MASTER’S THESISumu.diva-portal.org/smash/get/diva2:1209845/FULLTEXT01.pdf · Collateral...

Umea University

Master of Science in Engineering Physics

Department of Physics

MASTER’S THESIS

Collateral Optimization

Johanna Bylund

August 2017

Supervisor: Johan Sundberg, Cinnober Financial Technology, Umea

Examiner: Martin Rosvall, Department of Physics, Umea University

Collateral Optimization is a project done in the course Master’s Thesis in Engineering Physics,

30.0 ECTS at the Department of Physics, Umea University.

Master’s Thesis, Engineering Physics, Umea University.

Johanna Bylund, [email protected]

Supervisor: Johan Sundberg, Cinnober Financial Technology.

Examiner: Martin Rosvall, Department of Physics, Umea University.

mailto:[email protected]

Abstract

The financial crisis in 2008 has led to regulations and increased demand for high-quality collaterals,

which has forced many financial institutions, such as dealer banks, to improve their collateral managing

techniques. Optimizing the collateral pool in a dealer bank is known as Collateral Optimization.

However, not all banks and financial institutions use and manage their collateral portfolios in an

optimal way, which can lead to unnecessary costs. Banks can prevent this problem by investing in a

system that optimizes the collateral allocation or transforms to higher valued collateral. In this thesis,

we describe the basics of such a system in the case where a bank clears trades by several clearinghouses

with different collateral demands, constraints, and margin requirements. Therefore, the system must

allocate collateral in various ways to cover the risk margin at the different clearinghouses. We present

a Java implementation that takes as input collateral properties such as valuation, cost, type, risk, and

quality and outputs collateral quantity to cover risk margin. As constraints, the underlying model

takes clearinghouse requirements and collateral valuation, and a user can choose between two different

linear programming optimization methods, Simplex method and Branch and Bound. When using

the Simplex method, the user is either given a way to allocate the existing collateral or a suggestion

for how to transform to achieve a better solution. The Branch and Bound method always finds the

optimal collateral allocation and the optimal solution when collateral is missing. Besides proposing an

approach, we have examined other ways to look at collateral preference, cost and allocation, because

there is no right answer to the question on how to manage collateral, and every institution may need

an individual solution.

i

Sammanfattning

Finanskrisen 2008 gav upphov till regelverk som i sin tur ledde till okad efterfragan pa finansiella

sakerheter med hog kvalite. Detta har tvingat manga finansinstitut, sasom kommersiella banker,

att forbattra hanteringen av sakerheter. Att optimera anvandningen av sakerheter kallas Collateral

Optimiztion. Detta ar nagot som inte alla banker och finansinstitut gor pa ett optimalt satt, vilket

kan leda till oonskade kostnader. Detta kan dock forhindras genom att investera i ett system som

optimerar anvandningen av sakerheterna eller omvandlar existerande sakerheter till hogre varderade.

I denna rapport beskriver vi grunderna for ett sadant system i det fall en bank clearar handel hos flera

clearinghus med olika krav, begransningar och riskmarginaler. Detta gor att systemet maste fordela

sakerheter pa olika satt for att tacka riskmarginalen vid de olika clearinghusen. Arbetet resluterar i

en Java-implementation med inputparametrar som ror de finansiella sakerheternas vardering, kostnad,

typ, risk och kvalitet dar kvantiteten for varje sakerhet ar det som modellen resulterar i. Som begran-

sningar anvander sig den underliggande modellen av clearinghusens krav och vardering av sakerheterna,

dar anvandaren av systemet kan valja mellan tva olika linjara programmeringsoptimeringsmetoder,

Simplex-metoden och Branch and Bound. Nar Simplex-metoden anvands far anvandaren antingen ett

satt att allokera de befintliga sakerheterna eller ett forslag pa hur man omvandlar dessa for att uppna

en battre losning. Branch och Bound-metoden finner alltid den optimala sakerhetsallokeringen och den

optimala losningen i det fall dar finansiella sakerheter saknas. Forutom att foresla ett tillvagagangssatt

har vi granskat andra satt att titta pa sakerhetspreferenser, kostnader och allokering, vilket har gjorts

pa grund av att det inte finns nagot ratt svar pa fragan om hur man optimerar finansiella sakerheter,

och varje institution behover en individuell losning.

ii

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Collateral optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Disposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory 6

2.1 Collateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Credit quality rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Collateral markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2.1 Repurchasing agreement (Repo) . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2.2 Securities Lending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2.3 The derivatives market . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2.3.1 Exchange-traded derivatives . . . . . . . . . . . . . . . . . . . 9

2.1.2.3.2 Over-the-counter derivatives . . . . . . . . . . . . . . . . . . . 9

2.1.3 Legal agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Collateral management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Collateral transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Centralized clearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2.1 Risks of Central Clearing . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2.2 Requirements and constraints . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2.2.1 Initial margin and Variation margin . . . . . . . . . . . . . . . 16

2.2.2.2.2 Haircut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2.2.3 Credit quality . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2.2.4 Lower and upper concentration limits . . . . . . . . . . . . . . 17

2.2.2.3 Central counterparty charges . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Collateral cost model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3.1 Funding costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.3.2 Additional costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3.2.1 Transaction cost analysis . . . . . . . . . . . . . . . . . . . . . 20

2.2.3.2.2 Custodial Fees and Safekeeping Fees . . . . . . . . . . . . . . . 22

2.2.4 Risk management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

iii

CONTENTS iv

2.2.4.1 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4.1.1 GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4.1.2 EGARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.4.2 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.4.3 Liquidity Adjusted Value at Risk . . . . . . . . . . . . . . . . . . . . . . 24

2.2.4.4 Probability of default . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4.5 Exposure at default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4.6 Loss given default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4.7 Expected loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Collateral Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Optimizing cost models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1.1 Preference ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1.2 Market based ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.1.3 Economic based ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.2 Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2.1 Waterfall allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2.2 Numerical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Operational research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.1 Linear programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1.1 Simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.2 Integer linear programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.2.1 Branch and bound algorithm . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Method 37

3.1 Collateral Optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1.1 Collateral valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.1.2 Clearinghouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.2 Mixed integer linear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.2.1 Mathematical optimization problem . . . . . . . . . . . . . . . . . . . . 45

3.2 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Software implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Results 48

4.1 Ranking system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Collateral allocation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.1 Simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.2 Branch and Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.3 Analyzing the different methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Collateral Optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Discussion 51

5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

v CONTENTS

A Test cases - Revised Simplex method I

A.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI

A.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

A.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII

A.4 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

A.5 Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X

A.6 Case 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI

A.7 Case 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII

B Test cases - Branch & Bound XIII

B.1 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII

B.2 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVIII

B.3 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIX

B.4 Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XX

B.5 Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXI

C Result - Tables XXII

C.1 Revised Simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIII

C.2 Branch and Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXXI

Chapter 1

Introduction

Cinnober creates financial solutions to different kinds of financial parties, such as clearinghouses and

banks. A clearinghouse is a financial institution where one of its main function is to act as a central

counterparty (CCP), i.e handle the risk of a financial contract written between a buying and selling

part, later also called clearing members. If something unexpected were to happen, and one of the

parties goes into default, the clearinghouse will guarantee that the obligations of the contract will be

fulfilled. To do this the clearinghouse needs resources, which are gathered by demanding collateral

deposits from every clearing member. There are many different types of collateral, and since every

clearinghouse have different demands and requirements on these and every clearing member can be

connected to several clearinghouses, the optimization of the internal mix and usage of collateral for a

clearing member is of interest in order to use the collateral resourcefully.

This can be done in several different ways, however, the research on this are mainly done by finan-

cial institutions or companies selling a collateral optimization solution to financial institutions. Every

institution wants the best solution possible and needs this solution to be protected from the competi-

tion, which leads to a low amount of published solutions. Due to this lack of public information, the

optimization approach may not be obvious and this thesis is written in order to diminish this gap of

knowledge.In this thesis the essential parts needed to optimize a collateral portfolio are established by

investigating the internal optimization of collateral usages in a dealer bank that have contracts cleared

at several clearinghouses. Using this information, the main factors are determined and used to create

a mathematical model that can be optimized by using an operational research technique.

1.1 Background

In finance and banking, clearing is the process of all activities from the time a financial contract

is entered until it is settled, i.e. at the time the securities to fulfill the contractual obligations are

delivered.

In the market there are several market participants, here called parties, that can trade with each other.

If party P1 enter a contract to exchange financial flows with a party P2, the trade is registered between

the counterparties as illustrated in Figure 1.1.

1

1.1. BACKGROUND 2

P1 P2B

A

Figure 1.1 – The figure illustrates the flow of assets between two parties P1 and P2 where A and Bare the units of assets.

In a more real-life example each party have several counterparties and by this, several contract agree-

ments to fulfill. This can be rather messy, but the parties can handle and sort this by processing

the payment by themselves through bilateral netting. The downside with this is that each party still

suffer from the counterparty risk. The solution could be to use a clearinghouse acting as a central

counterparty (CCP).

P3

P4

P1

P2

P1 P3

P4

P2

P1 CCP P3

P4

P2

3

1

75

8

3

12 3

4 7

20

12

58

153

8

5

6

27

14

Figure 1.2 – From left, an example of a network of parties that have engaged in a trade can be seen,and at some fixed time transfers different units of assets to each other. The middle picture describes howthe network looks when applying bilateral netting, and the right depicts the network of using a centralcounterparty as an intermediate.

The clearinghouse is a financial institution that take the opposite position of each side of a trade,

acting as an intermediate and standing between two counterparties that are connected to the clearing-

house. These counterparties, also called clearing members, uses the clearinghouse in order to reduce

the risk that one of these will default, or in any other way, fail to honor its trade settlement obligations.

This means that the clearinghouse are responsible for the contract, and the clearing members expects

that all transactions will be fulfilled. In many cases, the clearing members are obligated to use the

clearinghouse due to regulatory demands. As can be seen in Figure 1.2, the initial trades between the

parties are simplified to one trade per party, with the clearinghouse in the middle.

To be able to fulfill obligations that a clearing member can’t, the clearinghouse requires the clearing

members to hold margin to cover unsettled positions. This margin is also described as collateral

deposits. Its crucial that the amount are correct considering that the clearinghouse don’t want the

members to pay too much and risk to loose the member to another clearinghouse, and it don’t want to

be in a position where it can’t fulfill its obligations. It is the clearinghouses responsibility to monitor

the margin levels and make sure that it covers the outstanding trades. Here, collateral can be whatever

a clearinghouse can convert to cash on short notice and with low liquidity risk. The most usual types

of collateral are cash, government bonds, corporate bonds, bank guarantees and equities.

3 CHAPTER 1. INTRODUCTION

1.1.1 Collateral optimization

Cinnober has a system for clearing of financial transactions, TRADExpressTM RealTime Clearing that

is used by clearinghouses, inserting itself as the counterparty to both the buyer and seller. Cinnober is

also currently in the process of exploring Client Clearing, a back-office system targeted for banks that

are connected to several clearinghouses, which is the outset of this thesis. In this case it is interesting to

emulate several of the clearinghouses processes of covering risk and optimize the banks total collateral

requirements against all of its connected clearinghouses.

Considering a case where a financial institution, party P1, have contracts that are cleared at several

different clearinghouses (see Figure 1.3), where each clearinghouse have different collateral requirements

and calculates the risk margin value for each contract differently. This means that P1 needs to have

a pool of different types of collateral, later also called a collateral portfolio. The issue here is to use

the existing collateral in the best way possible, and finding a good allocation process to cover the risk

values in the most efficient way.

P1

CCP1(cash)

CCP2(bonds)

CCP3(equities)

CCP4(cash)

Figure 1.3 – The figure describes a simplified example of how it can look when a financial institution P1trades at four clearinghouses CCP1, CCP2, CCP3 and CCP4. Every clearinghouse demands differenttypes of collateral to cover the risk associated with the trades.

1.2 Aim

The term collateral optimization includes all actions done by a financial institution to make a more

effective use of its existing portfolio of collateral assets. The aim of this thesis is to investigate how

a financial institution can optimize the collateral mix in this portfolio, that are used to cover the risk

margin value of a contract cleared through a central counterpart. The research regarding this is mainly

done by financial institutions where there are barely any public research published, which is one of the

factors that contribute to the fact that the solution to the collateral optimization problem isn’t very

obvious.

1.3. GOAL 4

A common fact of collateral optimization is that one of its purpose is to survey the use of collateral

and manage the risks in a more efficient way. This can in turn lead to lower funding costs of collateral,

and by this, give higher profits. Also, optimizing the collateral usage may boost an institution’s ability

to attract extra liquidity, and lower the need for costly and potentially risky collateral transformation

services. [1]

A condition for collateral optimization is that the financial institutions, in this case a bank, have

good understanding about their existing collateral, including knowledge of the cost, use and settle-

ment procedures for each collateral. In order to optimize the collateral usage, the portfolio allocation

of the collateral could be improved by considering the requirements and demands of the connected

clearinghouses and improving the effective parts that the market controls. [1]

To summarize, one can say that the parameters that controls the optimization of collateral are

• Collateral requirements

• Collateral positions

• Settlement procedures

• Collateral costs.

In this thesis, these factors are studied and elaborated to identify the importance in a collateral

optimization model.

1.3 Goal

The goal of this thesis is to determine the factors that affect collateral optimization, create a mathe-

matical model and use an operational research technique/algorithm to solve the collateral optimization

problem. To do this, one needs to present tools and guidance on how to manage and optimize the

internal collateral portfolio of a financial institution. The main goal of these tools is to minimize the

cost of collateral, maximize the funding and/or liquidity capacity of the collateral inventory and mini-

mize the funding costs.Another goal is to automate the allocation process which is solved by using an

operational research algorithm on the created mathematical model, where it all will be implemented

in Java.

1.4 Limitations

In order to make the perfect collateral optimization model for a specific financial institution, one would

need to get an accurate inventory of the collateral positions. However, in this thesis the work and cost

for this isn’t considered in the implementation due to lack of reference data. Another limitation of this

thesis is that even though some data is gathered as reference, the thesis doesn’t handle any type of real

life example. This makes it hard to fully investigate how the exactness of the collateral inventory and

the costs that follows affects the collateral optimization model, which is the reason for only composing

this and not fully implementing it into the collateral optimization model.

5 CHAPTER 1. INTRODUCTION

1.5 Disposition

This thesis starts with this chapter called Introduction followed by the four chapters Theory, Method,

Results and Discussion. The chapter Theory contains the relevant theory to create a collateral opti-

mization model starting by the fundamental facts (the quality of different collateral types, the collateral

markets, legal agreements and regulatory institutions and documents). This is followed by some theory

specific for this thesis called collateral management. Here, central clearing is presented followed by

collateral requirements, costs and some tools to manage the risk of a collateral portfolio. After this,

the collateral optimization problem is described as an issue with two parts: collateral valuation and

the collateral allocation to cover the risk of trading by a clearinghouse. The Theory chapter ends by

presenting the most common techniques to solve a linear optimization problem. After this comes the

chapter called Method that contains the method to optimize the collateral usage, the input data to

the collateral optimization model as well as the implementation approach and the output that comes

with it. The results of this thesis is gathered in chapter 4, which later are discussed in chapter 5 where

suggestions of future work also are presented.

Chapter 2

Theory

This chapter starts by specifying different types of collateral and its properties. With this established,

the collateral market and regulations are defined, which is done to describe the components and usage

of collaterals. It is also evaluated to identify the means that can be used to transform collateral

and to investigate the settlement procedures in order to identify possible costs. After this, a dealer

bank’s collateral management and relation with the collateral market is illustrated together with the

connection to centralized clearing where risks, costs and collateral requirements also are presented. The

chapter continues with describing the risks and costs of acquiring, transforming and using collateral in

order to produce a legitimate cost model. By this, the optimization of the cost model and optimization

of allocation to clearinghouses are presented along with the operational research.

2.1 Collateral

Collateral can be described as an easily priced liquid asset used in an agreement to provide cover

against credit risk exposure, and serves to diminish loss in case of a counterpartys’ default. A liquid

asset is defined as cash or something that quickly can be converted into cash with minimum affect

on the market price, e.g. government bonds, corporate bonds, bills, equities, metals commodities,

etc. Depending on how usable a specific collateral is to a collateral taker, it can either be labeled

acceptable or unacceptable. If a clearinghouse labels the collateral as acceptable, it means that i

accepts the collateral to be used. The acceptable collaterals are further divided into High Quality

Liquid Assets (HQLA) and High Quality Assets (HQA). To determine if a specific collateral is a

HQLA, one need to consider the Level 1 and Level 2 definitions of the Basel III regulations regarding

Liquidity Coverage Ratio [2]. HQA on the other hand has a wider definition where the collateral takers

normally decides which collateral types that belongs here by considering the usability of these. Some

examples of collateral holders are central counterparties, central banks, banking institutions, Central

Securities Depositories (CSDs) etc. The collateral market is further described in Section 2.1.2 below.

2.1.1 Credit quality rating

A way to compare and evaluate the credit risk of a specific business or government is to use credit

ratings. These ratings can further be used to distinguish the quality of different collaterals.

Sovereign credit ratings are used to compare different government bonds, where the credit risk level is

used to investigate the level of risk of the investing environment of a country. To compare the credit

6

7 CHAPTER 2. THEORY

quality of the financial instruments of a corporation, such as corporate bonds or stocks, one can look

at credit ratings from different credit rating agencies (Standard&Poor’s, Moodys and Fitch Ratings).

These rating agencies use letter designations such as A, B and C where A is highest and C lowest.

Higher grades implies that the probability of default is lower, and the lower grades implies a somewhat

higher default risk.

2.1.2 Collateral markets

The collateral markets are here described to originate from a dealer bank’s perspective. As earlier

mentioned, collateral is used by financial market participants to protect themselves against credit

exposures and are especially used for repurchase agreements, secured lending, derivatives transactions

and by the central bank market. The usage in these markets are further described in the subsections

below.

2.1.2.1 Repurchasing agreement (Repo)

A repurchasing agreement, repo, is a loan secured against collateral where the transaction involves one

party selling assets to a counterparty and at the same time commits to repurchase the same or similar

assets from the counterparty at maturity. [3] In this case the assets can be defined as securities or

other securities collateral, where the payment can be in cash or a cash equivalent collateral. This is

further illustrated in Figure 2.1. Due to the fact that the value of a sold security and any securities

collateral will change on a daily basis, repo transactions are subject to mark-to-market. This means

that one needs to account of daily accruals on both the securities and the cash.

Between the sale and the repurchase, the seller gets use of the cash and the buyer gets legal title to

the securities and can re-use them during the term of the repo by selling the assets outright, repo-ing

them or pledging them to a third party. However, the buyer needs to buy back the assets before the

end of the original repo in order to be able to sell them back to the seller. If the seller defaults, the

buyer can liquidate the securities to repay some or all of the cash. [4]

BuyerSeller

1a. collateral securities

1b. cash

2b. return on cash

2a. returned collateral securities

2b. returned cash

Figure 2.1 – Repurchase agreement. At (1a), the Seller sells assets to the Buyer that (1b) uses cash orcash equivalent to mitigate the seller’s risk of not being able to return its assets. At maturity, the Seller(2a) repurchase the assets sold and to account for an eventual change in the securities value, the sellerpays the mark-to-market price, (2b).

2.1. COLLATERAL 8

2.1.2.2 Securities Lending

The collateral in securities lending and borrowing is used to mitigate the lender’s risk of not being

able to return borrowed securities. In the same way, the borrowed securities act as protection against

the case where the lender fails to return the collateral, which is typically in the form of cash or other

securities. This is further shown in Figure 2.2. Securities lending and borrowing operates the same

way as repo agreements when securities lending is against cash with the difference that different legal

agreements apply and the securities lent here are often equities. [4]

SecuritiesLender

SecurityBorrower

1a. collateral securities

1b. cash or collateral

Fee

2a. returned collateral securities

2b. returned cash or collateral

Figure 2.2 – Securities Lending. At (1a), the Lender lends securities to the Borrower that (1b) uses cashor cash equivalent to mitigate the lenders’s risk of not being able to return its securities. At maturity,the Lender (2a) repurchase the securities sold and to account for an eventual change in the securitiesvalue, pays the mark-to-market price, (2b).

As can be seen in Figure 2.2, the borrower pays a fee to the lender for the use of the loaned security.

The securities lending market is especially popular to use when dealing with less attractive collaterals

that one wants to transform into more liquid securities.

2.1.2.3 The derivatives market

A derivative is a contract between a buyer and a seller concerning a transaction to be completed in a

future point of time. The main types of derivatives are forwards, futures, options and swaps where the

main categories of underlying assets are interest rates, foreign exchange, equity and commodities. In

the derivatives market, fully standardized products are traded on exchanges, while more idiosyncratic

products are traded bilaterally over the counter (OTC), which means that most derivatives are on the

OTC market.

To describe the derivatives market one can say that it works as a large professional wholesale market,

called inter-dealer market, where trading occurs between large broker-dealers such as banks, invest-

ment firms or securities houses.

Due to the fact that the OTC derivatives market have non-consistent infrastructure services, several

third-party service providers are available at all the steps. Exchange-traded derivatives are cleared

by central counterparties, where OTC derivatives can be cleared on both bilateral basis and through

central counterparties. [5] However, due to the fact that the derivatives market has grown, the impor-

tance of counterparty risk has grown with it and the regulatory connected to the OTC market most

of the time demands centrally cleared derivatives.

9 CHAPTER 2. THEORY

2.1.2.3.1 Exchange-traded derivatives

Exchange-traded derivatives (ETD) are derivatives traded through derivatives exchanges. A derivatives

exchange can be described as a market where the parties trade standardized contracts that are pre-

defined by the exchange. This means that the derivatives exchange acts as an intermediary to all

related transactions, and takes initial margin from both parties as a guarantee.

