148

Hiroshi Fujiki, Edward J. Green, and Akira Yamazaki

Sharing the Risk of Settlement Failure*
Hiroshi Fujiki
Bank of Japan

Edward J. Green
Federal Reserve Bank of Minneapolis

Akira Yamazaki
Hitotsubashi University

Abstract
Two policies toward payments-system risk are common, but superficially appear
to be contradictory. One policy is to restrict the exposure to risk generated by one
participant to other participants who are, by one measure or another, directly
concerned with the risky participant. The other policy is to provide a “safety net,”
typically provided by government and funded by taxes collected from all participants and even from non-participants, to share losses due to “systemic risk.” In
this paper, we provide a model in which both of these policies can be constituentsof an economically efficient regime of payments-risk management.

1

Introduction

Large-value payments are typically made through continuing, multi-party, contractual, clearing and settlement arrangements. During the past several decades,
there has been progressively increasing awareness of the importance of risk
management in such arrangements. Because a very large loss can potentially be
incurred if settlement of a payment fails, how such a loss would be shared should
be a matter of substantial concern for the participants in an arrangement. Moreover, to the extent that complete contingent-claims markets do not exist for insurance against settement failures and that there are political pressures for govern*
The full paper is available, as a Federal Reserve Bank of Minneapolis Working Paper, from(http://woodrow/mpls.frb.fed.us). Any views expressed in this paper are solely those of the authors,and do not necessarily represent those of the Bank of Japan, the Federal Reserve Bank of
Minneapolis or the Federal Reserve System, or Hitotsubashi University.

Sharing the risk of settlement failure149

ments or central banks to assume losses from such failures, management ofsettlement risk is also a public policy issue.
Some specific questions regarding risk management in a settlement arrangement are the following. If there is some risk of failure to settle a payment fromone party to another, should the payment be settled through that arrangement?
(For example, in a net-settlement arrangement, what is the level of risk at which apayment ought to be made instead through an alternative, real-time-grosssettlement, arrangement?) If so, then what considerations are relevant to determining whether third parties ought to share that risk? Are there conditions underwhich the general public or the central bank (in the case of a private arrangement)
ought to bear some risk and, if so, what level of compensation would it be approprate for them to receive? If a third party possesses private information that would
be of value in determining how best to settle a payment, how does the exposure ofthat party to the settlement risk affect the quality of information that the party
chooses to provide? In this paper, we address these questions by analyzing aschematic, formal model of a settlement arrangement.
Settlement-arrangement designers, managers, and policy makers are wellaware that the rules governing an arrangement can affect users’ decisions about
which transactions to make through the arrangement. Thus, to set the rules of anarrangement is implicitly to decide which payments will be settled through it, andwhich payments people will decide to settle in alternative ways. (In fact, rules
governing an arrangement that lacks stringent risk controls are sometimes designed deliberately to make the arrangement infeasible or unattractive for use in
making very large-value payments.) By modelling the cooperative setting of rulesby participants in a settlement arrangement, and by participants in the economy asa whole, from this perspective, we are able to analyze welfare questions in aconceptually satisfactory way. Rather than taking that approach of specifying
transactions exogenously as previous researchers have typically done, what wetake to be exogenous are traders’ utility functions, which we specify in a way thatprovides scope for welfare-improving transactions among some of the traders tooccur. We also specify a settlement technology that imputes risks and costs to
those potential transactions. Having specified the model in these terms, we areable to characterize the patterns of transactions that the traders would cooperatively choose to make.
This approach provides answers, for the class of model economies that westudy, to the questions posed above. Not surprisingly, risk considerations play arole in determining which payments ought to be made. The specifics of that role
can be quite surprising, though. For instance, under some conditions it is not
optimal for a risk-neutral third party to share risk with the principals to a payment,even when the third party is a party to the more comprehensive transaction ofwhich the payment is a constituent. Yet, under other conditions, even the generalpublic (that is, traders who would not have transactions with the members of thesettlement arrangement if risk were not present) ought to share settlement risk, as
can happen in practice when a central bank serves as guarantor of a settlementarrangement. Private information regarding risk, even when it is possessed by athird party rather than by a direct party to a payment, is likely to be untruthfully
reported unless the settlement arrangement is deliberately designed to elicit thetruth. While these results about a schematic model economy are far from constituting definitive advice regarding actual settlement arrangements, we hope that
this analysis may at least provide a helpful framework within which to think in an
organized way about the issues involved in practical cases.

