Mixed Integer Linear Programming for Allocation of Collateral within Securities Lending

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IN

DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2020

Mixed Integer Linear Programming

for Allocation of Collateral within

Securities Lending

MARTIN WASS

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Mixed Integer Linear

Programming for Allocation of

Collateral within Securities

Lending

MARTIN WASS

Degree Projects in Optimization and Systems Theory (30 ECTS credits) Master’s Programme in Applied and Computational Mathematics (120 credits) KTH Royal Institute of Technology year 2020

Supervisor at Skandinaviska Enskilda Banken AB: Fredrik Niveman Supervisor at KTH: Per Enqvist

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Royal Institute of Technology

School of Engineering Sciences KTH SCI

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Abstract

A mixed integer linear programming formulation is used to solve the prob-lem of allocating assets from a bank to its counterparties as collateral within securities lending. The aim of the optimisation is to reduce the cost of allo-cated collateral, which is broken down into the components opportunity cost, counterparty risk cost and triparty cost.

A solution consists of transactions to carry out to improve the allocation cost, each transaction consisting of sending a quantity of some asset from a portfolio to the bank or from the bank to some portfolio.

The optimisation process is split into subproblems to separate obvious transactions from more complex optimisations. The reduction of each cost component is examined for all the subproblems. Two subproblems transform an initial collateral allocation into a feasible one which is then improved by the optimisation.

Decreasing opportunity cost is shown to be an easier task than decreasing counterparty risk and triparty costs since the latter costs require a compara-tively large number of transactions.

The optimisation is run several times in a row, performing the suggested transactions after each solved iteration. The cost reduction of k optimisation iterations with 10 transactions per iteration is shown to be similar to the cost reduction of 1 optimisation iteration with 10k transactions. The solution time increases heavily with the number of iterations.

The suggested transactions may need to be performed in a certain order. A precedence constrained problem takes this into account. The problem is large and the execution time is slow if a limit is imposed on the number of allowed transactions.

A strategic selection of portfolios can limit the number of suggested trans-actions and still reach a solution which comes close to the optimal one. This can also be done by requiring that all suggested transactions must reduce the cost by a minimum amount.

The final model is ready to be used in a semi-automatic fashion, where transactions are verified by a human who checks if they are sound. A fully automated process requires further testing on historical and recent data.

Keywords: collateral, collateral management, optimisation, mixed

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Svensk titel: Blandad heltalsprogrammering för optimal allokering av pant inom värdepapperslån

Sammanfattning

Ett blandat-heltal linjärt optimeringsproblem används för att lösa uppgiften att tilldela värdepapper från en bank till dess kunder som pant för värdepap-perslån. Målet med optimeringen är att minska kostnaden av den tilldelade panten. Kostnaden bryts ned i komponenterna alternativkostnad, motpartsrisk och tripartykostnad.

En lösning består av föreslagna transaktioner som ska genomföras för att förbättre den nuvarande säkerhetstilldelningens kostnad. En transaktion består av att ta hem eller skicka ut en kvantitet av ett visst värdepapper från eller till en av bankens kunders portföljer.

Optimeringsproblemet bryts ned i flera delproblem med syfte att särskilja uppenbara föreslagna transaktioner från mer komplicerade sådana. Två pro-blem omvandlar en initial tilldelning av säkerheter till en godkänd tildelning som sedan blir en startpunkt för optimeringen.

Att minska alternativkostnad visar sig vara enklare än att minska motparts-risk och tripartykostnader på så sätt att de sistnämnda kostnaderna kräver fler transaktioner för att minskas.

Optimeringen körs flera gånger i rad, där alla föreslagna transaktioner från en iteration genomförs innan nästa iteration körs. Kostnadsminskningen av k körningar med 10 transaktioner visar sig vara väldigt nära, om än något mind-re, än en körning med 10k transaktioner. Exekveringstiden ökar drastiskt med antalet iterationer.

De föreslagna transaktionerna kan behöva genomföras i en viss ordning. En problemformulering konstrueras som tar höjd för detta, men exekverings-tiden är extremt lång när antalet transaktioner begränsas.

Ett strategiskt urval av portföljer kan begränsa antalet föreslagna transak-tioner utan att försämra lösningen särskilt mycket. På ett liknande sätt kan an-talet föreslagna transaktioner minskas genom att lägga till ett villkor som säger att lönsamheten av en transaktion måste överskrida en given minsta tröskel.

Den slutgiltiga modellen är redo att användas om de föreslagna transak-tionerna granskas manuellt innan de genomförs. En helt automatisk process ligger längre fram i tiden efter ytterligare tester på historisk och nuvarande data.

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Acknowledgements

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Chapter 1 Introduction

When a borrower enters an agreement with a lender, there is a risk involved that the lent assets may never be returned. Quantifying this risk involves taking into account the value of the lent assets as well as the probability of the borrower not being able to return the loan. To give incentive for a lender to enter into an agreement, the borrower can post (transfer) collateral to the lender. Collateral consists of assets which are given to the lender from the borrower to enter the loan and which is returned to the borrower when the loan is repaid. If the borrower defaults (fails to repay the loan), the lender keeps the collateral. The required value of the collateral, as well as the acceptable assets which may be posted as collateral, is decided in a bilateral agreement between lender and borrower.

This thesis was carried out in collaboration with Skandinaviska Enskilda

Banken AB, which will henceforth be referred to as SEB. This thesis focuses

on collateral in the form of cash, bonds and stocks posted by SEB in different portfolios with the goal of automating the handling of collateral in a way which keeps valuable collateral in the bank and mitigates the risk of losing money in the event of defaulting clients. The large number of clients (order of magnitude 102

) and assets (order of magnitude 103) the bank deals with makes this a large problem which requires sophisticated techniques to be solvable fast enough for practical use.

The goal of the optimisation is to reduce the cost of allocated collateral. The cost can be split into three components; opportunity cost, counterparty risk and triparty cost. These costs are explained in the Background section of the report. The solution variable of the optimisation problem consists of the quantity of which collateral assets should be allocated to which portfolios.