Exchange Traded (Standardized) = Market Risk

2.1.2.3.2 Over-the-counter derivatives

The OTC derivative market is the largest market for derivatives and mainly involves parties such as

banks or hedge funds. Comparing the bilateral and centrally cleared OTC derivatives markets (similar

to ETD market), the participants in bilateral markets are more exposed to the default risk and capital

charges. In bilateral markets the costs arise from counterparty risk, funding and capital, where the

costs in the central clearing market mainly is the funding cost with smaller capital charges. [6]

Bilateral OTC Traded (Customized) = Market Risk + Counterparty Risk

The market for selling and purchasing currencies can be described as an OTC-market where there is

no organized exchange on which currencies are traded. The most common participants here is the

commercial banks who provide two-way quotes for a number of currencies. This means that each bank

will quote a bid/ask rate for buying/selling a specific currency. The difference between these is called

spread and can be called the source of profit for the dealer. Considering the foreign-exchange market,

one always use the words buy, sell, purchase and sale from the perspective of a dealer. This means

that if a dealer buys a foreign currency, the payment is equivalent in terms of the domestic currency,

and if the dealer sells a foreign currency, the equivalent amount in terms of the domestic currency will

be taken. [7][6]

2.1. COLLATERAL 10

2.1.3 Legal agreement

There are several different legal agreements that applies to the collateral markets and due to its

complexity and comprehensive documentation, gathered in Table 2.1 to be further investigated when

charting the collaterals in a dealer bank’s collateral portfolio.

Table 2.1 – The legal agreements required to use the different collateral markets.

Collateral market Legal agreement

Repurchase Agreement(Repo)

Global MasterRepurchase Agreement

(GMRA)

Securities LendingGlobal Master

Securities LendingAgreement (GMSLA)

OTC-Derivatives(cleared)

andExchange TradedDerivatives (ETD)

Product specific operationallegal agreement for

clearing withcentral counterparties and

exchange specific documents

OTC-Derivatives(uncleared)

International Swaps andDerivatives Association (ISDA)Credit Support Annex (CSA)

to the Master Service Agreement (MSA)

2.1.4 Regulation

The financial crisis in 2008 resulted in stricter regulations which is very much applied on the usage

of collaterals. Examples of this are mandatory clearing of standardized derivatives and higher capital

requirements for transactions without clearing requirements. These regulations applies on every part

of the collateral market and are globally coordinated by Financial Stability Board (FSB) and Basel

Committee on Banking Supervision (BCBS). A short description of these rules and regulation can be

seen below. [6][8]

Liquid Coverage Ratio (LCR) and Net Stable Funding Ratio (NSFR) are developed to achieve

separate, but complementary objectives and are a part of BCBS package of reform measures

(Basel III). LCR helps a bank to ensure that it has a sufficient amount of HQLA to survive a

stress scenario lasting for a month forward. The NSFR however, helps the bank to maintain a

stable funding profile with the help of longer-term funding strategies. [8]

Shadow banking rules. To minimize risk for the shadow banking sector (the collection of non-bank

financial intermediaries), the FSB have decided to apply mandatory haircuts1 for securities that

isn’t cleared by central clearing, e.g. repo transactions and securities lending. [8]

1Further described in Section 2.2.2.2

11 CHAPTER 2. THEORY

The European Market Infrastructure Regulation (EMIR) is a part of Europe’s actions to in-

crease the transparancy in the OTC derivatives market and reduce counterparty risk. EMIR

requires that liquid swaps (liquid derivative contracts) should be cleared at a central counterpart

such as a clearinghouse, meaning that both the sell and buy side needs to post initial margin

(IM) and variation margin (VM). The IM and VM can typically be in the form of HQLA (cash,

AAA-rated government bonds). [8]

The second Markets in Finacial Instruments Directive (MiFID II) requires that all trading

of derivatives that are adequately liquid and suitable for clearing take place on regulated trading

venues such as organised trading facilities (OTFs), regulated markets or multilateral trading

facilities (MTFs). In the US, the Dodd-Frank Act establish all these new measures and structures.

The Basel Committe on Banking Supervision/International Organisation of Securities Com-

missions (BCBS-IOSCO) framework establish further steps to be taken between parties

which handles with non-centrally cleared OTC-derivative instruments. Some examples of steps

that are either seen as optional or not performed at this moment are calculation of exposure and

resulting exchange of VM, exchange of two-way IM (requiring amendments within the ISDA CSA)

and establishment of policies for minimize disputes by reconciling portfolios, risk sensitivities,

risk factors and margin calls with counterparties. [8]

The applied regulation of the different markets are summarized in Table 2.2. [9]

Table 2.2 – Regulations that should be taken into consideration for Collateral management.

Collateral market Regulation


FSB frameworkto standardizerepo haircuts

Securities LendingFSB framework

on shadow bankingrequirements



EMIRor Dodd-FrankAct in the US


BCBS-IOSCO

2.2. COLLATERAL MANAGEMENT 12

2.2 Collateral management

Collateral management is a key function with close links to trading, treasury risk, liquidity manage-

ment and capital optimization.[10] The easiest form of collateral management is that the financial

institution have several separate desks that each have a collateral portfolio to use. But in order to

optimize the collateral usage in a bank, one should centralize the management of this collateral, mean-

ing that instead of having several different desk that each handle collateral there can be a central one,

see Figure 2.3. The pros of having it this way is that if one desk lacks of a certain collateral, where

another one have excess of this collateral, the bank doesn’t need to buy more or transform any of the

other collaterals in order to cover the need in the own institution. [8]

Desk 2Derivatives

(OTC cleared/unclearedand ETD )

Desk 3Equities

(Securities Lending)

Centralized Collateral Management

Desk 1Fixed Income

(Repo)

Figure 2.3 – An illustration of centralized collateral management in a financial institution. Withoutcentralized management, each of the three desks uses separate collateral pools for each market. By onlyhaving one collateral pool for all collateral usages, the collateral usage can be optimized.

In short, with centralized management, the bank can select and trade the collaterals across all products

and trading. Also, all requirements are centralized monitored which simplifies the funding, making it

firm-wide. Since all collaterals are gathered in the same pool, this can be handled as one large port-

folio of collaterals. To optimize the collateral usage, there needs to be a comprehensive investigation

over all collaterals available in the institution. The inventory needs to cover both existing collateral

and collateral kept in other institutions such as custodians and central securities depositories (CSDs),

together with an overview of the assets that can be converted into eligible collateral by securitisation.

In short, all collateral positions in the bank needs to be monitored.[8]

After this, the bank needs to keep this collateral pool to be as liquid as the regulatory institutions

demands, and also follow other rules and guidelines regarding these collaterals. The collateral pool

composition and size should further be allocated to optimally match the needs, and effectively allocate

to cover risk margin values set by the counterparties. [11]


2.2.1 Collateral transformation

Due to recent regulations it will be mandatory to use a CCP for standard OTC-derivatives, which

implies an increased demand for high-quality collateral assets. From the regulations regarding the

Liquidity Coverage Ratio (LCR), every financial institution should hold a buffer of HQLA on their

balance sheets as well as other assets that can be used as collateral. In the case where a financial

institution doesn’t hold the desired quality or types of collateral that a clearinghouse demands, the

problem can be solved by collateral transformation. [1]

Collateral transformation (also called collateral upgrades or collateral swaps) are services where a

dealer bank can be offered to upgrade lower quality assets into CCP eligible collateral. Collateral

transformation can also be used to adjust the portfolio of collaterals to use the portfolio in the most

efficient and least costly way. In order to transform collateral, a financial institution can e.g. trade

on the repo or securities lending market or use an external services provider (securities settlement

systems, payment systems, internal systems of custodians etc.), see Table 2.3 below. [5] To further

illustrate the collateral market and its connection to the dealer bank, see Figure 2.4.

Table 2.3 – The collateral usage in the market from a dealer bank’s perspective.

Collateral market Description


To securecash replacements,collateral in form

of bonds or equity are used

Securities Lending

To securea loaned security,

cash or other collateralis delivered



Collateral is deliveredto or from a CCP

to cover risk margin value.Includes initial margin (IM)and variation margin (VM)


Collateral is movedbetween tradingcounterparties.

Mainly variationmargin (VM)

Repo transactions and securities lending are earlier discussed, and since the actions taken by hedge

funds and central banks reminds of these, it is only described briefly. Here a prime brokerage is de-

scribed as a special group of services that many brokerages give to special clients. The services provided

under prime brokering include securities lending, leveraged trade executions and cash management,

among other things. The main function of a central bank is to control the nation’s money supply. Most

central banks undertake repo transactions in the market to control short-term interest. The dealer

bank can also trade uncleared OTC-derivatives, however this is not depicted in Figure 2.4 due to the

fact that only the collateral transformation and central clearing are of interest here.


Central Banks

DEALER BANKwith

Centralized CollateralManagement

CustodiansMoney Market

Funds

CCPs

Hedge FundsCollateral Cash

Collateral

Cash

SHORT-TERM

(REPO)FUNDING

Collateral

Cash/Collateral

SECURITIES

LENDING

RISK TRANSFEROTC + Exchange

derivatives

Cash/Collateral

Collat

eral

Cash

PRIME

BROKERAGE

Figure 2.4 – This is the usages of collateral for a dealer bank. The bank uses central counterparties(CCPs) to transfer the risk with OTC derivatives or ETD, and by this, the bank needs to pledge collateralto cover the risk margin value of these contracts. In order to transform collateral to cash or the otherway around the bank can use different financial organizations to borrow (transform) the collateral/cash.With the transformation comes also a fee the dealer bank needs to pay. [12]


2.2.2 Centralized clearing

Clearing through clearinghouses acting as CCP’s are created to manage and reduce counterparty risk

in bilateral markets, illustrated in Figure 2.5.

The legal process of replacing the original counterparties and becoming the single counterparty for all

participants is generally called novation. Novation gives the clearinghouse a huge portfolio that always

are balanced and not affected by market risk due to its offsetting positions, but the clearinghouse do

experience a huge counterparty risk, and faces the earlier mention risk; that one of the parties will fall

into default. The CCPs main function is to be between counterparties acting as the buyer to every

seller, and seller to every buyer, accepting all obligations and rights that comes with it. By this the

counterparty and systemic risks will be reduced and the CCPs reallocate the default losses through

different methods such as demanding collateral (margin) requirements from their clearing members.

P1 P3

P4

P2

P1 CCP P3

P4

P2

3

1

75

8

3

12 3

4 7

Figure 2.5 – The illustration to the left describes an example of bilateral netting where the figure to theright illustrates centrally cleared markets. P1, P2, P3 and P4 can here be described as counterpartiesin the left illustration and direct clearing members to a clearinghouse in the right illustration.

The general role of a CCP is to take responsibility for closing out all the positions of a defaulting

clearing member, but also to set rules and standards for the clearing members. In order to cover the

losses in the case of a clearing member defaulting, the clearinghouse maintains a default fund and other

financial resources required from the clearing members in form of initial margin and variation margin.

The initial margin is to cover the worst case liquidation or close-out costs where the variation margin

trails the market movement. [6]

2.2.2.1 Risks of Central Clearing

The derivatives that are centrally cleared can only be liquid, standardized, non-exotic products. With

a derivatives contract comes several risks such as market risk, credit risk, operational risk, legal risk,

liquidity risk, counterparty risk and integration of risk types. However, in central clearing the clearing

member isn’t as exposed as in bilateral clearing since the CCP takes responsibility for the risks and

asks for collateral to cover these. Even if a clearinghouse offers advantages such as risk reduction and

operational efficiencies, they also demands significant collateralisation, centralization of counterparty

risk and loss-mutualisation. Also, if the clearinghouse would default or fail the impact would appear as a

systematic disturbance, meaning that the whole market would get affected. By this, the clearinghouses

creates arising operational, liquidity and systematic risks. [6]


2.2.2.2 Requirements and constraints

Considering that the exposure of a clearinghouse is very high, one can think that a clearinghouse

should demand a lot of collateral. But on the other hand, if the clearinghouse demands too much, the

clearing members will take their business elsewhere. Since this is a pretty delicate balance board, the

clearinghouses need a great approach of calculating the risk of the contracts it clears. The most crucial

when deciding the collateral deposit of a contract is that the clearinghouse needs to understand the

risks, i.e. decide which risks are acceptable, manage these and avoid the unacceptable risks.

There are several ways to calculate the risk where the most used ones are Value at Risk (VaR),

standardized portfolio analysis of risk (SPAN) and Expected Shortfall (ES), where the clearinghouses

always have their own twist on these. The risk algorithm comes up with a risk margin value per

portfolio, also called margin requirement or initial margin, where the collateral should be valued at

at least this amount after haircuts. Mutual for these risk calculations, one can say that the expected

value of collateral should be equal to the expected loss given the case of default.[5]

Expected loss given default = Expected value of collateral

As earlier mentioned, clearinghouses have different requirements on different types of collaterals, in

the subsections below are the most general requirements gathered and furthered described.

2.2.2.2.1 Initial margin and Variation margin

The clearinghouses usually demands Initial Margin (IM) and Variation Margin (VM) to protect them-

selves from the fluctuations of market movements in case of a clearing members default. Securities and

cash collaterals can be used to meet these margin calls, though securities collateral may be subject to

haircuts from CCPs. If a clearinghouse were to demand collaterals above the actual risk value, the

collaterals will be returned at the end of the transaction.

2.2.2.2.2 Haircut

When collaterals are being pledged to a clearinghouse, the collateral suffer from a reduced valuation

by some percentage called haircut. The percentage the collateral being reduced to is determined by

the amount of risk associated by the lender. If the collateral are cash and in domestic currency, the

clearinghouses normally always accept and don’t apply a haircut, however if its in a foreign currency

the haircut normally is high and it is more seldom accepted as collateral. Government bonds is often

accepted as collateral and the haircut is low, where corporate bonds have a higher haircut and is

seldom accepted. The haircut for a bank guarantee collateral is low, or even none, but only sometimes

accepted by the clearinghouses.

Haircuts exist to allow collaterals to have price variability. Theoretically, the haircut is driven by the

volatility and liquidity of the specific collateral but in practice the haircut levels are predetermined by

the clearinghouses and don’t change as the market changes. Haircut are primarily created to account

for market risk stemming from the price volatility. When deciding upon a haircut, the clearinghouses

consider the default risk, maturity and liquidity of the collateral security. It also considers the time

taken to liquidate the collateral, the volatility of the underlying market variables defining the value


of the collateral and wrong-way risk, meaning the relationship between the value of collateral and the

exposure or default of the counterparty.

Example: Haircut

Considering a contract going through CCP, where the calculated risk margin value is $10 000.

The clearing member will cover the risk margin value using a collateral security with a haircut

of 95%. This means that the clearing member needs to use collateral worth $10 526.36 to cover

the risk margin value.

Collateral with market value ·Haircut = Risk value

Collateral with market value =Risk value

Haircut=

$10 000

0.95∼ $10 526.36

2.2.2.2.3 Credit quality

The credit quality influence the importance of the collaterals due to its relation with haircuts, e.g.

for a high rating, the haircut becomes lower than for a collateral with lower rating. This also applies

to the clearing members explicit, if a clearing member has a high rating, the initial margin becomes

somewhat lower than for a clearing member with a lower credit rating.

2.2.2.2.4 Lower and upper concentration limits

Besides the properties mentioned above, clearinghouses can also have upper and lower concentration

limits on certain collaterals. The lower limits usually can be due to their preferences, certain collaterals

is simply considered more valuable and usable than others. The upper limits is mainly due to the

opposite, that a certain clearinghouse doesn’t prefer a certain collateral. The lower and upper limit

can also depend on this preference system with the combination of the risk margin value, if the risk

margin value is relatively large and the clearinghouse feels strongly about a certain collateral, this

collateral might get a higher or lower limit.

2.2.2.3 Central counterparty charges

In short, the main costs for a dealer bank when using a CCP is the funding and lower capital costs,

but to clarify the cost components connected to central clearing it is summarized below:[13]

• Inital Margin (IM) and Variation Margin (VM): Described earlier, mainly in cash andsecurities collaterals.

• Clearing fees: Paid in cash with usually a volume-based transaction, but can also be monthlypayments.

• Settlement charges: CPPs acquire settlement charges from clearing members on transactionsgoing through the clearinghouse.

• Default fund contribution: All clearing members needs to contribute to the clearinghouse’sdefault fund.

• Other charges: Such as service provider charges and the cost to have operations staffs, involvedto support central clearing. Technology used in the banks can also be large portion of theoperating cost.


Besides the costs directly related to the clearinghouse, there are several other costs that needs to taken

into consideration to fully manage the collateral usages. A hidden charge with central clearing is that

most CCPs demands that its clearing members needs to set up bank accounts at approved settlement

banks to secure payments. These banks can charge the clearing members for different services such

as wire transfers, reports, account access etc. However, if the clearing member actually is a bank, the

these costs can be ignored.

Other indirect charges can be in case of transferring securities collateral, the clearing members can

be charged settlement charges that arises from the fact that a CCP needs to go through a Central

Securities Depository (CSD).

2.2.3 Collateral cost model

When the collateral inventory and transformation actions are defined one can continue by identifying

the costs regarding the collaterals. In order to model the cost in a comprehensive manner, it is recom-

mended trying to truly understand the specific cost structure of the bank, and translate this into cost

per trade and by this also identifying the costs of collaterals.

However, to optimize the use of collaterals, one could also consider the total costs involved when

using different assets as collateral. Even though most costs are rather explicit, it can be difficult to

fully survey all in a satisfying way. Costs that are relatively easy to study is for example the cost of

acquiring collateral and transferring different collaterals to a counterparty. Costs that can be harder

to be protected from are costs such as opportunity cost.

There are different costs for different financial instruments, and in this section we’re going to try to

scrutinize the most general costs where the prior to this is to minimize the collateral cost.

To measure the costs and value the different collaterals in a consistent way, Basis Points (BPS) are

used.

Basis Points (BPS)

BPS is a common unit of measure for interest rates and other percentages in finance where one

basis point refers to 1100 th of 1%, i.e. 0.01%. This means that a change of 1% can be translated

as 100 basis points. Basis point is mainly used in order to minimize the confusion. For example,

if a rate of 10% that has an increase of 10% it can sometimes be hard to interpret this correctly

and its better to say that the rate has an increase of 1000 basis points, i.e 10%·(1 + 0.1) =

11%.[14]

2.2.3.1 Funding costs

Funding costs can either be the price tag of a specific collateral, or by looking at the active contracts

the funding value adjustment can be calculated together with the collateral value adjustment to know

how much this transfer is expected to cost.

Earlier, banks and other financial institutions weren’t handling funding costs when dealing with deriva-

tive contracts. This was however changed after the Lehman Brothers bankruptcy in 2008, where the

wholesale markets were drained, creating a huge funding problem for banks and forcing them to rely

on central bank liquidity. Now, the factors that regulate the funding costs are the clearing mandate,


bilateral collateral rules, liquidity coverage ratio, net stable funding ratio, increased capital require-

ments and leverage ratio. This have led banks to be more aware of the need to manage the funding

costs, which have created the concept of funding value adjustment (FVA) and margin value adjustment

(MVA). FVA is associated with the funding of derivatives, where MVA concerns the need to post the

initial margin.

Funding costs is asset-specific and an example of costs concerning a high quality treasury bond is

haircut and spread due to the fact that this bond can easily be used on the repomarket in order to

transform its value into cash. Here, the cost of borrowing money on an unsecured basis to buy the

bond isn’t considered. Derivatives assets (such as stocks bonds, commodities, currencies, interest rates

and market indexes) can be used as collateral, but can not be repoed. For assessing the underlying

funding costs for derivatives, the bank fund itself through many different sources. There is no right

way in doing this, but typically the banks treasury department will generate a blended cost of funds

curve that consist of all major currencies, this is also called funds transfer pricing curve (FTP curve). [6]

A difficulty with funding costs is that it can be hard to track down every cost that affect the final

funding cost of a collateral. However, there also exists costs that may be obvious, such as initial and

variation margin. Due to the fact that variation margin is posted against a Mark-to-Market (MTM)

loss, one cannot consider this an explicit funding cost. Initial margin on the other hand is not posted

against MTM losses, and by this it is a funding cost, which is another good reason for splitting funding

costs into MVA and FVA. [6] The variety of funding costs is also affected by the type of collateral. This

is due to the fact that if posting non-cash collateral, the haircut may be lower and by this, one doesn’t

need as much funding for this collateral. Also, the return paid on the collateral is of interest since

variation margin in a Credit Support Annex (CSA) generally is remunerated at the overnight indexed

swap (OIS) in the relevant currency, which is often viewed as a reasonable proxy for the risk-free-rate.

In this case, the funding cost is the cost above the risk free rate, and if collateral is less than OIS then

the funding cost should be higher.

Shortly, the funding costs represents the cost of use for a collateral asset. In order to apply funding

costs to individual positions, a common approach is to group them into different categories based on

credit quality, internal availability or ownership, eligibility at central funding venues and maturity.

An acceptable way to measure the credit quality is to use some well-established credit rating system,

such as the credit rating agencies Standard & Poors’, Moody’s or Fitch Ratings. Another parameter

connected to the funding costs could be the region in which the assets is mostly connected to. This

is interesting due to the risks connected to systematic risk and other risk such as wrong way risk also

may affect.

2.2.3.2 Additional costs

When selecting appropriate collateral one needs to consider the liquidity, volatility, avoiding strong

correlation with exposure, avoiding positive correlation to collateral giver and keep the collateral

portfolio well diversified. These cost factors are considered additional costs and are costs to the

counterparty such as the moving, settling and/or safekeeping of the collateral assets. In the subsections

below are a description of TCA, custodial fees, and safekeeping fees.


2.2.3.2.1 Transaction cost analysis

Transaction cost analysis (TCA) exists in order to help traders, investment managers and firms to

better understand how well they traded and how it can be improved.

Investment related transaction costs is the costs the comes from a time delay between the investment

decision and the order being released to the market. The longer it takes for the manager and trader

to resolve these issues, the risk of an adverse price movement arises, and with it, the cost.