2

Modelling a transaction

Our first task is to formulate a model of a transaction that involves a risky asset
transfer. The model should be rich enough to describe such a transaction recognizably, but simple enough to be analytically tractable.
Consider what sort of model could satisfy both the requirements of richness
and simplicity. A transaction is a related set of asset transfers between traders.
The assets involved might be either commodities or financial assets. An asset
transfer involves two traders, the donor and the recipient, but a transaction can
generally involve more than two traders. Therefore, at the very least, a model of atransaction involving a risky transfer should include three traders, so that a distinction can be drawn between a participant in the broad transaction and a participant (that is, the donor or the recipient) in the specific transfer where the risk
occurs. In order for the third-party participant in the transaction—that is, theparticipant who is neither the donor nor the recipient of the risky transfer—to be
essential to making a mutually beneficial transaction, there should be no “double
coincidence of wants” between the donor and the receiver. This considerationsuggests modelling the three participants as a “Wicksell triangle.”
There is a distinction between the two types of third party (or potential third
party) that a good model ought to capture. A third party to the risky transfer in a
Wicksell triangle might be intrinsically necessary in the sense that the donor andrecipient of the risky transfer would have no double coincidence of wants, even if
the transfer did not involve risk (that is, if the recipient would receive the expected value of the transfer with certainty). Alternatively, the riskiness of the transfer
might impair a double coincidence of wants that would exist under certainty
between the donor and the recipient, and the third party might be needed solely torestore that double coincidence by serving as a guarantor or insuror of the transfer.
For characterizing the differences between the roles of these two types of third
party, a four-trader model (including both an intrinsic third party and a trader
whose only involvement would be to share risk) can be useful. On the basis ofthese considerations, we will specify the set of traders to be {1, 2,…, N}, where
either N = 3 or N = 4. In either case, we will assume that trader 1 is essential to amutually beneficial transaction but that trader 2 is the donor and trader 3 is therecipient of the risky transfer. When a four-trader economy is considered, theattributes of trader 4 will be specified in such a way that trader 4 can only partici-

