementof wealth for survivors at the end of the game. In the table, the row represents theclosure policy and the column represents the capital enforcement.

Exit by Insolvency
Early Closure

151
+16.8
+26.6

186
+9.8
+18.4

245
+2.5
+7.6

278
0
+2.8

Each level of wealth in the first row represents respectively 10 bankruptcies, 5bankruptcies, 1 bankruptcy and 0 bankruptcy, under the benchmark case with theexit rule specified in the first column. Obviously, by adopting the early closure
rule, the number of bankruptcies in each level would rise. The general trend is,the larger the level of wealth banks hold, the less magnitude of improvement theyexperience. Also, the early closure rule, because it excludes a larger number ofailing banks, will improve the financial position of the survivors even further.

3.4

Policy: Frequency of netting operation

The next possibility we will explore is how the netting operation will affect the
number of defaulting banks. Let us first show how the number of defaults isaltered by changing the frequency of netting while maintaining not only the total
number of coin tosses but also the initial amount of wealth in 151 units.
Our conjecture is that the number of bankruptcies will rise as netting takes
place more frequently (i.e. larger values for n indicate that netting of the scores

Risk accumulation, contagion and the rules for bank failure

127

will take place for a lower number of coin tosses; for example, sim=5 andn=2,000 means that netting is done every 5 tosses). This follows basic intuition—
the more frequently the banks net their scores, the more often they face a situation
where they must pay their liabilities if they end up with negative scores after around, the higher level of wealth they must hold ex ante in order to avoid bankruptcy.20

Table 7: Number of defaults for 151 units of wealth
sim

n

5
10
20
50
100
200
500
1,000
2,000

2,000
1,000
500
200
100
50
20
10
5

Number of
Defaults
13
15
14
15
10
11
10
10
8

However, Table 7 shows results which are somewhat mixed. In the figure, we
expected the number of defaults to fall as sim rose. The difference in the numberof defaults between 13 in sim=5 and n=2,000 and 15 in sim=10 and n=1,000 may
be marginal, but we are not inclined to make a concrete statement on how tointerpret this result. The table does not seem to support our conjecture that the
number of defaults will rise as netting takes place more frequently. In the nexttable, we show another result.

20
In an extreme case, netting can take place every time the banks toss the coin (sim=1 and
n=10,000 represent the case). This “real time” netting could not be addressed in this paper, because
it exceeds our computer capacity.

128

Yukinobu Kitamura and Shuji Kobayakawa

Table 8: Minimum amount of wealth for various combinations of sim and n
sim
5
10
20
50
100
200
500
1,000
2,000
5,000

n
2,000
1,000
500
200
100
50
20
10
5
2

0
290
288
285
283
278
269
269
269
269
269

Number of Defaults
1
5
250
194
279
190
267
191
271
188
245
186
245
186
245
178
245
179
245
175
245
165

10
164
169
156
168
151
154
151
147
139
130

The table presents the minimum amount of wealth that causes four different
numbers of default under specified numbers for rounds and coin tosses. Again,
our benchmark (sim=100 and n=100 with 151 units of wealth) is shadowed. Inorder to reduce the number of bankruptcies, it is best to increase wealth to a level
where there are no more defaults. This is the same policy implication we derived
in the last section. We know, however, from Table 7, that there may be anotherroute, namely that if additional capital accumulation is too costly (for example,
each bank is asked to add 127 units of capital ex ante to reach a wealth position of
278 if they want to avoid any bankruptcy in the sim=100 and n=100 situation),they could prolong the interval of netting.
Of course, our numerical result in the table did not produce a case where there
were no bankruptcies with an initial wealth equivalent to 151. The best they could
achieve is to allow 8 bankruptcies by extending the interval of netting from one
netting out of sim=100 to one netting out of sim=2,000 (see Figure 7). Nonetheless, our analysis in this section seems to suggest that the interaction between thecapital enforcement rule and the frequency of settlement has to be considered. In
other words, it is easy to see a paradigm where the capital is strongly enforced
and the netting operation takes place in “real time” (for example, 290 units ofwealth in sim=5 and n=2,000 results in no bankruptcy). Yet if we face the situation where capital is limited or is simply too costly for banks to hold the required
amount of wealth by raising additional capital derived in the simulation, then amore feasible solution is to find a compromise which requires less wealth, but
provides a lower number of defaults. Our simulation seems to have this feature; itsuggests that it is not just increasing the amount of initial level of wealth, but
once combined with the frequency of settlement procedure, it may achieve better
results.
We also carried out the simulation with different seed numbers for banks to
see if there were any differences in our results. The next table is based on theRisk accumulation, contagion and the rules for bank failure