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1.1 Research Questions

• When splitting the optimisation procedure into subproblems, how is the cost reduction split between the subproblems?

• How does the number of allowed transactions affect the cost reductions? • How does the cost reductions look when solving the optimisation

prob-lem several times in a row?

• Allocation cost is comprised of three components; opportunity cost, counterparty risk and triparty costs. Explore the cost reduction of each component in different steps and varying configurations of the optimi-sation.

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Chapter 2 Background

2.1 Collateral

The term collateral, as used in this paper, refers to assets used within lending agreements to reduce the lender’s risk of losing what they have lent. In a col-lateralised loan, the borrower has to transfer assets to the lender when entering the loan. These assets will belong to the lender until the loan has been repaid and the collateral is transferred back to the borrower. If the loan is not repaid, the collateral is kept by the lender and the value of the collateral will reduce the lender’s loss.

One example of a lending agreement where collateral is used is mortgage loans, where money is borrowed in order to buy property and the property acts as collateral. In the event that the agreements of the loan are broken, the lender may take possession of the property to recoup the loss.

2.1.1 Types of Collateral

In theory, collateral could consist of any asset deemed as valuable by the lender. In this thesis, collateral consists of either cash or securities in the form of stocks or bonds. The types of acceptable assets can be seen below.

1. Equity (stocks)

2. Bonds of the following types

• Sovereign government bonds (issued by a country’s government) • Municipal government bonds (issued by a municipality, state or

county)

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• Covered bonds (bond value is secured by underlying assets) • Corporate bonds (issued by a corporation)

• Financial institution bonds (issued by a financial institution) 3. Cash

Further details regarding acceptable collateral is explained in section 2.1.5.

2.1.2 Value of Collateral Assets

The value that a collateral receiver deems an asset to be worth will be referred to as the collateral value. To meet the demand of the collateral receiver, the collateral giver must post assets with a sufficient total collateral value. The collateral giver and the collateral receiver must determine what value they put on different assets in order to reach an agreement where both parties feel like the posted collateral sufficiently covers the value of the loan.

To ensure that the lender feels properly secured by the collateral received, it is common practice to put some requirements in place on the assets transferred by the borrower.

Initial Margin

The lender can require that the value of collateral assets posted exceeds the value of the lent assets. The extra value required is known as an initial margin. Given an initial margin of 5 per cent, the borrower would have to post collateral assets totalling to a value of 105 per cent of the value of the loan.

Haircuts

The total required collateral value is dependent on the individual valuation of specific assets. An assets market value is the price that the asset could be sold for if put up for sale in the market. The market, however, is dynamic, constantly changing. As the market moves, the total market value of posted assets may diverge from the total market value of lent assets. If the total value of collateral assets falls below the total required collateral value 1 then the lender is no longer adequately secured.

To avoid this situation, it is important for the collateral receiver to put a value on received assets which accounts for potential situations where the mar-ket price of collateral assets falls below the marmar-ket price of lent assets. To do

1

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CHAPTER 2. BACKGROUND 5

this, the collateral should consist of assets with a price which tends to move in correlation with the price of the lent assets. The specifics of how to estimate this risk falls outside of the scope of this report, but it is important to know that the risk exists and must be managed.

Given the estimated risk of certain assets posted as collateral, the lender can put what is referred to as haircut restrictions on the borrower. A haircut is a factor which reduces the market value of an asset to determine its collateral value. A haircut of 0.1 on an asset means that the collateral value of the asset will be 90 per cent of its market value. The lender can put different haircuts on the same asset depending on the borrower.

Assume that the lender puts a haircut of 0.2 on all Swedish assets. If the collateral giver wants to post Swedish assets to a total collateral value of 1 500 000 SEK the required market value of the assets is 1 875 000 SEK2. Foreign Exchange Rates

When receiving cash in a foreign currency as collateral it is important to make sure the value of the cash sufficiently covers the collateral value requirement. Since exchange rates are subject to change, the collateral value may deviate from the collateral requirement as time goes on.

2.1.3 Cost of Posting Collateral

The cost for the borrower of an asset posted as collateral depends on two main factors.

1. Opportunity cost: How valuable is the asset to the party posting it, supposing that it is not posted as collateral?

2. Counter-party risk cost: What is the risk of the collateral receiver not being able to return the asset?

3. Triparty fees: Fees related to triparty collateral management, discussed in detail in section 2.3.1

Opportunity Cost

The opportunity cost of an asset is related to the value of bringing the asset from a portfolio to the bank. As the name suggests the opportunity cost con-cerns the opportunity for the bank to profit from the asset. A brought home

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asset may be used as collateral for another portfolio or it might be useful in other areas of the bank.

Possessing borrowed assets which are not used for anything is costly for the bank. The opportunity of repaying a loan by bringing home some asset is reflected in the opportunity cost of the asset.

The opportunity cost of an asset to any counter-party is a factor multiplying the market value of the asset.

Reason for counter-party Cost

The counter-party risk cost is relevant only when the market value of posted collateral is higher than the market value of borrowed assets. The difference between these market values is called the exposure and a positive exposure means the lender is at risk. If there is no exposure the collateral giver will not suffer a loss if the borrower fails to pay back the loan, since the value of the collateral will cover the value of the lent assets.

The Swedish Financial Supervisory Authority (sv. Finansinspektionen) puts capital requirements on banks to regulate the risks they take on [1]. Cap-ital requirements are put in place to regulate the amount of risk by requiring that a bank must keep a certain amount of capital in relation to the value of their risk weighted assets. A capital requirement of eight per cent means the bank capital needs to be worth eight per cent of the risk weighted assets. The banks need enough capital to repay their customers even if they experience losses. The capital requirements are part of an agreement known as Basel III which was constructed following the 2008 financial crisis. In addition to this, it is in a bank’s interest to hold more capital in order to be more trustworthy to its customers.

The risk weighting is applied in relation to the estimated risk of dealing with a certain counter-party. Having exposure to a counter-party deemed as high risk will be associated with a high weight.

By having more exposure towards counter-parties deemed as risk worthy, the capital requirement increases. The less capital a bank has to keep, the more return on capital can be given to the shareholders.

