FOREIGN EXCHANGE NETTING AND SYSTEMIC RISK/… · exchange contracts netting, and present an...

FOREIGN EXCHANGE NETTING ANDSYSTEMIC RISK *

Akira Yamazaki tDepartment of EconomicsHitotsubashi University

Kunitachi, Tokyo 186, JAPAN

May 1996

Abstract

As a natural consequence of a major development in financial ac-tivity since the introduction of flexible exchange rates in early 1970's,it is felt that payment systems have become a potential source of seri-ous financial crises. Payment system is a set of arrangements made forthe purpose of discharging obligations assumed by economic agents intheir economic transactions. There are two basic ingredients of pay-ments systems: they are settlement arrangements, and netting arrange-ments. Netting is an offsetting of a similar type of financial obligations,and only the net difference is settled. Foreign exchange transactionsaccount for a large share of all payment flows in major financial centers.

The purpose of this paper is to introduce a formal model of foreignexchange contracts netting, and present an analysis of multilateral andbilateral netting from the view point of credit risk reduction. In par-ticular, we are interested in comparing these two different forms ofnetting arrangements with respect to inherent systemic risks involved.The point of our present paper is to show that when more than twobanks defaults, indirect loss sharing of participants could harm theparticipants to multilateral netting beyond the potential risk level ofbilateral netting arrangements. The concept of systemic credit expo-sure is used for this purpose.

Keywords: Payment system; multilateral netting; bilateral netting;systemic risk; credit risk; foreign exchange.

JEL classification numbers: GI5,F31,F33,D40.

• I owe special thanks to Nobuyuki Oda for various comments on the earlier developmentof this draft. I am also indebted to Toshio Kanazawa, Makoto Ohsawa, and Hideaki Ohnofor various comments and for enlightening me concerning payment systems.

t E-mail address: akira©hitcc.cc.hit-u.ac.jp•

2 Netting and Systemic Risk •1 Introduction

Since the introduction of flexible exchange rates in early 1970's we have seena major development in financial activity. The introduction of new financialinstruments, deregulation of activities of different types of financial insti-tutions. remarkable advances of computer and communication technologies.the growing internationalization of financial markets and volatile movementsof assets prices including foreign exchange rates have all contributed to anunprecedented growth of financial activity. It has induced conspicuous in-creases both in the volume and the value of payment flows in major industrialcountries as well as across national borders.

As a natural consequence of this development. it is keenly felt that pay-ment systems have become a potential source of serious financial crises. Forthis reason. the central banks of the Group of Ten countries have study-ing the nature of risks and possible policy problems concerning paymentand settlement systems. See, for example. the Angell Report [11 in whichthe nature of various risks and policy problems under different netting sys-tems are discussed and the Lamfalussy Report 121 in which the minimumstandards for the design and operation of cross-border and multi-currencynetting and settlement schemes are proposed.

Payment system is a set of arrangements made for the purpose of dis-charging obligations assumed by economic agents in their economic trans-actions. There are two basic ingredients of payments systems: they aresettlement (or fund transfer) arrangements, and netting (or clearing) ar-rangements. Netting is an offsetting of a similar type of financial obligations.and only the net difference is settled. Thus arrangements for the bindingnetting of financial obligations provide a service that is a very close substi-tute for the function of money as a medium of payment. Netting betweentwo parties is called bilateral netting, and netting among multiple parties iscalled multilateral netting. In the latter case netting is also referred to asclearing.

Foreign exchange transactions account for a large share of all paymentflows in major financial centers. Because financial institutions have strongincentives to lower credit risk and payment flows, there have been activemovements toward structural innovations in interbank clearing and settle-ment procedures and a number of proposals have been presented to estab-lish multilateral foreign exchange netting arrangements in recent years. As amater of fact, the first foreign exchange multilateral netting scheme, ECHO.started its operation in Europe last August. (See ECHO l6].)

In this paper we would like to introduce a formal model of foreign ex-change contracts netting. and present an analysis of multilateral and bilat-eral netting from the view point of credit risk reduction. In particular, we areinterested in comparing these two different forms of netting arrangementswith respect to inherent systemic risks involved. In our analysis we will

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• Netting and Systemic Risk 3

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introduce a comparative statics type argument in intrinsically stochastic en-vironments. The approach might be termed as a comparative contingencies.By looking at an event, given by a possible finite sequence of defaultingbanks, we shall compare credit risks of a bank between the two differentnetting arrangements.

In multilateral netting arrangements in general. a clearing house will be-come the ultimate counterparty to every participant to the arrangements.Thus, one could argue that credit risk of each participant must be judgedin the light of the financial resources of the clearing house. In fact, theAngell report explicitly states this point. Okina 112) emphasizes it in hisdiscussion of prudential rules in multilateral netting. In our analysis of mul-tilateral netting in this paper, however, we have not followed this suggestionin order to simplify our analysis. Instead, in our model all the losses thatthe clearing house will incur due to failures of participating banks will bedistributed among surviving banks in proportion to their profit levels withrespect to defaulters. It is our view that if one wants to introduce collateraland margins in a model of multilateral netting arrangements, then partici-pants' credit exposure to the clearing house must be taken into account inconsidering their credit risks.

The paper is organized as follows. After a brief introduction to theliterature in the sequel, we present in Section 2 a formal model of foreignexchange contracts netting based upon individual transactions and contractsof financial institutions. In Section 3 the concepts of bilateral exposuresand multilaterally netted bilateral exposures are formally defined and theircomparison is made in Proposition 2 which is a basis of a casual argumentstating that if netting is done among many, then credit risk reduction is moreeffective than in the case of netting between two parties. The point of ourpresent paper, however, is to show that this is not the whole story. Whenmore than two banks defaults, indirect loss sharing of participants couldharm the participants to multilateral netting beyond the potential risk levelof bilateral netting arrangements. In Section 4 systemic risk is analyzedfollowing a very simple scenario of two defaulting banks, and some of itsimplications are discussed. Although this section takes up a very simplecase of systemic risks, results obtained here can be extended to generalcases. In fact, it is done in Section 5 which treats a general case of anarbitrarily given sequence of defaulters. By analyzing general properties ofloss share coefficients, we deduce cases in which multilateral netting giveshigher systemic credit exposures than the sum of bilateral credit exposures.Finally, in Section 6 we summarize some of the results obtained in this paper.

In the literature there have been very few studies that present formalmodels in discussing payment systems. It appears that serious studies onpayment systems have just begun. Schoenmaker 114), and Rochet and Ti-role [13] present a formal model of interbank settlement system. Eisenberg[71, Chakravorti [5], and Schoenmaker 115) present an analysis of systemic

4 Netting and Systemic Risk

risks in settlement systems. Humphrey 110) gives a simulation study of sys-temic risk. We are not aware of formal models in which different nettingarrangements are analyzed. However, the Angell report In gives a verynice discussion and a verbal analysis of risk structures in different nettingschemes. Borio and Van den Bergh 14) also gives an extensive verbal discus-sion of payment system in general.

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2 A Model of Foreign Exchange Contracts Netting

We shall consider a finite set I of financial institutions or banks that en-gage in cross-border or off-shore foreign exchange transactions over discretepoints in time that may be referred to as dates also. A foreign exchange con-tract or obligation specifies a contractual duty to deliver a defined (foreign)currency in exchange for another (foreign) currency between two institutionson an agreed date or value date. The words — transactions, contracts, andobligations — are used synonymously in this paper. Institutions that engagein a financial contract are called parties. The opposite party to a financialcontract is a counterparty. They are counterparties to each other. In thissection we will build up a model of foreign exchange contracts netting basedupon individual transactions of financial institutions.

2.1 Individual Transactions and Contracts

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Given a set of financial institutions I, typical elements of I will be writtenas

h,i,j E I.

The number of financial institutions, ;I, to be considered for the purpose ofour analysis is assumed to be at least three so that 3 < <

Now, let us consider foreign exchange transactions between banks j E Iat time t i , and collect all the transactions that take place on the same date2 1 with identical value date t 2 (> t i ). Denote by

Ji2( 2 1, 22)

the set of all such transactions or contracts. With this notation above. wehave

t2) Jj i (t , t2)

for any i and j and Jo(t i , t2) = 0 if i = j. For convenience, a contractk E Jii (t 1 ,2 2) in this paper represents a "matched pair" of promises todeliver foreign currencies to counterparties. A foreign exchange contractk E Jo(ti, t2) at t i obligates bank i to pay bank j at time t2 the amountYjik(t1,22,e) of currency c' and receives the amount yok(t i , 2 2 , c) in currencyc for some pair of currencies c and cf . For notational convenience we putyijk((2,t2, et) = 0 if the contract k does not involve an exchange of cur-rency ei. In the notation yijk (t i , 1 2 , c) , suffix i refers to a party, j to itscounterparty, and k to a contract. If y ijk (t i .t 2; c) is positive, then it meansthat party i is to receive the amount yijk (t i ,t2 ,c) from its counterparty j,and the counterparty is to pay the amount —y jik (t i , 2 2 . c) to the party i.If yijk (t i , t2, c) is negative, then i is to pay to j the amount —y uk (t i . t2. C),and j is to receive the amount yjik (t i , 2 2 ,c) from 1. Thus. for any contract

6 Netting and Systemic Risk •k E ..10 (t 1 ,1 2 ), we have

Yti k (t 1, t2, c) y)ik (t 1 , t2 , c) = 0 (1)

which states that the amount to be received or paid (= yijk (t 1 ,t 2 ,c)) by aparty is equal to the amount to be paid or received (= —yiik (t i . t 2 . k)) byits counterpart) . respectively. For expositional convenience we will use thefollowing mathematical notation: for any real number v

v + ▪ max{v, 0}max{—v,0}

v ▪ v+ — tr.