129

same GAUSS programme, but it features different seed numbers (we added 5,000
to each seed). The overall result of the simulation does not seem to show significant change.

Table 9: Minimum amount of wealth for various combinations of sim and n
sim
5
10
20
50
100
200
500
1,000
2,000
5,000

4

n
2,000
1,000
500
200
100
50
20
10
5
2

0
285
284
280
291
281
270
282
270
270
270

Number of Defaults
1
5
256
171
248
179
244
185
244
185
244
190
245
190
244
186
244
153
244
162
244
153

10
151
153
149
152
168
161
140
140
136
130

Simulation: Capital injection

In this section, we will add another dimension to the fundamental set-up of thegame. In the previous section, it was simply assumed that a bank is driven out of
the system once the loss in a round of play reaches a level that cannot be coveredby the available funds (position). This is consistent with the classical Gambler’s
Ruin problem. Then we extended the analysis by adopting the early closure rule:
if the summation of the bank’s wealth and the resulting score of the game fail toreach the minimum standard the regulator has set in advance, then the bank isclosed. We saw how this mitigates the contagion effect because the winners
manage to cover larger shares of their winning positions. Nonetheless, what wehave recently seen is not just how to get rid of the broke players, but also how tocontain them in the system without causing moral hazard. Of course, our framework is too primitive to incorporate any incentive to assume an additional risk
factor in investment activities, but our point is, even without generating moral
hazard, the system is exposed to further risk by an additional injection policy of
the regulator.
Here we ask the following question, what if a bank, faced with the situation
where the loss exceeds the position, has been injected with capital rather thanbeen forced to go bankrupt? We will see that, although the new capital is injected,
the bank will repeatedly face the threat of going bankrupt, where in each case theregulator needs to add new capital. We believe this is due to the fact that as longas the bank maintains the same portfolio (the same Markov chain in our pro-

130

Yukinobu Kitamura and Shuji Kobayakawa

gramme), it is more likely that the bank will continue to produce poor scores.
Perhaps, a more reasonable solution is to close these banks before injecting
additional capital.

4.1

Capital injection at insolvency

In the following tables, we summarise the total amount of capital injection thatgenerates a specific number of bankruptcies (10 bankruptcies in Table 10 and 5
bankruptcies in Table 11). Here the amount of injection is defined as a net-debit
position; hence, in each case the bank’s position will become zero after both thecapital injection and the settlement of the scores with other players take place.

Table 10: Total amount of capital injection with an initial wealth position
generating 10 bankruptcies (as specified in Table 2)
n=1
10
20
50
100
150
200

sim=1
n.a.
9.5
21.8
12.7
22.5
35.8
60.4

10
14.4
38.6
59.4
84.5
85.4
128.2
275.1

20
16.3
57.7
92.8
97.0
224.4
144.2
153.2

50
18.9
112.3
64.9
294.8
371.6
514.5
423.7

100
33.4
104.3
330.5
340.2
533.6
453.1
714.8

150
51.0
112.3
276.4
455.5
326.6
970.6
922.5

200
75.7
221.0
177.5
461.7
796.0
664.1
495.9

We need to compare the results in Table 10 with those in Table 8 where theminimum amount of wealth to generate different numbers of default is shown. In
the benchmark case, we know from Table 8 that if the system has to generate thenon-bankrupt case, each bank has to raise 127 (i.e. 278–151) additional units ofcapital; hence, the total cost for the system to reach a risk-free environment would
be 12,700. In Table 10, however, we derive that the total cost represented by theamount of capital injection is 533.6. Although we should be careful in comparing
the cost derived in these two figures,21 our result seemingly suggests that any exante measure against bankruptcy is more costly than an ex post measure, because
the latter pinpoints the exact amount of funds necessary to avoid bankruptcy,
while the former requires a large risk premium.
The next table also conducts the same simulation which results in 5 bankruptcies. Because banks initially hold higher levels of wealth, the total cost of injection is much less, compared with the amounts specified in Table 10.