Sharing the risk of settlement failure

151

pate in a risk-sharing capacity.
The risky transfer will be formalized in terms of a probability space of eventson which a probability measure Pr is defined.
There is a distinguished event S, with 0 < Pr(S) < 1. Assume that the risky
transfer from trader 2 to trader 3 succeeds in S, and that it fails in the complementary event F. When we say that the transfer succeeds, we mean that trader 3
receives the entire quantity of the asset that is transferred. When we say that thetransfer fails, we mean that the quantity of the asset that was intended to betransferred disappears irretrievably from the economy.1
Later, to analyze incentive issues, we will specify that trader 1 privately
observes an event that is statistically relevant to the outcome of that risky transfer.
Assume that each trader i has an endowment consisting solely of a type ofcommodity that only he possesses. We denote that type of commodity also by i.
Intuitively, trader i is endowed with one unit of commodity of type i with certainty.
In general, a commodity bundle provides a random amount of each of the Ntypes of commodity in the economy. That is, each trader can acquire commodities
by transactions with others, and randomness is introduced by the riskiness of thetransactions technology. We use the letter g to denote such a random commodity
bundle.
Each trader’s preference between commodity bundles conforms to expected utility.
Trader i has a von Neumann-Morgenstern utility function U i: R+N ® R È {–¥}.
Trader i’s expected utility of consuming a commodity bundle g is the expectation®
of the random variable U i (g ).
The sequence of economic activities in this economy is as follows.
Initially, before knowing whether the actual state of nature is in S or F, traders
make an agreement for transfers of goods among them. The agreement among thetraders is binding.
With one exception, the transfers are safe. That is, everything sent out reaches
its intended recipient in its entirety and with certainty. The exception is the transfer of trader 2’s endowment to trader 3. Recall that this transfer reaches trader 3
in its entirety in event S, but is completely and irretrievably lost in event F.
The traders also agree ex ante on a second round of transfers, to be made after
the first transfers have been completed and the result of the risky transfer has
become known. Thus the transfer to be made in the second round can be madecontingent on which of the events S and F has occurred.2
1
Failure of an actual transfer seldom involves such an irretrievable loss, although there are somecontemporary examples and many historical examples of that type of failure.
2
Strictly speaking, this sentence describes a different information structure from the preceding
one. If traders can only distinguish between events S and F on the basis of observing the success or
failure of a transfer, then they can not make any distinction unless a (non-zero) transfer has been
attempted. To assume that they can make a state-contingent transfer in the second round even if nofirst-round transfer from 2 to 3 has been attempted neglects this limitation of their opportunity for
inference. In the case where there is no private information, this ambiguity is harmless because riskaverse traders would not cooperatively choose to make a state-contingent transfer in the secondAll second-round transfers, including the one from trader 2 to trader 3, arenonstochastic. However, second-round transfers are costly. Only a proportionr < 1 of the goods that a trader sends in the second round are received. 3
Traders consume their stocks of goods after these two rounds of transfers havebeen completed. To simplify the characterization of traders’ consumption resulting from settlement, we make two assumptions: that a trader is able to transfer
only his own endowment good, and that only a few of the possible flows of those
goods are feasible. Specifically, trader can make a transfer to 2, 2 to 3, and 3 to 1.
(Traders 1, 2, and 3 together will constitute the Wicksell triangle to which reference was made earlier.) In addition, in the version of the model where there is afourth trader, traders 3 and 4 can each transfer their endowment good to the other.
As described above, either all, a proportion r, or none of the goods sent may
be received. A transaction is a sequence t = (t1, tS, tF) of rounds of transfers. The
elements t1, tS, and tF specify the initial round of transfers, the round of transfersin event S, and the round of transfers in event F, respectively.

Figure 1: Round of transfers
1

f1

f3
f4

2

f2

3

f5

4

A transaction is feasible if no trader is ever required to send a cumulative
amount that would exceed his endowment. That is, transaction t is feasible if"i t1i + max {tSi , tFi} £ 1.

(1)

Let t denote the set of feasible transactions.4

round unless they had exposed themselves to settlement risk in the first round. How the ambiguity is
resolved is important in the private-information case, though, and we will discuss this issue further
when we analyze private information.
3
This assumption, sometimes called “iceberg cost,” can be viewed as a crude way of reflecting
various intuitive considerations including time preference and exposure to business loss due todelayed availability of transferred funds.
4
As noted in the footnote above, the informational constraint that, if t21 = 0, then tS = tF, may or
not be added to the definition of feasibility for a transaction. If all traders are risk averse, then theconstraint is never binding when traders have common information.

Sharing the risk of settlement failure153

Now we provide an explicit definition of traders’ consumptions resulting froma transaction. To begin, informally let tc be the random net trade that results fromtransaction t, which depends on whether event S or F instead occurs. By adding
the positions of a particular trader in all such net trades in which that trader isinvolved to the trader’s endowment, the random consumption of the trader isdetermined. Specifically the consumption vector ci(t) that trader i receives as aconsequence of transaction t is as follows.
c1(t)

=

(1 – (t11 + tc1 ) z1 + (t13 + rtc3 ) z3

c2(t)

=

(1 – (t12 + tc2 ) z2 + (t11 + rtc1 ) z1

c3(t)

=

(1 – (t13 + tc3 ) z3 – (t15 + tc5 ) z5)(2)+ (cSt12 + rtc2 ) z2 + (t14 + rtc4 ) z4
c4(t)