The largest subset of transaction costs is the transaction cost related to trading, which includes the

costs that occur during the time period from the start, to the end of the trading. [15] Since these costs

cannot be eliminated, they need to be managed in a satisfying way, and the first step in doing so is

to analyze the components in question. The components that have the largest effect of the trading

related transaction costs are market impact and timing risk. [16]

If the order aren’t fully executed within the allotted period of time, the forgone profit or loss is

represented by the opportunity cost, which can be measured by the number of unexecuted shares

multiplied by the price change during which the order was in the market. [16]

To summarize, there are three classifications of transaction costs; Investment- , trading- and opportu-

nity costs, and every classification have some cost components that belongs to, which can be seen in

Figure 2.6.

Trading CostsCommission

FeesRebatesSpreads

Price AppreciationMarket Impact

Timing Risk

Opportunity CostOpportunity Cost

Transaction Costs

Investment CostsTaxes

Delay Cost

Figure 2.6 – Transaction costs classification. [16]

In order to categorize these costs further, it’s stated that there are some general properties that these

costs can possess such as fixed, variable, visible and hidden. The fixed cost are those that don’t depend

on the investment strategy which means that they cannot be changed. Variable costs on the other

hand are the costs where money managers, traders and brokers control the variable components in

the investment process to be consistent with the overall investment objective of the fund by adding a

substantial value.


The visible costs, also called transparent or explicit, are those that can be analyzed in advance. An

example of a visible cost can be a percentage of something such as $USD/share. The hidden, also

called non-transparent or implicit, are those costs whose fee structure is unknown. These costs usually

are estimated using statistical models. A summary of which properties the components possess can be

seen in Figure 2.4.

Table 2.4 – Unbundled transaction costs. [16]

Fixed Variable

VisibleCommission Spreads

Fees TaxesRebates

Hidden

Delay CostPrice Appreciation

Market ImpactTiming RiskOpportunity

In order to learn more about the transaction cost components, see [17]. The transaction cost compo-

nents that can be identified are the following:

Commission is the payment made to broker-dealers for executing trades. Normally expressed on a

per share basis or based on total transaction value. The payment is normally a fixed, explicit

cost.

Fees, also called exchange fees, are also an explicit cost (”visible”) and is charged when the order is

executed. This cost includes the clearing and settlement costs, the securities exchange transaction

fees, as well as the ticket charges assessed by floor brokers.

Taxes are considered as a visible, explicit investment related cost and varies by type of earning.

Rebates is a fairly new transaction cost. [16] An example of this is that if an investor posts liquidity,

the investor is provided with a rebate where the party receiving the liquidity is charged a fee.

The fee should always be larger than the rebate in order to ensure that the trading venue will

earn a profit.

Spreads is an explicit cost connected to the trading costs. The definition of a spread is the difference

between the best ask and best bid price, and it is used to compensate market makers for the

risks of holding and acquiring an inventory while waiting to offset the position in the market.

Delay cost is the cost in investment value that comes from a time delay between the manager making

the investment decision and the time the order is released to the market.

Price Appreciation, also called price trend, drift, momentum or alpha, and can be described as a

”natural price movement” that demonstrates how the price would evolve in the market without

uncertainty.


Market Impact is the movement in the price of a equity caused by a specific trade. This is one of

the more influential in the transaction cost components and can mathematically be described

as the difference between the price trajectory of the equity with the order and what the price

trajectory would have been if the order had not been released into the market. Due to the fact

that none of these factors can be measured with satisfying precision, the market impact have

been called the ”Heisenberg uncertainty principle of trading”.

Timing Risk consists of three components; price volatility, liquidity risk and parameter estimation

error. All components suffers from the risk that the estimated transaction cost wont be adequate

with the real ones. Price volatility causes the underlying equity to be either higher or lower

than the estimated one. The liquidity risk drives the market impact cost due to fluctuations in

the number of counterparties in the market and depends of volumes, intra-day trading patterns

and the buying and selling pressure of all market participants. The estimation error is here the

uncertainty surrounding the market impact parameters.

Opportunity Cost is used to measure the avoided loss or forgone profit of not being able to transact

the entire order. The main reasons of rising opportunity costs is due to adverse price movements

and insufficient liquidity.

2.2.3.2.2 Custodial Fees and Safekeeping Fees

The custodian fee is the fee charged by a broker or financial institution in order to be able to offer

safekeeping services. This means that the broker or financial institution holds securities or assets safe,

collects the dividend and interest income and gives a monthly or quarterly account statement for the

owner. Safekeeping Fees are applied when a brokerage holds a client’s securities or other assets on the

clients behalf.

2.2.4 Risk management

The word risk is most commonly associated with the risk of loosing money and have been mentioned

earlier in the thesis, but to summarize; The main types of risks connected to collateral management

are: credit risk, market risk, operational risk and liquidity risk.

Credit risk, also called default risk, is the risk that one party in a financial agreement will fail to meet

its obligations. One can divide credit risk into credit default risk, concentration risk and country risk.

All these risks of a contract can, as earlier mentioned, be reduced by a central counterparty. However,

in order to calculate the total risk of a specific collateral, the credit risk is of interest. There are several

different approaches to measure and manage risk, such as exposure at default (EAD), expected loss

(EL), loss given default (LGD), probability of default (PD) and Value at risk (VaR).

Market risk is the risk of losses due to change in the valuation of a financial asset. By modeling the

volatility and measuring this with with VaR, the market risk of a collateral can be determined.

Operational risk is the risk of losing money if the process of handling financial assets are inadequate.

This risk is harder to model and interpret than the earlier mentioned one, but can be a huge factor


when dealing with collaterals.

Liquidity risk is the risk of a financial asset not being able to be traded fast enough to prevent a loss.

There are two types of liquidity risk, asset and funding liquidity. To measure this risk, one can use

liquidity-adjusted value at risk, which is further described in Section 2.2.4.3. [18]

The main purpose of a risk measure is to determine how much of a financial asset is needed to make

the risks taken by financial institutions acceptable and accordance with the regulations.

2.2.4.1 Volatility

Volatility, σ, is the variation of the return of a security and is derived from the market price over time.

Here, either the volatility of a specific collateral can be determined, or the volatility of the market

where the collateral belongs.

In order to measure the stock and bond market volatilities, the theory from the thesis ”Sovereign credit

ratings market volatility and financial gains”, [19], are used. For every region i at time t, the stock

market return rstocki,t are defined as

rstocki,t = ln(stocki,t)− ln(stocki,t−1), (2.1)

which is the difference in logarithmic price at time t and t − 1. Equation 2.1 also holds for other

assets, such as currencies and commodities. However, the bond market return rbondi,t is defined as the

difference in logarithmic yield at time t and t− 1,

rbondi,t = ln(yieldi,t)− ln(yieldi,t−1).

Using these returns, a volatility model can be applied in order to simulate the volatility in the market.

Observe that the same described approach can be used to get the volatility of a specific collateral by

ignoring the region, i.

2.2.4.1.1 GARCH(1,1)

The generalized autoregressive conditional heteroskedasticity (GARCH) model is a univariate volatility

model where the volatility can be modeled by

σ2t = ω +

q∑j=1

αjY2t−j +

p∑j=1

βjσ2t−j

and ω, α and β are estimated parameters that needs to be positive to ensure positive volatility forecasts,

α, ω, β > 0. Also, α+ β < 1 to ensure covariance stationarity. [20]

The most common version of GARCH is with one lag, GARCH(1,1),

σ2t = ω + αY 2

t−1 + βσ2t−1.


2.2.4.1.2 EGARCH(1,1)

There are several different forms of the GARCH model, and one of these is exponential generalized

autoregressive conditional heteroskedasticity (EGARCH) which was introduced by Nelson and Cao

(1991).[19] With one lag, the model can be written as

log(σ2i,t) = ωi + αi(zi,t−1) + βi log(σ2

i,t−1) + δi(|zi,t−1| − E(|zi,t−1|)),

where the parameter δi is estimated along with αi and βi, and i in this case is related to the earlier

mentioned regions. Here, the E(zi,t−) is the expected value of the standardized residuals.[19]

2.2.4.2 Value at Risk

Value at Risk, VaR, is a measure of market risk and measure the worst expected loss that a firm

can suffer during a period of time and normal market conditions at a specified confidence level. One

can also say that VaR is a quantile of the profit/loss distribution and can be used to determine the

probability of loosing a certain amount of cash given a period of time. A common way to write this

risk measure is V aRα·100%(L) where L is the sorted losses and α ∈ (0, 1) is to the probability level,

usually set to 0.99 to be compliant with regulatory demands.

One way of calculating the V aR of an asset is to use historical observations and a statistical model to

estimate the probability distribution. From the historical observations, possible future scenarios can

be generated, and from this the profit and loss vector are created at a specific time t,

PnLi = V − Vi, (2.2)

where V is the value of the asset and i corresponds to the scenario i and i = 1, .., N . This vector is

sorted in decreasing order where the α ·N :th element the worst expected loss,

V aRα·100%(L) = PnL(α ·N). (2.3)

However, this value, V aRα100%(L), does not capture risk exposures such as operational risk, liquidity

risk, regulatory risk or sovereign risk. [21]

To determine if the V aR forecast is eligible, violation ratios can be used, and is defined by

V R =Observed number of violations

Expected number of violations=

ν

(1− α)(WT )

where ν is the number of V aR-violations in the time period WT .

2.2.4.3 Liquidity Adjusted Value at Risk

The liquidity-risk adjusted value at risk, LAdj−V aR, can be calculated by assuming that the liquidity

risk can be described by a bid-ask spread and added on the conventional VaR measure.[22] Assume

that the bid-ask spread is stochastic and the relative spread, S can be used for modeling,

S =bid-ask spread

mid-price.


The average relative spread, S, plus a multiple of the volatility, σ, of the relative spread to cover most,

say 99%, of the spread distribution, gives the liquidity-risk adjusted VaR,

LAdj-VaR = V aR+1

2Mid · (S + aσ),

where Mid is the mid-price and a is a scaling factor such that one achieve 99% probability coverage of

the change in the relative spread.[22]

2.2.4.4 Probability of default

Probability of default (PD) is defined as the likelihood that a counterparty can’t repay its debts and

fall into default. To calculate PD, one need to take the credit history into consideration. PD take

values between 0-100% where a higher percentage implies more risk than a low percentage. There

are many appraoches to calcuate the probability of default but if not calculated internally, PD can be

gathered from rating agencies such as Standard and Poors’, Moodys and Finch.

2.2.4.5 Exposure at default

Exposure at default (EAD) can be defined as the gross exposure in the case of default of a counterparty.

The EAD parameter is mainly used in calculations regarding capital regulated by Basel for a financial

institution, e.g a bank.

2.2.4.6 Loss given default

Loss given default (LGD) is a common parameter in risk models and can also be used in calculations

regarding capital regulated by Basel. LGD is most commonly defined as a share of the asset that is

lost in the case of a borrowers’ default.

2.2.4.7 Expected loss

As mentioned in the Section 2.2.2.2, the expected value of collateral should be equal to the expected

loss (EL) given default. To calculate EL for a single asset, we can used the earlier mentioned PD, EAD

and LGD

Li = PDi · EADi · LGDi, (2.4)

where i = 1, 2, ..., n and the total expected loss for a portfolio simply are calculated by adding EL for

every asset.

EL =

n∑i=1

ELi (2.5)

2.3. COLLATERAL OPTIMIZATION 26

2.3 Collateral Optimization

The term collateral optimization includes all actions done by a financial institution in order to make a

more effective use of its existing portfolio of collateral assets. One purpose of collateral optimization is

to survey the use of collateral, and by this manage the risks in a more efficient way. This can in turn

lead to lower funding costs, and by this, higher profits. Also, by optimizing the collateral in banks,

it may boost the institution’s ability to attract extra liquidity if needed, and lower its need for costly

and potentially risky collateral transformation services. [1][23]

A condition for collateral optimization is that the financial institutions, in this case a bank, have good

understanding about their existing collaterals, where they need to both know the cost and use for

each collateral. In order to optimize the collateral usage, the allocation of different collateral and

the effective parts that the market controls can be improved. To summarize, one can say that the

parameters that controls the optimization of collaterals are:

• Collateral positions

• Collateral costs


• Settlement procedures

To specify the optimization infrastructure, the institution needs to keep track on inventory, market

data, requirements and other agreement terms. If all data is gathered appropriately and the cost model

can be defined in a satisfying way. [24]

2.3.1 Optimizing cost models

The collateral cost model is defined to identify all potential risks and costs with a specific collat-

eral. Also, since different collateral holders have different priorities and preferences, this leads to an

individual cost model for every collateral holder.

In this section, there are three different approaches ranking the collaterals internally within a dealer

bank.

2.3.1.1 Preference ranking

As earlier mentioned have every institution different preferences e.g. a bank in a specific country can

value this country’s currency to more than its actual market value due to its other usage areas. The

easiest way to optimize the collateral usage is that the financial institutions creates its own preference

ranking system. These ranking rules will of course vary, but if the institutions wishes to create a

versatile ranking model it should take the following into consideration:

• credit quality

• rating

• liquidity profile

• asset class

• market segment


Also, the ranking should change as the inventory, risks and economic outlook changes. Even though

the own preference ranking system can be a good way to start, it isn’t perfect. An example of this is

that it is hard to ensure that the ranking is updated in a satisfying way. To take this into consideration,

market based ranking should be applied.

An example of a ranking system can be seen in Figure 2.5. [25]

Table 2.5 – A simplified cost category definition depending on a combination of asset class, issuercountry and rating.

Categories Asset Class Country Rating Funding SpreadCategory 1 Cash - - +40 BPCategory 2 Government Bonds Core EU& US - +35 BPCategory 3 Government Bonds Non-Core EU& US BBB or higher +30 BPCategory 4 Suprabond Core Europe - +23 BPCategory 5 Corporate Bonds - BBB or higher +18 BPCategory 6 Corporate Bonds - BBB or lower +12 BPCategory 7 Equities - BBB or higher +5 BP

Analyzing Figure 2.5, one can see that if a core EU government bond were to be substituted with an

equity of the same rating it would lead to a fund saving of 30 basis points.

2.3.1.2 Market based ranking

Marked based ranking can take different approaches, but normally is a complex system where daily

data of a collaterals’ volume, cost and utilization are taken into account. This makes the most desirable

collateral to the highest valued one, which in this case is a bit problematic. By always using the best

according to the market and don’t take such things as haircut into consideration, one can actually

end up with using a collateral that costs more to use, and the complexity of this problem grows when

considering other operational factors.[26] To forecast market risk factors such as price, return, credit

ranking and volatility can be used.

2.3.1.3 Economic based ranking

The most precise approach is to base the ranking on the economic cost of use for collaterals. By

doing this, the full cost together with the collateral requirements and haircuts should be taken into

consideration. This is a great challenge and individually done for every institution. A great factor in

the economic cost model is the spread between the return on the asset in the repo or securities lending

market compared to the internal funding desk. [26] The economic model should also consider the

relative values of different collateral types and the inventory with balance sheet in order to make the

best use of the collaterals in the portfolio, e.g. should the model acknowledge that a certain collateral

type would be adequate to offload into the collateral market due to concentration risk or RWA reasons.

Another large factor is the operational cost of moving a piece of collateral. This is equivalent with

the earlier mentioned transaction cost and is defined by some internal and external operational costs

that depends on the market and depository/custodians. The operational costs affect the willingness

to transform/substitute collaterals, and also influence the settlement risk.

2.3. COLLATERAL OPTIMIZATION 28

2.3.2 Allocation

Next step to build a well functioning collateral optimization model is to choose and create the collateral

allocation method. Here we focus on the allocation of collaterals used as margin to clearinghouses,

however, the same allocation methods can be used for the banks whole collateral usage. This is where

the different requirements of the different clearinghouses is taken into consideration to choose how to

post the collaterals. There are two common approaches with somewhat different finesses.

P1

CPP1

CPP2

CPP3

CPP4

Figure 2.7 – An illustration of how a financial institution is connected to different central counterparties.

2.3.2.1 Waterfall allocation

The ”natural” approach to optimize the collateral is to use the sequential or waterfall allocation. By

this, the method will iterate through the requirements and allocate the worst ranked collateral first

to the highest ranked agreements. Another approach of this is to match the highest quality collateral

together with the central counterparty that is hardest to please. The approaches described above

doesn’t need an optimization algorithm to run and is quite simple to implement by a simple rule set

iterating through the requirements. Even if this method result in a more efficient allocation, there are

better ways to optimize the allocation.[26]

2.3.2.2 Numerical optimization

To optimize the collateral allocation to different CCPs, all variables and requirements should be con-

sidered in a single process. This can be done using a mathematical techniques applied in numerical

optimization. There are several optimization algorithms that can be used, where the choice of algorithm

must be carefully selected and calibrated. By using a numerical optimization algorithm, constraints

such as eligibility, concentration limits and haircuts can be taken into consideration. Building a tool

where the cost model is implemented together with a numerical algorithm demands further knowledge

in optimization algorithms, please see Section 2.4.


2.4 Operational research

Operational research (OR) is a field of study that uses a scientific approach for decision making. Below

is a simplified scheme that describes the solution process in applied optimization.[27]

Problem Modeling Algorithm Solution Decision

In the beginning of every OR procedure, one needs to identify the problem with all factors that can

affect the outcome and also determine how these interact with each other.

The most optimal solution to the problem can be found using different problem solving techniques

applied in simulation, mathematical optimization, queuing theory and other stochastic-process models.

Optimization models are often used to describe and analyze technical and economical problems where

the goal is to find the most optimal solution. One outset to be able to use an optimization model is

that the problem consists of something that can be regulated, the problems decision variables. In order

to optimize these decision variables, one need to have a specific goal, where the goal can be described

by a function that depends on the variables that will be maximized or minimized. The restrictions

of this decision function is given by some constraints that describes the limits and interaction of the

decision variables. [27]

General model

An optimization problem (∗) can generally be described as

(∗) min f(x) when x ∈ X (2.6)

where the function f(x) depends on the variables x = (x1, x2, ..., xn)T . The set X defines the

allowed solutions, and is defined with some constraints. An alternative formulation of (∗) is

then

min f(x) when gi(x) ≤ bi, and i = 1, ...,m (2.7)

where the functions g1(x), ..., gm(x) depends on x, and b1, ..., bm is given constants.

A solution x ∈ X that minimizes f(x) is usually labeled as x∗, where the optimal value of the

function f(x) is z∗. To change this from a minimization problem to a maximization problem

one can say that to maximize z1 = f1(x) is the same as minimizing z2 = f2(x) = −f1(x) and

z∗2 = −z∗1 . The constraints of these problems can also be described generally by

g1(x) ≥ b1 ←→ −g1(x) ≤ −b1 (2.8)

and

g2(x) = b2 ←→ g2(x) ≤ b2 and g2(x) ≥ b2 (2.9)

And from this, there are different classes of problems depending on how to specify the functions

f, g1, ..., gm and what the allowed values on x could be.

When the mathematical model is identified, an algorithm is applied to solve the max/min problem. By

assuming that the problem have a large number of different solutions, one needs specific optimization

2.4. OPERATIONAL RESEARCH 30

methods to decide the best solution. If the solution satisfies all constraints, is feasible, and renders

the maximum/minimum value of the objective function, it is an optimal solution. From the solution,

decisions regarding the problem and its variables can be made.

2.4.1 Linear programming

If all functions f, g1, ..., gm in the general model are linear functions and all variables are real continuous

values, x ∈ Rn, the problem (∗) is a Linear Programming (LP) problem which can be written as

minmin z =

n∑j=1

cjxj

when

n∑j=1

aijxj ≤ bi, i = 1, ...,m

xj ≥ 0, j = 1, ..., n (2.10)

where cj is a known coefficient to the function that depends on x, aij is the constraint-coefficient to

variable xj in constraint i and bi is the right-hand-side to constraint i.

The most common way to solve this kind of problems is by the Simplex method, or revised Simplex

method. [27]

2.4.1.1 Simplex method

The Simplex method, first described by George Dantzig, solves LP problems by testing adjacent vertices

of the feasible set. The method is relatively fast, where the amount of iterations is at most two or

three times the number of constraints.[28]

The Simplex method is a distinguished, well-defined algorithm that can be described by some simple

steps.

Simplex method

1. Determine the starting basic feasible solution.

2. Determine the entering basic variable. (The most negative non-basic variable for maxi-

mization, and most positive for minimization).

If there is no entering variable: Stop! The last solution is optimal.

3. Select a leaving variable using the feasibility condition.

4. Determine the new basic solution using Gaussian elimination.

5. Repeat from Step 2.

By looking at Example 1 below, one can more clearly see the actual steps that are done in order to

get a feasible solution. [29]


Example 1: Simplex method

Solve the LP problem

max x1 + x2

2x1 + x2 ≤ 4

x1 + 2x2 ≤ 3

x1 ≥ 0, x2 ≥ 0

Convert the problem into standard form by adding the slack variables x3 ≥ 0 and x4 ≥ 0. Also,

let z denote the objective function value in such way that z − x1 − x2 = 0.

z − x1 − x2 = 0 Row 0

2x1 + x2 + x3 = 4 Row 1

x1 + 2x2 + x4 = 3 Row 2

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0

The variables which only appear in one equation are the basic variables, where the basic solution

is when the nonbasic variables are zero. In this case the basic solution is x1 = x2 = 0, x3 = 4,

x4 = 3 and z = 0.

Due to the fact that there are negative coefficients in Row 0, this is not optimum. Pick an enter-

ing variable (here x1) and pivot in order to make this non basic variable to a basic one.Remember

that the pivot element should be chosen in a way that doesn’t create infeasible solutions. Here,

by choosing the pivot element in Row 1, the basic solution becomes x2 = x3 = 0, x1 = 2, x4 = 1

and z = 2. So far, the two main rules for solving are identified:

Rules

1. The basic solution is optimal if all variables have a nonnegative coefficient in Row

0. If this doesn’t hold, choose a negative variable xj in Row 0.

2. For each Row i, (i ≥ 1) where there is a strictly positive entering variable coeffi-

cient, computeRight hand side

Entering variable coefficient

and choose the pivot row to be the one with the minimum ratio.