21
The comparison requires care because the simulation has not incorporated heterogeneity ofbanks. If each bank is treated differently (e.g. riskier banks are required to raise additional capital toachieve higher levels of wealth while sound ones are not), the total cost is expected to decline.

Risk accumulation, contagion and the rules for bank failure

131

Table 11: Total amount of capital injection with an initial wealth position
generating 5 bankruptcies
n=1
10
20
50
100
150
200

4.2

sim=1
n.a.
4.0
8.7
6.4
4.4
29.9
25.6

10
3.6
5.0
33.2
61.7
31.6
37.7
117.4

20
5.5
32.2
50.6
33.8
108.0
111.2
69.2

50
7.9
74.1
29.3
101.0
241.3
168.4
269.1

100
19.4
43.9
87.0
281.9
222.0
191.6
417.9

150
34.5
29.6
77.3
219.9
140.1
270.2
212.8

200
41.6
93.9
70.7
209.1
421.9
306.1
262.9

Capital injection at 8% level

We have also carried out another simulation. In section 4.1, the amount of capitalnecessary for injection is designed to cover a net-debit position. Here we adopt a
more generous rule, namely that the amount of capital injection not only covers
the net-debit position but also raises the next round’s position equivalent to 8% of
the initial wealth. Under this rule, banks which accepted the capital injection will
have some buffer to start with from the next round of play.

Table 12: Total amount of capital injection with an initial wealth position
generating 10 bankruptcies
n=1
10
20
50
100
150
200

sim=1
n.a.
13.4
27.1
15.9
34.7
44.1
72.2

10
17.2
48.6
75.0
100.4
118.1
174.3
296.5

20
19.6
71.7
108.9
130.7
276.0
202.6
201.4

50
24.2
134.1
94.4
343.8
434.7
597.5
511.5

100
45.7
137.4
378.5
394.9
608.2
572.8
852.3

150
63.8
155.4
334.6
548.2
459.7
1,068.5
1,115.4

200
88.6
275.6
247.5
568.8
904.9
876.5
760.0

Table 12 summarises the total amount of capital injection that generates 10bankruptcies. Because each time capital is injected, it raises the bank’s position to
8% of the initial wealth, the total amount is much higher than the one in Table
10.22

22
Admittedly, the analysis in this section is still incomplete. Several further studies remain to be
done. We would like to know what the optimal level of capital injection is and how efficient is thecapital injection.

132

5

Yukinobu Kitamura and Shuji Kobayakawa

Summary of simulation results

This section sums up broad simulation results. By construction of the game, wefind the contagion effect is rather weak although our results still support our
proposition that the contagion will weaken the financial positions of the soundplayers. On the other hand, the arc-sine theorem shows that those banks which aredestined to perform poorly continue to perform poorly. From this result, it is quite
costly to leave the poor performing banks; a wait-and-see policy will allow thesebanks to perform even worse and the contagion effect will become more prevalent.
Thus it is better not to wait until they recover in the unforeseeable future.

5.1

Ex ante measures

Three types of ex ante policy measures are considered. Although we cannot
provide an overall evaluation of which policy rule is the optimal, we show ample
evidences that combined use of these rules would enforce the policy implementation more effectively.
(a) Early closure rule

We study what if banks are asked to close operation before they become insolvent.
We study the case where the threshold level of closing its operation is 8% of theinitial wealth. We compare the performance of players and see how effective thisearly closure policy is to isolate the contagion. From the benchmark case, we findthe early closure results in more bank closures, but the average wealth position of
the surviving banks improves further. This seems to suggest that the early closure
policy is effective in the sense that the contagion is restricted. Comparing theearly closure rule with the exit by insolvency rule, the former, because it excludes
more ailing banks, would improve the financial position of the survivors evenfurther than the latter. We therefore conclude that the policy would be beneficial
to the financial system as a whole.
(b) Capital enforcement rule