3

=

(1 – (t14 + tc4 ) z4 + (t15 + rtc5 ) z5

The core

We modify the core of an exchange economy to serve as the solution concept tocharacterize the set of mechanisms to which the traders might agree. A core
allocation is one that can be obtained (according to (2)) by a feasible transaction,and such that no coalition of traders can implement another allocation that its
members unanimously prefer—with at least one of them having a strict preference—by using an alternative transaction that is feasible for its members. Definea core transaction to be a feasible transaction from which a core allocation is
obtained via (2).
To formalize the notion of unanimous preference within a coalition, for eachnonempty C Í {1,…, N}, define q Î t to C-dominate t Î t if"iÎC

E [U i (ci(t))] £ E [U i (ci(q))]$iÎCE [U (c (t))] < E [U (c (q))].
i

i

i

and

i

(3)

Also define q Î t to be feasible for C if, with certainty,"iÏCci(q ) = zi

(No participation of other traders is required).

(4)

Finally, define t Î t to be a core transaction if there exist no C Í N and q Î tsuch that q is feasible for C and q C-dominates t.
Let us say that transaction t is individually rational if it is weakly preferred toautarky by every i Î N, and that t is Pareto-undominated if it is undominated forN.

Proposition 1 Let each trader’s utility function be locally nonsatiated, at all
points, in his own endowment good. Then a feasible transaction t is a core transaction if and only if the following conditions hold: t is individually rational,
Pareto-undominated, and not either {1, 2, 3}-dominated or {3, 4}-dominated.

4

Analysis of a public information environment

It will be useful to carry through our analysis using specific utility functions toshow why the preference and private information do matter in the settlement
system.
To this end, we study core transactions in some parametric versions of theeconomic environment defined above. We begin with a simple environment,
where N = {1, 2, 3, 4} and there is no private information. 1, 2, and 3 are theessential parties and 4 is the stand-by party to transactions. We specify the traders’ utilities as follows.
U1(c)

=

ln(c1 + bc3)

2

U (c)

=

ln(c2 + bc1)

3

U (c)

=

ln(c3 + bc2 + yc4)4=ln(c4 + jc5)

U (c)

(5)

with b > max{s–1, r–1}, 0 < jy < 1.
Here, goods received in trade are “better” substitutes for endowment goods foressential participants 1, 2, and 3. Trader 4 considers trader 3’s good to be a“worse” substitute for his own endowment good, and trader 3 considers 4’s good
to be a “worse” substitute for trader 2’s good or even for his own endowment
good. We assume that the transfer technology to satisfy 0 < r £ s <1 and s >1/2,
and we assume that br ³ 3 2 .5
Analysis of this model shows that a core allocation exists, and that it has thefollowing characteristics.
Proposition 2 A core transaction t always specifies state contingent transfers. A
typical core transaction t specifies transfers such that:
All essential traders send first-round transfers within the Wicksell triangle. In
addition, trader 3 sends a first-round transfer (which might be considered to be
an insurance premium against settlement failure) to the stand-by party, trader 4.
The first-round transfer sent from 2 to 3 does not supply all of trader 3’s con5
The condition for a small transfer at the endowment allocation, using the safe technology, toincrease the sum of utilities of the two traders is that br > 1. Thus, the intuitive meaning of the latterassumption is that traders would have clear willingness to use the safe technology if it were the onlytransfer technology available.

Sharing the risk of settlement failure

155

sumption of good 2. Rather, even in event S, 2 also sends a second-round transfer
to 3. Thus, despite its costliness, trader 2 uses the safe transfer technology tominimize exposure to the risk of settlement failure in the first round. However,except for this risk-mitigating transfer, no other second-round transfers are madein event S.
In event F, both traders 2 and 4 make second-round transfers to 3. The transferfrom 2 to 3 in this event is larger than the second-round transfer in event S. Thus,
settlement risk in round 1 implies consumption risk for trader 2 as well as for
trader 3. Trader 1 shares this risk by making a second-round transfer to 2.
Moreover, trader 3 acts to minimize this induced consumption risk for trader 1 bymaking a second-round transfer. (Although 3 has suffered a loss himself, it isefficient for him to help 1 because of the difference in their marginal rates ofsubstitution between goods 1 and 3.) Thus settlement failure in round 1 triggers
economy-wide transactions in round 2.