In these notations y,;fr(ti,t2, c) + and iiiik (t 1 ,12 ,c)4 represent the receipts,and ytik(ti : t2, c) -- and Iho(t i .t 2 , c)- the payments of / and j respectivelyconcerning a contract k E .10 (1 1 , 1 2) . In order to avoid misunderstandingsas regards to our notation, let us record the following obvious relations:

Yiik( t i, t2. C) ' = Yi ik(tj. t2. C) -

Ihjk(t1 t2, C) = Viik( t 1, t2,C)•

which simply state that a receipt of a party is a payment of its counterpartyand vice versa.

We assume that there is a finite set of (foreign) currencies traded inthe markets. It is denoted by C. We assume 2 < ::C < r It is alsoassumed that there is a "base currency'', denoted by b . that is used toexpress values of all other currencies or that is the standard of value amongdifferent currencies.

2.2 Bilateral Netting of Contracts

Netting is an offsetting of receipts and payments to be made for a similartype of financial contracts. Bilateral netting is a netting between two partiesIt is considered to be a most natural form of netting in foreign exchangetransactions as large number of "matched trades" with same currency, samevalue date and same counterparties exist.

Consider two banks i and j and the set of contracts Jo(t ) . t 2 ) betweenthem. Without a netting of contracts, banks i and j face payments of

E yuk (t i , t2 , and

E ynk (1 1 , 1 2 , c)ke .7„(ti .i2)

respectively at time 1 2 as a result of contracts in Ju(l1.12) = J.it(ti.12)

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Netting and Systemic Risk

In order to discuss several distinct forms of netting that can be appliedto conventional foreign exchange contracts, we shall define, for any t i , t2 , cwith ti < t2.

Yij(tl, C) :"E E Yijk(tl: t2, C)•kEJ,)(ti,t2)

Here, summing the amounts yijk (t i , t 2 , c) of payments and/or receipts in-stead of the amounts yijk(t i , t 2 . c)- of payments and the amounts yijk (ti, t 2 , c)+of receipts separately over all the contracts in J ij (t i .t 2) means that the vari-ous payments yok (t i , t 2 , c)- are offset against various receipts y ijk (t i , t 2 . c)+for k E Jii (1 1 ,1 2) to arrive at one net amount yii(ti, t 2 , c). It therefore repre-sents the amount of currency c that bank i is to receive from bank j at timet 2 as a result of foreign exchange transactions taking place between them attime t 1 under a bilateral netting arrangement between the two patties. Asbefore a positive yii (t I .t2 , c) means a net receipt and a negative yi i (t i , t 2 , c)a net payment on the part of bank i. With this notation, it follows from theequality in (1) that we have for each t i , t 2 , c

t2 c) + yii (t i ,t 2 , c) 0, (2)

which means that the amount that i is to receive from j is equal to theamount that j is to pay to i for each currency c under bilateral netting.

For notational convenience, if there are no transactions in ..111 (t 2 , t 2) forwhich yok(t1,12,c) $ 0, then we put

t 2 , c) = 0. (3)

Thus, if Jii(ti, t2) = 0, we have

(ti 7 12 C) = 0

for all c E C, and in particular

vii(t 1, t2, c ) = 0

for all c E C.We assume that all the forward foreign exchange contracts take place

during a definite time span that is given by time periods between T 1 andT2. Let us define for each t. £ 2 , c with

1

^ t2

X tj(t. i2. C) E (ti t2, c),to<t,

where to is the date at which an initial transaction of currency c with valueat t 2 took place between i and j. As foreign exchange contracts betweenparties i and j with same currency and same value date accumulate as timeelapses toward the value date, the amount that i is to pay to or receive

8 Netting and Systemic Risk •from j in currency c at time t2 changes. This net amount as of time t isexpressed by xv(t,1 2 . c). Before we proceed to a description of types ofbilateral netting. we like to check two basic equalities relating to xv(t. t 2 . c).

First, it is immediate from the definition of xv(t,1 2 . c) that it is the"accumulated - bilateral position at time t of bank i with respect to j forcontracts in currency c with value at t2 , that is. for any 1.1 2 , c with t < t2.we have

xi, (1, t 2 . c) = x,,(1 – 1.1 2 .c) y,3 (t. t 2 . c)

Secondly, we have again a fundamental "mirror image - equality for anyt. 1 2 , c with t < 12

xv(t 1 2 . c) + x p (1, t 2 , c) 0 (4)

because

x,„(t,t2. c) + x„,(t.1 2 . c)

= E 111;( 1 1- t2. E (t t 2 c)t.0 < t I Ct LaCti <t

E ilfu(tl.t2.C) t2 C))to<11<t

= 0

where the last equality follows immediately from (2). •2.3 Different Forms of Bilateral Netting

Position netting or payment netting is a form of offset under which twoparties informally make one net payment between themselves for each cur-rency and value date. But there is no change in their contractual obliga-tions. Thus. in this form of netting the amount of payment in currencyc due for bank i at time 1 2 is x,;(t,4 2 , c)- (if this is equal to 0. thenxii(t,12, c) - > 0). But the contractual obligation requires bank i to payj the amount Et l <t Ek Xijk(tI . 12, C)—.

Netting by novation or obligation netting between two parties refers tothe replacement of contracts between them for delivery of a specified cur-rency on the same date by one single net amount for that date in such away that original contractual obligations are satisfied and discharged. Thus.under netting by novation between banks i and j, bank i is obligated to paythe amount x„(t,1 2 . c)" or bank j is obligated to pay x (1, 1 2 . c)- ratherthan the amount Et < t Ek zijk (t i , t 2; cr and Et, .c t Ek Xjik(11,12, c) - re-spectively.

Under the netting by novation between banks i and j, the amountx,,(t, i2 , c) which is identical to -x,;(1.1 2 , c) represents a "novated" (i.e..new) contract for the net amount as of time t. In this sense xv(t,t2. c)(and xii (t, / 2 , c)) expresses the bilateral novation process through time t be-tween parties i and j with respect to currency c and value date t2

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Under the bilateral netting, whether it is position netting or netting bynovation, bilaterally netted bilateral settlement position in currency c ofbank i with respect to j'at time t is given by the amount xo(t.t.c).

In close-out netting all the outstanding contracts are replaced by cur-rent contracts with same value dates and same currencies to compute theirpresent market values. That is, they are all marked to market. Using thesepresent market values, payments and receipts are offset to arrive at onesingle net amount in the base currency which represents the bilateral posi-tion between two parties in close-out netting. This procedure is equivalentto computing the current "replacement costs" of outstanding contracts be-tween the parties. This type of netting arrangement is invoked in case of adefault of a counterparty.

In our comparison of bilateral netting with multilateral netting in thispaper, we will assume bilateral netting to be netting by novation for clear-ing and settlement and to be close-out netting in the events of defaults byparticipants. However, since we will be mainly concerned with credit riskin a narrow sense (see Section 3 below), we will be uniquely dealing withclose-out netting.

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2.4 Market Value of Forward "fransactions

Let q(t,t2 .c) denote forward exchange rates of currencies c E C quotedin terms of the base currency b at time t with value at time t 2 . Now,for a foreign exchange transaction k E .1,i (t 1 ,1 2) which took place at datet 1 , there is a pair of currencies (c.c') for which we have yuk(ti, t 2 . c) $ 0,Yok(t i. £ 2 . c') 0 0 and yok(t i , t 2 . c") = 0 for c" c. c'. For the pair (c.c') onemust have

g(t 1 , t 2; c)yi ik (t i , t2 , c) + g(t i , (2, ci)Yijk(ti. t 2 . c') = 0

meaning that one unit of currency c is to be exchanged for

9(ti t2. t2. CI))q ( t 1 t2, 90k(t1 t 2 , c)

units of currency c' at time t 2 which is the market forward exchange rate attime t l . It thus follows that

E q(t i , t 2 , t2. C) = 0.cEC

This simply says that, at origination, the value of an at-market foreignexchange contract is zero. Since this equality holds for any contract k EJij (t i . t2 ). one must have

E q(i i , t2, c)Ys)(t1,t2,c)cEC


E E q(ti,t2.c)y,,k(ti,t2.0kEJ,)(ti.ti)cEC

= 0.

However, for t 1 t < t2 , one may have

E (7(1 t 2 . t2, c) 0cEC

due to fluctuations in forward exchange rates q(t. t 2 , c). If this amount isnegative, bank i's forward book shows a loss concerning the transactionsat t; with value at t 2 . The amount Ec q(t, t 2 , c)y,„ (t t , 1 2 , c) represents themark-to-market value or net present value at time t of the transactions attime t; between banks i and j with value at time t 2 . We denote this valueby

vil (t , t 2 )(t) E E q(t, C2. Ch/,)( t 1, t2,

cEC

We now calculate the mark-to-market value or net present value of theforward book of bank i with respect to bank j as of time t which will bewritten as v„(t). It is given by

v (t) = E vs, (ti , t2)(t)t, <t

t<ticr2

for each t. Note that by our notational convention in (3) we have t .9 (t) = 0if i = j or if J,j(ti.t2 ) = 0 for all t 1 <

It will become convenient to rewrite this expression for the computationof net present values of the forward book.

E Vtj(tl, i2)(i)

t, <tt<t2<T2

EED(t, t2. c)y,i (tl,t2,c)t 2 t,

E= ED(ci2,oyucti,i2•01 2 C

= EEO, t2, c)E t2,t2 /2

= EEq ( t, t2 , c)x,i (t, t2 , c).C

It thus follows that one has

vu (t) = E E q(t, t2,C)2,,(t.t2,c)t<c 2 <2-7 cEC

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vii (t) is the bilateral position of bank i with bank j at time t concerningi's forward book s . The equality above shows that the bilateral position canalso be calculated on the basis of the accumulated bilateral forward positionin each currency for each value date, i.e. xo(t,1 2 ,c), instead of going backto all the original contracts. We note that

t.„ ( t ) EL 0,1 2 . c)x,,(t .1 2 . c)t2

= -EE q( t,t2,c)Xn(t.t2,C)t2 C

= —vii(t)

so that we havevii (t) + vii(t) = 0. (5)

Since vii (t) is the net present value of all forward transactions of i with jevaluated by forward prices at time t, the equality (5) is interpreted to saythat in forward transactions both two parties cannot win at the same time— a party or its counterparty must loose between the two.