We examine what if banks start with higher levels of initial wealth by raising
additional capital. We identify four cases that generate no bankruptcy, 1 bankruptcy, 5 bankruptcies and 10 bankruptcies, and the minimum levels of initial
wealth to generate an assigned number of bankruptcies are computed. In short, thegeneral trend is that the larger the amount of wealth banks hold, the less likely
they face the threat of going bankrupt. In order to achieve the “risk free system,”they have to hold a substantial amount of wealth, perhaps by means of raising
additional capital. In this system, none of the banks having sufficient capitalwould go bankrupt, and the rescue operation as specified in the last section would
not have to be implemented. Needless to say, this rule imposes a high cost on theparticipants.

Risk accumulation, contagion and the rules for bank failure

133(c) Frequency of netting operation

We examine what if the clearing house changes the frequency of the netting
operation. In an extreme case, the operation takes place every time banks toss thecoin. We calculate how many banks will go bankrupt if we change the frequencyof the operation while maintaining both the total number of coin tosses and theinitial wealth constant. In general, the more frequently the banks net out theirscores, the more often they face the situation where they are asked to pay their
liabilities if they end up with negative scores in the round of the game, and thehigher the level of wealth they need to hold ex ante in order to avoid bankruptcy.
Our simulation did not provide strong support to this conjecture, but the combination of this measure with others seems to be a promising as well as practicalsolution to the contagion effect. If we face the situation where the capital islimited or simply it is just too costly for banks to hold the required amount ofwealth by raising additional capital derived in the capital enforcement rule, then amore feasible solution is to find a compromise which requires less amount ofwealth with more frequent settlement.

5.2

Ex post measures

Although the new capital is injected, the bank will face the threat of going bankrupt repeatedly, where in each case the regulator needs to add new capital. This isdue to the fact that as long as the bank maintains the same portfolio (the same
Markov chain in our programme), it is more likely that the bank will continue to
produce poor scores.
Our result also contends that any ex ante measure against bankruptcy is morecostly than any ex post measure, because the latter pinpoints the exact amount offunds necessary to avoid bankruptcy, while the former requires a large risk premium.

6

Conclusion

This paper examined how we view the cost associated with negative externality,
often referred to as “systemic risk” in the financial markets and how effectively
ex ante discipline can prevent this externality from spreading to other participantsin the market. The ultimate objective for the market participants, the policy
makers and the operators of the clearing house is to keep the market open, and forthat purpose we identified how the regulator sets the rule which prevents thecontagion effect, while making as many participants as possible take part in thetransactions.
By construction of the game, we find the contagion effect is rather weakalthough our results still support our proposition that the contagion will weaken
the financial positions of the sound players. Moreover, the arc-sine theorem
shows that those banks that are destined to perform poorly continue to perform134

Yukinobu Kitamura and Shuji Kobayakawa

poorly. This result has a remarkable policy implication that no matter how wellpolicy rules are implemented, the finanical market game will not converge to any
stable equilibrium where no player wants to play the game any longer. By construction, the game generates big winners and big losers constantly. It is the big
losers that make the financial system unstable. As long as this financial market
game keeps going, potential prevalence of systemic risk or contagion exists.
Practical solutions to prevent the contagion effect from spreading are twofold:market discipline and regulatory discipline. Nonetheless, the sheer existence ofmarket discipline has been under controversy in various studies.
In this paper, we focused on regulatory discipline. We examined ex ante rules
on how to resolve the problematic banks. We scrutinised three scenarios: earlyclosure rule, capital enforcement policy and frequent settlement system. In thefirst scenario, banks which have performed very poorly but are not yet insolvent,are asked to close down. In the second scenario, the participants are asked to holdadditional capital at the beginning of the game in order to prevent insolvency ofthe participants. In the third scenario, we examine how the amount of wealthnecessary to generate the same number of exits is altered by changing the frequency of settlement operations.
Our simulation results indicate that each policy rule works with its own costs.
Although we do not intend to identify which policy rule is most appropriate,through our analysis under the arc-sine theorem, our results seem to suggest thatearly closure and capital enforcement are effective devices for preventing thecontagion effect. In addition, the combination of various policy rules seems toderive a more practical resolution mechanism for preventing the contagion effect.
We also studied the capital injection rule whereby the clearing house injects
money into a participant facing potential bankruptcy, in order to improve the bank’s
debt position. In such a situation, the clearing house is a “lender of last resort.” We
concluded that the ex post measure is less costly than any ex ante measures, because
the former pinpoints the exact cost necessary to prevent contagion.