5

Preliminary analysis of a private-information environment

In the general discussion above, we have contemplated that a third party within
the coalition might have some private information, not possessed by either the
payor or the payee, about the level of risk. In such a case, the information ispotentially relevant to how the transaction should be conducted and even to howlarge a transaction ought to be undertaken.
If the privately informed third party were involved solely as the reporter ofthat information to the coalition, then there would be no problem about ensuring
the truthfulness of the report. In particular, if compensation were required toinduce reporting, that compensation could be made in the form of a flat fee. If thepossessor of information functions in the payments process as an agent for one ofthe principals in the transaction, though, then there will generally be an issue of
whether there is incentive for truthful reporting.6 One might think, for example,
that efficiency would generally require a payments coalition to penalize an information provider when a payment would fail without a warning of particularly
risky circumstances having been given.
In this section we will show that there is indeed an incentive-compatability
issue for the payments coalition to resolve, but that there is no simple generalization about how to resolve it. The incentive for truthful revelation of information
6
This idea, that a dual role of privately informed members of a payments coalition is critical forunderstanding how the institutional design of a payment arrangement is related to the attainment of
economic efficiency, has previously been studied by Rochet and Tirole. In their model, unlike thepresent one, traders’ information can only be revealed through their trades, and not by making
explicit reports. In many actual payments networks, the limited opportunities for traders to makeexplicit reports seem to fall between the absence of opportunity modelled by Rochet and Tirole andthe completely adequate opportunity modelled here. When traders are required to set prior limits(which will not necessarily ever be binding in equilibrium) on their bilateral exposure tocounterparties, for instance, their choices of which limits to set can be regarded as partiallyinformative reports of their private information about those counterparties’ riskiness.

depends on the pattern of risk sharing within the payments coalition, the differences in risk attitudes among coalition members, and the distribution of rents thatis to be achieved by a core transaction mechanism, which generalizes the notionof a core transaction to a private-information environment. In fact, for theparametric environment that we study, some core mechanisms involve a binding
incentive-compatibility constraint for truthful revelation that a transfer is likely tofail (that is, revelation of event L), while other core mechanisms for the sameenvironment involve a binding constraint for truthful revelation that failure isunlikely (event H). As a practical matter, then, an implication of using the coretransaction mechanisms as an equilibrium concept for payment arrangements isthat supervisory authorities ought to accord substantial discretion to the governing
body of a payments coalition to establish rules aimed at eliciting accurate information from members.

5.1

Generalizing the model to encompass private information

To model private information, suppose that an event that is statistically relevant to
the outcome of that risky transfer will be privately observed by trader 1, who is
not directly involved in the risky transfer but who is an essential participant in amutually beneficial transaction among the traders. To consider the simplest caseof nontrivial private information, suppose that trader 1 observes a signal that takes
the value either ‘H’ or ‘L’. Suppose that ‘H’ and ‘L’ satisfy
Pr (S|L) < Pr (S|H).

(6)

The agreement among traders regarding the structure of the transaction,
described in section 2, is an ex ante agreement, made before trader 1 has received
any information. However, trader 1 will observe H or L before the first round oftransfers takes place. Thus it is natural for the agreement to specify that trader 1
will report what he observes, and that his report will determine which transaction
to make. That is, the agreement among the traders specifies a transaction mechanism rather than a single transaction. Formally, a transaction mechanism is amapping m: {‘H’, ‘L’} ® t .
Transaction mechanism m will elicit truthful reporting from trader 1 if thefollowing incentive-compatibility condition is satisfied.
"AÎR

"BÎP

E[U1(c1(m (B))) | A] £ E[U1(c1(m (A))) | A].

(7)

Let M denote the set of incentive-compatible transaction mechanisms. We restrict
attention to incentive-compatible mechanisms, as is justified by the revelation
principle.7 If m Î M, then the resulting transaction t and the consumption Gi foreach trader i Î N is defined as follows (with cA denoting the characteristic func7

Myerson (1991) provides an exposition of incentive compatability and the revelation principle.