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Unit for Allocation Costs

The unit of the cost is basis points per annum, where a basis point is equal to 0.01%3

and a cost of 1 basis point per annum means that allocating an asset is estimated to result in a cost of 0.01% of the allocated value each year. As an example, if the total cost of allocating 1 unit of a certain asset to a certain portfolio is 20 basis points per annum, then allocating 1 500 000 SEK of this security to the portfolio is estimated to cost 3 000 SEK every year4.

It is important to note that the costs are stochastic in nature. The cost of posting a particular asset to some lender cannot be determined, only estimated. Minimising the Collateral Cost

Consider the problem of minimising the cost of posted collateral to a lender. The posted collateral may be subject to an initial margin and haircut factors. By posting assets with low haircuts the exposure becomes lower, which subse-quently lowers counter-party risk cost. The borrower must consider not only the opportunity cost of what is posted as collateral, but also how it affects the counter-party risk cost through increased exposure.

2.1.4 Exposure and Margin Calls

As the market moves over time, the value of different assets may increase or decrease. The required collateral value that the lender has demanded must be fulfilled during the lifetime of the agreement. The lender can request the borrower to post additional collateral if the required value is not met by the cur-rent posted assets. The process of demanding additional collateral is known as a margin call. The borrower can similarly request collateral assets to be returned if market movements result in the posted collateral exceeding the re-quirement.

The lender reduces exposure by requesting collateral from the borrower and both lender and borrower manage the exposure by executing margin calls. If either party fails to make margin calls they risk ending up in a situation where they lose their assets and the value of the assets they have received, either as collateral or as a loan, does not cover their loss.

3_{0.01% = 0.0001}

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2.1.5 Concentration Limits

In addition to meeting the total required value and the haircut factors imposed on posted assets as collateral, there are further restrictions which can limit the allowed collateral posted in an agreement.

These additional constraints are called concentration limits and their pur-pose is to mitigate the risk of received collateral losing its value by diversifying the assets posted.

Each limit concerns a certain group of assets defined by the following fields.

• Fields concerning all assets:

– The asset type: equity, bonds or cash – Which country the asset belongs to

– Industry sector of stock company (healthcare, financial technology, construction etc.)

• Fields concerning equities:

– Which indices the stock belongs to • Fields concerning bonds:

– Who issued the bond?

– The bond rating and which institute determined the rating (for in-stance rating A+by S&P)

The fields serve as a way of defining which assets a certain limit concerns. For instance a limit could be put on Norwegian equities in the healthcare sector

which are on the OBX stock index.

The concentration states that the value (market or collateral) of the con-strained assets must exceed or fall below a certain limit. The concentration can concern either the market value or the collateral value. It may also be ex-pressed as a relative limit by putting a minimum or maximum limit on how many per cent of the stated value is made up from the constrained assets.

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American bond may not exceed five per cent of the total required collateral value. An example of a concentration concerning a sum of assets on the other

hand would be for instance the total collateral value of stocks from the OMX

index must exceed 15 000 000 SEK.

2.2 Securities Lending and Borrowing

Securities lending and borrowing refers to transactions where securities are lent against collateral to be returned at a future point in time.

2.2.1 Incentive of Lending Securities

The incentive for a firm to lend securities is to generate profit on assets which would otherwise not be used for anything. Additionally, received collateral may be invested. Another reason is to avoid custodian charges from the bank where the securities are stored by sending the assets to the collateral receiver. Securities lending improves the flow and availability of assets in the mar-ket, facilitating deals which would otherwise be cumbersome or impossible.

2.2.2 Incentive of Borrowing Securities

Borrowing securities could be a way of avoiding settlement failure when assets must be delivered to a party but cannot be bought quickly enough.

Borrowing securities could also be a way of trying to make a profit on the market. The borrower can make a profit from the borrowed securities by selling them and buying them back later at a lower price.

More complicated ways of profiting from borrowed securities involve find-ing and utilisfind-ing arbitrage opportunities where price differences between dif-ferent markets allow an investor to make a profit.

A market maker facilitating trades between buyers and sellers of assets may need to borrow securities to fulfil buy orders from their customers.

2.3 Collateral Management

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Collateral management can be looked at as a closed system with an inven-tory which only changes when assets are either posted to or returned from some portfolio. This system is only sustainable when the existing assets can suffi-ciently cover any exposure resulting from market value fluctuations or from additional collateral requirements arising due to new transactions occurring.

If the situation arises where existing assets cannot fulfil existing require-ments then additional assets must be acquired from the market. This open system can be seen as a modified version of the original problem where there exists no inventory limit but additional assets posted will come at a cost. Due to operational costs and time delays of getting assets from market it is preferred to operate in the closed system as much as possible.

2.3.1 Triparty Collateral Management

Triparty collateral management agents handle the distribution of collateral to the banks counter-parties. Each agent handles a number of portfolios. Assets which are sent to the agent will be used as collateral for the portfolios. The agent suggests collateral allocation to the counter-parties based on the assets which are available in the agent inventory.

Something important to note is that if an agent receives collateral from a counter-party then those assets can only be sent to other portfolios handled by the agent.

Some counter-parties may demand that the collateral management of their portfolios is handled through a triparty agent. This allows them to make trans-actions with their securities without having to handle collateral management on their own.

Posting collateral via a triparty agent incurs a cost on the collateral giver, not the receiver. The giver pays a fee which is a fraction of the total market value of assets posted by the triparty agent. Using a collateral agent with a fee of 5 basis points per annum which on average has posted assets to a value of 1 500 000 000 SEK will result in a cost of 750 000 SEK each year5. Note that the agent does not charge per transaction. The cost of using the agent is dependent on the total market value of assets which are posted at the end of each day.

Additionally, the fee may depend on which country the posted assets be-long to. E.g. posting Swedish assets may incur a different fee than posting Japanese ones.