Appendix 1: Diffusion processes of random walk
If the length of one-step (d) is made smaller and each step is spaced so close intime, the resultant change appears practically as a continuous Brownian motion.
Suppose that an unrestricted random walk starts at the initial endowment position x 0 and that the n-th step leads to the position w n where
wn = x0 + x1 + x2 +....+ xn is the sum of n-independent random variables each
assuming the values ±d with equal probability of 1/2.
The variance of wn is expressed as v(wn) = tD (D=d2/T, d= length of one-step,T = unit time interval), and D is a diffusion coefficient. The mean of wn isE(wn)=tC where C is a drift coefficient.
Considering the position of the particle at the n-th and the (n+1)-st trial, it isobvious that the stochastic process p(z, t+T) satisfies the difference equations,Risk accumulation, contagion and the rules for bank failure11
p ( z − 1, t ) + p ( z + 1, t )22
p (0,0) = 1, p ( z,0) = 0( Z ≠ 0)p ( z,t + T ) =135(A1-1)

Since p has continuous derivatives we can use the Taylor series expansion at thelimit,∂p( z, t )∂p( z , t ) 1 ∂ 2 p ( z , t )
= −C
+ D
2
∂t
∂z
∂z 2

(A1-2)

This is a special diffusion equation known as the Fokker-Plank equation for
diffusion. In our system, drift coefficient C is zero and thus (A1-2) becomes,∂p( z, t ) 1 ∂ 2 p( z, t )
= D
∂t
2
∂z 2

(A1-3)

This is known as Brownian motion or Wiener process.
Stochastic process p(z,t) can be obtained by integration over (−∞, ∞), h enceby Fourier transformation with initial condition p(0,0)=1,p( z, t ) =

1

−

z2

e 2 Dt ≡ N (0, tD)
1(A1-4)(2πDt ) 2

This turns out to be the normal distribution with mean = zero and variance = tD.
Figure A1-1 shows changes in the probability density of Brownian motion overtime (t1< t2< t3).

Figure A1-1: The probability density of Brownian motion without driftp(z, t)t=0p(z, t)=N(0, Dt)

t1
t2
t3

0

z

136

Yukinobu Kitamura and Shuji Kobayakawa

Appendix 2: Relationship between the mean reversion process
and the arc-sine theorem
Under a fair and infinitely repeated game, a player will return to the initial position with probability one. This is known as the mean reversion process (see Feller(1957), pp.347-348). The arc-sine theorem, on the other hand, implies that it ismore likely for players to stay continuously on the winning side or the losing sidethan to frequently switch between the two sides.
The former states that a player will return to the initial position for sure, while
the latter states that a player tends to stay on one side for most times in the gameor the rounds of games.
These seemingly contradictory phenomena occur in the same diffusion process of a random walk as shown in Appendix 1 and they are, in fact, not contradictory at all, if we understand the nature of a random walk, i.e. non-stationarity.
We can restate the random walk process discussed in Appendix 1 in terms oftime series econometrics. According to Granger and Newbold (1986, pp.38-41), a
random walk without drift has the following characteristics.
If the process starts at t=0 with initial endowment, x0, thent −1

xt = x0 + ∑ ε t − j(A2-1)

j =0

where ε t is a zero-mean white noise process.