We record the above two properties as:

Fact 1 [Mark-to-Market Value) The mark-to-market value of the forwardbook of a bank with respect to one of its counterparties can be calculated onthe basis of its accumulated bilateral forward position in each currency withseparate value dates, i.e.

vii( t) = E E q ( t , t2 . 020(t.12. c).t<t2<mcEC

Fact 2 [Zero-Sum Property] Profits and losses arising from forward for-eign exchange transactions have the zero-sum property, that is, for any banksi and j and for any T1 < t < T2 one has

vii (t) + vii (t) = 0.

2.5 Multilateral Netting by Novation and Substitution

In netting by novation and substitution a contract between two parties isamended in such a way that a third party, i.e. a central clearing agent, isinterposed as a central counterparty, to each of the two parties, and two new(= "novated") contracts, where the central clearing agent is substituted forthe counterparty in the original contract, are created. The original contract

1 For concreteness, in calculating the positions of forward books at time t, value datesstart from t+ I. i.e. the first business day from the date t. in this paper. This could bet +n with some n = 1.2.... depending upon institutional rules of clearing. For example,in case of the rule of the ECHO, one has rt = 2.•

12 Netting and Systemic Risk •between the two parties is then satisfied and discharged. It can be regardedas a prototypical multilateral netting arrangement where financial contractsamong a given group of financial institutions, composed of more than threeinstitutions, are netted.

More specifically, there is a central clearing agent – a clearing house– and a number of participating banks to the netting arrangement. Par-ticipants submit their foreign exchange contracts between any two of theparticipants to the clearing house. The clearing house is substituted as thecentral counterparty to each party for a foreign exchange contract submittedby a pair of participating banks. The obligations between the participantsare discharged as a result of binding multilateral netting among the partic-ipants. At each point in time the clearing house maintains a current andnew (or novated) net position for each currency and for each value date withrespect to each participant. These net positions determine the amounts dueto each bank from the clearing house, or vice versa, for each currency and foreach value date. But once a default of participating banks occurs, close-outnetting takes place.

3 Risk Assessment

Financial contracts are subject to different types of risk during the course oftheir lives. First. two types of risk can be distinguished: credit risk in broadsense and market risk. Credit risk in broad sense arises from the possibilityof default by counterparties whereas market risk arises from the possibilityof adverse movement in market variables such as foreign exchange rates.Market risks or foreign exchange risks can be hedged by individual financialinstitutions. Credit risks, however, cannot be fully hedged by individualbanks alone. It is here that institutional arrangements and regulations needto play an important role.

Credit risk in broad sense may again be divided into two types: set-tlement risk and credit risk in a narrow sense. Settlement risk refers to apossibility of a counterparty's failure to make settlements during the courseof a business day whereas credit risk in a narrow sense refers to a possibilityof a counterparty's failure to honor its forward obligations.

Since there have been several studies in recent years concerning set-tlement risks (for example, see Rochet and Tirole [13) and Schoenmaker114),1151), we would like to focus on credit risk in a narrow sense.

The cost to a bank of a counterparty's default on a forward contract isthe cost of replacing the cash flows specified by the contract. Thus, a default-induced loss will occur only if the mark-to-market value of a contract riseto a positive value. One is thus led to define actual credit exposure usingthe bilateral position of individual banks.

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3.1 Bilateral Credit Exposures

We first consider bilateral netting by novation between two banks i and j.The amount xij (t, t, c), c E C, exhibits the bilateral settlement position ofbank i with respect to bank j in the sense that it represents the amount ofcurrency c E C that bank i is to receive from bank j at time t. Potential lossof bank i may arise during the course of business day t from bank j's failureto make settlements. It will be called the bilateral settlement exposure incurrency c E C of bank i with respect to j.

Let us denote the bilateral settlement exposure at time t by eii (t, a).Then, by definition we have

a) E xij(t,t,c)+

for j E I and c E C. Note that settlement exposures at time t derivefrom contracts that are due to be settled on that day. Thus, it is naturalfor exposures to be defined currency by currency.

But in the event of a default of a counterparty, there is another potentialloss. It arises from the couterparty's failure to honor its forward obligations.When forward obligations are dishonored, it is natural to assume that eachbank replaces them by similar forward obligations in current markets toresume its forward positions in each currency. Thus, the idea of close-outnetting, which is invoked in case of a default, is to compute replacementcosts of dishonored forward obligations. Loss arising from the failure of acounterparty to honor its forward obligations will be termed as actual ex-posure of a party with respect to its counterparty. It may also be calledmark-to-market exposure or net present value exposure. Following the no-tation of our model, the (actual) bilateral exposure at time t of bank i withrespect to bank 5, denoted by cii (t), is thus defined by

c1 (t ) • vij(t)fr

▪ E Eq(t,t2,c)xij(t.t2.c)t<, <T2 c

3.2 Multilateral Credit Exposures

3.2.1 Notional Bilateral Position

Let the set of financial institutions I represent the group of participatingbanks to an arrangement of multilateral netting by notation and substitutionfor clearing and settlement, and that of multilateral close-out netting in caseof a default. The clearing house will be denoted by the letter H. Define for

14 Netting and Systemic Risk •

each i E 1 and t. t2. c

X % H (t. t2, E E x,j (t. t2 . c)..7E1

X H I (t, t2. E E x,,(t, t2:3E1

Other notations with respect to H are introduced similarly. For example.for E I and 1.

In essence suffix H indicates that sum is taken over all participating banksto netting and clearing arrangements.

Under a multilateral netting arrangement by novation and substitution,multilateral settlement position in currency c at time t is given by

xof (1. t. c) E xu (t, t, c):7E1

E E E vok(ti. t, c).IEI tiS kEJ., (0.0

for each participating bank i E I. Here, for every contract k E

t i < t, the clearing house is substituted for the counterparty j so that thebilateral settlement position z,;(t, t. c) of i with respect to j becomes onlya part of settlement position of i with respect to the clearing house H.Taking the sum of xii (t, t, c)'s over j E I means totalling all those partsof bilateral settlement positions of counterparties of bank i for which theclearing house is substituted. Thus, the multilateral settlement position ofi given by xill (t, t. c) is the bilateral settlement position of i with respectto the clearing house after substitutions and notations of obligations areeffected.

We are now concerned with the amount of forward book positions ofbank i that is exposed to a default risk. If bank j( � i) will default at time t,potential loss of bank i is not necessarily given by the amount t .,H (0 + norvii (t) + . It is because original contracts of bank i with bank j are replaced bythose with the clearing house. Hence, possible losses that each bank mustface in multilateral netting depend upon how losses are allocated amongparticipating banks in netting arrangements. Without going into details ofvarious loss allocation rules, we would like to make one basic assumptionon a loss allocation rule which seems to us general enough to be adopted inthis context. (See Subsection 3.2.3 below.)

Bilateral position under a multilateral netting arrangement will be callednotional bilateral position. It is notional because all the counterparties in

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Netting and Systemic Risk 15

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the contracts presented for netting are substituted by the clearing house andbilateral positions between two participating banks no longer exist. Thenotional bilateral forward position of bank i with bank j at time t is

vii(t) = E E 9(t,t2.6xii (t, t2,c).t<t 2 <T2 cEC

The multilateral (forward) position of bank i at time t is

tiffl (t) = E vii(t).jEl

The amount v in(t) is the bilateral position of bank i with respect to theclearing house at time t under close-out netting. We say bank i at time thas a multilateral profit position if yin (t) > 0 and a multilateral loss positionif vin (t) < 0.

3.2.2 Foreign Exchange Risk Hedging and Clearing Efficiency

Let us introduce a few more concepts before going into definitions of forwardbook credit exposures. We say that bank is forward position is perfectlyhedged or that bank i's foreign exchange risk is perfectly hedged at time t if

E ,t1t2 5 T2 ) E rij (t, t2, C) = 0.jEl

If bank i's forward position is perfectly hedged, then for all T1 < t C T2,we have

Ej

• EEl

E E 9(t,t2,c)xii(t,t2,c)jEl t<t2 <T2 cEC

• E E g(t,t 2 , E .0( t, i2,t<t2<T2 cEC jEl0

so that for all t

viy(t) = 0.

That is, if the forward position of a bank is perfectly hedged, then its forwardposition with the clearing house is nil. From a traditional banker's point ofview it represents an ideal situation for foreign exchange transactions and,moreover, it will correspond to the most ideal case for contracts nettingefficiency as all the credits exactly offset all the debits in each currency atevery due date.

Fact 3 'Clearing Efficiency] If a participating bank's forward position isperfectly hedged, then no settlements are needed at due dates as credits anddebits are fully matched.

16 Netting and Systemic Risk •3.2.3 Bilateral Default Risk under Multilateral Netting

Even if bank is foreign exchange risk is perfectly hedged so that v, H (t) = 0,it faces default risks of counterparties. Assume that one of the counterpar-ties. say bank j, of forward contracts that bank i had, defaulted at time t.Then, a general rule of the clearing house is to allocate default induced lossesamong the participants, called concerned participants or concerned banks:who have contracts with the defaulting bank maturing at time t. Lossesare allocated pro rata to the profit levels of the concerned participants. Byassuming this loss allocation rule, we define multilaterally netted bilateralexposure or indirect bilateral exposure of bank i with respect to bank j attime t, denoted by mi (t), to be

Thj( t ) (rvij(AJ( ) till (t)+

It simply says that the loss t Ri (t)* of the clearing house caused by thedefault of bank j is allocated among those participants i having notionalprofit position with bank j, i.e., v,,(0 + > 0, according to the ratio

vo (0+ Ehu vni(0+

of its profit level to the total of profits made by concerned banks in trans-action with the defaulting bank j.