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2.4 Thresholding Transactions

The bank may request collateral to be returned from a counter-party if the posted assets exceed the portfolio requirement. By bringing home collateral, the bank reduces the total cost of posted collateral. Bringing home collateral requires contacting a counter-party. For a counter-party to be worth contact-ing, the cost reduction must exceed a predetermined threshold. This thresh-old varies depending on the size and type of the portfolio in question. If the potential cost reduction is very small, bringing the collateral back from the portfolio may not be worth the hassle. The total cost reduction of all assets brought home from a counter-party must exceed the threshold value.

The threshold generally decreases as the clients get larger. Larger firms often have people working full time on managing collateral and requesting changes in the current collateral positions is therefore not seen as a big issue.

2.5 Mixed Integer Linear Programming

A mixed integer linear programming problem is an extension of linear pro-gramming where one or more variables are required to be integer valued [2].

2.5.1 Optimisation Solver

The solver used in this thesis is CBC, Coin-OR Branch and Cut. It is an open source solver written in C++. It solves mixed integer linear programming prob-lems through a branch and bound method utilising cutting planes to improve the running time.

2.5.2 Branch and Bound

Consider the integer programming problem

min f (x) s.t. Ax = b

x ∈ Z

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If the optimal solution to the linear relaxation happens to be integer valued, it is also optimal to the integer program.

Denote the described integer programming problem by I(x) and the linear relaxation as L(x). Further, let Iceil(x, k) be I(x) with the added constraint

x ≥ dˆ_{xe and let the problem I}_floor_{(x, k) be I(x) with the added constraint} x ≤ bˆ_xc.

Begin the branch and bound method by solving L(x). If the solution is integer valued, it is optimal to I(x). If the solution, let’s call it ˜x is not in-teger valued, the optimal solution is defined as max(Ifloor(x, ˜x), Iceil(x, ˜x)).

I_floor(x, ˜_{x) and I}_ceil(x, ˜_{x) are solved using branch and bound. The method has} a recursive nature where solving a subproblem may create two new subprob-lems with added constraints.

An important part of the method is to not solve unnecessarily many sub-problems. By solving the linear relaxation in a node one achieves a lower bound of all solutions in children of that node. If this lower bound is larger than the current best solution then the whole branch can be discarded.

The problem described below considered x ∈ Z. When x is a vector with some integer components, the branching is performed on one variable at a time.

The curious reader can study the branch and bound method in detail in Sierksma and Zwols [3].

2.5.3 Cutting Planes

The cutting planes method is a way of restricting the feasible region of an integer problem without removing any potential optimal integer solutions. The restricted region excludes the current optimal point if the point does not fulfil the integer constraints. The feasible region is restricted by adding additional constraints to the problem. The additional constraints aim to remove non-integer valued points from the search region.

Further readings on cutting planes can be found in Sierksma and Zwols [3].

2.5.4 Branch and Cut

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2.5.5 Big M Method

The Big M Method is used in the method to avoid infeasible, and thus unsolv-able, problems. The big M method modifies a problem by introducing artificial variables which can be used to put large penalties on any infeasibilities arising. The artificial variables are non negative. Setting artificial variables to a value larger than zero is discouraged by multiplying them with large penalty con-stants in the objective function. By making these concon-stants sufficiently large one can make sure that artificial variables will only be used if the original problem would otherwise be infeasible. The factor multiplying an artificial variable must be so large that no optimal solution will include an artificial variable if it can be avoided [3].

2.6 Previous Research

Collateral management services are offered by companies including Nasdaq, Euroclear and JP Morgan. The methods used by these companies are confi-dential, although some information can be found in white papers. The use of numerical optimisation for collateral management is highlighted by Sapient

Global’s Schiebe et al. [4], stating that the large amount of possible solutions

creates a need for sophisticated solving methods.

Academia does offer some papers on the area of mathematical optimisation for collateral management, but the area is not widely researched.

Bylund [5] formulated and solved the problem with an integer linear pro-gramming model using a branch and bound solver. The model covers many aspects of the problem but could be improved by considering additional as-pects such as limits on the number of transactions and concentration limits on groups of assets. Further, the test data of less than 20 assets and less than 10 assets is small and the collateral positions are not real life data.

Reynolds [6] has formulated a very comprehensive integer linear program-ming model accounting for concentration limits on individual assets as well as groups of assets. The formulation uses integer variables to create constraints on the maximum number of different assets which may be sent to a counter-party and to avoid low quantity transfers of certain assets. Large input in-stances are tested but the data is not sourced from real collateral positions.

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inventory. The model is tested on data consisting of 68 loans and 65 assets. Reducing the problem to a known one, such as a network flow or a knapsack problem, could be useful in finding new solution methods.

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Chapter 3 Methods

3.1 Optimisation Problem Formulation

A mixed integer linear programming optimisation problem is used to deter-mine optimal collateral allocation.

3.1.1 Formulating the Optimisation Problem

We consider the problem of optimal allocation of NA assets as collateral to

NP portfolios.

The nonempty finite sets

P = {0, 1, . . . , NP − 1}

A = {0, 1, . . . , NA− 1}

contain the portfolios and assets respectively. Let the variable x such that

xi,j ∈ R, i ∈ P, j ∈ A

be used to denote transactions in each portfolio. The value xi,j gives the quantity of asset j which should be posted to portfolio i in the optimal solution. Transactions are real valued, where a negative value indicate that assets should be brought back from a portfolio.

Similarly let the constant vector x0where x0_i,j _{∈ R, i ∈ P, j ∈ A}

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denote the current asset allocation. The total allocated assets in each port-folio will be given by the sum of x and x0.

3.1.2 Objective Function

As described in section 2.1.2, the cost of posting collateral is split into oppor-tunity cost, counterparty risk cost and triparty cost.