Sharing the risk of settlement failure

157

tion of an event A).
G i ( m ) = å c Ac i ( m ( A)).
AÎP

(8)

A core transaction mechanism can be defined in a way that is straightforwardly analogous to the definition of a core transaction.8 Specifically, to formalize the notion of unanimous preference within a coalition, for each C Í N, definen Î M to C-dominate m Î M if"iÎC

E[U i (Gi(m))] £ E[U i (Gi(n))] and$iÎC

E[U i (Gi(m))] < E[U i (Gi(n))].

(9)

Also define n Î M to be feasible for C if, with certainty,"iÏCGi(n) = zi

(7) holds if

1ÎC

(No participation of other traders is required)(Incentive compatability).

(10)

Finally, define m Î M to be a core transaction mechanism if there exist no C Í N
and n such that n is feasible for C and n C-dominates m.
With the core of a transaction mechanism so defined, proposition 1 has thefollowing straightforward generalization.
Proposition 3 Let each trader’s utility function be locally nonsatiated, at all
points, in his own endowment goods. Then m Î M is a core transaction mechanism if and only if the following conditions hold: m is individually rational, Pareto-undominated, and optimal for payments-system participants, and m is not{3, 4}-dominated.

5.2

A parametric environment with accurate private information

Consider the three-trader environment in which trader 1 receives a private signal
about the success or failure of a transfer from trader 2 to trader 3. To simplify thispreliminary analysis, we assume that success and failure have equal probability, and that trader 1’s signal is perfectly accurate, That is, we assume that
Pr (H) = Pr (L) = 1/2, Pr(S|H) = 1, and Pr (S|L) = 0.
In this section, we work with piecewise-linear utility functions for the traders.
Their utilities will be defined in terms of parameters d and e, which are assumed
to satisfy 0 < e < d < 1/4. Utility functions are defined in terms of the following
functions on the nonnegative real numbers.

8
Allen (undated) has previously used incentive-compatible mechanisms in this way to define acore equilibrium concept for environments with private information.

V(x)

=

min{1, x} + e max{0, x – 1};

W(x)

=

min{1/2, x} + e max{0, x – 1/2}.

(11)

Define the agents’ utilities as
U1(c)

=

c1 + c3;

2

U (c)

=

V(c1 + c2);

3=

W(c2) + d c3.

U (c)

(12)

Because trader 1’s utility function is linear and information is perfectly accurate, the incentive-compatability constraint reduces to the following two equations. The first and second equations state that trader 1 has incentive to report
truthfully in event H and L, respectively.
(m13(H) + mS3(H)) – (m11(H) + mS1(H)) ³ (m13(L) + mS3(L)) – (m11(L) + mS1(L))(m13(L) + mF3(L)) – (m11(L) + mF1(L)) ³ (m13(H) + mF3(H)) – (m11(H) + mF1(H))

5.3

(13)

IC constraint can bind in either H or L

It has been proved that, in the three-trader environment, a transfer mechanism isin the core if and only if it implements a Pareto efficient (subject to both technological and incentive constraints) allocation that is individually rational foreach trader. An allocation that maximizes the weighted sum of traders’ expectedutilities, with all weights strictly positive, is Pareto efficient. Therefore, for a Î
R+3+ , consider
3

U ( m ,a ) = åa iEU i (Gi ( m )).
i =1

(14)

Regarding the transfer technology, suppose thatr = 1/2.

(15)

In this environment, it can be shown that the incentive-compatability constraint can bind in either event H or L, depending on which value of the utilityweight vector a is used for maximization of U.

6

Conclusion

The rules in a settlement system must encourage the participants to take optimal
degrees of risk in accordance with their attitude towards risk. If some participants
have socially useful private information, the rules in a settlement system must beSharing the risk of settlement failure

159

constructed such that it does not give participants adverse incentives to mask theirinformation. Policy makers can achieve this objective if they think about the rules
in a settlement system as a mechanism design problem. If policy makers ignore
those points and introduce new rules into a settlement system, the equilibrium
allocation of goods might be distorted.
If we regard the crucial issue in the settlement system as efficient risk sharing
among the participants in the presence of private information, this view allows
policy makers to consider the rules governing a settlement system as a kind ofsocial safety net. Efficient risk sharing under the default of banks might involve
the transfer of resources from the agents who normally are not directly involved
with the settlement network. The central bank plays such a role as a lender of lastresort, by transferring the resources of the general public into the banking sector
during a period of financial panic.