Now, assume bank Fs forward position is not perfectly hedged so thatat time t

# 0

and assume that the net present value of its forward books show a great dealof losses, a typical situation for a defaulting bank. Then, the actual creditexposure of the clearing house with respect to hank j at time t is..

j (0 + = vill(t)-

= t

l

(t)(El i

= (

E v„ (0 + - E ,,, ( t ) -)

ieiriet

= max {0,- (EvIi (04 -Evii(ty)}

iEl la

= max {0. - (E vo(t) - - E vw(t)+)

'El set

= max {0. E tit) (0 + - E vu(t)-}.iEl Iv

(6)

•

•

( v,; (0+ shEi 1)113( t ) + VH3 (tYk

th3 (0+ E 0,;(0 + - tin; (0+

LhEI Vhj(0+

e ii (t) — 77, (t) = vii(0+

hE/

vii(t)-E E Vhj(0–)

LhEI Vhj (t)+ h El(9)

•

•


We thus obtain

E vii(o+ - ,„; (0 + E vii (t) + — max {0, E v,;(0+ - E t•ii(t)—}

min {

E Vij (0 + , E vii (t) .El iE/ (7)

Since we assumed rin(t) <<0 so that

E v.,;( 0+ < EiE/ iE/

we obtainvll, (0+ = E vij (0 + — E Vij(t)–.

'El iE/

The above equality shows that the total sum of the actual (forward book)exposures of participating banks with respect to bank j, i.e., LE/ t'sj (0+,is reduced by the amount, Lie/ vii( t ) — , of the losses of the forward booksof the participating banks with respect to bank j. Let us record this fact.

Proposition 1 [Credit Exposure of the Clearing House] Suppose thatbank j has a multilateral loss position at time t. Then, the sum of the bilateralcredit exposures of other participants with respect to the bank j at time texceeds the actual credit exposure of the clearing house with respect to j by theamount of the sum of losses that the participants are making in transactionswith j i.e.

E vii (0 + — v j (0 + = E vii (t) (8)iE/ 1E/

One can immediately compare direct bilateral exposure with indirectbilateral exposure (i.e., multilaterally netted bilateral exposure) using theequality (8). In fact,

17

iE/ iE/ iE/ iE/

where the third equality follows from the equation (8). Hence, we obtain

•the following:

18


Proposition 2 Bilateral Exposure vs. Multilaterally Netted Bilateral Ex-posure; The difference between direct bilateral exposure and indirect bilat-eral exposure of a bank i with respect to any other bank j i having amultilateral loss position at time t is exactly equal to the sum of losses ofindividual banks in transaction with the bank j multiplied by the proportionof bank is profits to the total profits of all the participating banks in trans-actions with the bank.. i.e.

E v)

h (0-)LAE/ Vh.7(0 + hEl

Let us introduce a notion which indicates the extremity of market risk.We say that the foreign exchange risk of bank i at time t is extreme if for allh z

vv.,(t)+ = 0.

and for some h iva(t)- 0.

One may decompose the reduction of total credit exposures in (8) intothat of individual banks. We define (apparent) bilateral benefits of multilat-eral netting. denoted by au (0,2 as the residual of direct bilateral exposuresover indirect bilateral exposures, that is,

8,,(t) a ev (t)- 77„(t)

vv(1)+LAEI uh,(0+

It follows from the definition of the bilateral benefits of multilateral nettingand the equation (9) that

vu(0+ DIE/ VhJ(t)

t9(0+ EhEl Vh)(t)

E vhj(t)-hE I

E Vpi(t)+hEl

Thus, if the foreign exchange risk of bank j is not extreme. there will bepositive (apparent) bilateral benefits (i.e.,i(3,j(t) > 0) of multilateral nettingon the part of concerned participant i, having a positive bilateral exposure.i.e., v,j (t) + > 0. But if the foreign exchange risk of bank j is extreme, thenthe above equation for ,3,j (t) shows that 09 (0 = 0, that is. there are nobilateral benefits of multilateral netting over bilateral netting. Thus oneobtains the following proposition.

•

•

2 0ne should not be misled by the term benefit. This need not be an actual benefitwhen one considers a possibility of systemic risk. •

•

•


Proposition 3 [Bilateral Benefits/No–Benefits of Multilateral Nettingl Sup-pose that bank j is making a loss on its forward book at time t. If its foreignexchange risk is not extreme, then for any concerned participant having apositive (notional) bilateral credit exposure, the bilateral benefit of multilat-eral netting is positive. But if bank j's foreign exchange risk is extreme, thenthere will be no benefits of multilateral netting.

The implication of this proposition is that as the loss-making bank'sforeign exchange risk increases the bilateral benefit of multilateral nettingdeclines 3 .4 It has a policy implication as well: in order for a multilateralnetting system to have increased efficiency for reducing credit risks, extremeforeign exchange contract positions should be controlled.

4 Systemic Risk Assessment — A Simple Case

The failure of one participant in a multilateral netting system to meet itsrequired obligations when due may cause other participants unable to meettheir obligations which in turn could trigger a chain reaction of defaultsamong participants. The risk of this type is known as systemic risk. Thecredit exposure of a participant faced with a possibility of systemic risk willbe analyzed in this and the following section.

4.1 A Scenario of Systemic Risk

The approach followed here is to consider a "scenario" of a possible chainreaction of defaults in multilateral netting arrangements and to find actualcredit exposures of individual participants in the given scenario of bankdefaults.

3 0f course, admittedly, we are very loose here in the statement of an increasing foreignexchange risk. It is not entirely impossible to give a formal statement of this fact but wedo not think it is worth while to further complicate our setting of the model.

*Financial contracts netting systems provide de facto payment systems for financialcontracts in the sense that a financial contract is used for a partial or entire discharge ofanother financial obligation. This means that a netting system makes financial contractsmeans of payment or medium of exchange. It may be called "quasi-money". However, thisquasi-money is not supplied by any of the monetary authorities, i.e. central banks. Thus,it raises the traditional issue of whether and how the supply of money can be controlledby the supervisory authorities.

In a multilateral netting system the maximal amount of money being supplied corre-sponds to the case where all the participating banks do not have any foreign exchangerisks so that for each i e I

viii(t)= 0.

Then, no additional settlements will be due and the amount of money m(t) "created" atsettlement time t would be

E zu(t.f,c)-1..,E1

for each currency c E C.

20 Netting and Systemic Risk •In this subsection we describe a scenario of a most simple systemic risk.

A typical scenario is as follows:

There is an original defaulter j i E I that fails to meet its obli-gations at time t. Bank j i faces extreme foreign exchange riskat time t and its mark-to-market value of its forward positionsshows a big loss by the amount L 1 > 0. This situation is de-scribed as

(1) VH); (0 + = L t > 0,

v.00+ = 0.

Due to the big loss suffered by j 1 , it fails to meet its obligationsat time t. The bank j i is the original defaulter.

Assume that all the other participants can meet their obligationsat time t. But according to the assumed multilateral nettingarrangement of this paper the loss-share rule becomes effectiveonce there is a defaulter. The remaining participants must makeadditional settlement payments following the general loss-sharerule of the netting arrangement. Now, assume, for the purposeof making the very nature of systemic risk more transparent,there is a participant j 2 # j i that has perfectly hedged positionsso that it faces no foreign exchange risk at time t, i.e.

t;j2H (t) = 0,

but has a huge profit position with respect to the original de-faulter j i so that

tlYb (0 - = t2v2221 ( t) - = 0 )

To sum up, the situation is such that there are at least two defaultingparticipants, one initial and the other subsequent. The initial defaulter,in a sense, forced itself to default by taking a huge foreign exchange riskwhereas the second and subsequent defaulter had perfectly hedged foreignexchange positions. But it has a huge bilateral profit position with theoriginal defaulting bank.

4.2 Systemic Credit Exposure in a Simple Case

The possibility of a chain reaction of defaults among participants to a mul-tilateral netting scheme clearly depends upon multilateral credit exposuresof participants. We will compare below the credit exposures of participatingbanks under bilateral netting system with those under multilateral nettingsystem by following the above scenario.

•

•


•

•

Let us introduce further notation to express credit exposures under mul-tilateral netting system with possibility of multiple defaults. Given a finitesequence of banks (in our present case a sequence of two banks) 14tjnin=1.2:the systemic credit exposure at time t of bank i with respect to the sequence,written aiith (0, is defined by

(r j1+ rt, (Ori2j , (0) cm, (0+ + T i2j2 (t)t, Hj2 (0+'700 2

where rij (t) and r .iin (t), n = 1.2 are given by

vii (0+ vii(0+ von (0+ rii (t) = r?in (t), .

EhE/ lihj(0+ Ehoi Vhj(0+ LhOji,j2 thj y, (0+

The benefit of multilateral netting of bank i with respect to the given se-quence {j,i }n= 1,2 of banks is defined by

/3ini2(t ) Er- Ein + eii2 (t) cli2i2 (0 •

This can be rewritten as follows:

QijiJ2 ( t ) = (Eiji — 77i11 (0) + (e.i3-2 (0 — p2(t)no2(0)

W ri2i1 (O vHj i (0+ (10)

with p2 (t) defined by

p2(t) = EhEI Vhj2 (t)+

Vizi2 (0+

It follows that we have

Oini2(1) = Ai, + (t) - [(p2(t) - 1) /hi, + r ii (Orj2;,(Oviii,(0+1

Since p2 (t) > 1 when vi,j,(04 > 0, one obtains

Siili2 (t ) < fii:71 + 002 (t)*

We would like to record this property as follows:

Proposition 4 (Strictly Decreasing Benefits of Multilateral Netting] Givena possibility of defaults by a sequence of two banks with the original defaulterhaving a multilateral loss position and the second and subsequent defaulterhaving a bilateral profit position with the original defaulter, it is always thecase that the benefit of multilateral netting of a concerned participant isstrictly less than the sum of bilateral benefits of multilateral netting, i.e.

4.'2(0 <1301 (t) -I- 0020).


If we apply the equality (9) to j = j I , we get

e t)) ( t ) - (t) = rth (t) E vith (t)-

hEl

Since we have tinH (t) = 0 so that

E vto2 (0 + = E titin (0 + = 0,AEf AEI

it follows that

ee ,72( t ) - P2(0% (0 = ru (2(t) E Uh)2( tr - Th(Oni) 2 (04

he!