Denote the market value of asset j as vj. Also denote the opportunity cost

of an asset j as cOj, expressing the total opportunity cost of posted assets as

X

i∈P

X

j∈A

cO_j · vj · xi,j

Denote the counterparty risk cost in portfolio i as cCi . The counterparty

risk affects the exposure. Let Ri be the required collateral, excluding initial

margin, in portfolio i. The expression X

j∈A

vj· xi,j

gives the total market value of collateral allocated to portfolio i. The ex-posure is then given as

(X j∈A vj· xi,j) − Ri and finally X i∈P cC_i · ((X j∈A vj· xi,j) − Ri)

gives the total counterparty cost of all allocations. To simplify the prob-lem, the constant term Ri may be disregarded during optimisation. The new

expression for the counterparty cost is then X

i∈P

X

j∈A

cC_i · vj · xi,j

With the triparty cost of an asset j denoted as cTrij , express the total triparty

cost of posted assets as

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CHAPTER 3. METHODS 17

The total allocation cost becomes X i∈P X j∈A (cC_i + cO_j + cTri_{) · v} j· xi,j

which, by denoting ci,j = (cOj + cCi ) · vj, enables us to write the final objective function as f (x) =X i∈P X j∈A ci,jxi,j = cTx

The vector c will always have positive elements. Negative transactions will therefore decrease the cost whilst positive transactions will increase it.

3.1.3 Constraints

Positivity Constraint

Collateral posted to a portfolio must be positive. This is ensured through the constraint

(x0_i,j + xi,j) ≥ 0

where (x0i,j+ xi,j) denotes the final amount of posted asset j in portfolio i.

Inventory Constraints

Denote the inventory of asset j as Ij, j ∈ A. The constraint

X

i∈P

(x0_i,j+ xi,j) ≤ Ij, ∀j ∈ A

makes sure the total posted collateral of an asset type j does not exceed the available inventory.

Collateral Value Constraints

Denote the market value of an asset j as vj, the haircut of asset j in portfolio

i as hi,j and the required collateral value of portfolio i as Ri. Let the initial

margin in portfolio i be denoted by MiThe constraint

X

j∈A

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makes sure the required collateral value is met in all portfolios. The asset values are calculated as described in section 2.1.2.

Concentration Limit Constraints

To formulate concentration limit constraints, we need an expression for the collateral value and market value of an asset.

Concentration limits are either imposed on the total value of a set of at least one asset and they are either lower or upper limits. Limits on individual assets are always upper limits. The limits either concern the market value or the collateral value.

Denote Ckas the set of portfolio-asset-transactions concerning

concentra-tion limit k. Denote lkas the numerical value associated with the limit. Then

X

k∈Ck

hi,j · vj · (x0i,j + xi,j) ≥ lk

puts a lower limit on the total collateral value and X

j∈Ck

vj· (x0i,j + xi,j) ≥ lk

puts a lower limit on the total market value of the concerned assets. For upper limits the constraints are identical with the exception of the ≥ symbol being changed into a ≤ symbol. The constraint

X

k∈Ck

hi,j · vj · (x0i,j + xi,j) ≤ lk

puts an upper limit on the total collateral value and X

j∈Ck

vj· (x0i,j + xi,j) ≤ lk

puts an upper limit on the total market value of the concerned assets. The constraints for upper limits of asset values in a portfolio are not strictly correct in a real world sense. A more true-to-life constraint, which is not used in the model for reasons elaborated on later, would be

X

j∈A\Ck

hi,j· vj · (x0i,j+ xi,j) ≥ (Ri− lk)

which puts a lower limit on assets not in Ck rather than an upper limit

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much of an asset. If a concentration limit regulates that 500 SEK out of a

required collateral value of 1 500 SEK can be comprised of Swedish stocks, a feasible solution could be to post 2 000 SEK of Swedish stocks and 1 000 SEK of non Swedish stocks. The reason for regulating assets of an arbitrary category C is to make sure a sufficient portion of the collateral value is made up of assets which do not belong to C. Whether or not we get additional stock from category C does not matter; nobody will complain about receiving extra assets.

The reason it is preferable to express the limit as an upper limit is that it puts a greater restriction on the allowed solution space. This allows for greater execution speed both when constructing and when solving the problem. Fur-thermore, there will never be a situation where it is advantageous to post assets which do not add to the total collateral value. Thus, putting this additional re-striction will not affect the quality of the solutions obtained.

To differentiate between the different concentration limits, let CLC be the set of concentration limits which are lower limits on the collateral value, CLM are lower limits on the market value, CU C are upper limits on the collateral value and CU M are upper limits on the market value.

Transaction Constraints

Transaction constraints are used to ensure that the number of transactions do not exceed a reasonable workload. The incentive of regulating the number of transactions is that each transaction costs both time and money.

The optimisation aims to find an optimal ˆx reachable within a fixed number of transactions from the initial allocation.

The transaction limit is imposed by regulating the number of non-zero el-ements in x. To count the number of non-zero elel-ements

yi,j =

0 xi,j 6= 0

1 xi,j = 0

as a binary variable where element (i, j) is 1 if xi,j 6= 0. Ensuring this

condition is done by the constraint

|xi,j| ≤ yi,j · Ij

which must be rewritten as the equivalent condition −yi,j· Ij ≤ xi,j ≤ yi,j · Ij

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The constraint ensures that |xi,j| > 0 ⇒ yi,j = 1 but it does not ensure

|xi,j| = 0 ⇒ yi,j = 0. This however does not create any issues since an

additional constraint limits the total number of allowed transactions.

Limiting the number of transactions to some positive constant Y is done by constraining the sum

X

i∈P

X

j∈A

yi,j ≤ Y

Since the total number of transactions is limited, yi,jwill not needlessly be

set equal to 1.

Direction Constraints The optional constraint

xi,j ≤ 0

states that no additional postage of collateral is allowed for asset j in port-folio i.