These rules indicate that the current basic solution isn’t optimal; Rule 1 instruct to choose x2as entering variable and Rule 2 advise to pivot in Row 2. By pivot on 3

2x2, the following occurs

z − 1

2x1 +

1

3x2 = 2 ⇒ z +

1

3x3 +

1

3x4 =

7

3Row 0

x1 +1

2x2 +

1

2x3 = 2 ⇒ x1 +

2

3x3 −

1

3x4 =

5

3Row 1

3

2x2 − x3 + x4 = 1 ⇒ x2 −

1

3x3 +

2

3x4 =

2

3Row 2

where the basic solution is x3 = x4 = 0, x1 = 53 , x2 = 3

3 and z = 73


All the computations in Example 1 can be represented in tableau form, please see below.

Table 2.6 – Tableau of the computations made in Example 1

z x1 x2 x3 x4 RHS Basic solution1 -1 -1 0 0 0 basic x3=4 x4=30 2 1 1 0 4 nonbasic x1= x2=00 1 2 0 1 3 z=21 0 − 1

212 0 2 basic x1=2 x4=1

0 1 12 − 1

2 0 2 nonbasic x2= x3=00 0 3

2 − 12 1 1 z=0

1 0 0 13

13

73 basic x1=5

3 x2=23

0 1 0 23 − 1

353 nonbasic x3= x4=0

0 0 1 − 13

23

23 z=7

3

Due to the fact that Example 1 only have two decision variables, it is easy to illustrate the feasible

solution graphically, see Figure 2.8. The Simplex method starts from x1 = x2 = 0 and as x1 increases,

so does z. The red and blue line are constraints and beyond these, the solution will become infeasible.

This is the reason for Rule 2 since it identifies the first constraint to be encountered. When reaching

the red constraint line, the slack variable becomes zero, and z = 2. After this, Rule 1 discovers that if

x3 = 0, z can be increased by increasing x2. After pivoting, the optimal solution is found to be z = 73 .

[29]

−2 −1 1 2 3 4

2

4

(z = 73 )

(z = 0)

(z = 2)

x1

x2

Figure 2.8 – An illustration of the solution of Example 1 using the Simplex method.


2.4.2 Integer linear programming

An integer linear programming (ILP) problem can be described in the same way as a LP problem,

with the difference that the decision variables should be integers.

min z =

n∑j=1

cjxj

when

n∑j=1

aijxj ≤ bi, i = 1, ...,m

xj ≥ 0, j = 1, ..., n

xj integer

If only some of the decision variables are constrained to be integers, it is a mixed integer linear

programming (MILP) problem. [30]

MILP problems can be solved using linear programming relaxations, meaning that the integrality

constraints is relaxed in order to obtain a LP problem and solving this with the Simplex method. The

reason for doing this is that solving integer problems is NP-hard, where LP problems can be solved in

polynomial time. A common approach solving MILP problems is to use Branch and Bound (B&B),

presented in Section 2.4.2.1.

2.4.2.1 Branch and bound algorithm

The Branch and Bound (B&B) algorithm, first described by A.H. Land and A.G. Doig, was developed

for solving general mixed, or pure, integer linear programming problems. The basic concept of the

B&B technique is to divide and conquer. The original problem is hard to solve directly, and is divided

(branched) into smaller and smaller subproblems until these can be conquered, where the conquering

is called fathoming. Fathoming is done by first giving a bound for the best solution of the subset and

then discarding the subset if the bound indicates that it can’t possibly contain the optimal solution.

The B&B algorithm begins with a LP relaxation of the original problem. If the LP relaxation gives

an optimal solution where the decision variables are integers, this solution is also optimal for the ILP

problem without LP relaxation. However, if this doesn’t hold, the solution can be found using the

B&B techniques. [31] [32] [33]

Using the fact that the optimal value of the LP-relaxation are known, the upper bound of the optimal

solution can be specified for ILP minimization problem (or the lower bound for a ILP maximization

problem).[30]

Optimal solution from LP-relaxation = z ≥ Optimal solution for minimization problems

Optimal solution from LP-relaxation = z ≤ Optimal solution for maximization problems

After this, B&B seeks to find out the position of the optimal solution by partition the feasible region

of the LP-relaxation. By choosing a fractional variable and looking at its closest integers, the B&B

algorithm divides the feasible set into two sub groups. Solving these two groups, and also future

subgroups, the restriction creating the subproblem as well as the optimum value and optimal solution

of the LP problem can be found.


If the branching of a subproblem won’t yield useful information, it can be fathomed into subproblems.

The subproblems that yield integral solution will be stored as incumbent, which will be replaced if

another solution value is larger than the first. When there is no unfathomable solutions left, the

algorithm have found the optimal solution. An example of when the B&B algorithm is applied and

finds the solution in a ILP problem can be seen in Example 2 where the algorithm is further described

below. [34] [28]

Branch and Bound algorithm

For a maximization problem, set initial lower bound to z = −∞. For a minimization problem,

set initial upper bound to z = ∞. Remove the integrality restrictions from the ILP problem

in order to get a LP relaxation. Initialize i = 0 in order to keep count of how many times the

LP-relaxation are done. [34]

1. Fathoming/bounding: Select the next subproblem to be examined, LPi. Solve this by

LP relaxation and if one of these three conditions holds, the subproblem can be fathomed:

• If the current optimal solution cannot yield a better value the lower/upper bound.

• If the subset yields a better feasible integer solution than the current lower/upper

bound.

• If the subset has no feasible solution.

Expect two cases:

• If the subset LPi is fathomed and a better solution is found the lower/upper bound

will be updated.

If all subsets have been fathomed: Stop!

(Optimum ILP = current finite lower/upper bound).

If no finite lower bound exists → no feasible solution.

Else, set i = i+ 1 and repeat step 1.

• If the subset LPi isn’t fathomed, go to Step 2.

2. Branching Select one of the integer decision variables xj whose optimum value xj∗ isn’t

integer in the current solution. Use the closest integer values of xj∗ in order to create two

subsets xj ≤ [xj∗] and xj ≥ [xj∗] + 1. After adding these subsets to the waiting list and

updating i = i+ 1, repeat Step 1.


Example 2: Branch and Bound algorithm

Solve the ILP problem

max 5x1 + 4x2

x1 + x2 ≤ 5

10x1 + 6x2 ≤ 45

x1, x2 ≥ 0

x1, x2 integer

Start by making a LP relaxation on the ILP problem, one get a solution optimum of x1 =

3.75, x2 = 1.25 and z = 23.75. Due to the fact that this solution doesn’t satisfy the integer

requirements, the region 3 < x1 < 4 is eliminated, and the two subsets x1 ≤ 3 and 4 ≤ x1are created. These constraints are added and examined individually. Starting with LP2 gives

the solution x1 = 3, x2 = 2, z = 23, which satisfies the integer requirements for x1 and

x2. LP2 is fathomed due to the fact that it can’t yield any better ILP solution. The lower

bound is z = 23 on the maximum objective value of the original ILP. This means that any

subproblem that cannot yield a better objective value than the lower bound must be discarded

as non promising. Due to the fact that the optimum solution of LP1 is 23.75 and there only

are positive integer coefficients to the decision variables, it is impossible for LP3 to produce a

better solution, and LP3 is fathomed.

If one have started by examining LP3, the road to optimum would be longer, and the worst

case scenario could have been that one would have traveled through all nodes, solving seven

LP problems (LP1 → LP3 → LP5 → LP4 → LP7 → LP6 → LP2) before the B&B algorithm

could be terminated, which is one of the cons using this method.

LP1

x1 = 3.75, x2 = 1.25, z = 23.75

LP2 (x1 ≤ 3)

x1 = 3, x2 = 2, z = 23

Lower bound (optimum)

LP3 (x1 ≥ 4)

x1 = 4, x2 = 0.83, z = 23.33

LP4 (x2 ≤ 0)

x1 = 4.5, x2 = 0, z = 22.5

LP6 (x1 ≤ 4)

x1 = 4, x2 = 0, z = 20

LP7 (x1 ≥ 5)

No feasible solution

LP5 (x2 ≥ 1)

No feasible solution


As can be seen in Example 2, the integer optimal solution is found by splitting the linear possible

solution into different areas, and then examining these separately. To get a better illustration of this,

see Figure 2.9.

−1 1 2 3 4 5 6

2

4

6

8

(z = 2375 )

x1

x2

(a) Final solution for linear optimization.

−1 1 2 3 4 5 6

2

4

6

8

x1

x2

(b) First decision in integer linear optimization.

Figure 2.9 – An illustration of how the Branch and Bound algorithm first solves the problem linearly,then divides into two areas in order to find the optimal integer solution. Every grey point is a possiblesolution to the ILP.

Chapter 3

Method

This chapter starts with presenting the layout of the created collateral optimization model with its

input parameters. After this is a short presentation of the ranking system, collateral requirements and

allocation methods, followed by the optimization problem in form of the mathematical model that is

solved by an optimization algorithm. At the end of the section the output of the structure is shown

and after that is a section regarding the software implementation.

3.1 Collateral Optimization model

Since the collateral optimization problem often is solved differently for different institutions and this

thesis is meant to investigate the most optimal collateral mix of a specific dealer bank, the method of

solving is here designed to be general and flexible.

To solve the collateral optimization problem using a relatively general approach, we need to specify the

factors that always will influence the method. As the theory describes, it is the dealer banks’ ranking

system of different collaterals and the clearinghouses’ requirements of collaterals that mainly affects

the usage. In this model, we consider three different types of collaterals; cash, government bonds and

equities, where the currency USD is used to price and value these collaterals.

The first step in optimizing the collateral usage is to know how much of each collateral there is. This

optimization model has two different ways of getting the quantity of the collaterals in the collateral

pool. The user can either know the quantity by inventory and manually type each collateral quantity, or

the user knows how much the total value Vtot of the collateral pool are and the fixed portfolio allocation

weights. The quantity input in the optimization model in other words are either integers (quantity q),

or a portfolio weight w (0 ≤ w ≤ 1) determined by some pre-determined portfolio allocation.

As earlier mentioned, the ranking system can differentiate a lot between different institutions. To make

this method as general and compliant as possible, the ranking system here results in an adjustment

of the valuation, vi, for a specific collateral i, where a large adjustment implies that the collateral is

more valuable for the dealer bank than a collateral with a small valuation adjustment.

37

3.1. COLLATERAL OPTIMIZATION MODEL 38

Example: Adjusted Valuation of Collateral

Considering a collateral i with the market value 100 USD, which is adjusted with 100 BP, the

collateral adjustment is:

Collateral adjustment = 100 USD · 100 BP = 1 USD

and the bank valuation of this collateral is vi = 101 USD, which indicates that the bank thinks

that the properties of this collateral makes it more valuable than the market values it to.

Each clearinghouse values the collaterals differently, which is easy to see when comparing the require-

ments; haircut, upper concentration limit and lower concentration limit. In this model, we don’t

consider the actual contracts, only the current risk of clearing the contract at a clearinghouse. The

risk of a contract is represented by a value, and if this model where to be used in a real life example

one would need to get this risk value from the clearinghouses before using the model. To summarize,

input parameters of the optimization model are

• Collateral quantity (same as portfolio weights together with the total portfolio value)

• Collateral valuation (the adjusted valuation determined by the dealer bank is included)

• Collateral requirements from the clearinghouses

• Total risk margin value of the contracts

which is depicted together with the operational flow in Figure 3.1.

39 CHAPTER 3. METHOD

Model:Creatingdecisionfunction

&constraints

CHsCollateral Pool

Dealer Bank

MILP solverSimplex method

orBranch & Bound

Solution: Optimal Collateral allocation- Calculates cost for covering risk margin values.

- Determines if the dealer bank will gainon reallocating the portfolio,

proceeds with Alternative 1 or 2.

RANKINGSYSTEM

- Exchange rate- Credit rating

- Historical data- Collateral costs

- Data for riskcalculation

INPUT

- CollateralValuation

- CollateralQuantity

INPUT

- Collateralrequirements- Risk values

1. Collateral to cover riskwith Optimal Collateralallocation for every CH

1. Unused collateralin Collateral Portfolio2. Reallocate portfolio

by transformingcollateral using thecollateral market.

Redo previous steps.

Figure 3.1 – The figure describes the flow of the model. The collateral in the collateral pool is valuedby a ranking system depending on the exchange rate, the credit rating, historical data and collateralcosts. Using this together with the collateral requirements, quantity and the risk margin value of thecontracts, the constraints of the model is created. After the constraints is created, the MILP solveruses an optimization method to give the optimal allocation of collaterals to cover the risk margin value(margin call) from the connected clearinghouses. The optimization model also calculates the cost of thisto determine if a better, and cheaper, solution exists.

After the constraints have been created, the model uses a MILP solver to solve the optimization

problem, also called collateral allocation problem. Here, the model can choose between solving by the

Simplex or Branch&Bound method. This is done in order to compare the different methods. Naturally

B&B should be used here, but the Simplex method is also implemented to study how the model is

affected when not considering the fact that only the collateral USD cash can be partial, where the

other collateral quantities needs to be integers.

For every time the optimal solution is found, we investigate if there is any room for improvement, i.e.

if there exist a cheaper way to allocate the collaterals by transforming some of the collaterals in the

collateral pool.


3.1.1 Input Data

To create the constraints needed in the mathematical optimization model, we need input data from

both the bank and the clearinghouse. The data from the bank are:

• Collateral valuation

• Collateral quantity

As earlier mentioned, the valuation and quantity is to be chosen for each case. However, in this thesis

I’ve chosen to work around one specific portfolio where the prices were gathered at a specific date with

a fix value adjustment. Also, the quantities were predetermined and only changed in the case where

the Simplex method acquired a reallocation (transformation) of collateral, or if there exists a more

optimal solution to the problem.

In Table 3.1 below are the collaterals in the portfolio. The ticker symbols for the currencies and equities

correspond to Yahoo! Finance, and the tickers of the bonds correspond to Bloomberg.

Table 3.1 – The table contains the collateral type, name, ticker, price and quantity of the consideredcollateral portfolio.

CollateralType

Collateral Name TickerPrice[USD]

PriceAdjustment

[USD]Quantity

Cash

United States dollar CCY: USD=X 1 1,005 2000Euro CCY: EURUSD=X 1,0864 1,091832 2000Great Britain Pound

(pound Sterling)CCY: GBPUSD=X 1,2806 1,287003 2000

Swedish crown(krona)

CCY: SEKUSD=X 0,1125 0,113063 2000

Danish crown(krone)

CCY: DKKUSD=X 0,1461 0,146831 2000

Norwegian krone CCY: NOKUSD=X 0,1172 0,117786 2000Canadian dollar CCY: CADUSD=X 0,7424 0,746112 2000Japanese Yen CCY: JPYUSD=X 0,0091 0,009146 2000

GovernmentBonds

UK Gilt 5 Year Yield Bloomberg: GTGBP5Y:GOV 99,97 100,2699 15Germany Bund

5 Year YieldBloomberg: GTDEM5Y:GOV 131,48 131,8744 15

UK Gilt 10 Year Yield Bloomberg: GTGBP10Y:GOV 95,48 95,76644 15Germany Bund10 Year Yield

Bloomberg: GTDEM10Y:GOV 101,64 101,9449 15

UK Gilt 30 Year Yield Bloomberg: GTGBP30Y:GOV 98,54 98,83562 15Germany Bund30 Year Yield

Bloomberg: GTDEM30Y:GOV 130,94 131,3328 15

EquitiesAlfaLaval AB Yahoo Finance: ALFA.ST 19,83375 19,84367 100AB Electrolux Yahoo Finance: ELUX-B.ST 30,42 30,43521 100AB Volvo Yahoo Finance: VOLV-B.ST 16,07625 16,08429 100


I have chosen to consider three types of collateral in my model; cash, government bonds and bills, and

equities. The reason I chose these was that I wanted to be able to compare how the model handles

collaterals that widely are highly valued versus low valued. We also need data from the clearinghouses,

such as:

• Risk margin value


See Table 3.2 for the collateral requirements used in this implementation.


Table 3.2 – The table describes the collateral restrictions and requirements of every clearinghouse.

Collateral PriceAdjusted

priceAdjustedby [BP]

Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0

GTGBP5Y:GOV 99,97 100,2699 30 0,99 99,26721 0,99 0GTGBP10Y:GOV 131,48 131,8744 30 0,99 130,5557 0,99 0GTGBP30Y:GOV 95,48 95,76644 30 0,99 94,80878 0,99 0GTDEM5Y:GOV 101,64 101,9449 30 0,99 100,9255 0,99 0GTDEM10Y:GOV 98,54 98,83562 30 0,99 97,84726 0,99 0GTDEM30Y:GOV 130,94 131,3328 30 0,99 130,0195 0,99 0

ALFA.ST 19,83375 19,84367 5 0,98 19,44679 0,98 0ELUX-B.ST 30,42 30,43521 5 0,98 29,82651 0,98 0VOLV-B.ST 16,07625 16,08429 5 0,98 15,7626 0,98 0

Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0EURUSD=X 1,0864 1,091832 50 0,95 1,03724 1 0GBPUSD=X 1,2806 1,287003 50 0,95 1,222653 1 0SEKUSD=X 0,1125 0,113063 50 0,95 0,107409 1 0DKKUSD=X 0,1461 0,146831 50 0,95 0,139489 1 0NOKUSD=X 0,1172 0,117786 50 0,95 0,111897 1 0CADUSD=X 0,7424 0,746112 50 0,95 0,708806 1 0JPYUSD=X 0,0091 0,009146 50 0,95 0,008688 1 0



Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0EURUSD=X 1,0864 1,091832 50 1 1,091832 0,9 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 0,9 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 0,9 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 0,9 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 0,9 0CADUSD=X 0,7424 0,746112 50 1 0,746112 0,9 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 0,9 0



Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0,2DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0



Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0SEKUSD=X 0,1125 0,113063 50 0,97 0,109671 1 0DKKUSD=X 0,1461 0,146831 50 0,97 0,142426 1 0NOKUSD=X 0,1172 0,117786 50 0,97 0,114252 1 0CADUSD=X 0,7424 0,746112 50 0,97 0,723729 1 0JPYUSD=X 0,0091 0,009146 50 0,97 0,008871 1 0

GTGBP5Y:GOV 99,97 100,2699 30 0,9 90,24292 1 0GTGBP10Y:GOV 131,48 131,8744 30 0,85 112,0933 1 0GTGBP30Y:GOV 95,48 95,76644 30 0,8 76,61315 1 0GTDEM5Y:GOV 101,64 101,9449 30 0,9 91,75043 1 0GTDEM10Y:GOV 98,54 98,83562 30 0,85 84,01028 1 0GTDEM30Y:GOV 130,94 131,3328 30 0,8 105,0663 1 0

ALFA.ST 19,83375 19,84367 5 0,75 14,88275 1 0ELUX-B.ST 30,42 30,43521 5 0,75 22,82641 1 0VOLV-B.ST 16,07625 16,08429 5 0,75 12,06322 1 0


3.1.1.1 Collateral valuation

The collateral valuation is here comprised of several different factors such as the banks preferences, the

market and also the cost regarding different collaterals. Each and every collateral should be ranked

individually but due to the fact that all ranking factors doesn’t apply to every collateral, and in order

to simplify the system, the collaterals are looked upon as a member of one of the three different col-

lateral types: cash, government bonds and equities.

Starting with the valuation of the collateral, the daily price of the cash and equities can easily be

gathered from Yahoo Finance!. The government bonds however is a bit harder to get and can be gath-

ered manually from Bloombergs website. The main reason for the more problematic data collection of

government bonds is that historical data of government bonds are valuable and can only be bought at

some institutions. The difficulty of collecting historical data also reflects on other ranking factors such

as forecasting market risk. However, when taking credit rating and maturity into consideration, we

can at least get a probability of the worst case scenario, i.e. the default of the collateral holder. Here

it is also of value to identify the liquidity of every collateral, usually reflected in large ask-bid spreads

or large price movements, but due to lack of data this is somewhat difficult to show.

Also looking at where the collateral comes from, i.e the country or market segment, makes it possible

to say something about whether there exists some systematic risks in the collateral portfolio, which

also can be a factor when ranking.

One of the main factors when dealing with collaterals are the costs, and especially the funding cost.

The easiest approach to determine how much funding is needed is to look at the initial margin to-

gether with the haircut and operational/transaction costs, which all are defined in chapter 2. In order

to compare these assets further, the cost definition needs to be structured in a way that cash, bonds

and equities are directly comparable, convertible into a base currency and within the same time horizon.

I’ve simulated each cash and equity collateral volatility using GARCH, where the volatility for each

collateral type can be simulated using EGARCH.[19] Since these results only are created to confirm

that the preference system correspond to how the market values these collaterals types it will not be

presented as result in this thesis. However, to see an example of the output from the implementation,

see Figure 3.2a and 3.2b.


(a) Volatility forecast for the GARCH-model. (b) Figure describing VaR-forecast using the GARCH-model with the negative returns.

Figure 3.2 – Example output of the MATLAB implementation to confirm the preference system.

Considering the above, I’ve ranked the different collateral types where the following adjusted valuation

can be seen gathered in Table 3.3.

Table 3.3 – Collateral types with adjusted valuation used in the optimization model.

Collateral Type Adjusted ValuationCash +50 BPSGovernment bonds +30 BPSEquities +5 BPS

This can also be compared with the valuation adjustment in Figure 2.5. [25]


3.1.1.2 Clearinghouse

In order to make the modeling as realistic as possible, I have studied 4 different clearinghouses and their

collateral requirements, please see reference in Table 3.4. Evaluating these clearinghouses I concluded

that the main types of collateral is cash, government bonds and bills, other types of bonds, bank

guarantees, commodities and equities.

Table 3.4 – The table contains a summary of clearinghouses and references that was used to determinesome common requirements clearinghouses had on collaterals.