Using these values of co, (0 - (t) and co2 (t) - p2 (0r),12 (t) along with theassumed property 15271 (1) - = 0 and (10). we can rewrite the expression for.3,3132 (t) as below:

0,1112 (1) = rth (t) E viz') (tr ry2 (t) E vt,j2 (J) - 2- 1 (/)75 211 (Ovj,he/ he/

In order to make the nature of systemic exposures transparent we con-sider the systemic exposure of bank i which is a concerned participant withrespect to the initial defaulter bank j i but not with respect to the secondarydefaulter bank j 2 . Denoting the set of concerned participants with respectto bank by Jn . for t E \J2 , we have • ••

- 202)2.22( 1 ) = ru l (t) E t'h,1 (0 - r), (01:721 ,( )

hE!

Thus, if we put

pj (0-

L i = vl,H(t) -1

EAEJ VA), (t) ,n = EhEl Viul (0 + r2 = I 2271 M ?)

we have/

auit2 (t ) = r212 C - P2(t) n2)11

where L 1 is the net loss of defaulting bank j i , I the total of losses of banksfrom contracts with j 1 , T1 the total of profits of banks from contracts with

and r2 the profit of the secondary defaulter j2 from contracts withThus.

1;2Quir2(0 < 0 <=t• < p2(t).

Li

hEl

= ro2 (t)E viu2(0-

•

•


Since the loss L 1 suffered by the initial defaulting bank j 1 is assumed to bemuch larger than the total 1 of losses suffered by the other banks in trans-actions with the bank j i and since the profit 7; 2 of the secondary defaulterj2 in transactions with the initial defaulting bank jl is a relatively largecomponent of the total II of the profits made by banks in their transactionswith the bank j i , it follows that we have

13;j02 (t) c 0

as p2 (t) > 1. In other words the systemic credit exposure of a concernedparticipant with the initial defaulter but not with the secondary defaulterunder a multilateral netting system is greater than its bilateral credit expo-sure under a bilateral netting arrangement.

One could give a general statement of this fact as below:

Proposition 5 [Systemic Credit Exposures] Consider a possibility of achain reaction of defaults at time t given by a sequence of defaulting banks{j4„,. 1.2 . Assume that the initial defaulting bank j i faced with high foreignexchange risk suffers a huge mark-to-market loss by an amount L 1 and thatthe secondary defaulting bank j 2 with or without the perfect hedged positionhas a large profit amount r2 arising from transactions with the initial de-faulting bank. Then, if the proportion of the secondary defaulter's profit tothe total of profits made by participants in forward transactions with theinitial defaulting bank j i is greater than the proportion of the total forwardbook losses suffered by the participants in transactions with the initial de-faulter j i to the large amount of the forward book loss suffered by the initialdefaulter 5 1 due to its huge foreign exchange risk, then for any participatingbank having a net profit position with the initial defaulter but not with thesecondary defaulter, it is always the case that its systemic credit exposureunder a multilateral netting arrangement is greater than the sum of bilat-eral credit exposures with respect to the defaulting banliS under a bilateralobligation netting arrangement.

With the cost of increased complexity we can arrive at expressions forsystemic credit exposures in a more general case where a given finite se-quence of defaulting banks N has any length N, 1 < N < ti — 1.We will discuss it in the next section and extend Proposition 4 and Propo-sition 5 to a geneal case. But note that the simple cases in this sectionillustrate the essence of the nature concerning the stability of netting ar-rangements.

5 Systemic Risk Assessment — A General Case

We now turn to a general case of systemic risk and consider a possibility of• chain reactions of multiple defaults of participating banks to a multilateral


netting system. If a finite sequence of defaulting banks { in}, 1 at timet is given, a bank j E is called a defaulter or a defaulting bankand a bank j f Ijithi=1.....N is called a survivor or a surviving bank. Givena defaulting bank j, a surviving bank i is called positively concerned if

> 0.

and is called negatively concerned if

> 0.

5.1 Default Sequence and Loss Share Property

Let a finite sequence of banks { 3,}„=1 N with 1 < N < :I — 1 is a possibleset of defaulters at time t. For simplicity we assume that all the defaultsat time t occurs in a time stream matching the order in the sequence f.i4nduring the date t but before the beginning of the next date t + 1.

We first discuss some of the basic properties of the loss share rule for agiven sequence of defaulting banks at time t. We then proceed to describesystemic credit exposures in general.

Now, our assumed loss share rule is to distribute the losses of the clearinghouse, due to a default of a participant, among positively concerned surviv-ing banks according to the ratio of their notional bilateral profit levels withrespect to the defaulter at each round of default. Let us first define

vu„ (Oa71,„

m—irm„(t)E- E r.Lon (t)r„(/)

s=n

with a convention thatr„ft (t) 1

for n = 1, N — 1. In particular, we have

r2i(0 = r2I (02.11

r31 (t) = r7132i (t) + 523)1 r21(t)

. -4- rIt il ( r (N — altriv ( = (t) + 52sh (t)r22 (t) +

Eh� .71 (0+

for i and n < rn < N. rtmin (t) represents the proportion of theloss, vii),(t) + , caused by the n-th defaulter j„ at time t, which the bank ihas to share immediately after the m-th default. Coefficients, r,""m (t)S maybe called direct bilateral loss share coefficients at m-th round. We will neednotation to express proportions of losses shared by subsequent defaulters fortheir preceding defaulters

Given a finite sequence of banks { j r,},,= 1 N at time t, define for rn =2, , N, n = 1,...,N — 1, m > n + 1

•

•


•

•

In this notation, r21 (t) is the proportion of the loss of the initial defaulterat time t which must be shared by the second defaulter, and is equal tothe proportion of its notional bilateral profit among all the positively con-cerned survivors after the initial default. Similarly, r 31 (t) is the total sumof the proportions of the loss of the initial defaulter at time t which mustbe shared by the third defaulter by the time of its own default, and is equalto the sum of the proportion of the loss which the third defaulter mustshare immediately after the initial default plus the proportion of the pro-portional loss share of the second defaulter with respect to the loss causedby the initial defaulter, which the third defaulter must share immediatelyafter the second default. In general, r,,.,, i (t) is the total sum of the directand indirect proportional loss shares of the m-th defaulter for the loss ofthe n-th defaulter accumulated as the sequential defaults continued up tothe (m — 1)-st round. In this sense, coefficients rrn,„(t)'s may be called totalbilateral loss share coefficients among defaulters.

We can prove the following basic property satisfied by above loss sharecoefficients.

Proposition 6 [Property of Loss Share Coefficients] For any n = 1the following equality holds:

E• ?t in (Ors,(t) = rt in (0

(12)5=n

form=n+1 N andk=n,...,m— 1.

Proof : We first prove the equation (12) for n = 1. For k 1, (12) is truefor any in = 2 , N as both the left-hand side and the right-hand side ofthe equation (12) are trivially given by rL ii (t). By induction argument,assume that the equation (12) is true for all m = 2, , N up to k — 1,we shall show that (12) is true for all m = 2, ... ,./V for k. Indeed, for anym = 2, ... , N, we have

^k

(t)rn, (t)s=1

k- 1 k—iE (Ors' (t) + (t) E 1, 11 (Ors i (t)s=-1 5=1

= 71:111 ± ( ▪ (t)

V jrn j 1 (0 + V./mil (0 + (0+ - Eh�ii ik —3 Vhi 1 (t)

++

titiji (0 + ) (Eh�ii .... V hil(0+

tij,„j1 (t) EhOji ....,jk Vhj i (0 + + V7ki 3 (0+

Vhi l (0 ±EhOji,... Vhj, (0+


V.11,131(0+ Eh� ji it Vt131(0+

• r k (0+3m.11

where the second equality follows from the induction hypothesis. This com-pletes the proof of the equation (12) for all m 'V and k = 1.... , m-1for the case of n = 1.

By the induction hypothesis. assume that the equation (12) is true forall m = n + 1 .IV and k = n, , m – 1 up to the case of n – 1. We shallshow (12) for all m n + 1, N and k = n,...,m – 1 for the case of n.We use again an induction argument on k.

If k = ri, then for all rn = n + 1,... , N both the left-hand side and theright-hand side of (12) are equal to rt,n),,(t). Assume it is true up to k – 1.Then, we have

E r 3S ,in (Or „(t)S=Il

k-1 k-1E r.Ljn(t)r,„(t) + r,„on (t)E 50,(Orn(t)S=71 s=n

• r rk-1 ( t ) k (ork-1 (t)3-.7n .70n1n I 20n

viunco+ )4 _ Vtu 1 (0 + Ehoji ja V hi. (0 4 Eholi ..... Vhin (0'

„

Eh�Ji .1k-1V/11"W+

(Eh0.11 .741)11.1n(tr V.1kin ( t ) )

Vhin(t)".

t5„0 (0+

i;j_ j„ (0+

thin ftYr

= (o +,,„,„

where the second equality follows from the induction hypothesis. This com-pletes the steps in our induction arguments of a proof of the proposition.

■When the n-th bank j, in the sequence {j,)„„1 , defaults at the n-

th round, the bank j,,,. which itself defaults at a later m-th round, mustshare the burden of the loss of the clearing house due to the default ofthe bank jn. The proportional burden of the bank j„, from this defaultis rjtin (t). But at the next round of defaults, the (n + 1)-st bank jn+1in the sequence defaults. It brings in additional burdens of loss shares tosurviving banks in two ways. A direct additional burden of loss share mightcome from the loss of the clearing house in case the new defaulter 2, 41 hasa multilateral loss position. A more important indirect additional burdenof loss share comes from an increase in the proportions of loss share for

•

•

•

•


losses caused by each of its preceding defaulters. Thus, in this manner theproportional burden of the bank j,„ for loss caused by a preceding defaulter

j, accumulates as additional banks default prior to the default of the bankIn itself. Now, what the above Proposition 6 says is that at k-th round ofdefault this accumulated proportional burden of the bank j,„, for loss causedby its preceding defaulter .5, simply amounts to the ratio of its notionalbilateral profit level with respect to the bank j,, among all the positivelyconcerned survivors at the k-th round (r, < k < m).