We may similarly regulate

0 ≤ xi,j

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3.1.4 Sets, Variables and Constants

Notation Explanation

NP Number of portfolios

NA Number of assets P _{Set of portfolios} A _{Set of assets}

xi,j Transaction of asset j to portfolio i x0_i,j _{Current allocation of asset j to portfolio i} ci,j Cost of transferring asset j to portfolio i

Ij Available inventory of asset j, including posted assets hi,j Haircut factor of asset j in portfolio i

vj Market value of asset j

Ri Required total collateral value in portfolio i Mi Initial margin in portfolio i

Ck Set of portfolio-asset-transactions involved in concentration limit k

lk Numerical limit associated with concentration set Ck

yi,j Binary variable: 1 if asset j has a transaction to / from portfolio i

Y _{Upper limit on allowed number of transactions}

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3.1.5 Complete Problem

The complete problem becomes

min x P i∈P P j∈A ci,j· xi,j

s.t. (x0i,j + xi,j) ≥ 0 ∀i ∈ P, ∀j ∈ A (1)

P

i∈P

(x0_i,j + xi,j) ≤ Ij ∀j ∈ A (2) P

j∈A

hi,j· vj· (x0i,j + xi,j) ≥ (1 + Mi) · Ri ∀i ∈ P (3)

P

k∈Ck

hi,j· vj· (x0i,j + xi,j) ≤ lk Ck ∈ CLC (4) P

k∈Ck

vj· (x0i,j + xi,j) ≤ lk Ck ∈ CLM (5)

P

k∈Ck

hi,j· vj· (x0i,j + xi,j) ≥ lk Ck ∈ CU C (6) P

k∈Ck

vj· (x0i,j + xi,j) ≥ lk Ck ∈ CU M (7)

−yi,j· Ij ≤ xi,j ∀i ∈ P, ∀j ∈ A (8)

xi,j ≤ yi,j· Ij ∀i ∈ P, ∀j ∈ A (9) P i∈P P j∈A yi,j ≤ Y ∀i ∈ P, ∀j ∈ A (10) yij ∈ {0, 1} ∀i ∈ P, ∀j ∈ A (11) (1) Positivity of allocation (2) Inventory

(3) Required collateral value

(4) Minimum collateral concentration (5) Minimum market concentration (6) Maximum collateral concentration (7) Maximum market concentration (8) Conditional constraint (1/2) (9) Conditional constraint (2/2) (10) Transaction limit

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3.2 Subproblems and Optimisation Flow

3.2.1 Problems with the Optimisation Model

Pruning Trivial Suggestions

A large problem which may arise when running the optimisation is that the suggested transactions are trivial when the number of transactions are limited. Increasing the transaction limit will solve the issue, but the problem complex-ity will increase.

A trivial transaction may for instance be bringing home assets from a port-folio where the total posted collateral value is larger than the total required col-lateral value. Such a suggestion is easy to find through manual investigation and therefore the suggestion does not add much value to the user.

Another trivial suggestion is the inverse situation, where the posted col-lateral value does not meet the required colcol-lateral value. Posting additional assets is a relatively easy task to do manually.

The goal of the optimisation is to find intricate ways of replacing posted collateral, ways which would be difficult to find manually. The optimiser could for instance find collateral posted in one portfolio and suggest that it should be moved to another portfolio.

By performing trivial transactions before the optimisation problem is solved, the hope is to obtain more valuable suggestions.

A big part of why it is important to keep trivial transactions out of the op-timisation results is that part of the problem formulation is that the number of allowed transactions is limited. If the problem formulation only allows for 10 transactions based on a given starting allocation of collateral, these 10 transac-tions might be wasted on trivial transactransac-tions. An even more serious situation is that the transaction limit results in an unsolvable problem, for instance because we need to post assets to 20 portfolios with insufficient posted collateral.

By performing the trivial transactions in a separate step before the opti-misation, it can be ensured that the initial allocation of collateral is feasible such that no transactions must be performed to fill requirements in portfolios. This ensures that the allowed transactions are only used in order to improve an already accepted solution.

Unreachable Solutions

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Some asset A1 cannot be brought home from a portfolio if it results in

the portfolio required collateral value or minimum concentration limits are broken. To bring home A1some other asset A2must first be posted to create a value surplus in the portfolio. Multiple assets may have to be posted to bring home certain assets, depending on which concentration limits are affected by the asset which is brought home.

Consider the suggested collateral transactions as presented below1.         x0,0 = −300 x0,1 = 200 x0,2 = 0 x1,0 = 300 x1,1 = −200 x1,2 = 0        

Asset 0 should be brought home from portfolio 0 in quantity 300 and asset 1 should be brought home from portfolio 1 in quantity 200. Then asset 0 should be posted to portfolio 1 in quantity 300 and asset 1 should be posted to portfolio 0 in quantity 200.

Asset 0 cannot be brought home from portfolio 0 unless a substitution is posted, in this case the substitution is asset 1. Similarly asset 1 cannot be brought home from portfolio 1 until the substitution asset 0 is posted.

This situation could cause a problem. Assume we have no existing inven-tory of neither asset 0 nor asset 1. No asset may be brought home without a substitution and thus the proposed allocation is unreachable. A possible so-lution is to use an intermediary asset as substitution. Suppose we have asset 3 in inventory. Asset 3 may be usable as a substitution to allow asset 0 to be brought home from portfolio 0. Asset 0 is then posted to portfolio 1 to bring home asset 1. Asset 1 is finally posted to portfolio 0 and asset 3 is brought home. Note that asset 3 is not part of the optimal solution, it is simply used as a way of facilitating transactions to move from the original allocation to the suggested one.

3.2.2 Subproblem 1 - Homebringing

The first subproblem defines a restricted version of the optimisation where assets may be brought home but not posted. The aim of this problem is to bring home assets from portfolios where an excess of collateral is posted in order to reduce allocation costs.

1

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The first restriction of this problem is to include only portfolios with ex-ceeded collateral value and no broken minimum concentration limits. The sec-ond restriction is to state that all components of the vector x are non-positive. Since collateral only moves in one direction, a given suggested transaction of this subproblem will never be dependent on another transaction. This means that the suggested new allocation will always be reachable.

3.2.3 Subproblem 2 - Posting

This subproblem is a restricted problem where assets may be posted freely but only brought home if they must be brought home due to broken maximum concentrations. Bringing home assets is extremely expensive and will only be done if some limit is broken. To easily differentiate between assets being brought home and assets being sent away, the optimisation variable is split into a positive and a negative part.

The subproblem is intended to be solved after subproblem 1 and should result in a feasible, although possibly expensive, allocation of collateral.