Clearinghouse ReferencesTMX CDCC [24] [35]EUREX clearing [36] [37] [38] [39] [40]NASDAQ [41]LCH [42] [43]

For variation margin, daily collateral calls is pretty standard, where initial margin is at the time the

deal is made. The clearinghouse takes several things into consideration when determining which col-

laterals are eligible to cover the margin calls and adjust the valuation as well as the concentration in

order to mainly get high quality collateral. An example of such adjustment is the applied haircut.

The haircut is calculated by considering the liquidation (sale) of collateral securities, meaning that if

we have a low haircut, the collateral is more liquid compared to a collateral with a high haircut. The

clearinghouse can also apply a upper and lower concentration limit to protect themselves from not

getting a diversified collateral portfolio. The most common regarding concentration limits is that a

clearinghouse demand a certain lower percentage limit of the collateral it thinks is a HQLA, and the

upper limit is to protect themselves from only getting low liquid collateral.

In the collateral optimization model, I’ve determined the collateral requirements of five clearinghouses.

The requirements is not accurate with any of the studied clearinghouses but are randomly generated

using real preference data, validating the choice somewhat. However, the point of these requirements

is not to perfectly reflect the reality in this case, but to be able to set and analyze the properties of

the optimization methods.

3.1.2 Mixed integer linear problem

To allocate the collaterals in the portfolio in an optimal approach, we need to solve a minimization

problem. This problem can be solved using the same algorithms as when solving a MILP problem, due

to the fact that the collaterals that will be optimized only is considered as integer values, except for

the base currency USD that can be continuous values. However, the collateral optimization model also

considers the LP case where we can have fractions of a government bond or stock, which also makes it

easier to calculate the value of the collaterals and the whole portfolio.

3.1.2.1 Mathematical optimization problem

The optimization problem (P ) is here described as

(P ) min f(x) when x ∈ X (3.1)

where the linear function f(x) depends on the variables


x = (x11, x21, ..., xn1, x12, x22, ..., xn2, ..., x1m, x2m, ..., xnm) (3.2)

where n is defined as the number of collaterals, and m is the number of clearinghouses. The function

f(x) contains the banks valuation vi of each collateral i = 1, 2, ..., n together with the quantity of a

specific collateral to a specific clearinghouse j.

f(x) =

m∑j=1

n∑i=1

vi · xij (3.3)

The set X defines the allowed solutions, and is determined by some constraints. In this case, we

consider a MILP-method, meaning that the solution will give integer values to the variables in x that

we’ve specified. An alternative formulation of the problem (P ) is

min f(x) when gk(x) ≤ Fk, and k = 1, ..., p (3.4)

where g1(x), ...,gp(x) is some linear functions that depends on x, and F1, ...,Fp vectors containing

given constants. In this model the amount of constraints depends on the number of collaterals and

clearinghouses, but we can divide the constraints into 2 different types, which means that p = 2. The

first type of constraint can be described as

F1 = qi

g1i(x) =

m∑j=1

xij ≤ qi, i = 1, ..., n (3.5)

where the linear function g1i is the sum of the variables concerning a specific collateral should be less

or equal to the quantity, qi, available in the portfolio. If the model is given a portfolio weight wi, the

quantity is calculated using

qi = wi · Vtot

where∑ni=1 wi ≤ 1 and Vtot is the total value of the collateral portfolio. The second type of constraints

is described by

F2 = rj

g2j(x) =

n∑i=1

vixij · hij ≥ rj , j = 1, ...,m (3.6)

where the linear function g2j concerns that the sum of each clearinghouse haircut, hij , of every collateral

should be greater than the risk margin value rj in order to cover the risk. The clearinghouses also

specifies an upper, uij , and a lower, lij , concentration limit for every collateral they can use.

lij ≤ xij ≤ uij


3.2 Output

For every time we run the collateral optimization model we get the result gathered in excel documents.

The first the model does is to use a collateral portfolio with an infinite amount of collateral, i.e. the

weights winfinite. This is done in order to get a reference point to the best solution, i.e. the most

optimal collateral allocation solution xbest solution among the clearinghouses due to their requirements

and restrictions.

After this, we consider a collateral portfolio with weights, w, which initially is allocated in a way that

makes it impossible to solve the problem as the portfolio with the infinite amount of collaterals can.

x 6= xbest solution

This non-optimal initial portfolio allocation may result in an unsolvable mathematical model, a non-

optimal collateral allocation or an optimal solution allocated in a different way. If the latter occurs,

the optimization model deals with this in two different ways. The model either reallocates the whole

initial portfolio to be have the same weights as the optimal allocated collateral portfolio, or the model

uses the excess collateral that weren’t used to cover the risk, and reallocates this to have the same

weights as the optimal collateral allocation. To summarize:

1. Initial portfolio allocation gives the best optimal solution

2. Initial portfolio allocation can’t give the same optimal solution as in 1. If the solution is far from

1, we reallocate the portfolio and solve again. This is done in two different ways:

(a) Reallocate the initial portfolio using the weights from the optimal solution

(b) Reallocate the excess collateral using the weights from the optimal solution

All this is done using both the Simplex and Branch & Bound method.

3.3 Software implementation

MATLAB is used in order to simulate the market volatility using historical data and explore some risk

management tools.

The implementation of the collateral optimization problem was done in Java using a MILP solver

called lp solve, which uses the revised Simplex method and Branch&Bound method to solve linear

and mixed linear optimization problems. The solver is free to use when following the rules of GNU

LESSER GENERAL PUBLIC LICENSE.

Chapter 4

Results

In this chapter the reasoning behind the chosen ranking system and an analyze of the allocation

methods are presented. After this, the main results are analyzed from a case where the dealer bank is

connected to five clearinghouses and uses the collateral optimization model in the Java implementation

to solve the problem.

4.1 Ranking system

As mentioned, the ranking system is individual for every financial institution and should preferably be

built on preferences, the market and the different costs connected to the collateral usage.

To strengthen the preference system, a GARCH(1,1)-model was fit using return data of the cash and

equities for the time period from 2003-01-01 to 2012-12-31 in order determine the GARCH parameters

and to perform a volatility forecast using the time period 2012-01-01 until 2015-12-31. Comparing the

volatility forecast of cash with the volatility of the stocks made it clear that the stocks were more likely

to vary, which implies that the collateral type cash should be more highly valued than the collateral

type equity (the stocks).

Also, when comparing credit ratings between the government bonds and the equities, there is a distinct

difference where the government bonds as a rule have a lower volatility and higher credit rating than

the stocks. This also strengthen the assumption that the government bonds should be higher valued

than the equities.

With the help of ”Sovereign credit ratings, market volatility, and financial gains”, [19], the connec-

tion between credit ratings and financial markets volatility are shown and implies that the reasoning

regarding valuation of the different collateral types are adequate.

48

49 CHAPTER 4. RESULTS

4.2 Collateral allocation methods

The two optimization methods used here for allocation of the collateral to cover the risk margin value

of each clearinghouse is the Simplex method and Branch&Bound method.

4.2.1 Simplex method

Due to the fact that this problem actually is a MILP problem, and that the Simplex method assumes

that all the variables are continuous, the solution is not necessarily the most optimal one. In order to

get a better solution, we reallocate the initial portfolio with respect to the most optimal solution, see

Section 3.2.

4.2.2 Branch and Bound

Since the mathematical model can be solved optimally using a MILP solving method such as Branch

and Bound, we expect to always get an optimal solution.

4.2.3 Analyzing the different methods

I’ve created some test scenarios in order to see if the different optimization methods handles added and

removed constraints in an expected and satisfying way. In these cases we have one clearinghouse and

15 types of collateral. The collateral are priced using prices in an unchanged cvs-file and the initial

quantity in the collateral portfolio is always the same. To analyze the methods, we consider 7 different

test cases

1. No adjusted value, no haircut, no upper limit, no lower limit

2. No haircut, no upper limit, no lower limit

3. No adjusted value, no upper limit, no lower limit

4. No adjusted value, no haircut, no upper limit

5. No adjusted value, no haircut, no upper limit, and a lower limit exceeds the total value in the

portfolio

6. No adjusted value, no haircut, no upper limit, no lower limit, and the risk margin value exceeds

the total portfolio value.

7. No adjusted value, no haircut, no upper limit, no lower limit, but transformation costs is calcu-

lated as the portfolio is reallocated in 2 different ways

The result of these test cases are gathered and further analyzed in Appendix A and Appendix B.


4.3 Collateral Optimization model

Considering the case when the dealer bank needs to cover a risk margin value to five clearinghouses,

and each and every clearinghouse have different requirements such as haircuts, upper concentration

limits and lower concentration limits.

As mentioned in the method, the model gives us several solutions for every time we run the program:

1. Initial portfolio allocation gives the best optimal solution

2. Initial portfolio allocation can’t give the same optimal solution as in 1. If the solution is far from

1, we reallocate the portfolio and solves again. This is done in two different ways:

(a) Reallocate the initial portfolio using the weights from the optimal solution

(b) Reallocate the excess collateral using the weights from the optimal solution

This result is gathered in Tables found in Appendix C, but the main result can be seen in Table 4.1.

Table 4.1 – The table contains the solutions for adjusted valuation on the collaterals and solving theproblem using different methods.

ClearinghousesRiskvalue

Simplex method Branch&

BoundBest

solutionInitial

portfolio

Reallocateinitial

portfolio

Reallocateexcess

collateralClearinghouse 1 10000 10050 10050.08 10050.03 10050.08 10050Clearinghouse 2 1300 1306.5 1306.5 1306.5 1306.5 1306.5Clearinghouse 3 1000 1005.902 1005.761 1005.688 1006.81 1005Clearinghouse 4 2000 2010.319 2010.05 2010.138 2010.05 2010.05Clearinghouse 5 1500 1507.5 1507.5 1507.5 1507.82 1507.5Sum 15800 15880.221 15879.891 15879.856 15881.26 15879.05Difference 80.221 79.891 79.856 81.26 79.05

As the test cases showed, when using the Simplex method we sometimes got the best solution to be

exactly the same as the most optimal solution. However, it was more likely to end up with a worse

solution than the optimal. The Branch & Bound method gave the same allocation as the most optimal

solution did, which is the reason for comparing the solutions the Simplex method gave with this.

Chapter 5

Discussion

One can look at the collateral optimization model as two parts; cost model and allocation method.

These parts can be executed in different ways, where Figure 5.1 are created to illustrate this further.

Cost model

Allocation method

Waterfall Numerical Opti. Non-linear Opti.

Preferences based ranking

Dynamic ranking

Full economic cost model

A

B

Figure 5.1 – The figure describes how well the collateral optimization methods are when combined.This means that the waterfall allocation is the worst and non-linear optimization is the best, as well asthe preference ranking is the worst, where the full economic model is the best. The point A describeshow good the collateral optimization model could be when studying this thesis and point B describeshow good the implementation of the optimization model really was.

As can be seen in Figure 5.1, the goal was to find an almost fully developed economic cost model

together with the allocation problem solved with numerical optimization. There is no true way or gen-

eral approach to decide upon a ranking system, but as suggested in the Theory, the volatility of every

collateral can be modeled using GARCH. However, due to the fact that this thesis only has historical

data for the cash and stocks, it was harder to find a specific approach to compare the collaterals to each

other and create a good cost model. This is the reason I’ve chosen to rank the three different collateral

types by preference. In order to somehow verify that my preferences are a bit compliant with how the

market values them, I’ve compared them using the properties a full economic cost model would have.

However, due to the fact that an economic model is hard to find when lacking in consistency in data,

51

5.1. CONCLUSION 52

this goal was hard to reach. If the collateral portfolio was chosen more carefully or the thesis was built

on a real-life example, this goal could maybe have been reached. Even though the thesis never resulted

in an implementation of a full economic cost model, it still contains a thorough study of the costs, risks

and regulation connected to collaterals, which later can be used to create the economic cost model.

However, by using the preference system in the collateral optimization model we got a careful analysis

of the optimization methods.

As was expected, the Branch & Bound method was a better method for this problem due to the fact

that our optimization problem contained collateral that one couldn’t take fractions from, meaning that

we can’t give a clearinghouse half of a stock just because the optimization method tells us to. When

the Simplex method suggested such as an solution, the program was forced to round off this to the

closest integer, causing the solution to be non-optimal. This meant that the Simplex method gave a

more costly solution compared to the Branch & Bound method that took the integer constraint into

consideration, and by this, always gave an optimal solution. The point of reallocating the portfolio,

was to minimize this difference somewhat.

Even though the B&B was giving the best solution, it was a bit slower than the Simplex method.[44]

Also, in this thesis I’ve decided to use a solver but if one were to implement the methods by hand, the

B&B method are considered to be harder to implement compared with the Simplex method. However,

when examining the pro’s and con’s, I’ve come to the conclusion that the B&B is better to apply to

this problem. Another reason why B&B is the better choice is due to the fact that it is applicable with

non-linear programming problems, meaning that if one were to evolve the existing MILP problem into

a non-linear programming problem, this would be less of a problem than if the collateral optimization

model had the Simplex method as solving approach. The reason for not creating a non-linear math-

ematical model was that this wasn’t necessary when considering the requirements and constraints in

this case. However, it would be necessary if one were to take the variation and correlation into account

which is a good approach to develop the existing model. Another way to develop the model is not only

to optimize the collateral of today, but also trying to foresee and manage the risks ahead.

Besides discussing the content of this thesis, it is also of interest to discuss the sources. Contemporary

sources have been used as much as possible, however, some might be outdated which may cause some

inconsistency in the thesis considering the fact that present regulations and laws may not have been

applicable.

5.1 Conclusion

The goal of this thesis was to investigate how to optimize the internal collateral mix of a dealer bank.

This has been done by charting the collateral market and the collateral costs, and giving tools to

control and foresee the behavior of the factors that impact the optimization. The thesis also presents

allocation approaches where numerical optimization have been fully investigated. Comparing the two

methods, Simplex and Branch&Bound, we can conclude that the Branch&Bound method is better for

this purpose.

53 CHAPTER 5. DISCUSSION

5.2 Future work

Collateral management can be improved further by also incorporating future collateral needs and dy-

namic optimization where the collaterals can be adjusted during the life of the transaction. Therefore,

for future work, we shouldn’t only look at how to optimize the current collateral usage, but also try to

foresee the future collateral needs. This might be done by doing a full risk analysis of every contract

cleared by a clearinghouse. Also, it would be of interest to investigate how to build a model that

optimize the collateral differently depending on the time of the contract, i.e. one approach to optimize

before the trade, one optimization approach when the trade is present and one after the trade is done.

The idea of this is further described in [26]. If one were to analyze the contracts to manage the collat-

eral quantity, the same risk measure methods can be used as mentioned in this thesis to measure the

risk of collaterals.

It would also be of interest to investigate how much a non-linear programming optimization method

would improve the collateral allocation, and the impact of a full economic cost model. If one would like

to improve the existing preference system, it would have been interesting to decide each adjusted val-

uation individually and also incorporating the effect of maturity time in government bonds for example.

In this thesis I haven’t fully investigated how the wrong-way risk affects the collaterals, and it would

be interesting to study the collateral portfolio diversification.

To get a even better optimization model, it would also be of interest to incorporate the regulatory

demands into the model. If one were to expand the existing model in this thesis and e.g. wants the

collateral portfolio to have a specific liquidity, I’ve would suggest to add such calculations together

with constraints regarding when to transform and how much.

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Appendix A

Test cases - Revised Simplex

method

Starting to look at the case where the price isn’t adjusted, and we have no upper or lower concentration

limits and no haircut. I expected that the method would choose to use the USD dollar collateral such

as it have done in the optimal allocation, however this isn’t done, which can be seen in Figure A.1.

Due to the fact that the Simplex method primarily choose the collaterals with the highest price and

the lowest amount, we can suspect that the Simplex method in the solver prioritize these to be entering

variables. The optimal solution to this problem is that the bank gives the clearinghouse exactly the

amount that covers the risk margin value. However, this isn’t the case. Here the risk margin value

and optimal solution is 10 000, but the Simplex method gives the value 10 483.78 where the portfolio

allocation and collateral usage can be further seen in Table A.1 in Section A.1. The reason for this

relatively large difference is partly due to the fact that the Simplex method doesn’t consider the fact

that we can’t handle fractions of a bond or a stock, but it is also due to the fact that the method

choses the largest decision variables as entering variables, which make it impossible to exactly cover

the risk margin value.

Figure A.1 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, theclearinghouse have no upper/lower concentration value limits and no haircut, but the bank adjusts thevalue of the collaterals to consider the funding costs.

Looking at the case where the clearinghouse have no upper/lower concentration value limits and no

haircut, but the bank adjusts the value of the collaterals to consider the funding costs. To make

I

II

the different collateral types comparable I chose the same adjustment percentage for every type. The

optimal value here is 10 050, where the optimal allocation is fully invested in USD that is adjusted with

50 BP. However, the best allocation looks similar to Figure A.1 which should be able to be motivated

by the same argument as before.

For the case where the bank don’t adjust the value, and the clearinghouse only have a haircut on the

value, with no upper and lower concentration limits, the best solution value is here 11 111.17 due to

the fact that the bank needs to but out more collateral as the clearinghouse values it less. Having a

portfolio and optimizing by Simplex method, the best value becomes 11 617.6794. In Figure A.2, we

can see that the allocation differs from Figure A.1, and the Simplex method finds it more important

to allocate the cash before the Government bonds.

Figure A.2 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, theclearinghouse only have a haircut on the value, with no upper and lower concentration limits.

One could expect that the method would behave in the same way as when the bank adjusts the value,

however, in this case we can’t possibly compare these two cases. An example of why we can’t compare

these is that the haircut of cash adjusts the value with 1000 basis points, where the bank only adjusted

the value with 50 basis points. In the same way the haircut of government bonds and equities was

2000 and 3000 BP respectively, and the adjusted value was about 30 and 5 basis points. Looking at

the optimal case when the clearinghouse have haircuts applied, one can see that the Simplex method

suggest to fully invest in SEK, which is surprising, when we expected to see an full investment in USD.

Figure A.3 – The optimal usage of the portfolio in order to cover the risk margin value 10 000. Here,the clearinghouse only have a haircut on the value, with no upper and lower concentration limits.

When applying an upper concentration limit without a a lower concentration limit and haircut, and

also not considering that the bank adjust the collateral values, we have that the best solution value

III APPENDIX A. TEST CASES - REVISED SIMPLEX METHOD

should be 10 115.32 where the value with our portfolio is 10 483.78. Looking at the usage of the

portfolio to cover the risk, see Figure A.4 we can see that the Simplex method once again focus on

distributing the government bonds first. In the same way, the optimal allocation suggest that we

should primary use the most expensive government bond.

Figure A.4 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, we havean upper concentration limit without a a lower concentration limit and haircut, and also not consideringthat the bank adjust the collateral values.

Figure A.5 – The optimal usage to cover the risk margin value 10 000. Here, we have an upperconcentration limit without a a lower concentration limit and haircut, and also not considering that thebank adjust the collateral values

We can also see that in Table A.11, the upper concentration limit is crossed. This is due to the fact

that the Simplex method doesn’t consider that we can use half a bond or so. If the method took this

into consideration, it could have been able to remove some of the government bonds units.

When considering the case where we have a lower concentration limit without an upper limit, haircut

and adjusted value for the bank, the best solution value is 10 096.80 where the optimal allocation can

be seen in Figure A.6.

IV

Figure A.6 – The optimal usage to cover the risk margin value 10 000. Here, we have a lower concen-tration limit without an upper limit, haircut and adjusted value for the bank.

When using the portfolio, the best solution is 10 419.94 and the optimal allocation can be seen in

Figure A.7

Figure A.7 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, we havea lower concentration limit without an upper limit, haircut and adjusted value for the bank.

With the same argument as before, the Simplex method chooses the entering variable to be the one

with highest value and lowest amount, resulting in an non-optimal solution.

When having the same conditions as in the other case, but having a lower concentration limit that

exceeds the total value in the portfolio, we get the following result. As one can see in Table A.11,

the implementation handles this situation in a satisfying way by suggesting that we need to buy the

amount that is missing. However, the Simplex method continues to suggest that the bonds should

be prioritizes when covering up for the risk margin value, and again this must be due to the added

constraints regarding the amount the bank has of each collateral.

V APPENDIX A. TEST CASES - REVISED SIMPLEX METHOD

Figure A.8 – The optimal usage to cover the risk margin value 10 000. Here, we have a lower concen-tration limit that exceeds the total value in the portfolio. The model is without restrictions such as anupper limit, haircut and adjusted value for the bank.

When using the portfolio, the best value is 10 419.94 and the optimal allocation can be seen in Figure

A.7

Figure A.9 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, we have alower concentration limit that exceeds the total value in the portfolio. The model is without restrictionssuch as an upper limit, haircut and adjusted value for the bank.

If we consider the case where we don’t have haircuts, lower and upper concentration limit, and adjusted

valuation from the bank but the risk margin value exceeds the total portfolio, the optimal allocation

would be to pledge USD. However, the model can’t handle this type of problem and the model becomes

infeasible.

A.1. CASE 1 VI

A.1 Case 1

No haircut, no upper or lower concentration limit and no adjusted value.

Table A.1 – Portfolio used to test if the code do the expected. The total risk margin value is 10 000where the pledged value is 10 483.78.