As an immediate corollary to Proposition 6 one obtains

m—irmn( t) = E r;mjn(t)rsn(t)

Sr-71

7, (71-- t)ltnin •

We record it as the following proposition.

Proposition 7 [Loss Share Coefficients and Direct-Indirect Loss Share]Given two defaulters jn, and IT, with j,„ preceding In in the default sequenceat time t, the total sum of proportional shares of loss, due to the default ofin , that the bank j„, has to share directly or indirectly, is exactly equal tothe direct proportional loss share of 5„ excluding the first m — 1 defaulters,that is,

r,„„,(t) = rZi:s(t)

We are now ready to show the similar property as in Proposition 6 foran arbitrary participating bank to multilateral netting arrangements.

Proposition 8 [Loss Share Coefficients in General) For any participatingbank i E Ito the multilateral netting, the direct loss share coefficients at thern-th round of defaults with respect to the loss of the clearing house due tothe n-th defaulter (n < m), (t), is exactly equal to its accumulated directand indirect loss shares up to the m-th round of defaults, that is, for anyi , j„,n = 1, ... ,N or for any i = j„,,, with n' > rn > n, n' n,one has

rm (t) E (t)r (t)snson

(13)

for m n, , N.

Proof: By Proposition 6 it is sufficient to prove (13) for i j•.We use the result of Proposition 7 and an induction argument on in exactlyalike the one in Proposition 6.

Let i jN be given arbitrarily. Given any n = 1 N, if in = n,both the left-hand side and the right-hand side of (13) are given by ?Tin(t).


Assume, by induction. the equality (13) is true up to in — 1 for anyn = 1 N. Then, we have

E rf (Orri (t)1)n

Sr-11

In-,• Eijn (Orm (t) + r,";„ (t)r„(t)

s,1

• en-1 (0 +

t•un (0 tun (t)t (0+ • Eno' , VA)n(t)+ (Eh� VIv

a

(0+ ) th)n,- 1 (0+

i .,

Vt) , (0 + Eh�31 3,, ttv, (0+ + (0+

h*J Eh#31. 4, % JO +• E N (0+

rm1.7n

where the second equality is a consequence of the induction hypothesis andProposition 7. This completes the proof. •

Let us introduce further notation to express surviving banks' total directand indirect loss shares. For each i ...,24v and for each n = 1, , N,define

lm(t) E r,;(0r,(t).

l,n (t) is the total sum of proportional shares by bank i of the loss caused bythe n-th defaulter, accumulated at the end of the default sequence 1 1t4n ) n=1 at time t Note the similarity as well as the difference between 1,n (t) andr„(t). /m (t) is the total bilateral loss share of a surviving bank z withrespect to the loss of the n-th defaulter whereas r„(t) is the total bilateralproportional loss share of the m-th defaulter with respect to the loss of then-th defaulter. As an immediate corollary to Proposition 8, we obtain

1,„(t) • E rm (t)r,„,(t)111=71

= r;;:, (t).

Proposition 9 [Survivors' Total Loss Share Coefficients] For any surviv-ing bank i, the total sum of direct and indirect proportional shares of the lossof the clearing house due to the n-th defaulting bank is identical to the directproportional share of the loss of the &army house due to the n-th defaulterwhen all the N defaulters are excluded in the calculation of its proportionalshare, that is.

tin(0 = rij„(t)

•

•

•


5.2 General Systemic Credit Exposure

Given a finite sequence of banksy at time t with 1 < N < — 1,{inthe systemic credit exposure of bank i at time t with respect to the sequence{.it}n=1,...,N, written criji...j),(t), is defined by

( E,22. 0 y (t) E Ec7 ra (t)r„(t)) vii,„ (0 +(14)n=1 m=n

•

for i j i , jN and n < m < N.In order to understand the meaning of the definition given by (14), let

us consider a sequence of three defaulting banks 17M.1 at time t. Then,‘,, m=1.2,3we have

n(3j2j1 (t) + 71j, (t)r21 (t) + r ij1 (t)r31 (0) twi, (0+aii3+ (rFi2 (t) + rii,(0r32 (0) veil (t) + + 71i3(t)vni3(t)+

The meaning of the last term r53 (Ovah (t) + is straightforward. It is thebank i's direct share of the loss of the clearing house due to the third de-faulting bank j 3 . The second term consists of two parts. The first partri2j2 (t)vm2 (0 + is comparable to the last term except for the defaulting bank.It is the bank i's direct share of the loss of the clearing house caused by thesecond defaulting bank 12 . The second part rfj3 (0r32 (t)till12 (0 + representsthe bank i's indirect share of the loss that the third defaulting bank j 3 hadto share for the loss caused by the second defaulting bank. The first term iscomposed of three parts. The first part rIn (t)vlii ,(0 + is again straightfor-ward. It is the bank i's direct share of the loss of the clearing house due tothe failure of the first bank j i . The second part r? l (Orzi (O vtin (0 + is thebank i's indirect share of the loss that the second defaulting bank 12 hadto share for the loss caused by the first defaulting bank j i . Similarly, thethird part 4,(0r31(Ovnj, (0 + represents the bank is indirect share of theloss that the third defaulting bank j 3 had to share for the loss caused by thefirst defaulting bank 5'.

Thus, in general, in the expression (14), n-th term is composed of N —n+1 parts. The first part rZin (Owij„(0 + represents the bank i's direct shareof the loss of the clearing house due to the default of n-th bank. The m-thpart It, (t)r,(t)t.Hin (t) + , n < m < N, represents the bank is indirectshare of the loss that the m-th defaulting bank ju, had to share for the losscaused by the n-th defaulting bank jn.

With the interpretation of each term in the expression (14) as above, thesystemic credit exposure of a survived bank i represents the sum of its shareof losses that are directly or indirectly related to its forward book profitsposition with respect to defaulting banks. In order to see that a participantmay have to share a part of the losses that are indirectly related to its profitposition, let us take for example the term r;11 (t)r 31 (t)t• Hii (04 in a sequence


of three-bank defaults. It represents the part of the losses of the initialdefaulting bank j i which came to be shared by the surviving bank T becauseit was originally due to be shared by bank )3 that subsequently defaulted.

Let us rewrite the expression of systemic credit exposure given by (14)using the properties of loss share coefficients derived in the previous section.By . Proposition 9 we obtain

,.(t) = E E r:77'„ (t)r„(0) t .H.7„ (04n=1 rn=n

A'• E Itn(t)vm,(0+n=1

• r;;,‘ (Ovnir, (0+n=1

▪ I\ rtun (04

(0+.vh3, (0+The systemic credit exposure of a surviving bank is defined to be the total

of sequentially accumulated proportional shares of losses of the defaulters,that the survivor has to share in proportion to its notional bilateral profitlevel with respect to each defaulter. It can be calculated in two mutuallyequivalent ways: One natural way is to calculate the amount

NN

E E r;(t)r„,,,(0) t: H)„ (0+n=1 fn=n

as in the definition (14) which. at each round of an additional default, oneproportionally divides the total direct and indirect loss shares of the de-faulter among the positively concerned survivors at that round. Second wayis to calculate the amount

E v (0+1.1n ) v11(0+

which, at each round of default, simply divides the loss of the clearing housedue to the multilateral loss position of the defaulter of that round amongthe positively concerned survivors according to their notional bilateral profitlevel with respect to the defaulter of that round. Let us record this as follows:

Proposition 10 'Calculation of Systemic Credit. Exposure) The systemiccredit exposure of a surviving bank i can be calculated :n the following twoequivalent ways:

N N

u.n = E E run (0r,,‘„(0) 3„ (0+n=1 M

•

•

•

•

•


= rtilYn (t)vitin (0+.n=1

5.3 Benefit Function

We are ready to define the benefit of multilateral netting of bank i withrespect to a given finite sequence {j,„}„.=1 v of banks at time t. It isdefined by

ath .iN (t) E E Eij„(i) — (t)

(15)n=1

The benefit of multilateral netting is thus the difference between the sum ofnotional bilateral credit exposures and the systemic credit exposures withrespect to a given sequence of banks. So to speak, our approach is that of thecomparison by contingencies in assessing the possible benefit of multilateralnetting vis-a-vis bilateral netting.

It is convenient to rewrite (15) as below:

(t)

= kii1( t ) — (t) + (t)r21 (t) + . + /Z1 (OrN i (t)) (01 + • • •

+ [eiin (t) — (r7;,,(t) + eii: 1 ( t)rfri+iin(t) + . + r it (t)rv,i(t))t , lli„(t)1 +Eij iv (t) — rj)N (07.,H iN (0+

N-1 (N-n n+rnE (e ijn (t) — pn(t)min (t)) — E run (Or(n+n)t(t) 1 vxj.„ (0 + (16)n=1 n=1 m=1

whereE no,

Pi(t) El and pn (t) = (t) for n--> 2 .

Ehoi, vein (t)When there is a possibility of multiple defaults, a simple sum of bilateralbenefits of multilateral netting usually overstates benefits of multilateralnetting. This fact was pointed out in Proposion 4 in a most simple caseof systemic risk. We shall proceed to show that the statement is true in ageneral case also.0

It follows from (16) that

N-1 (N-n+ E E 1":),rn(t)r(n+rn)n (t)) t Hi(1)±1

n=1 m=1

(t)

E (sun (t) — nt),,(0) — [E(pn (t)— )tjtjn (t)n=1 n=1


• E Oi„„(t) — [ E (p,i ( t ) — 1)770„(t)n=1 n=1

Therefore, we obtain

a,,,

3, ( 0 <Efi„.„(t)n=1

if one of the following conditions hold:

• (37n)v).0.,.-1/04- > 0 and v,,„ (0 + > 0, or

• (a(n, m) with n < m)vu 1„(0 + > O. vi„,„„ (0 + > 0, and v,„„ (0 + > 0.