µ t = E ( xt ) = x0

(A2-2)

λ0,t = var( xt ) = tσ ε2(A2-3)

λτ ,t = cov( xt , xt −τ ) = (t − τ )σ ε2 ,τ ≥ 0(A2-4)

where σε2 is the variance of ε t and is finite. Thus,ρ τ,t = corr ( xt , xt −τ ) =(t − τ )t(t - τ )

=

t −τ
t

(A2-5)

Provided t is large compared to τ , all ρ τ ,t approximate unity. The sequence of xt
is smooth but nonstationary since its variance is increasing with t as shown in(A2-3). This result implies that the xt process tends to stay on one side (above or
below the initial position x0) and thus it satisfies the arc-sine theorem. Of course,
this process will reverse the course. That is, if it continues infinitely, the process
will return to the initial position. The statement “all roads lead to Rome” is justified in a random walk of less than two dimensions. The mean reversion is fulfilled.

Risk accumulation, contagion and the rules for bank failure

137

Nevertheless, within a practical length of period, the arc-sine process dominates the mean reversion process. Furthermore, if the random walk is defined in
more than two dimensions, the mean reversion process is not satisfied (Feller(1957), pp.359-360).

Appendix 3: Coin tossing game and arc-sine theorem
The purpose of this appendix is to formally derive the arc-sine theorem in amathematical context, which nevertheless contradicts a general belief based on
the law of averages.
The starting point to resolve the misunderstanding based on the law of averages is stating the derived result that a large number of independent coin tosses at
one time does not share the same statistical nature as the results from a single cointoss played over a long time period. The former game possesses the characteristics of the law of averages, while the latter is what we will pursue in the arc-sinetheorem.
With these differences in mind, imagine the following situation. An individual
is involved in a coin-tossing game. If he gets heads, he adds + 1 to his score, andif he gets tails, he adds – 1 to his score. Now the question is, how likely it is that
his score reverts to zero, namely that the number of heads and tails up to the timeof counting is identical, as he continuously plays the game over a long period oftime. The intuitive answer to this question is, because the game itself is fair in thesense that both heads and tails appear with the same probability (i.e. probability0.5), the resulting score tends to stay around zero. The arc-sine theorem, however,states that this is not the case. The startling message of the theorem is, unlike
basic intuition, that the score breaking even is the least likely outcome.
To demonstrate, we need to follow several steps.23 We first examine the ballot
theorem based on the reflection principle, from which we will learn the number ofpaths which do not cross the break-even point (position value of zero). Second,
we show the lemma which states that the probability that the paths will not go
through position 0 at all is identical to the probability that the paths will return toposition 0 eventually. Third, we explain the arc-sine theorem which can be derived from the ballot theorem and the lemma.
Notation
Let us denote P{(0,0)→(n ,r)} as the probability that the path leads to (n ,r).
In the following analysis, we write each position as (n ,r) which corresponds
to the position r after n coin tosses. The starting position is obviously (0,0).
For simplicity any probability starting from (0,0) will be written as
P(n,r)≡P{(0,0) →(n,r)}. Let S i be the position after time period i. Let us denote23
We owe deep gratitude to Cox and Miller (1965), Feller (1957, 1966), Loeve (1977) and Ross
(1970, 1983, 1996).

138

Yukinobu Kitamura and Shuji Kobayakawa

N(n,r) as the number of paths with Sn = r, namely after n tosses, the player’s
position is r.
Suppose now coin tossing results are p heads and q tails. Consequently, wederive the following relationship,p + q = n,

and we write

and its probability is

p−q =r

(A3-1)

 n 
n 

N ( n, r ) =   =  n + r  p  2 

(A3-2) n P( n, r ) =  n + r  × 2− n.
 2 



(A3-3)

Ballot Theorem
The number of paths finishing with Sn = r after n coin tosses is proportional tothe difference between the number of paths reaching r–1 in n–1 tosses and thenumber of paths reaching r+1 in n-1 tosses:
N ( n, r ) =n× [ N ( n − 1, r − 1) − N (n − 1), r + 1)].
r

Sketch of Proof
Let us define (n,–r) as the “reflection position” of (n,r). It implies that n cointosses will yield an opposite outcome, which can be achieved by reversing thenumbers of heads and tails.
First, we know that the number of paths from (1,1) to (n,r) which nonetheless
do not go through (t,0) where t=1,..., n-1 is equal to the number of paths from(0,0) to (n,r) which do not go through (t,0) either. Second we also know that the
number of paths from ( α,β), where 0≤α <n a nd β >0, to (n,r), which cross (t,0)
where t= α +1,..., n-1 is equal to the number of paths from (α,–β), which is thereflection position of ( α,β), to (n,r).
With these features, the number of paths from (0,0) to (n,r), which do not
cross (t,0) where t=1,..., n-1, is given as N(n–1,r–1), the number of paths from(1, 1) to (n,r), minus N(n–1,r+1), the number of paths from (1, –1) to (n,r). Fromequations (A1-1) and (A1-2),