The brought home assets should either be allowed to be sent to other port-folios or be forced to remain in inventory. The more strict version of the prob-lem, where brought home assets may not be sent to another portfolio, could be preferable because any suggested transaction may be ignored since no trans-action will be dependent on another. There may arise a situation where excess collateral should not be brought home for some reason. The downside of this restrictive version is that the current inventory may not be sufficient to post assets to all portfolios.

Depending on the available inventory, the subproblem may not be solv-able. The inventory must contain enough collateral to fill concentrations and requirements in the portfolios. To stop the problem from being infeasible when supplied with insufficient inventory, artificial variables are added to the formu-lation. As described in 2.5.5, using the artificial variables is extremely expen-sive, which means they will only be used when absolutely necessary. The us-age of artificial variables could be seen as sourcing additional inventory from the market.

3.3 Triparty Constraints

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posted the asset to the bank. If the bank receives an asset with ISIN I from portfolios which belong to different triparty agents T1and T2then these assets will be assigned new ISIN codes I_T1 and I_T2. The new assets I_T1 and I_T2are identical in almost all aspects such as price and cost of posting. They

differ only in what haircuts they are subject to. An asset I_T will have a zero haircut on all portfolios not belonging to the triparty T , stating that it cannot be used to fulfil these portfolio requirements. Since assets are assigned dif-ferent ISIN codes, the available inventory of asset I_T will be determined by how many assets with ISIN I have been received by portfolios in T .

By assigning assets new ISIN codes in this manner, no further modifica-tions have to be made to the problem formulation.

3.4 Problem Formulation with Precedence

Con-straints

As before, x0 is the initial allocation, which is assumed feasible.

Now we will successively post and bring home assets. Posting and bringing home will be done in a sequence of swaps.

Every swap consists of posting and bringing home assets, Si,jw states the

quantity of asset j posted to portfolio i in swap number w. Swap w + 1 is always performed after swap w.

Swap Allocation after swap 1 x0+ S1

2 x0+ S1 + S2 3 x0+ S1 + S2+ S3 4 x0+ S1 + S2+ S3+ S4 5 x0+ S1 + S2+ S3+ S4+ S5

The resulting allocation after each swap must be feasible. Consider the constraint A(x0+ x) = b in the original problem. In the problem formulation with precedence constraint, this results in the following series of constraints.

A(x0+ S1) = b A(x0 + S1+ S2) = b A(x0_{+ S}1 _{+ S}2_{+ S}3₎ _{= b}

A(x0_{+ S}1_{+ S}2 _{+ S}3_{+ S}4₎ _{= b}

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The total number of assets transferred across all waves is limited.

We want to minimise cT(x0+ S1+ S2+ S3+ S4+ S5) which is the final cost of posted collateral.

The variable Sijw has a lower bound based on how many assets of type j

have been posted to portfolio i. We may not bring home more than has been posted.

As in subproblem 2, there is a need of differentiating between incoming and outgoing transactions. The swap variable is therefore split into two vari-ables, one non negative and one non positive. This is necessary because in a single swap it is not allowed to bring home assets from one counterparty and post them to another. The reason this is disallowed is that it can create cycles. The problem quickly becomes very large as the number of swaps increase.

3.5 Thresholding Transactions

Thresholding transactions require binary variables ziwhich states whether the

cost reduction of all transactions xi,j for portfolio i is larger than some

de-termined threshold. The thresholding will only concern assets brought to the bank from the portfolios, therefore the constraint assumes xi,jis non-positive. With a minimum cost reduction threshold of K, this constraint may be written as follows, where the constant M is a value which must be larger than the largest possible cost reduction.

X

j∈A

cT_i,jxi,j + K ≤ Mzi

The constraint implies that if ziis zero, then

P

j∈A c T

i,jxi,j +K is at most

zero. This only holds ifPj∈A cTi,jxi,j

is larger than or equal to K.

In other words, if ziis 0, then the transaction exceeds the minimum

transac-tion threshold and is therefore deemed acceptable to perform. By multiplying each zivalue by a large constant in the objective function there is an incentive

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Results

4.1 Input Data and Confidentiality

The data used for the optimisation is from February 2019. It is real data con-sisting of SEB’s collateral allocations, customer agreements and inventory at that point in time. The data is confidential and for this reason no portfolio or asset names will be presented and any monetary values will be normalised.

4.2 Presented Values

The goal of the optimisation is to reduce the collateral allocation cost. The fol-lowing values will be presented in the experiments to gauge the effectiveness of the cost reduction.

∆C _{The total cost change from the transactions} ∆COpp

The change in opportunity cost from the transactions ∆CRisk

The change in counterparty risk cost from the transactions ∆CTri

The change in triparty cost from the transactions

All figures will illustrate the cost reductions rather than the cost changes. A cost change of ∆C implies a cost reduction of −∆C. This decision was made purely for aesthetic reasons.

4.3 Subproblems

The initial allocation of collateral is infeasible. Infeasibilities consist of un-met portfolio requirements due to insufficient posted collateral value or

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CHAPTER 4. RESULTS 29

ken concentration limits. Some portfolios suffer from excess posted collateral, i.e. the collateral value exceeds the portfolio requirement. Excess collateral is bad because it means the bank is taking on unnecessary risk and also because bringing home excess assets may lead to new transaction opportunities.

Infeasibilities arise due to market changes, such as certain stocks losing value, or due to changes in contracts. As new loans are made, or old loans repaid, the current collateral allocation can become infeasible or unnecessarily costly.

To transform the infeasible initial allocation into a feasible allocation the following subproblems, as described in sections 3.2.2 and 3.2.3, are ran in the following order.

Subproblem 1 (hometaking) To bring home excess collateral Subproblem 2 (sending) To fulfil portfolio requirements

Subproblem 1 (hometaking) To bring home remaining excess collateral

Subproblem Time (s) ∆C ∆COpp ∆CRisk ∆CTri Hometaking 2.065 -3.95% -0.42% -3.29% -0.23% Sending 2.271 2.96% 0.49% 2.68% -0.22% Hometaking 1.985 -0.63% -0.01% -0.6% -0.02% Total -1.62% 0.06% -1.21% -0.47% Table 4.1: Allocation cost changes from solved subproblems

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Figure 4.1: Cost reduction per subproblem

Figure 4.2: Cumulative cost reduction per subproblem

4.4 Varied Transaction Limits

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trans-CHAPTER 4. RESULTS 31

ferring one asset to or from one portfolio.