Appendix: Understanding central banks as a risk-sharing device
The discussion in this paper is based on the view that the standard microeconomic
theory could be useful to analyze the settlement network. We try to understand theflow of funds between the parties as endogenous phenomena. We emphasis thatboth private information and preference could play independent roles to determine the optimal risk sharing in equilibrium. The approach could be useful todesign regulations free from unintended distortions caused by the lack of necessary consideration on the preference and information set of participants.
We believe our strategy is consistent with the recent development of microeconomics of banking (see Dewatripont and Tirole (1994), Freixas and Rochet(1997)), and hope to address the following issues more explicitly than the former
literatures in a formal model.Those include the distinction of private information
held by risky agents versus that held by other agents, the role of lender of lastresort in the settlement network, the interpretation of additive linear preference as
an amount of cash flow, and so forth.
However, some readers might wonder to what extent our analysis is of practical relevance since our model looks too formal. Therefore, in this appendix weshow several heuristic examples that support our argument.
We will discuss the following topics in turn: (i) the role of linear aggregatorand piecewise linear preference, (ii) the role of central bank as a lender of lastresort, and (iii) historical examples of barter trade that exactly match our model.

The role of linear additive aggregator
The literature of home bias points out that the proportion of assets invested indomestic assets is substantially high in many industrialized countries comparedwith that suggested by the internationally diversified optimal portfolio based on
the modern portfolio theory (see French and Poterba (1996)). Moreover, the socalled country-specific cash in advance constraint, although it is hard to explain

endogenously, is commonly assumed in the literature of modern international
finance (see Obstfeld and Rogoff (1996) for a review of the literature). Those twolines of research justify treating two goods as if imperfect substitute currencies,
and an additive linear aggregator is a useful way to measure the sum of two goods.
More specifically, the literature of home bias suggests that the risk of exchange
rate fluctuation, the asymmetry in the tax systems across countries which could
encourage investment in domestic assets rather than foreign assets, and incomplete information regarding foreign countries as potential explanations for thehome bias. On the other hand, diminishing liquidity in the foreign exchange
market could potentially explain the country-specific cash in advance constraint.
Such ideas shown in the literature seem to justify our discussion on the linear
aggregator on two goods, which puts asymmetric weight on the goods transmitted
from other participants once we regard those goods as currencies. Moreover, thestrong tax and institutional bias might tend to exaggerate the choice of currencies,
as French and Poterba (1991) have documented. That evidence might be consistent with our piecewise linear utility function, which tends to put the agents in thecorner solution.

The role of the central bankThroughout this paper, we argue that the fourth trader who serves as a lender oflast resort, can be viewed as a central bank only if there are no gains from tradewith this agent in the absence of risk. We argue that the fourth trader should bejust treated as the general public if there are gains from trade without risk.
More formally, trader 4, who obtains goods at the usual time and sends goods
to trader 3, looks like a central bank if bg < 1. Note that the fact that this inequality does not hold implies that a unit of exchange of goods between trader 3 andtrader 4 is Pareto improving, hence it is no wonder that trader 4 transfers his own
goods to trader 3. The implication of our model is that since the inequality does
not hold ex ante, there is no gain from trade in the absence of risk. The fact thatthe marginal utility of trader 3, which becomes substantially high given the shipment failure from trader 2, induces trader 4 to serve as a lender of last resort,because trader 4 sends goods to trader 3 only under the situation of settlement
failure. The fact that trader 3 sends some goods to trader 4 without shipment
failure could be understood that trader 3 pays some fee in order to obtain insurance against shipment failure. Such pattern of trade between those two traders issimilar to that in which a bank submits collateral and obtains discount window
lending from the central bank (see original ideas on lender of last resort Bagehot(1906), and Bordo (1990) for a recent survey).
We stress the view that the fourth trader plays the role of central bank, but it is
not different from the usual traders intrinsically, and there is no particular reason
to believe there are gains from trade between the central bank and others without
risk consideration. We further argue that if there were a risk in a settlement network without a central bank, then a private institution would serve such a role.