Quantity inPortfoliobeforepledge

Quantity inPortfolio

afterpledge

Value ofPortfoliobeforepledge

Value ofPortfolio

afterpledge

Concentrationvalue

How muchof the totalportfolio

thatis used

USD=X 1 1 0 2000 2000 2000 2000 0 0EURUSD=X 1,0864 1,0864 0 2000 2000 2172,8 2172,8 0 0GBPUSD=X 1,2806 1,2806 0 2000 2000 2561,2 2561,2 0 0SEKUSD=X 0,1125 0,1125 0 2000 2000 225 225 0 0DKKUSD=X 0,1461 0,1461 0 2000 2000 292,2 292,2 0 0NOKUSD=X 0,1172 0,1172 0 2000 2000 234,4 234,4 0 0CADUSD=X 0,7424 0,7424 0 2000 2000 1484,8 1484,8 0 0JPYUSD=X 0,0091 0,0091 0 2000 2000 18,2 18,2 0 0

GTGBP5Y:GOV 99,97 99,97 0 15 -1 1499,55 -99,97 0,159952 0,062745GTGBP10Y:GOV 131,48 131,48 0 15 -1 1972,2 -131,48 0,210368 0,082522GTGBP30Y:GOV 95,48 95,48 0 15 0 1432,2 0 0,14322 0,056182GTDEM5Y:GOV 101,64 101,64 0 15 0 1524,6 0 0,15246 0,059806GTDEM10Y:GOV 98,54 98,54 0 15 -1 1478,1 -98,54 0,157664 0,061848GTDEM30Y:GOV 130,94 130,94 0 15 -1 1964,1 -130,94 0,209504 0,082183

ALFA.ST 19,83375 19,83375 0 100 100 1983,375 1983,375 0 0ELUX-B.ST 30,42 30,42 0 100 95 3042 2889,9 0,01521 0,005966VOLV-B.ST 16,07625 16,07625 0 100 100 1607,625 1607,625 0 0

Table A.2 – Clearinghouse 1 with no haircut, no upper/lower concentration limits and no adjustedvalue. risk margin value 10 000 and the total pledged value is 10 483.78. Here we can see that thecollaterals that have the largest value per unit is the ones that are mainly used in order to cover the riskmargin value. This is due to the fact that here the bank and clearinghouse values each collateral thesame and it becomes natural to use the smallest amount possible.



Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused

Value ofquantityused

Concentration

USD=X 1 1 0 1 1 1 0 0 0 0EURUSD=X 1,0864 1,0864 0 1 1,0864 1 0 0 0 0GBPUSD=X 1,2806 1,2806 0 1 1,2806 1 0 0 0 0SEKUSD=X 0,1125 0,1125 0 1 0,1125 1 0 0 0 0DKKUSD=X 0,1461 0,1461 0 1 0,1461 1 0 0 0 0NOKUSD=X 0,1172 0,1172 0 1 0,1172 1 0 0 0 0CADUSD=X 0,7424 0,7424 0 1 0,7424 1 0 0 0 0JPYUSD=X 0,0091 0,0091 0 1 0,0091 1 0 0 0 0

GTGBP5Y:GOV 99,97 99,97 0 1 99,97 1 0 16 1599,52 0,159952GTGBP10Y:GOV 131,48 131,48 0 1 131,48 1 0 16 2103,68 0,210368GTGBP30Y:GOV 95,48 95,48 0 1 95,48 1 0 15 1432,2 0,14322GTDEM5Y:GOV 101,64 101,64 0 1 101,64 1 0 15 1524,6 0,15246GTDEM10Y:GOV 98,54 98,54 0 1 98,54 1 0 16 1576,64 0,157664GTDEM30Y:GOV 130,94 130,94 0 1 130,94 1 0 16 2095,04 0,209504

ALFA.ST 19,83375 19,83375 0 1 19,83375 1 0 0 0 0ELUX-B.ST 30,42 30,42 0 1 30,42 1 0 5 152,1 0,01521VOLV-B.ST 16,07625 16,07625 0 1 16,07625 1 0 0 0 0

VII APPENDIX A. TEST CASES - REVISED SIMPLEX METHOD

A.2 Case 2

No haircut, no upper or lower concentration limit, but with adjusted value

Table A.3 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the pledged value is 10 514.85109 due to the fact that we only can have integers as units ofintegers. The optimal solution are here 10 050.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1,005 50 2000 2000 2010 2010 0 0EURUSD=X 1,0864 1,091832 50 2000 2000 2183,664 2183,664 0 0GBPUSD=X 1,2806 1,287003 50 2000 2000 2574,006 2574,006 0 0SEKUSD=X 0,1125 0,113063 50 2000 2000 226,125 226,125 0 0DKKUSD=X 0,1461 0,146831 50 2000 2000 293,661 293,661 0 0NOKUSD=X 0,1172 0,117786 50 2000 2000 235,572 235,572 0 0CADUSD=X 0,7424 0,746112 50 2000 2000 1492,224 1492,224 0 0JPYUSD=X 0,0091 0,009146 50 2000 2000 18,291 18,291 0 0


ALFA.ST 19,83375 19,84367 5 100 100 1984,367 1984,367 0 0ELUX-B.ST 30,42 30,43521 5 100 95 3043,521 2891,345 0,015218 0,005969VOLV-B.ST 16,07625 16,08429 5 100 100 1608,429 1608,429 0 0

Table A.4 – Clearinghouse 1 with no haircut, no upper/lower concentration limits. risk margin value10 000 and the total pledged value is 10 514.85109. Due to the fact that the equities are the least costlycollateral, the bank naturally will give these away first, which can be seen and confirmed in the tablebelow. We can also look at it as the bank values the equities the least and want to pledge these first.



Valueafter

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Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1,005 50 1 1,005 1 0 0 0 0EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



A.3. CASE 3 VIII

A.3 Case 3

No adjusted valuation, no upper or lower concentration limit, but with haircut.

Table A.5 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the pledged value is 11 617.6794 and the best solution value is 11 111.175





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1 0 2000 1 2000 1 0,2 0,078455EURUSD=X 1,0864 1,0864 0 2000 -1 2172,8 -1,0864 0,217389 0,085276GBPUSD=X 1,2806 1,2806 0 2000 -1 2561,2 -1,2806 0,256248 0,10052SEKUSD=X 0,1125 0,1125 0 2000 0 225 0 0,0225 0,008826DKKUSD=X 0,1461 0,1461 0 2000 -1 292,2 -0,1461 0,029235 0,011468NOKUSD=X 0,1172 0,1172 0 2000 -1 234,4 -0,1172 0,023452 0,0092CADUSD=X 0,7424 0,7424 0 2000 0 1484,8 0 0,14848 0,058245JPYUSD=X 0,0091 0,0091 0 2000 -1 18,2 -0,0091 0,001821 0,000714

GTGBP5Y:GOV 99,97 99,97 0 15 15 1499,55 1499,55 0 0GTGBP10Y:GOV 131,48 131,48 0 15 -1 1972,2 -131,48 0,210368 0,082522GTGBP30Y:GOV 95,48 95,48 0 15 15 1432,2 1432,2 0 0GTDEM5Y:GOV 101,64 101,64 0 15 15 1524,6 1524,6 0 0GTDEM10Y:GOV 98,54 98,54 0 15 15 1478,1 1478,1 0 0GTDEM30Y:GOV 130,94 130,94 0 15 11 1964,1 1440,34 0,052376 0,020546

ALFA.ST 19,83375 19,83375 0 100 100 1983,375 1983,375 0 0ELUX-B.ST 30,42 30,42 0 100 100 3042 3042 0 0VOLV-B.ST 16,07625 16,07625 0 100 100 1607,625 1607,625 0 0

Table A.6 – Clearinghouse 1 with no haircut, no upper/lower concentration limits. risk margin value10 000 and the total pledged value is 11 617.6794.



Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 0,9 0,9 1 0 2000 2000 0,2EURUSD=X 1,0864 1,0864 0 0,9 0,97776 1 0 2001 2173,886 0,217389GBPUSD=X 1,2806 1,2806 0 0,9 1,15254 1 0 2001 2562,481 0,256248SEKUSD=X 0,1125 0,1125 0 0,9 0,10125 1 0 2000 225 0,0225DKKUSD=X 0,1461 0,1461 0 0,9 0,13149 1 0 2001 292,3461 0,029235NOKUSD=X 0,1172 0,1172 0 0,9 0,10548 1 0 2001 234,5172 0,023452CADUSD=X 0,7424 0,7424 0 0,9 0,66816 1 0 2000 1484,8 0,14848JPYUSD=X 0,0091 0,0091 0 0,9 0,00819 1 0 2001 18,2091 0,001821

GTGBP5Y:GOV 99,97 99,97 0 0,8 79,976 1 0 0 0 0GTGBP10Y:GOV 131,48 131,48 0 0,8 105,184 1 0 16 2103,68 0,210368GTGBP30Y:GOV 95,48 95,48 0 0,8 76,384 1 0 0 0 0GTDEM5Y:GOV 101,64 101,64 0 0,8 81,312 1 0 0 0 0GTDEM10Y:GOV 98,54 98,54 0 0,8 78,832 1 0 0 0 0GTDEM30Y:GOV 130,94 130,94 0 0,8 104,752 1 0 4 523,76 0,052376

ALFA.ST 19,83375 19,83375 0 0,7 13,88363 1 0 0 0 0ELUX-B.ST 30,42 30,42 0 0,7 21,294 1 0 0 0 0VOLV-B.ST 16,07625 16,07625 0 0,7 11,25338 1 0 0 0 0

IX APPENDIX A. TEST CASES - REVISED SIMPLEX METHOD

A.4 Case 4

No adjusted valuation and haircut, no lower concentration limit, but with upper concentration limit

Table A.7 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the pledged value is 10 483.78. The best solution is 10 115.32. Difference is due to the factthat my best solution assumes that we can have parts of equities and bonds, which is not true.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used




Table A.8 – Clearinghouse 1 with no haircut, no upper/lower concentration limits. Risk margin value10 000 and the total pledged value is 10 483.78, where the best solution value is 10 115.32.



Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 1 1 0,9 0 0 0 0EURUSD=X 1,0864 1,0864 0 1 1,0864 0,9 0 0 0 0GBPUSD=X 1,2806 1,2806 0 1 1,2806 0,9 0 0 0 0SEKUSD=X 0,1125 0,1125 0 1 0,1125 0,9 0 0 0 0DKKUSD=X 0,1461 0,1461 0 1 0,1461 0,9 0 0 0 0NOKUSD=X 0,1172 0,1172 0 1 0,1172 0,9 0 0 0 0CADUSD=X 0,7424 0,7424 0 1 0,7424 0,9 0 0 0 0JPYUSD=X 0,0091 0,0091 0 1 0,0091 0,9 0 0 0 0

GTGBP5Y:GOV 99,97 99,97 0 1 99,97 0,8 0 16 1599,52 0,159952GTGBP10Y:GOV 131,48 131,48 0 1 131,48 0,8 0 16 2103,68 0,210368GTGBP30Y:GOV 95,48 95,48 0 1 95,48 0,8 0 15 1432,2 0,14322GTDEM5Y:GOV 101,64 101,64 0 1 101,64 0,8 0 15 1524,6 0,15246GTDEM10Y:GOV 98,54 98,54 0 1 98,54 0,8 0 16 1576,64 0,157664GTDEM30Y:GOV 130,94 130,94 0 1 130,94 0,8 0 16 2095,04 0,209504

ALFA.ST 19,83375 19,83375 0 1 19,83375 0,7 0 0 0 0ELUX-B.ST 30,42 30,42 0 1 30,42 0,7 0 5 152,1 0,01521VOLV-B.ST 16,07625 16,07625 0 1 16,07625 0,7 0 0 0 0

A.5. CASE 5 X

A.5 Case 5

No adjusted valuation and haircut, no upper concentration limit, but with lower concentration limit.

Table A.9 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the pledged value is 10 419.93825. The best solution is 10 096.79825. (Difference is due to thefact that my best solution assumes that we can have parts of equities and bonds, which is not true. Thenegative value is due to the fact that some clearinghouse needs more of this collateral than is availablein the portfolio.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1 0 2000 1000 2000 1000 0,1 0,039227EURUSD=X 1,0864 1,0864 0 2000 2000 2172,8 2172,8 0 0GBPUSD=X 1,2806 1,2806 0 2000 2000 2561,2 2561,2 0 0SEKUSD=X 0,1125 0,1125 0 2000 2000 225 225 0 0DKKUSD=X 0,1461 0,1461 0 2000 -4845 292,2 -707,855 0,100005 0,03923NOKUSD=X 0,1172 0,1172 0 2000 2000 234,4 234,4 0 0CADUSD=X 0,7424 0,7424 0 2000 2000 1484,8 1484,8 0 0JPYUSD=X 0,0091 0,0091 0 2000 2000 18,2 18,2 0 0

GTGBP5Y:GOV 99,97 99,97 0 15 9 1499,55 899,73 0,059982 0,023529GTGBP10Y:GOV 131,48 131,48 0 15 -1 1972,2 -131,48 0,210368 0,082522GTGBP30Y:GOV 95,48 95,48 0 15 15 1432,2 1432,2 0 0GTDEM5Y:GOV 101,64 101,64 0 15 0 1524,6 0 0,15246 0,059806GTDEM10Y:GOV 98,54 98,54 0 15 4 1478,1 394,16 0,108394 0,04252GTDEM30Y:GOV 130,94 130,94 0 15 -1 1964,1 -130,94 0,209504 0,082183

ALFA.ST 19,83375 19,83375 0 100 100 1983,375 1983,375 0 0ELUX-B.ST 30,42 30,42 0 100 100 3042 3042 0 0VOLV-B.ST 16,07625 16,07625 0 100 37 1607,625 594,8213 0,10128 0,03973

Table A.10 – Clearinghouse 1 with no haircut, no upper/lower concentration limits. risk margin value10 000 and the total pledged value is 10 419.93825.



Valueafter

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Haircut

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limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 1 1 1 0,1 1000 1000 0,1EURUSD=X 1,0864 1,0864 0 1 1,0864 1 0 0 0 0GBPUSD=X 1,2806 1,2806 0 1 1,2806 1 0 0 0 0SEKUSD=X 0,1125 0,1125 0 1 0,1125 1 0 0 0 0DKKUSD=X 0,1461 0,1461 0 1 0,1461 1 0,1 6845 1000,055 0,100005NOKUSD=X 0,1172 0,1172 0 1 0,1172 1 0 0 0 0CADUSD=X 0,7424 0,7424 0 1 0,7424 1 0 0 0 0JPYUSD=X 0,0091 0,0091 0 1 0,0091 1 0 0 0 0

GTGBP5Y:GOV 99,97 99,97 0 1 99,97 1 0 6 599,82 0,059982GTGBP10Y:GOV 131,48 131,48 0 1 131,48 1 0 16 2103,68 0,210368GTGBP30Y:GOV 95,48 95,48 0 1 95,48 1 0 0 0 0GTDEM5Y:GOV 101,64 101,64 0 1 101,64 1 0 15 1524,6 0,15246GTDEM10Y:GOV 98,54 98,54 0 1 98,54 1 0,1 11 1083,94 0,108394GTDEM30Y:GOV 130,94 130,94 0 1 130,94 1 0 16 2095,04 0,209504

ALFA.ST 19,83375 19,83375 0 1 19,83375 1 0 0 0 0ELUX-B.ST 30,42 30,42 0 1 30,42 1 0 0 0 0VOLV-B.ST 16,07625 16,07625 0 1 16,07625 1 0,1 63 1012,804 0,10128

XI APPENDIX A. TEST CASES - REVISED SIMPLEX METHOD

A.6 Case 6

No adjusted valuation and haircut, no upper concentration limit, no lower concentration limit, but the

risk margin value is larger than the portfolio value.

Table A.11 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the pledged value is 10 329.56375 and the best solution is 10 012.80375.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1 0 2000 -2000 2000 -2000 0,4 0,15691EURUSD=X 1,0864 1,0864 0 2000 2000 2172,8 2172,8 0 0GBPUSD=X 1,2806 1,2806 0 2000 2000 2561,2 2561,2 0 0SEKUSD=X 0,1125 0,1125 0 2000 2000 225 225 0 0DKKUSD=X 0,1461 0,1461 0 2000 2000 292,2 292,2 0 0NOKUSD=X 0,1172 0,1172 0 2000 2000 234,4 234,4 0 0CADUSD=X 0,7424 0,7424 0 2000 2000 1484,8 1484,8 0 0JPYUSD=X 0,0091 0,0091 0 2000 2000 18,2 18,2 0 0

GTGBP5Y:GOV 99,97 99,97 0 15 15 1499,55 1499,55 0 0GTGBP10Y:GOV 131,48 131,48 0 15 -1 1972,2 -131,48 0,210368 0,082522GTGBP30Y:GOV 95,48 95,48 0 15 15 1432,2 1432,2 0 0GTDEM5Y:GOV 101,64 101,64 0 15 4 1524,6 406,56 0,111804 0,043858GTDEM10Y:GOV 98,54 98,54 0 15 15 1478,1 1478,1 0 0GTDEM30Y:GOV 130,94 130,94 0 15 -1 1964,1 -130,94 0,209504 0,082183

ALFA.ST 19,83375 19,83375 0 100 100 1983,375 1983,375 0 0ELUX-B.ST 30,42 30,42 0 100 100 3042 3042 0 0VOLV-B.ST 16,07625 16,07625 0 100 37 1607,625 594,8213 0,10128 0,03973

Table A.12 – Clearinghouse 1 with no haircut, no upper/lower concentration limits.The total riskmargin value is 10 000 where the pledged value is 10 329.56375 and the best solution is 10012.80375.



Valueafter

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Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 1 1 1 0,4 4000 4000 0,4EURUSD=X 1,0864 1,0864 0 1 1,0864 1 0 0 0 0GBPUSD=X 1,2806 1,2806 0 1 1,2806 1 0 0 0 0SEKUSD=X 0,1125 0,1125 0 1 0,1125 1 0 0 0 0DKKUSD=X 0,1461 0,1461 0 1 0,1461 1 0 0 0 0NOKUSD=X 0,1172 0,1172 0 1 0,1172 1 0 0 0 0CADUSD=X 0,7424 0,7424 0 1 0,7424 1 0 0 0 0JPYUSD=X 0,0091 0,0091 0 1 0,0091 1 0 0 0 0

GTGBP5Y:GOV 99,97 99,97 0 1 99,97 1 0 0 0 0GTGBP10Y:GOV 131,48 131,48 0 1 131,48 1 0 16 2103,68 0,210368GTGBP30Y:GOV 95,48 95,48 0 1 95,48 1 0 0 0 0GTDEM5Y:GOV 101,64 101,64 0 1 101,64 1 0 11 1118,04 0,111804GTDEM10Y:GOV 98,54 98,54 0 1 98,54 1 0 0 0 0GTDEM30Y:GOV 130,94 130,94 0 1 130,94 1 0 16 2095,04 0,209504

ALFA.ST 19,83375 19,83375 0 1 19,83375 1 0 0 0 0ELUX-B.ST 30,42 30,42 0 1 30,42 1 0 0 0 0VOLV-B.ST 16,07625 16,07625 0 1 16,07625 1 0,1 63 1012,804 0,10128

A.7. CASE 7 XII

A.7 Case 7

No adjusted valuation and haircut, no upper concentration limit, no lower concentration limit with

calculated transformation cost.

Table A.13 – The risk margin value is 10000, where the solution is 10 483.78. When reallocating theexcess collateral the best solution value 10 307.66 plus a transformation cost of 10.47, reallocating theinitial portfolio to be equal to the optimal allocation will make the best solution to be 10 000 + 12.38in transformation costs.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used




Table A.14 – Clearinghouse 1 with no haircut, no upper/lower concentration limits. risk margin value10 000 and the total pledged value is 10 483.78. When reallocating the excess collateral the best solutionvalue 10 307.66 plus a transformation cost of 10.47, reallocating the initial portfolio to be equal to theoptimal allocation will make the best solution to be 10 000 + 12.38 in transformation costs.



Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 1 1 1 0 0 0 0EURUSD=X 1,0864 1,0864 0 1 1,0864 1 0 0 0 0GBPUSD=X 1,2806 1,2806 0 1 1,2806 1 0 0 0 0SEKUSD=X 0,1125 0,1125 0 1 0,1125 1 0 0 0 0DKKUSD=X 0,1461 0,1461 0 1 0,1461 1 0 0 0 0NOKUSD=X 0,1172 0,1172 0 1 0,1172 1 0 0 0 0CADUSD=X 0,7424 0,7424 0 1 0,7424 1 0 0 0 0JPYUSD=X 0,0091 0,0091 0 1 0,0091 1 0 0 0 0



Appendix B

Test cases - Branch & Bound

Starting to look at the case where the price isn’t adjusted, we have no upper or lower concentration

limit and no haircut. I expect that the method would choose to use the USD dollar collateral such as

it have done in the optimal allocation, however this isn’t done, which can be seen in Figure B.1.

Figure B.1 – The usage of the portfolio to cover the risk margin value 10 000. Here, the price isn’tadjusted, we have no upper or lower concentration limit and no haircut.

As can be seen in Figure B.1, the Branch&Bound method also uses mostly of the heavier weighted

collaterals, but with the difference that it minimizes the pledged value in a better way, using the large

collaterals to cover up the most of the risks and using the USD to cover the rest, which in this case is

a very small amount.

When looking at the case where the clearinghouse have no upper/lower concentration limits and no

haircut, but the bank adjusts the value of the collaterals to consider the funding costs. To make the

different collateral types comparable I chose the same adjustment percentage for every type. The

optimal value here is 10 050, where the optimal allocation is fully invested in USD that is adjusted

with 50 BP. However, the best allocation of the portfolio looks similar to Figure B.2 where the optimal

solution is 10 026.15, which actually is lower than the optimal solution, but due to the fact that it is

higher than the risk margin value, we can argue that the solution is optimal. This also strengthens the

assumption that only USD isn’t the most optimal solution when considering eventual funding costs.

XIII

XIV

Figure B.2 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, theclearinghouse have no upper/lower concentration limits and no haircut, but the bank adjusts the valueof the collaterals to consider the funding costs.

Figure B.3 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, theclearinghouse have no upper/lower concentration limits and no haircut, but the bank adjusts the valueof the collaterals to consider the funding costs.

For the case where the bank don’t adjust the value, and the clearinghouse only have a haircut on the

value, with no upper and lower concentration limits, the optimal value is here 11 111.11 (fully invested

in USD) due to the fact that the bank needs to but out more collateral as the clearinghouse values

it less. Having a portfolio and optimizing by Branch&Bound, the optimal value becomes 11 376.43,

which also is a better result than for the Simplex method. In Figure B.4, we can see what collaterals

is used to cover the risk.