In the former instance one has

77..i„( t ) > 0 and pn (t) > 1

for 5„, and in the latter instance one has

r1,7,,(t)r,,,,,(0vH,„(0+ > 0

so that in each of these instances a positive amount must be subtractedfrom End's:, 0.1„ (t) to arrive at the amount 01)1 . jA. (t). Vt'e thus obtain thefollowing proposition.

Proposition 11 [General Decreasing Benefits of Multilateral Netting] Sup-pose that there is a possibility of multiple defaults among participants to amultilateral netting system with the initial defaulting bank having a multtlat-eral loss position. Then, for any participating bank i to the netting system.the sum of bilateral benefits of multilateral netting always overstates actualbenefits of multilateral netting, i.e. 00,...1„(t) < Ena 1 Oth,(1), if. in a se-quence of possible multiple defaulters at time t.

• there is a subsequent defaulter with whom the participant i is posi-tively concerned and who is positively concerned with the immediatelypreceding defaulter, or

n=1 In=1

N-1 (N-nE E ry,;:m (t)rvii-m)n(t)) vm„ (01

•

•

• there are at least two defaulters with the preceding defaulter having amultilateral loss position such that the participant i is positively con-cerned with the preceding defaulter and that the succeeding defaulter ispositively concerned with the preceding defaulter. •

jN (0

= nE=, (E0,, co

E%tow

— 4 \'n (t)vni, (t)+)

t (0±vor, (0+

)n=1 Eit�31 ,,, jjv Vh)n (i)*

„E=, ( Eh #.72 ,

Von(N0+

hh, (i)+)E Vain (0+ — Vnin (0+)

I*, 1 ,,,,, iN


In comparing merits of multilateral netting with those of bilateral nettingfrom a view point of credit risk reduction one might be tempted to com-pare multilaterally netted bilateral exposures with (actual) bilateral creditexposures. However, if one does compare bilateral netting with multilateralnetting using the sum of bilateral benefits of multilateral netting in the sim-ilar spirits as above, then Proposition 11 above warns us that in possibleevents of multiple bank failures one overestimates the benefits of multilat-eral netting because the actual benefit of multilateral netting is strictly lessthan the sum of bilateral benefits of multilateral netting due to a possibilityof an indirect sharing of losses of defaulting banks:

One should probably stress the fact that there are possibilities of indirectsharing of losses in face of a chain reaction of multiple defaults that couldreduce the attractiveness of multilateral netting system over that of bilateralnetting system. Actual situation for multilateral netting may be worse inthe sense that a reversal of relative attractiveness of netting between multi-lateral and bilateral netting may occur when one allows for a considerationof multiple defaults. In order to show that there are possibilities of negativebenefits of multilateral netting over bilateral netting, we wish to proceed toexhibit conditions under which we would have

)30,...m( t) < 0

in general.For this purpose it will become convenient to rewrite the equation (15)

again using Proposition 10 as below:

•

The last term of the last equality above can be rewritten using the equality(6)

E V h (0+ — V H (0±hOli ) N

E l'hj„ (0 + — max E vhjn ( t) + — E vh;„ (t) , 0hO hEl hEI

(t)

EN Vu„ (0+

Eh� .71 1-1112, (0+n=1

34 Netting and Systemic Risk •min E titun (0 - - E E tit).,

he/ h=31 //: h� .1t

Hence. we obtain

(min

{

E vh,„ (0 - - E vhj „ (tr, E 14,2„ (0 + }) (17)he/ 11031 /1#/1,-JY

We shall come back to this expression after introducing a couple of con-cepts concerning hedging in the next subsection

5.4 Comparison of Credit Risks under Multilateral and Bi-lateral Netting

Recall that bank i has a perfectly hedged position at time t if

(Vc E C, V/2 S T2) E.„ ( t, t 2 . 0..3E 1

If bank i has a perfectly hedged position at time t, then

vai (t) = 0.

Even if a bank has a perfectly hedged position, it cannot avoid default risk ofa counterparty This motivates us to introduce a further concept of perfecthedging. Let us say that bank t has a palmist perfectly hedged position withrespect to bank j at time t if

(dc E C and Vt 2 < T2) zu(t, t 2 , c) = 0. (18)

A perfect pairwise hedging with respect to all j i clearly implies a perfecthedging. But note that if a party has a pairwise perfectly hedged positionwith respect to its counterparty, then it will not face default risk of its coun-terparty. Since our focus is the reduction of credit risk rather than thatof market or foreign exchange risk in comparing different netting arrange-ments, we shall analyze in this subsection participants who have a perfectlyhedged position but not a pairwise perfectly hedged position with an initialdefaulter for, otherwise, it will not face credit risk. However, in order tominimize the complexity of analysis we analyze cases of a chain of bankdefaults where a participant has pairwise perfectly hedged positions withsecondary and subsequent defaulters.

A scenario of systemic risk we consider in this subsection is as follows:

•

•


• The original defaulter j i E I fails to meet its obligations at time t.The bank j i faces foreign exchange risk at time t and its multilateralposition shows a huge loss by an amount L 1 > 0 so that

twi , (0 + = viili (0+ = > 0 .

• Bank i has a perfectly hedged forward position so that

vill (t) = 0,

but it has bilateral profit position with the initial defaulter j i . Thebank i. however, has pairwise perfectly hedged positions with the re-maining defaulting banks 52 , ... , jN.

Then, we have

viin CO + = 0 for n 2„ N .

To ease our notation, put

•EhEt Win (0+ •

is the notional bilateral profit of the bank i with respect to the initialdefaulter S i and H is the total sum of notional bilateral profits of participantswith respect to the defaulter ji . 1, on the other hand, is the total sum ofnotional bilateral losses of participants with respect to 5 1 . We have

L I = — 1

and since we are assuming that the initial defaulter j i is making a huge loss:put

= MI

for some "large" number M. Then,

Li = (M — 1)1

Now, it follows from the equality (17) that we have

- r:Y (0 min E co — — EhEi h=i2

h3, ( t ) 1E %JO+hOit iN

• • r„N, (t) E co — — E V h3i(t)+hEl i)N

(19)

E Vo l (0+ > 0 ,

1 E E vhi , (0,hEI

II E vh11 (t)+ , andhE/

Vhji (n+=

h mt.72.-- 'Jr ttl.h(tr

EhEl vh,, co+E vh) , (0+

1.0.32

1 EhEl Vh.7)(0-Al EhEl VhJa(0+

> E Tit" , (0-

hel

6=

36 Netting and Systemic Risk •where the first and second equalities follow from the assumed properties thati has pairwise perfectly hedged positions with respect to banks j2 ,. , j N sothat v,,„ (0 + = 0 for n = 2, and that

vii) ,(0 + = E v„„ (0 + — E v„„ (t) - > 0.

hEl hEl

Using simplified notation. one obtains

51.7)...n: = Tiliti ( t )( ( - en ) (20)

Since HI = 6.411 and since i is positively concerned with j i so that r; )C (t) > 0,we have

< 0 <=i. 6M > 1 . (21)

This gives a necessary and sufficient condition under which one has

< 0 .

Recall that Al indicates that the total of the notional bilateral losses of theinitial defaulter with its counterparties is Al times the total of its profits withits counterparties. Thus, the systemic credit exposure under a multilateralnetting system is always greater than the sum of bilateral credit exposureswith respect to the possible sequence of defaulting banks if and only ifM times the proportion of the sum of notional bilateral profits of all theremaining defaulters to the total of notional bilateral profits with the initialdefaulter is greater than 1, that is. 6A1 > 1; in other words. if and only ifM is large enough so that its inverse is strictly less than the proportion ofthe sum of notional bilateral profits of the remaining defaulters to the totalof notional bilateral profits of all the participants with respect to the initialdefaulter. This condition will be met if the total of bilateral losses of theinitial defaulter is very large relative to its profits at time t.

Since one has

•

still another way of stating the condition above is that the total of notionalbilateral profits of the remaining defaulters, with respect to the initial de-faulter, is strictly greater than the total of notional bilateral losses of all theparticipants with respect to the initial defaulter.

We thus obtain the following proposition which we regard as the mainresult of the present paper:

•


•

•

Proposition 12 [Multilateral vs. Bilateral Netting) Consider a possiblearbitrary finite sequence of defaulting banks linIn=i , 1 < N < t/ at timet. Assume that the initial defaulting bank j i faced with high foreign exchangerisk suffers a huge mark-to-market gross loss by an amount L 1 which is AItimes the amount of the sum of mark-to-market notional bilateral losses ofother participants with respect to the initial defaulter ji.

Consider any participating bank that has a net bilateral profit positionwith the initial defaulter. And assume that at time t it has a multilateralperfectly hedged position and pairwise perfectly hedged positions with respectto each of the defaulting banks except for the initial defaulter. Then, itssystemic credit exposure under multilateral netting is strictly greater than thesum of bilateral credit exposures with respect to each of the banks along thesequence of possible defaulters under bilateral netting if and only if the totalof notional bilateral profits of the remaining defaulters, with respect to theinitial defaulter, is strictly greater than the total of notional bilateral lossesof all the participants with respect to the initial defaulter. or differently put,if and only if AI is large enough so that its inverse is strictly less than theproportion of the sum of notional bilateral profits of the remaining defaultersto the total of notional bilateral profits of all the participants with respect tothe initial defaulter.

5.5 Loss Share by Unconcerned Participants and PairwisePerfect Hedging

We have pursued the comparison of credit risk under netting arrangementsbetween bilateral and multilateral netting with a basic assumption that allthe losses incurred by the clearing house due to defaults of participantsare shared by positively concerned participants only. Our intention was tofree our analysis from risk factors of the clearing house and unconcernedparticipants. In a general multilateral netting arrangement, however, thereare three further elements that are built into a system. First is the impositionof a part of losses incurred by the clearing house to itself. Second is arequirement of collaterals and/or margins on participating banks. Third isthe imposition of loss sharing on participating banks regardless of whetherthey are concerned or not.