Sharing the risk of settlement failure161

Our view is based on the U.S. history of banking. For example, take agent 2 as
New York banks, agent 3 as Boston banks, and agent 1 as Philadelphia banks.
Consider shipment failure of one bank as the default of the bank. Safer shipment
takes time as banks inspect the quality of banknotes. Note that the fourth trader
would act as a clearing house of New York that starts supervising banks with amembership fee. This suggests that if there is no central bank for political reasons,
there would be quasi central banks by private arrangement (see Gorton andMullineaux (1987)). That consideration justifies why our model captures some
important aspects of the free banking area documented by King (1983) and theemergence of central banks. Such interpretation suggests that our model is evenconsistent with the recent view that regards the role of the Federal Reserve as the
supplier of settlement services among private netting arrangements once weregard traders 1, 2 and 3 as private clearing networks (see Summers and Gilbert(1996)).
Note that Green (1997) shows that by allowing traders to issue so-called
novation securities, in equilibrium, both the initial securities and the novation
securities will trade at face value in a monetary economy a la Freeman (1996a, b).
This means the risk induced by trading-opportunity uncertainty will be fully
insured, and efficiency will be attained. The novation securities introduced below
bear a striking resemblance to the clearinghouse loan certificates that were issued
during those episodes in the absence of a central bank. Those certificates, and thecentral banking role played by U.S. clearinghouses at that time generally, aredescribed by Timberlake (1984).

Loss sharing rule in historical economies
Strictly speaking, we have discussed the implication of barter trade with risky
delivery of goods.
Given these limitations, one way of viewing our model is as a somewhat
realistic model of shipping insurance and other loss-sharing arrangements inhistorical economies. Early modern Europe and feudal Japan provide such examples. Regarding Europe, Lopez and Raymond (1955, p.259) reprint Genoesedocuments of 1191 and 1192, in which a merchant pays a premium to a shipper
who puts up security for the successful delivery of the merchant’s goods. Moreover, some examples of insurance contracts involving third-party underwriters canbe found as early as the fourteenth century. In such a contract, the underwriters
are supposed to purchase for a certain price a certain amount of goods from amerchant, but that the contract is to be void if the goods arrive safely at a certain
port.
Feudal Japanese sea law, dating back to at least 723 AC, also pertained to an
economy that exemplified the general features of our model.9 Most sea transportation was commissioned by governments. Specifically, local government offi9

The following discussion is based on Takeda (1992) and Toyoda and Kota (1970).

cials (owners of the cargo) hired sailors to ship goods from their regions to thecentral government as feudal tax payments.10 Sometimes bad weather forced thesailors to jettison the cargo in order to stay afloat. In that event, the central government asked the sailors and the local government to pay 40% and 60% of thedamage of the cargo respectively. (That is, the central government tried to induce
the sailors not to abandon too much cargo intentionally by introducing this rule.)
However, if the ship sank or more than half of the sailors were drowned despite
having jettisoned the cargo, then, the central government did not ask them (that is,either surviving sailors or drowned sailors’ heirs) to pay indemnity. These provisions were evidently designed to induce sailors to take appropriate actions contingent on the severity of the weather at sea—a situation regarding which they possessed private information relative to the senders and receivers of their cargo. 11
As well as serving as a fairly realistic model of such historical arrangements,
it also serves as a more schematic model of loss-sharing arrangements adopted byvarious payments systems today. It is noteworthy that the model explains thesearrangements in terms of standard concepts of insurance theory, without having to
invoke unproved assertions about a special, ill-defined kind of “systemic” risk.

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11
Presumably the provisions also served to prevent sailors from engaging in intentional fraud—a phenomenon from which our model abstracts—by surreptitiously stopping en route to sell the
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Sharing the risk of settlement failure

163

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