XV APPENDIX B. TEST CASES - BRANCH & BOUND

Figure B.4 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, the thebank don’t adjust the value, and the clearinghouse only have a haircut on the value, with no upper andlower concentration limits,

One could expect that the method would behave in the same way as when the bank adjusts the value,

however, in this case we can’t possibly compare these two cases. An example of why we can’t compare

these is that the haircut of cash adjusts the value with 1000 basis points, where the bank only adjusted

the value with 50 basis points. In the same way the haircut of government bonds and equities was

2000 and 3000 BP respectively, and the adjusted value was about 30 and 5 basis points. Looking at

the optimal case when the clearinghouse have haircuts applied, we can see that the Branch&Bound

chooses to only use USD to cover the risk, which was expected.

When applying an upper concentration limit without a lower concentration limit and haircut, and also

not considering that the bank adjust the collateral values, we have that the optimal value should be

10 000 where the value with our portfolio also is 10 000. We can’t possibly be fully invested in the

USD due to the upper limits and the most optimal allocation can be seen in Figure B.5

Figure B.5 – The optimal usage to cover the risk margin value 10 000. Here, an upper concentrationlimit is applied without a lower concentration limit and haircut and the bank doesn’t adjust the collateralvalues.

Comparing the solution with the Simplex method solution we can see that the Branch&Bound method

never crosses the upper concentration limit.

XVI

When considering the case where we have a lower concentration limit without an upper limit, haircut

and adjusted value for the bank, the optimal value is 10 096.80 where the optimal allocation can be

seen in Figure B.6.

Figure B.6 – The optimal usage to cover the risk margin value 10 000. Here, we have a lower concen-tration limit without an upper limit, haircut and adjusted valuation for the bank.

We can see that the Branch&Bound algorithm suggest to use as much USD as possible, and only use

minimum of the other collaterals. When using our portfolio, the optimal value is also 10 096.80 and

the optimal allocation can be seen in Figure B.7

Figure B.7 – The usage of the portfolio in order to cover the risk margin value 10 000. Here, we havea lower concentration limit without an upper limit, haircut and adjusted valuation for the bank.

As can be seen in Table B.9, the portfolio doesn’t contain enough DKK, and the value of this col-

lateral after pledge is a negative number, meaning that we need to buy collateral in order meet the

requirements.

If we consider the case where we don’t have haircuts, lower and upper concentration limit, and adjusted

valuation from the bank but the risk margin value exceeds the total portfolio, the optimal allocation

would be to only pledge USD. However, the Branch&Bound method can’t handle this type of problem

and the model is infeasible.

Due to the fact that the Branch&Bound algorithm is good at finding the optimal solution when no

haircut, upper/lower concentration limit and adjusted value is applied, it isn’t interesting to look at the

transformation costs because it isn’t necessary to transform the collateral when the optimal solution

is found.

XVII APPENDIX B. TEST CASES - BRANCH & BOUND

B.1 Case 1

No haircut, no upper or lower concentration limit and no adjusted value.

Table B.1 – The optimal value is 10 000, and the solution to the problem when using this optimizationtechnique is also 10 000.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1 0 2000 1999,495 2000 1999,495 5,05E-05 1,98E-05EURUSD=X 1,0864 1,0864 0 2000 1990 2172,8 2161,936 0,001086 0,000426GBPUSD=X 1,2806 1,2806 0 2000 2000 2561,2 2561,2 0 0SEKUSD=X 0,1125 0,1125 0 2000 2000 225 225 0 0DKKUSD=X 0,1461 0,1461 0 2000 2000 292,2 292,2 0 0NOKUSD=X 0,1172 0,1172 0 2000 2000 234,4 234,4 0 0CADUSD=X 0,7424 0,7424 0 2000 2000 1484,8 1484,8 0 0JPYUSD=X 0,0091 0,0091 0 2000 2000 18,2 18,2 0 0

GTGBP5Y:GOV 99,97 99,97 0 15 0 1499,55 0 0,149955 0,058824GTGBP10Y:GOV 131,48 131,48 0 15 0 1972,2 0 0,19722 0,077364GTGBP30Y:GOV 95,48 95,48 0 15 15 1432,2 1432,2 0 0GTDEM5Y:GOV 101,64 101,64 0 15 0 1524,6 0 0,15246 0,059806GTDEM10Y:GOV 98,54 98,54 0 15 0 1478,1 0 0,14781 0,057982GTDEM30Y:GOV 130,94 130,94 0 15 0 1964,1 0 0,19641 0,077047

ALFA.ST 19,83375 19,83375 0 100 97 1983,375 1923,874 0,00595 0,002334ELUX-B.ST 30,42 30,42 0 100 51 3042 1551,42 0,149058 0,058472VOLV-B.ST 16,07625 16,07625 0 100 100 1607,625 1607,625 0 0

Table B.2 – Clearinghouse 1 with no haircut, no upper/lower concentration limits and no adjustedvalue. risk margin value 10 000



Valueafter

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limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 1 1 1 0 0,50475 0,50475 5,05E-05EURUSD=X 1,0864 1,0864 0 1 1,0864 1 0 10 10,864 0,001086GBPUSD=X 1,2806 1,2806 0 1 1,2806 1 0 0 0 0SEKUSD=X 0,1125 0,1125 0 1 0,1125 1 0 0 0 0DKKUSD=X 0,1461 0,1461 0 1 0,1461 1 0 0 0 0NOKUSD=X 0,1172 0,1172 0 1 0,1172 1 0 0 0 0CADUSD=X 0,7424 0,7424 0 1 0,7424 1 0 0 0 0JPYUSD=X 0,0091 0,0091 0 1 0,0091 1 0 0 0 0

GTGBP5Y:GOV 99,97 99,97 0 1 99,97 1 0 15 1499,55 0,149955GTGBP10Y:GOV 131,48 131,48 0 1 131,48 1 0 15 1972,2 0,19722GTGBP30Y:GOV 95,48 95,48 0 1 95,48 1 0 0 0 0GTDEM5Y:GOV 101,64 101,64 0 1 101,64 1 0 15 1524,6 0,15246GTDEM10Y:GOV 98,54 98,54 0 1 98,54 1 0 15 1478,1 0,14781GTDEM30Y:GOV 130,94 130,94 0 1 130,94 1 0 15 1964,1 0,19641

ALFA.ST 19,83375 19,83375 0 1 19,83375 1 0 3 59,50125 0,00595ELUX-B.ST 30,42 30,42 0 1 30,42 1 0 49 1490,58 0,149058VOLV-B.ST 16,07625 16,07625 0 1 16,07625 1 0 0 0 0

B.2. CASE 2 XVIII

B.2 Case 2

No haircut, no upper or lower concentration limit, but with adjusted value

Table B.3 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the best solution is 10 026.15





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1,005 50 2000 1999,495 2010 2009,493 5,07E-05 1,99E-05EURUSD=X 1,0864 1,091832 50 2000 1990 2183,664 2172,746 0,001092 0,000428GBPUSD=X 1,2806 1,287003 50 2000 2000 2574,006 2574,006 0 0SEKUSD=X 0,1125 0,113063 50 2000 2000 226,125 226,125 0 0DKKUSD=X 0,1461 0,146831 50 2000 2000 293,661 293,661 0 0NOKUSD=X 0,1172 0,117786 50 2000 2000 235,572 235,572 0 0CADUSD=X 0,7424 0,746112 50 2000 2000 1492,224 1492,224 0 0JPYUSD=X 0,0091 0,009146 50 2000 2000 18,291 18,291 0 0


ALFA.ST 19,83375 19,84367 5 100 97 1984,367 1924,836 0,005953 0,002335ELUX-B.ST 30,42 30,43521 5 100 51 3043,521 1552,196 0,149133 0,058501VOLV-B.ST 16,07625 16,08429 5 100 100 1608,429 1608,429 0 0

Table B.4 – Clearinghouse 1 with no haircut, no upper/lower concentration limits. risk margin value10 000 and the total pledged value is 10 026.15.



Valueafter

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limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1,005 50 1 1,005 1 0 0,50475 0,507274 5,07E-05EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 10 10,91832 0,001092GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0

GTGBP5Y:GOV 99,97 100,2699 30 1 100,2699 1 0 15 1504,049 0,150405GTGBP10Y:GOV 131,48 131,8744 30 1 131,8744 1 0 15 1978,117 0,197812GTGBP30Y:GOV 95,48 95,76644 30 1 95,76644 1 0 0 0 0GTDEM5Y:GOV 101,64 101,9449 30 1 101,9449 1 0 15 1529,174 0,152917GTDEM10Y:GOV 98,54 98,83562 30 1 98,83562 1 0 15 1482,534 0,148253GTDEM30Y:GOV 130,94 131,3328 30 1 131,3328 1 0 15 1969,992 0,196999

ALFA.ST 19,83375 19,84367 5 1 19,84367 1 0 3 59,531 0,005953ELUX-B.ST 30,42 30,43521 5 1 30,43521 1 0 49 1491,325 0,149133VOLV-B.ST 16,07625 16,08429 5 1 16,08429 1 0 0 0 0

XIX APPENDIX B. TEST CASES - BRANCH & BOUND

B.3 Case 3

No adjusted valuation, no upper or lower concentration limit, but with haircut.

Table B.5 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000where the pledged value is 11 376.425564 and the optimal solution is 11 111.11111.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1 0 2000 0,004444 2000 0,004444 0,2 0,078455EURUSD=X 1,0864 1,0864 0 2000 0 2172,8 0 0,21728 0,085233GBPUSD=X 1,2806 1,2806 0 2000 0 2561,2 0 0,25612 0,100469SEKUSD=X 0,1125 0,1125 0 2000 0 225 0 0,0225 0,008826DKKUSD=X 0,1461 0,1461 0 2000 0 292,2 0 0,02922 0,011462NOKUSD=X 0,1172 0,1172 0 2000 0 234,4 0 0,02344 0,009195CADUSD=X 0,7424 0,7424 0 2000 0 1484,8 0 0,14848 0,058245JPYUSD=X 0,0091 0,0091 0 2000 0 18,2 0 0,00182 0,000714

GTGBP5Y:GOV 99,97 99,97 0 15 8 1499,55 799,76 0,069979 0,027451GTGBP10Y:GOV 131,48 131,48 0 15 15 1972,2 1972,2 0 0GTGBP30Y:GOV 95,48 95,48 0 15 14 1432,2 1336,72 0,009548 0,003745GTDEM5Y:GOV 101,64 101,64 0 15 9 1524,6 914,76 0,060984 0,023922GTDEM10Y:GOV 98,54 98,54 0 15 13 1478,1 1281,02 0,019708 0,007731GTDEM30Y:GOV 130,94 130,94 0 15 9 1964,1 1178,46 0,078564 0,030819

ALFA.ST 19,83375 19,83375 0 100 100 1983,375 1983,375 0 0ELUX-B.ST 30,42 30,42 0 100 100 3042 3042 0 0VOLV-B.ST 16,07625 16,07625 0 100 100 1607,625 1607,625 0 0

Table B.6 – Clearinghouse 1 with no haircut, no upper/lower concentration limits. The total riskmargin value is 10000 where the pledged value is 11 376.425564 and the optimal solution is 11 111.11111.



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limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 0,9 0,9 1 0 1999,996 1999,996 0,2EURUSD=X 1,0864 1,0864 0 0,9 0,97776 1 0 2000 2172,8 0,21728GBPUSD=X 1,2806 1,2806 0 0,9 1,15254 1 0 2000 2561,2 0,25612SEKUSD=X 0,1125 0,1125 0 0,9 0,10125 1 0 2000 225 0,0225DKKUSD=X 0,1461 0,1461 0 0,9 0,13149 1 0 2000 292,2 0,02922NOKUSD=X 0,1172 0,1172 0 0,9 0,10548 1 0 2000 234,4 0,02344CADUSD=X 0,7424 0,7424 0 0,9 0,66816 1 0 2000 1484,8 0,14848JPYUSD=X 0,0091 0,0091 0 0,9 0,00819 1 0 2000 18,2 0,00182

GTGBP5Y:GOV 99,97 99,97 0 0,8 79,976 1 0 7 699,79 0,069979GTGBP10Y:GOV 131,48 131,48 0 0,8 105,184 1 0 0 0 0GTGBP30Y:GOV 95,48 95,48 0 0,8 76,384 1 0 1 95,48 0,009548GTDEM5Y:GOV 101,64 101,64 0 0,8 81,312 1 0 6 609,84 0,060984GTDEM10Y:GOV 98,54 98,54 0 0,8 78,832 1 0 2 197,08 0,019708GTDEM30Y:GOV 130,94 130,94 0 0,8 104,752 1 0 6 785,64 0,078564


B.4. CASE 4 XX

B.4 Case 4

No adjusted valuation and haircut, no lower concentration limit, but with upper concentration limit.

Table B.7 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the solution is 10 000.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1 0 2000 1999,495 2000 1999,495 5,05E-05 1,98E-05EURUSD=X 1,0864 1,0864 0 2000 1990 2172,8 2161,936 0,001086 0,000426GBPUSD=X 1,2806 1,2806 0 2000 2000 2561,2 2561,2 0 0SEKUSD=X 0,1125 0,1125 0 2000 2000 225 225 0 0DKKUSD=X 0,1461 0,1461 0 2000 2000 292,2 292,2 0 0NOKUSD=X 0,1172 0,1172 0 2000 2000 234,4 234,4 0 0CADUSD=X 0,7424 0,7424 0 2000 2000 1484,8 1484,8 0 0JPYUSD=X 0,0091 0,0091 0 2000 2000 18,2 18,2 0 0


ALFA.ST 19,83375 19,83375 0 100 97 1983,375 1923,874 0,00595 0,002334ELUX-B.ST 30,42 30,42 0 100 51 3042 1551,42 0,149058 0,058472VOLV-B.ST 16,07625 16,07625 0 100 100 1607,625 1607,625 0 0

Table B.8 – Clearinghouse 1 with no haircut, no upper/lower concentration limits. The total riskmargin value is 10 000 where the solution is 10 000.



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Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 1 1 0,9 0 0,50475 0,50475 5,05E-05EURUSD=X 1,0864 1,0864 0 1 1,0864 0,9 0 10 10,864 0,001086GBPUSD=X 1,2806 1,2806 0 1 1,2806 0,9 0 0 0 0SEKUSD=X 0,1125 0,1125 0 1 0,1125 0,9 0 0 0 0DKKUSD=X 0,1461 0,1461 0 1 0,1461 0,9 0 0 0 0NOKUSD=X 0,1172 0,1172 0 1 0,1172 0,9 0 0 0 0CADUSD=X 0,7424 0,7424 0 1 0,7424 0,9 0 0 0 0JPYUSD=X 0,0091 0,0091 0 1 0,0091 0,9 0 0 0 0

GTGBP5Y:GOV 99,97 99,97 0 1 99,97 0,8 0 15 1499,55 0,149955GTGBP10Y:GOV 131,48 131,48 0 1 131,48 0,8 0 15 1972,2 0,19722GTGBP30Y:GOV 95,48 95,48 0 1 95,48 0,8 0 0 0 0GTDEM5Y:GOV 101,64 101,64 0 1 101,64 0,8 0 15 1524,6 0,15246GTDEM10Y:GOV 98,54 98,54 0 1 98,54 0,8 0 15 1478,1 0,14781GTDEM30Y:GOV 130,94 130,94 0 1 130,94 0,8 0 15 1964,1 0,19641

ALFA.ST 19,83375 19,83375 0 1 19,83375 0,7 0 3 59,50125 0,00595ELUX-B.ST 30,42 30,42 0 1 30,42 0,7 0 49 1490,58 0,149058VOLV-B.ST 16,07625 16,07625 0 1 16,07625 0,7 0 0 0 0

XXI APPENDIX B. TEST CASES - BRANCH & BOUND

B.5 Case 5

No adjusted valuation and haircut, no upper concentration limit, but with lower concentration limit.

Table B.9 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the best solution also is the most optimal 10 096.79825.





afterpledge


Value ofPortfolio

afterpledge

Concentrationvalue


thatis used

USD=X 1 1 0 2000 999,1145 2000 999,1145 0,100089 0,039262EURUSD=X 1,0864 1,0864 0 2000 1987 2172,8 2158,677 0,001412 0,000554GBPUSD=X 1,2806 1,2806 0 2000 2000 2561,2 2561,2 0 0SEKUSD=X 0,1125 0,1125 0 2000 2000 225 225 0 0DKKUSD=X 0,1461 0,1461 0 2000 -4845 292,2 -707,855 0,100005 0,03923NOKUSD=X 0,1172 0,1172 0 2000 2000 234,4 234,4 0 0CADUSD=X 0,7424 0,7424 0 2000 2000 1484,8 1484,8 0 0JPYUSD=X 0,0091 0,0091 0 2000 2000 18,2 18,2 0 0

GTGBP5Y:GOV 99,97 99,97 0 15 0 1499,55 0 0,149955 0,058824GTGBP10Y:GOV 131,48 131,48 0 15 0 1972,2 0 0,19722 0,077364GTGBP30Y:GOV 95,48 95,48 0 15 14 1432,2 1336,72 0,009548 0,003745GTDEM5Y:GOV 101,64 101,64 0 15 15 1524,6 1524,6 0 0GTDEM10Y:GOV 98,54 98,54 0 15 0 1478,1 0 0,14781 0,057982GTDEM30Y:GOV 130,94 130,94 0 15 0 1964,1 0 0,19641 0,077047

ALFA.ST 19,83375 19,83375 0 100 97 1983,375 1923,874 0,00595 0,002334ELUX-B.ST 30,42 30,42 0 100 100 3042 3042 0 0VOLV-B.ST 16,07625 16,07625 0 100 37 1607,625 594,8213 0,10128 0,03973

Table B.10 – Portfolio used to test if the code do what is expected. The total risk margin value is 10000 where the best solution also is the most optimal 10 096.79825.



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limit

Lowerconcentration

limit

Quantityused


Concentration

USD=X 1 1 0 1 1 1 0,1 1000,886 1000,886 0,100089EURUSD=X 1,0864 1,0864 0 1 1,0864 1 0 13 14,1232 0,001412GBPUSD=X 1,2806 1,2806 0 1 1,2806 1 0 0 0 0SEKUSD=X 0,1125 0,1125 0 1 0,1125 1 0 0 0 0DKKUSD=X 0,1461 0,1461 0 1 0,1461 1 0,1 6845 1000,055 0,100005NOKUSD=X 0,1172 0,1172 0 1 0,1172 1 0 0 0 0CADUSD=X 0,7424 0,7424 0 1 0,7424 1 0 0 0 0JPYUSD=X 0,0091 0,0091 0 1 0,0091 1 0 0 0 0

GTGBP5Y:GOV 99,97 99,97 0 1 99,97 1 0 15 1499,55 0,149955GTGBP10Y:GOV 131,48 131,48 0 1 131,48 1 0 15 1972,2 0,19722GTGBP30Y:GOV 95,48 95,48 0 1 95,48 1 0 1 95,48 0,009548GTDEM5Y:GOV 101,64 101,64 0 1 101,64 1 0 0 0 0GTDEM10Y:GOV 98,54 98,54 0 1 98,54 1 0,1 15 1478,1 0,14781GTDEM30Y:GOV 130,94 130,94 0 1 130,94 1 0 15 1964,1 0,19641

ALFA.ST 19,83375 19,83375 0 1 19,83375 1 0 3 59,50125 0,00595ELUX-B.ST 30,42 30,42 0 1 30,42 1 0 0 0 0VOLV-B.ST 16,07625 16,07625 0 1 16,07625 1 0,1 63 1012,804 0,10128

Appendix C

Result - Tables

In this chapter is the gathered output from the optimization model when five clearinghouses are

connected to one bank. Starting with the result when using the Simplex method as allocation approach,

and the the Branch&Bound method’s result are presented.

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C.1 Revised Simplex method

Using an infinite amount of collaterals the most optimal allocation can be found.

Table C.1 – The table contains the prices with price adjustment of the different collaterals, where the adjustment is meant to represent the banks’valuation of the collaterals. The table also contains the collateral usage when applying the Simplex method to a collateral portfolio with no upperquantity limit in order to get the optimal allocation. Total value of the pledged value is 15 880.22 where the total risk value was 15800. Remark:Here the initial quantity seems to be 100 0000 for every collateral, however, there are no upper constraint when solving the optimization problem.




before pledge


after pledge


Value ofPortfolio

afterpledge

Concentrationvalue


that is usedUSD=X 1 1,005 50 1000000 993095,5 1005000 998061 0,439177 9,52E-06

EURUSD=X 1,0864 1,091832 50 1000000 998527 1091832 1090224 0,101789 2,21E-06GBPUSD=X 1,2806 1,287003 50 1000000 999300 1287003 1286102 0,057019 1,24E-06SEKUSD=X 0,1125 0,113063 50 1000000 996444 113062,5 112660,4 0,025446 5,52E-07DKKUSD=X 0,1461 0,146831 50 1000000 1000000 146830,5 146830,5 0 0NOKUSD=X 0,1172 0,117786 50 1000000 1000000 117786 117786 0 0CADUSD=X 0,7424 0,746112 50 1000000 1000000 746112 746112 0 0JPYUSD=X 0,0091 0,009146 50 1000000 340659 9145,5 3115,497 0,381646 8,27E-06

GTGBP5Y:GOV 99,97 100,2699 30 1000000 1000000 1E+08 1E+08 0 0GTGBP10Y:GOV 131,48 131,8744 30 1000000 1000000 1,32E+08 1,32E+08 0 0GTGBP30Y:GOV 95,48 95,76644 30 1000000 1000000 95766440 95766440 0 0GTDEM5Y:GOV 101,64 101,9449 30 1000000 1000000 1,02E+08 1,02E+08 0 0GTDEM10Y:GOV 98,54 98,83562 30 1000000 1000000 98835620 98835620 0 0GTDEM30Y:GOV 130,94 131,3328 30 1000000 1000000 1,31E+08 1,31E+08 0 0

ALFA.ST 19,83375 19,84367 5 1000000 1000000 19843667 19843667 0 0ELUX-B.ST 30,42 30,43521 5 1000000 1000000 30435210 30435210 0 0VOLV-B.ST 16,07625 16,08429 5 1000000 1000000 16084288 16084288 0 0

C.1. REVISED SIMPLEX METHOD XXIV

Table C.2 – The table describes the collateral restrictions and requirements of every clearinghousetogether with the risk value, the collateral quantity used to cover the risk value and how much of this iscompared to the dealer banks original quantity in the collateral portfolio.