Risk accumulation, contagion and the rules for bank failureN ( n − 1, r − 1) − N ( n − 1, r + 1) =139

( p + q − 1)! ( p + q − 1)!
−( p − 1)!q!
p!( q − 1)!

=( p − q ) ( p + q )!
×
( p + q)p!q!

=r
× N ( n, r).
n

Lemma
The probability that the paths will not go through the position 0 at all during
the period between 1 and 2n is identical to the probability that the paths will
return to the position 0 at period 2n.
Next, consider P(2n,2r), the probability that paths from (0,0) to (2n,2r) stay on
the positive side, or in other words, the position will always remain positive
during the period between 0 and 2n. The ballot theorem yields the equation:
N (2 n − 1, 2r − 1) − N ( 2n − 1,2 r + 1)
22n
1
= [ P( 2n − 1,2 r − 1) − P( 2 n − 1,2 r + 1)].
2

P( 2n, 2r ) =If we add all the paths with respect to r in order to derive the probability that thepath will return to (t,0), where t=1,..., 2n-1, during the periods we are concerned
with, we have∞

∑ P (2 n, 2r ) = 2 [ P( 2n − 1,1) − P( 2 n − 1,3) + P( 2 n − 1,3)1

r =1

−L + P (2 n −1,2r − 1) − P ( 2n − 1,2 r + 1)]
1= P ( 2n − 1,1).
2
Transforming the equation, we have

2 n − 1 −( 2n −1)  2 n  −2 n
P( 2n − 1,1) = =  ×2= P( 2n, 0).
×2
 n 
n

This implies that the probability that paths from (0,0) remain positive is equal tohalf the probability that the path will return to position 0 after a 2n time period.
The same applies when the position is negative. If we add them together, we will
derive the probability that the paths will not go through (t,0) while t=1,..., 2n-1 isequivalent with P(2n,0).

140

Yukinobu Kitamura and Shuji Kobayakawa

Arc-sine Theorem (Feller (1957, p.79))
The probability that in the time interval from 0 to 2n the position spends 2ktime units on the positive side and 2n–2k time units on the negative sideequals R{2k,2n}≡P(2k,0)× P(2n–2k,0), where n 
P ( n, r ) =  n + r  × 2 − n .


 2 
Hence,
 2k   2n − 2k  −2nR{2k ,2n} =   × 
×2 .
 k   n−k 
Sketch of Proof
For simplicity, let us use the equation:
 2v 
u 2v = P( 2v,0) =   × 2 −2 v ,(A3-4)
v
which states the probability that the return to the origin occurs at period 2v. We
also let f2v be the probability that the first return to the origin occurs at period 2v,
or in other words that S1≠0,…,S2v-1≠0, but S2v=0. From these two notations, wederive the following relationship:u 2 n = f 2 u 2 n −2 + f 4 u 2 n −4 + L + f 2 n u 0 .

(A3-5)

In other words, a return to the origin at period 2n may be the first return, orotherwise it occurred some time ago, such as 2k (k=1,..., n–1), followed by arenewed return at period 2n. Here, the probability of the latter is denoted by
f2ku2n-2k, because there are 22kf2k paths ending with a first return at period 2k, and
22n–2ku2n–2k paths from (2k, 0) to (2n, 0).
Next we denote b2k,2v as the probability that the position lies above the initial
position for a time period of 2k for a length of time of 2v. What we have to proveis:b2 k , 2v = R{2k , 2v}.