Admittedly, it is important to incorporate the role of the clearing housein reducing credit risk of participants as a management institution thatshares a part of the losses induced by defaulting participants, but it seemsto us that it will produce a considerable complication of our analysis in thatit will surely add the default risk of the clearing house itself to the credit riskof each participant. It means that one needs to incorporate credit exposureto the clearing house in calculation of credit exposures of each participant.We did not go into this aspect of credit exposures in this paper.

Collaterals and margins certainly reduce credit risk of participants of

38 Netting and Systemic Risk •any netting arrangements. Thus. it will not make a fair comparison betweenmultilateral and bilateral netting if we introduce requirements of collater-als and/or margins in a multilateral netting arrangement only. Hence. wealtogether left out these requirements in the present paper.

We wish to introduce in this subsection the third element pointed outabove — that is, loss sharing by unconcerned participants of netting arrange-ment. We shall describe a general loss share rule of unconcerned participantsby a given set of functions {9, : R+ R}tv that satisfy

• g,(L) > 0 for any L > 0 and E I

• LEI g.(L) < L for any L > 0

where L is the total losses of the clearing house.Given a finite sequence of possible defaulters N at time t. the

total sum of losses caused by defaulters in 6,2 )„ is denoted by

L(t) a E tw,„ (0+rt=1

If a set of functions {2,),E1 describe the part of loss sharing among par-ticipants, which is not necessarily related to their notional bilateral profitpositions with the defaulters, then out of L(t) the amount

Eg, (L(t)) (< L(0)2E!

is shared by the participants without any regard to their profit positions.Let us assume that the remaining part of the losses are allocated amongsurvived concerned participants pro rata to their notional bilateral profitlevels with respect to defaulters. Then. the loss of the clearing house due tothe defaulting bank is

VIiIn(tYP till,„(t) Le (t)

(t)+

where

L' (t) s L(t) — E g, (L(t))

Let us write

(0 + = vii)„(0 + L'(t)/211)•(t)+

Then, ti./2, (t) 4 represents the part of the loss due to the failure of thebank j,,. which is shared among the positively concerned participants. Thesystemic credtt exposure at time t of bank i •,7s with respect to

•

•

•

•


the given sequence q written cyrj,...5n(t), is defined as in (14)with vii ,„ (0 + replaced by tifin (t) + and an independent loss sharing part g,added, that is

(t) (

NE E

S ra(t)r,,i(t)) jn (t)+ + (L(0). (22)

n=1 m--=n

The benefit function (15) is redefined accordingly using the above (22).

alii...jiv (t)

E Co, (t) — Q (0. (23)nr=1

Now, it is intuitively clear that a pairwise perfect hedging gives the mostefficient case of bilateral netting. However, given Fact (3) which states thatperfect hedging implies the clearing efficiency, it would first seem natural toexpect that in case of a pairwise perfect hedging both bilateral and multilat-eral netting reduce credit exposures of a participant to nil. This expectation.however, is false. Contrary to the expectation, the case of a pairwise perfecthedging exemplifies the very nature of a weakness inherent in multilateralnetting. It is this: in face of a pairwise perfect hedging with respect to eachand every possible defaulter, bilateral credit exposures are reduced to nilwhereas systemic credit exposures remain due to indirect loss sharing undermultilateral netting.

We shall check the validity of the above statement below. Indeed, given

a finite sequence {in}n-1 N of banks at time t and a bank i { v,assume that the bank i had a pairwise perfectly hedged position with respectto bank j. Then, for any t, one has

Vij(t) =

E E q(t,tz,c)x,j(t,t2, c)t<t2 CT2 cEC

= 0.

by (18). Thus, if the bank i has a pairwise perfectly hedged position withrespect to every jrt .n = 1.... , N, then

(Vn)(Vt) eon (t) = vo„ (0+ = 0.

It therefore follows from the definition (23) that we have

4;1 . 4 !VW = Cr

Thus, this represents a worst possible case for multilateral netting if one has

• ar.„...,,,,(t)> 0, (29)


in that entire bilateral credit exposures vanish whereas the full systemiccredit exposures remain in multilateral netting. But, since rr, n (t) = 0 forall rz and tri > rt. it follows that

a. (t)

(E E r:;%,(t)rm,(0) 1, (t) + gt (L(t))n=1 fn=n

= g, (L(t)) > 0

when L(t) > 0. Hence : we obtain (24). Let us summarize this result in thefollowing:

Proposition 13 (Indirect Loss Share of Multilateral Netting) Considera multilateral netting system with a class of indirect loss share functions{gi } io: and compare it with bilateral netting. Given a possibility of a finitesequence of defaulting banks v.1 < N < ;I . at time t such thatthe initial defaulter has a multilateral loss position, assume that a bank i

j i, • • js had a patrwise perfectly hedged position with each of the banks

j i , Then,

I. The sum of bilateral credit exposures of bank i with respect to banks jAz is always zero.

2. Systemic credit exposure at time t of bank i with respect to the sequence

finln=1 v is strictly positive.

3. Benefit of multilateral netting at time t of bank i with respect to

tin n=1 A•

is strictly negative.

The implication of this proposition is that a bank may avoid defaultrisk in bilateral netting by maintaning the level of credit risk as well asmarket risk "very low". but it may not be able to avoid the default risk in amultilateral netting in cases where loss sharing by unconcerned participantsare required.

6 Concluding Remarks

In this paper we introduced a formal model of foreign exchange contractsnetting. and presented an analysis of multilateral and bilateral netting fromthe stand point of credit risk reduction. In particular, we compared the twodifferent forms of netting arrangements with respect to inherent systemicrisks involved. We would like to summarize some of the basic results of thepaper and consider implications for institutional arrangements and theirappropriate regulation.

•

•

•


•

1. If, in a very natural setting of a sequence of defaults, a chain reactionof which begins with a participating bank having a huge loss due to itsextreme foreign exchange risk taken by it, then for any participatingbank that is a positively concerned participant with respect to theoriginal defaulter is exposed to an increased credit risk in multilateralnetting system than in bilateral netting system provided that there isa subsequent defaulter that has a huge profit position relative to thetotal profits with respect to the original defaulter. (Proposition 5 andProposition 12)

2. The sum of bilateral benefits to a participant of multilateral nettingusually overstates its actual benefits if a systemic credit risk is to betaken account. (Proposition 4 and Proposition 11)

3. Consider multilateral netting arrangements which require loss sharingby unconcerned participants as well. If, in a sequence of multipledefaults, a participant has a pairwise perfectly hedged position witheach defaulter, then it is exposed to credit risk under multilateralnetting even if it is not exposed to credit risk under bilateral netting.(Proposition 13)

4. For those banks that are willing to take large foreign exchange risks.there will be not much gain in efficiency using a multilateral nettingsystem. Their incentives to join the multilateral netting system mustbe motivated reasons other than efficiency and reductions of transac-tion costs. One conceivable "motivation" of this might be related tomoral hazards and some sort of adverse selection which we did notdiscuss in this paper. (Proposition 3)

5. For efficient operation of a multilateral netting system, controllingand regulating foreign exchange risks of participating banks enhanceefficiency of the netting system. This implies desirability of settingprudent limits on each participant's mark-to-market value positionwith respect to the clearing house. (Fact 3)

•

92 Netting and Systemic Risk •References

11) Bank for International Settlements, Report on Netting Schemes, ("An-gell Report"), Group of Experts on Payment Systems of the CentralBanks of the G-10 Countries, February 1989.

(2) Bank for International Settlements, Report of the Committee on In-terbank Netting Schemes of the Central Banks of the Group of TenCountries, ("Lamfalussy Report"). Group of Experts on Payment Sys-tems of the Central Banks of the G-10 Countries, November 1990.

(3) P. Behof, Reducing Credit Risk in Over-the-Counter Derivatives,Economic Perspectives 17, Federal Reserve Bank of Chicago, Jan-uary/February 1993, 21-31.

I4J C.E.V. Borio and P. Van den Bergh, The Nature and Management ofPayment System Risks: An International Perspective, BIS EconomicPapers 36, February 1993.

(51 S. Chakravorti, Analysis of Systemic Risk in the Payment Systems.mimeo., Brents Group, LLC of KPMG Peat Marwick, LLP, November1995.

16) ECHO, Exchange Clearing House — Service Profile and ManagementOverview —, Exchange Clearing House Ltd., London, 1995.

Eisenberg, Connectivity and Financial Network Shutdown,mimeo., The Risk Control Co.. New York, 1999.

(8) P.M. Garber and S.R. Weisbrod, The Economics of Money and Bank-ing, Lexington, MA: D.C. Heath and Co., 1992.

191 J. Hull, Assessing Credit Risk in a Financial Institution's Off-BalanceSheet Commitments, Journal of Financial and Quantitative Analysts24, 1989, 989-501.

1101 D.B. Humphrey, Payments Finality and Risk of Settlement Failure. in:Technology and Regulation of Financial Markets, ed. by A. Saundersand L.J. White, MA: Lexington Books, 1986, 97-120.

III) N. Oda, A Private Communication. The Bank of Japan, Tokyo. Septem-ber 1995.

•

(12] K. Okina, On Rules to Reduce Credit Risks in Multiple Netting, (inJapanese). mimeo.. The Bank of Japan. Tokyo, September 1999.

113) Rochet and J. Tirole, Controlling Risk in Payment Systems,mimeo., March 1995. •

I Netting and Systemic Risk 43

I141 D. Schoenmaker, A Comparison of Alternative Interbank SettlementSystems, Discussion Paper 204, LSE Financial Markets Group, March1995.

(151 D. Schoenmaker, Measuring Systemic Risk in Interbank Payment Sys-tems, Discussion Paper, LSE Financial Markets Group, October 1995.

[161 A. Yamazaki. Heterogeneity of General Transaction Costs Technologiesand Equilibria with or without Money, Discussion Paper, HitotsubashiUniversity, Kunitachi, Tokyo, 1993.

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