Valueafter

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limit

Lowerconcentration

limit

Quantityused


ConcentrationRiskvalue

Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4 4000 4020 0,402

10000 (10050)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 659341 6030,003 0,603

GTGBP5Y:GOV 99,97 100,2699 30 0,99 99,26721 0,99 0 0 0 0GTGBP10Y:GOV 131,48 131,8744 30 0,99 130,5557 0,99 0 0 0 0GTGBP30Y:GOV 95,48 95,76644 30 0,99 94,80878 0,99 0 0 0 0GTDEM5Y:GOV 101,64 101,9449 30 0,99 100,9255 0,99 0 0 0 0GTDEM10Y:GOV 98,54 98,83562 30 0,99 97,84726 0,99 0 0 0 0GTDEM30Y:GOV 130,94 131,3328 30 0,99 130,0195 0,99 0 0 0 0

ALFA.ST 19,83375 19,84367 5 0,98 19,44679 0,98 0 0 0 0ELUX-B.ST 30,42 30,43521 5 0,98 29,82651 0,98 0 0 0 0VOLV-B.ST 16,07625 16,08429 5 0,98 15,7626 0,98 0 0 0 0

Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0 1300 1306,5 1,005

1300(1306,5)

EURUSD=X 1,0864 1,091832 50 0,95 1,03724 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 0,95 1,222653 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 0,95 0,107409 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 0,95 0,139489 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 0,95 0,111897 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 0,95 0,708806 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 0,95 0,008688 1 0 0 0 0



Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0 104,4776 105 0,105

1000(1005,902)

EURUSD=X 1,0864 1,091832 50 1 1,091832 0,9 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 0,9 0 700 900,9021 0,900902SEKUSD=X 0,1125 0,113063 50 1 0,113063 0,9 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 0,9 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 0,9 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 0,9 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 0,9 0 0 0 0



Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0 0 0 0

2000(2010,319)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 1473 1608,269 0,804134GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0,2 3556 402,0503 0,201025DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0 1500 1507,5 1,005

1500(1507,5)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 0,97 0,109671 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 0,97 0,142426 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 0,97 0,114252 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 0,97 0,723729 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 0,97 0,008871 1 0 0 0 0

GTGBP5Y:GOV 99,97 100,2699 30 0,9 90,24292 1 0 0 0 0GTGBP10Y:GOV 131,48 131,8744 30 0,85 112,0933 1 0 0 0 0GTGBP30Y:GOV 95,48 95,76644 30 0,8 76,61315 1 0 0 0 0GTDEM5Y:GOV 101,64 101,9449 30 0,9 91,75043 1 0 0 0 0GTDEM10Y:GOV 98,54 98,83562 30 0,85 84,01028 1 0 0 0 0GTDEM30Y:GOV 130,94 131,3328 30 0,8 105,0663 1 0 0 0 0


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Here the initial portfolio allocation (Table 3.1) gives the best optimal solution.

Table C.3 – The table contains the prices with price adjustment of the different collaterals, where the adjustment is meant to represent the banks’valuation of the collaterals. The table also contains the collateral usage when applying the Simplex method to a collateral portfolio with a specificallocation. Total value of the pledged value is 15 879.89 where the total risk value was 15 800.




before pledge


after pledge


Value ofPortfolio

afterpledge

Concentrationvalue


that is usedUSD=X 1 1,005 50 2000 -6504,48 2010 -6537 0,540949 0,335277

EURUSD=X 1,0864 1,091832 50 2000 1175 2183,664 1282,903 0,05701 0,035335GBPUSD=X 1,2806 1,287003 50 2000 2000 2574,006 2574,006 0 0SEKUSD=X 0,1125 0,113063 50 2000 -54890 226,125 -6206 0,407097 0,252316DKKUSD=X 0,1461 0,146831 50 2000 2000 293,661 293,661 0 0NOKUSD=X 0,1172 0,117786 50 2000 2000 235,572 235,572 0 0CADUSD=X 0,7424 0,746112 50 2000 2000 1492,224 1492,224 0 0JPYUSD=X 0,0091 0,009146 50 2000 2000 18,291 18,291 0 0

GTGBP5Y:GOV 99,97 100,2699 30 15 15 1504,049 1504,049 0 0GTGBP10Y:GOV 131,48 131,8744 30 15 15 1978,117 1978,117 0 0GTGBP30Y:GOV 95,48 95,76644 30 15 15 1436,497 1436,497 0 0GTDEM5Y:GOV 101,64 101,9449 30 15 15 1529,174 1529,174 0 0GTDEM10Y:GOV 98,54 98,83562 30 15 15 1482,534 1482,534 0 0GTDEM30Y:GOV 130,94 131,3328 30 15 15 1969,992 1969,992 0 0

ALFA.ST 19,83375 19,84367 5 100 100 1984,367 1984,367 0 0ELUX-B.ST 30,42 30,43521 5 100 100 3043,521 3043,521 0 0VOLV-B.ST 16,07625 16,08429 5 100 100 1608,429 1608,429 0 0

C.1. REVISED SIMPLEX METHOD XXVI




Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused



Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4 4000 4020 0,402

10000 (10050,08)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0 53334 6030,075 0,603008DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0 1300 1306,5 1,005

1300(1306,5)




Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0 104,4776 105 0,105

1000(1005,761)

EURUSD=X 1,0864 1,091832 50 1 1,091832 0,9 0 825 900,7614 0,900761GBPUSD=X 1,2806 1,287003 50 1 1,287003 0,9 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 0,9 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 0,9 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 0,9 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 0,9 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 0,9 0 0 0 0



Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0 1600 1608 0,804

2000(2010,05)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0,2 3556 402,0503 0,201025DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0 1500 1507,5 1,005

1500(1507,5)




XX

VII

AP

PE

ND

IXC

.R

ES

ULT

-T

AB

LE

S

Here the initial portfolio allocation (Table 3.1) are reallocated with respect to the optimal allocation found in Table C.1.

Table C.5 – The table contains the prices with price adjustment of the different collaterals, where the adjustment is meant to represent the banks’valuation of the collaterals. The table also contains the collateral usage when applying the Simplex method and reallocating the excess collateral inthe initial portfolio with respect to the optimal allocation. Total value of the pledged value is 15 879.86 where the total risk value was 15 800.




before pledge


after pledge


Value ofPortfolio

afterpledge

Concentrationvalue


that is usedUSD=X 1 1,005 50 12154,04 5354,04 12214,81 5380,81 0,432532 0,243993

EURUSD=X 1,0864 1,091832 50 2778,597 2778,597 3033,761 3033,761 0 0GBPUSD=X 1,2806 1,287003 50 2370,005 2370,005 3050,204 3050,204 0 0SEKUSD=X 0,1125 0,113063 50 58769,63 55213,63 6644,641 6242,591 0,025446 0,014354DKKUSD=X 0,1461 0,146831 50 0 -52736 0 -7743,25 0,490079 0,276456NOKUSD=X 0,1172 0,117786 50 0 0 0 0 0 0CADUSD=X 0,7424 0,746112 50 0 -1207 0 -900,557 0,056997 0,032152JPYUSD=X 0,0091 0,009146 50 350513,9 350513,9 3205,624 3205,624 0 0

GTGBP5Y:GOV 99,97 100,2699 30 0 0 0 0 0 0GTGBP10Y:GOV 131,48 131,8744 30 0 0 0 0 0 0GTGBP30Y:GOV 95,48 95,76644 30 0 0 0 0 0 0GTDEM5Y:GOV 101,64 101,9449 30 0 0 0 0 0 0GTDEM10Y:GOV 98,54 98,83562 30 0 0 0 0 0 0GTDEM30Y:GOV 130,94 131,3328 30 0 0 0 0 0 0


C.1. REVISED SIMPLEX METHOD XXVIII




Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused



Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4 4000 4020 0,402

10000 (10050,03)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 41068 6030,035 0,603003NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0 1300 1306,5 1,005

1300(1306,5)




Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0 0 0 0

1000(1005,688)

EURUSD=X 1,0864 1,091832 50 1 1,091832 0,9 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 0,9 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 0,9 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 0,9 0 716 105,1306 0,105131NOKUSD=X 0,1172 0,117786 50 1 0,117786 0,9 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 0,9 0 1207 900,5572 0,900557JPYUSD=X 0,0091 0,009146 50 1 0,009146 0,9 0 0 0 0



Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0 0 0 0

2000(2010,138)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0,2 3556 402,0503 0,201025DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 10952 1608,088 0,804044NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0 1500 1507,5 1,005

1500(1507,5)




XX

IXA

PP

EN

DIX

C.

RE

SU

LT

-T

AB

LE

S

Here the unused collateral are reallocated with respect to the optimal allocation found in subsection C, and solved.

Table C.7 – The table contains the prices with price adjustment of the different collaterals, where the adjustment is meant to represent the banks’valuation of the collaterals. The table also contains the collateral usage when applying the Simplex method and reallocating the initial portfolio withrespect to the optimal allocation. Total value of the pledged value is 15 881.26 where the total risk value was 15 800.




before pledge


after pledge


Value ofPortfolio

afterpledge

Concentrationvalue


that is usedUSD=X 1 1,005 50 11173,99 4273,989 11229,86 4295,359 0,438892 0,271173

EURUSD=X 1,0864 1,091832 50 2383,857 905,8567 2602,771 989,0434 0,102135 0,063105GBPUSD=X 1,2806 1,287003 50 1132,858 432,8579 1457,992 557,0894 0,057019 0,03523SEKUSD=X 0,1125 0,113063 50 5754,918 2198,918 650,6654 248,6152 0,025446 0,015722DKKUSD=X 0,1461 0,146831 50 0 0 0 0 0 0NOKUSD=X 0,1172 0,117786 50 0 0 0 0 0 0CADUSD=X 0,7424 0,746112 50 0 -8082 0 -6030,08 0,38165 0,235806JPYUSD=X 0,0091 0,009146 50 1067057 1067057 9758,767 9758,767 0 0



C.1. REVISED SIMPLEX METHOD XXX




Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused



Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4 4000 4020 0,402

10000(10050,08)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 8082 6030,077 0,603008JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0 1300 1306,5 1,005

1300(1306,5)




Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0 0 0 0

1000(1006,81)

EURUSD=X 1,0864 1,091832 50 1 1,091832 0,9 0 97 105,9077 0,105908GBPUSD=X 1,2806 1,287003 50 1 1,287003 0,9 0 700 900,9021 0,900902SEKUSD=X 0,1125 0,113063 50 1 0,113063 0,9 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 0,9 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 0,9 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 0,9 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 0,9 0 0 0 0



Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0 1600 1608 0,804

2000(2010,05)




Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0 0 0 0

1500(1507,82)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 1381 1507,82 1,005213GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 0,97 0,109671 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 0,97 0,142426 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 0,97 0,114252 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 0,97 0,723729 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 0,97 0,008871 1 0 0 0 0



XX

XI

AP

PE

ND

IXC

.R

ES

ULT

-T

AB

LE

S

C.2 Branch and Bound

Using an infinite amount of collaterals the most optimal allocation can be found.

Table C.9 – The table contains the prices with price adjustment of the different collaterals, where the adjustment is meant to represent the banks’valuation of the collaterals. The table also contains the collateral usage when applying the Branch and bound method to a collateral portfolio withno upper quantity limit in order to get the optimal allocation. Total value of the pledged value is 15 879.05 where the total risk value was 15 800.Remark: Here the initial quantity seems to be 100 0000 for every collateral, however, there are no upper constraint when solving the optimizationproblem.




before pledge


after pledge


Value ofPortfolio

afterpledge

Concentrationvalue


that is usedUSD=X 1 1,005 50 1000000 987094,6 1005000 992030,1 0,820881 1,78E-05

EURUSD=X 1,0864 1,091832 50 1000000 1000000 1091832 1091832 0 0GBPUSD=X 1,2806 1,287003 50 1000000 998052 1287003 1284496 0,158676 3,44E-06SEKUSD=X 0,1125 0,113063 50 1000000 996444 113062,5 112660,4 0,025446 5,52E-07DKKUSD=X 0,1461 0,146831 50 1000000 1000000 146830,5 146830,5 0 0NOKUSD=X 0,1172 0,117786 50 1000000 1000000 117786 117786 0 0CADUSD=X 0,7424 0,746112 50 1000000 1000000 746112 746112 0 0JPYUSD=X 0,0091 0,009146 50 1000000 1000000 9145,5 9145,5 0 0

GTGBP5Y:GOV 99,97 100,2699 30 1000000 1000000 1E+08 1E+08 0 0GTGBP10Y:GOV 131,48 131,8744 30 1000000 1000000 1,32E+08 1,32E+08 0 0GTGBP30Y:GOV 95,48 95,76644 30 1000000 1000000 95766440 95766440 0 0GTDEM5Y:GOV 101,64 101,9449 30 1000000 1000000 1,02E+08 1,02E+08 0 0GTDEM10Y:GOV 98,54 98,83562 30 1000000 1000000 98835620 98835620 0 0GTDEM30Y:GOV 130,94 131,3328 30 1000000 1000000 1,31E+08 1,31E+08 0 0


C.2. BRANCH AND BOUND XXXII




Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused



Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4 10000 10050 1,005

10000 (10050)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0 1300 1306,5 1,005

1300(1306,5)




Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0 104,8606 105,3849 0,105385

1000(1005)




Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0 0,5306 0,533253 0,000267

2000(2010,05)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 1249 1607,467 0,803733SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0,2 3556 402,0503 0,201025DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0 1500 1507,5 1,005

1500(1507,5)




XX

XIII

AP

PE

ND

IXC

.R

ES

ULT

-T

AB

LE

S

Here the initial portfolio allocation (Table 3.1) gives the best optimal solution.

Table C.11 – The table contains the prices with price adjustment of the different collaterals, where the adjustment is meant to represent the banks’valuation of the collaterals. The table also contains the collateral usage when applying the Branch and bound method to a collateral portfolio witha specific allocation. Total value of the pledged value is 15 879.05 where the total risk value was 15 800.




before pledge


after pledge


Value ofPortfolio

afterpledge

Concentrationvalue


that is usedUSD=X 1 1,005 50 2000 -12504,8 2010 -12567,3 0,922616 0,571832

EURUSD=X 1,0864 1,091832 50 2000 1176 2183,664 1283,994 0,056941 0,035292GBPUSD=X 1,2806 1,287003 50 2000 2000 2574,006 2574,006 0 0SEKUSD=X 0,1125 0,113063 50 2000 -1556 226,125 -175,925 0,025446 0,015771DKKUSD=X 0,1461 0,146831 50 2000 2000 293,661 293,661 0 0NOKUSD=X 0,1172 0,117786 50 2000 2000 235,572 235,572 0 0CADUSD=X 0,7424 0,746112 50 2000 2000 1492,224 1492,224 0 0JPYUSD=X 0,0091 0,009146 50 2000 2000 18,291 18,291 0 0

GTGBP5Y:GOV 99,97 100,2699 30 15 15 1504,049 1504,049 0 0GTGBP10Y:GOV 131,48 131,8744 30 15 15 1978,117 1978,117 0 0GTGBP30Y:GOV 95,48 95,76644 30 15 15 1436,497 1436,497 0 0GTDEM5Y:GOV 101,64 101,9449 30 15 15 1529,174 1529,174 0 0GTDEM10Y:GOV 98,54 98,83562 30 15 15 1482,534 1482,534 0 0GTDEM30Y:GOV 130,94 131,3328 30 15 15 1969,992 1969,992 0 0

ALFA.ST 19,83375 19,84367 5 100 100 1984,367 1984,367 0 0ELUX-B.ST 30,42 30,43521 5 100 100 3043,521 3043,521 0 0VOLV-B.ST 16,07625 16,08429 5 100 100 1608,429 1608,429 0 0

C.2. BRANCH AND BOUND XXXIV




Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused



Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4 10000 10050 1,005

10000 (10050)

EURUSD=X 1,0864 1,091832 50 1 1,091832 1 0 0 0 0GBPUSD=X 1,2806 1,287003 50 1 1,287003 1 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 1 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 1 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 1 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 1 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 1 0 0 0 0



Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0 1300 1306,5 1,005

1300(1306,5)




Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0 104,8064 105,3304 0,10533

1000(1005)

EURUSD=X 1,0864 1,091832 50 1 1,091832 0,9 0 824 899,6696 0,89967GBPUSD=X 1,2806 1,287003 50 1 1,287003 0,9 0 0 0 0SEKUSD=X 0,1125 0,113063 50 1 0,113063 0,9 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 0,9 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 0,9 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 0,9 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 0,9 0 0 0 0



Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0 1600 1608 0,804

2000(2010,05)




Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0 1500 1507,5 1,005

1500(1507,5)




XX

XV

AP

PE

ND

IXC

.R

ESU

LT

-T

AB

LE

S

Here the initial portfolio allocation (Table 3.1) are reallocated with respect to the optimal allocation found in Table C.9.

Table C.13 – The table contains the prices with price adjustment of the different collaterals, where the adjustment is meant to represent the banks’valuation of the collaterals. The table also contains the collateral usage when applying the Branch and bound method and reallocating the excesscollateral in the initial portfolio with respect to the optimal allocation. Total value of the pledged value is 15 879.05 where the total risk value was15 800.




before pledge


after pledge


Value ofPortfolio

afterpledge

Concentrationvalue


that is usedUSD=X 1 1,005 50 21326,12 12925,15 21432,75 12989,77 0,534365 0,301641

EURUSD=X 1,0864 1,091832 50 2000 1846 2183,664 2015,522 0,010642 0,006007GBPUSD=X 1,2806 1,287003 50 3029,641 2379,641 3899,157 3062,605 0,052946 0,029887SEKUSD=X 0,1125 0,113063 50 5435,57 1879,57 614,5591 212,5089 0,025446 0,014364DKKUSD=X 0,1461 0,146831 50 0 0 0 0 0 0NOKUSD=X 0,1172 0,117786 50 0 0 0 0 0 0CADUSD=X 0,7424 0,746112 50 0 -8081 0 -6029,33 0,381603 0,215409JPYUSD=X 0,0091 0,009146 50 0 0 0 0 0 0



C.2. BRANCH AND BOUND XXXVI




Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused



Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4 4000,666 4020,669 0,402067

10000 (10050)




Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0 1300 1306,5 1,005

1300(1306,5)




Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0 0,3044 0,305922 0,000306

1000(1005)

EURUSD=X 1,0864 1,091832 50 1 1,091832 0,9 0 154 168,1421 0,168142GBPUSD=X 1,2806 1,287003 50 1 1,287003 0,9 0 650 836,552 0,836552SEKUSD=X 0,1125 0,113063 50 1 0,113063 0,9 0 0 0 0DKKUSD=X 0,1461 0,146831 50 1 0,146831 0,9 0 0 0 0NOKUSD=X 0,1172 0,117786 50 1 0,117786 0,9 0 0 0 0CADUSD=X 0,7424 0,746112 50 1 0,746112 0,9 0 0 0 0JPYUSD=X 0,0091 0,009146 50 1 0,009146 0,9 0 0 0 0



Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0 1600 1608 0,804

2000(2010,05)




Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0 1500 1507,5 1,005

1500(1507,5)




XX

XV

IIA

PP

EN

DIX

C.

RE

SU

LT

-T

AB

LE

S

Here the unused collateral are reallocated with respect to the optimal allocation found in subsection C, and solved.

Table C.15 – The table contains the prices with price adjustment of the different collaterals, where the adjustment is meant to represent the banks’valuation of the collaterals. The table also contains the collateral usage when applying the Branch and bound method and reallocating the initialportfolio with respect to the optimal allocation. Total value of the pledged value is 15 879.05 where the total risk value was 15 800.




before pledge


after pledge


Value ofPortfolio

afterpledge

Concentrationvalue


that is usedUSD=X 1 1,005 50 20885,68 12380,15 20990,11 12442,05 0,541016 0,334296

EURUSD=X 1,0864 1,091832 50 0 0 0 0 0 0GBPUSD=X 1,2806 1,287003 50 3152,582 2453,582 4057,382 3157,767 0,056938 0,035182SEKUSD=X 0,1125 0,113063 50 5754,918 2198,918 650,6654 248,6152 0,025446 0,015723DKKUSD=X 0,1461 0,146831 50 0 0 0 0 0 0NOKUSD=X 0,1172 0,117786 50 0 0 0 0 0 0CADUSD=X 0,7424 0,746112 50 0 -8081 0 -6029,33 0,381603 0,235794JPYUSD=X 0,0091 0,009146 50 0 0 0 0 0 0



C.2. BRANCH AND BOUND XXXVIII




Valueafter

Haircut

Priceafter

Haircut

Upperconcentration

limit

Lowerconcentration

limit

Quantityused



Clearinghouse 1

USD=X 1 1,005 50 1 1,005 1 0,4 4000,666 4020,669 0,402067

10000 (10050)




Clearinghouse 2

USD=X 1 1,005 50 1 1,005 1 0 1300 1306,5 1,005

1300(1306,5)




Clearinghouse 3

USD=X 1 1,005 50 1 1,005 0,9 0 104,8606 105,3849 0,105385

1000(1005)




Clearinghouse 4

USD=X 1 1,005 50 1 1,005 1 0 1600 1608 0,804

2000(2010,05)




Clearinghouse 5

USD=X 1 1,005 50 1 1,005 1 0 1500 1507,5 1,005

1500(1507,5)




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