(A3-6)

We know b2v,2v=u2v=P(2v, 0)× P(0, 0)=R{2v, 2v}. Assume now a time period of
2k is spent on the positive side. Then there are two possibilities.
(1) A first return to the origin occurs at some time 2r < 2n. If the 2r time period isspent on the positive side, then 2k–2r must stay above the initial position as
well as during the remaining time period. The number of such paths is:Risk accumulation, contagion and the rules for bank failure

141

1 2r
(A3-7)( 2 f 2 r × 2 2 n−2 r b2 k −2 r , 2 n−2 r ).
2(2) A first return occurs at some time 2r < 2n. If the 2r time period is spent on thenegative side, then 2k must stay above the initial position during the remaining
time period. The number of such paths is,

1 2r( 2 f 2 r × 2 2n−2 r b2k , 2 n−2r ).
2(A3-8)

Adding up these numbers of paths in (A3-7) and (A3-8), we derive

b2k , 2 n =

n-k

1 k
∑ f 2 r b2 k −2 r , 2 n−2 r + ∑ f 2 r b2 k , 2 n−2r  .
2  r =1
r =1


(A3-9)

We know (A3-6) holds when v=1. By induction, suppose (A3-6) is also true
for v ≤n–1. This leads to:b2 k , 2 n =

n −k

1 k
∑ f 2 r R{2k − 2r ,2n − 2r} + ∑ f 2 r R{2k ,2n − 2r}2  r =1
r =1


k

∑ f 2 r P (2k − 2r ,0) × P ( 2n − 2k ,0)1
=  r =1

n −k
2

+ ∑ f 2 r P ( 2k ,0) × P( 2n − 2r − 2k ,0) (A3-10)
r =1


=

n− k

1 k
∑ f 2 r u 2 k − 2 r u 2 n − 2 k + ∑ f 2 r u 2 k u 2 n − 2 r − 2 k 
2  r =1
r =1


=

k
n− k

1
u 2 n−2 k ∑ f 2 r u 2 k −2 r + u 2 k ∑ f 2 r u 2 n−2 r −2 k 
2 

r =1
r =1

From (A3-5), the equation above can be written as:
b2 k , 2 n =1[u 2n−2k u 2k + u 2k u2n−2k ] = u 2k u2n−2k = R{2k ,2n}.
2

We therefore conclude (A3-6) also holds when v=n.

(A3-11)

142

Yukinobu Kitamura and Shuji Kobayakawa

References
Avery, R., T. Belton, and M. Goldberg, 1988, “Market Discipline in Regulating
Bank Risk: New Evidence from the Capital Markets,” Journal of Money,
Credit and Banking 20(4), 597-610.
Billett, M., J. Garfinkel, and E. O’Neal, 1998, “The Cost of Market versus
Regulatory Discipline in Banking,” Journal of Financial Economics 48, 333358.
Cox, D., and H. Miller, 1965, The Theory of Stochastic Processes. Methnen.
Diamond, D., and P. Dybvig, 1983, “Bank Runs, Deposit Insurance, andLiquidity,” Journal of Political Economy 91(3), 401-419.
Dixit, A., 1993, The Art of Smooth Pasting. Harvard Academic Publishers.
Duffie, D., 1996, Dynamic Asset Pricing Theory. Princeton University Press.
Feller, W., 1957, An Introduction to Probability Theory and Its Application, 3rd ed.
Vol. 1, Wiley.
Feller, W., 1966, An Introduction to Probability Theory and Its Application, 2nd ed.
Vol. 2, Wiley.
Gorton, G., and A. Santomero, 1990, “Market Discipline and Bank SubordinatedDebt,” Journal of Money, Credit and Banking 22(1), 119-128.
Granger, C., and P. Newbold, 1986, Forecasting Economic Time Series, 2nd ed.,Academic Press.
Loeve, M., 1977, Probability Theory I & II. Springer.
Maitra, A., and W. Sudderth, 1996, Discrete Gambling and Stochastic Games.
Springer.
Park, S., and S. Peristiani, 1998, “Market Discipline by Thrift Depositors,”
Journal of Money, Credit and Banking 30(3) Part 1, 347-364.
Ross, S., 1970, Applied Probability Models with Optimization Applications.
Dover.
Ross, S., 1983, Introduction to Stochastic Dynamic Programming. Academic
Press.
Ross, S., 1996, Stochastic Processes, 2nd ed., Wiley.