All costs expressed are relative to the initial allocation. A lower bound is found by solving the optimisation problem with no transaction limits. This reduces the initial allocation cost by 9.32 per cent.

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Figure 4.3: Cost reductions for optimisation with varying transaction limits

4.5 Repeated Optimisation

When the optimisation problem has been solved and all suggested transactions are carried out, it is of interest to see what happens if the problem is solved

again on the new allocation. In this experiment the optimisation problem is

solved with a transaction limit of 10 multiple times. All suggested transactions are carried out after each solution.

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Iteration Time (s) ∆C ∆CCumulative ∆COpp ∆CRisk ∆CTri 1 118.75 -1.89% -1.89% -1.90% 0.02% 0% 2 564.39 -1.19% -3.08% -1.21% 0.04% -0.03% 3 1891.94 -0.66% -3.74% -0.40% 0.01% -0.27% 4 692.39 -0.51% -4.25% -0.52% 0.01% 0 5 384.33 -0.41% -4.66% -0.41% -0.00% 0% 6 1725.83 -0.20% -4.86% -0.05% -0.14% 0% 7 7445.35 -0.21% -5.07% -0.21% -0.00% 0% 8 176154.77 -0.17% -5.25% -0.17% -0% 0% 9 238521.9 -0.74% -5.976% / -0.53% -0.11% -0.10% Total -5.97% -5.41% -0.16% -0.4%

Table 4.3: Allocation cost changes from repeated optimisation with transac-tion limit 10

Figure 4.4: Cost reductions for repeated optimisations with transaction limit 10

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The number of swaps (posting assets from inventory and bringing home assets from portfolios) varies and the cost reductions are expressed relative to the optimisation problem with no precedence constraints and no transaction limit.

Swap iterations Time (s) Transactions ∆C ∆COpp ∆CRisk ∆CTri

1 2.38 252 -5.34% -5.1% 0% -0.24%

2 6.21 628 -7.23% -6.48% -0.34% -0.42%

3 18.79 813 -8.47% -6.52% -0.69% -1.27% 4 67.14 1201 -8.89% -6.52% -0.69% -1.69% 5 140.61 1806 -9.32% -6.52% -0.69% -2.12% Table 4.4: Allocation cost changes from precedence constrained problem, no transaction limits, varied number of swaps

Figure 4.5: Cost reductions for precedence constrained optimisation with no transaction limit

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requirements. In this experiment the number of portfolios is increased in in-crements to examine how it affects the cost reduction.

Top portfolios Time (s) ∆C ∆COpp ∆CRisk ∆CTri 6.90% 2.414 -4.78% -3.77% 0.04% -1.05% 13.79% 2.594 -7.21% -5.56% 0.09% -1.73% 20.69% 2.623 -8.27% -6.46% -0.02% -1.79% 27.59% 2.997 -8.86% -6.48% -0.37% -2.01% 34.48% 2.83 -8.87% -6.48% -0.38% -2.01% 41.38% 3.13 -9.13% -6.51% -0.57% -2.06% 48.28% 2.673 -9.13% -6.51% -0.57% -2.06% 55.17% 2.58 -9.24% -6.51% -0.64% -2.10% 62.07% 2.691 -9.26% -6.51% -0.64% -2.10% 68.97% 2.594 -9.29% -6.51% -0.66% -2.11% 75.86% 2.602 -9.30% -6.51% -0.67% -2.12% 82.76% 2.764 -9.31% -6.52% -0.68% -2.11% 89.66% 2.689 -9.32% -6.52% -0.69% -2.12% 96.55% 2.697 -9.32% -6.52% -0.69% -2.12% Table 4.5: Allocation cost changes from subsets of top portfolios optimisation with no transaction limit

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4.8 Transaction Threshold

In this experiment the hometaking subproblem is solved with a transaction threshold. The threshold states that assets will only be brought home from a portfolio if the total cost reduction of all assets brought home from the portfolio exceeds a specified value. The costs, as before, are normalised. A threshold of T per cent means the assets brought home from a portfolio must reduce the initial allocation cost by a total of T per cent or more.

Threshold size Transactions ∆C ∆COpp ∆CRisk ∆CTri 0.0000% 113 -3.95% -0.42% -3.29% -0.23% 0.0013% 99 -3.94% -0.42% -3.29% -0.23% 0.0134% 53 -3.80% -0.40% -3.17% -0.23% 0.0402% 53 -3.80% -0.40% -3.17% -0.23% 0.00670% 32 -3.58% -0.37% -2.99% -0.21% 0.1341% 15 -2.97% -0.27% -2.50% -0.21% 0.4022% 6 -1.82% -0.10% -1.52% -0.21% 0.6703% 5 -1.19% -0.04% -0.94% -0.21% 1.3407% 0 0.00% 0.00% 0.00% 0.00%

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Chapter 5 Discussion

5.1 Subproblems

Running the subproblems to reach a feasible point for the optimisation results in a cost which is slightly smaller than the initial allocation cost. The cost of this allocation depends on the initial allocation. If a lot of excess collateral has been posted then the allocation cost can be reduced. Conversely if many requirements or concentrations are not met the allocation cost will increase.

The main cost reduction consists of reduced counterparty risk. The re-duction in counterparty risk costs come when excess collateral is brought to the bank from high risk portfolios. These margin calls are very important to reduce allocation costs of collateral. Counterparty risk costs are generally much higher than opportunity costs which is why the cost changes from the subproblems mainly consist of counterparty risk costs. The first subproblem which only brings home assets will decrease exposure in portfolios with ex-cess posted collateral which will decrease the counterparty risk. Portfolios with insufficient posted collateral will have no counterparty risk, so the sub-problem which sends assets to these portfolios naturally lead to an increase in counterparty risk costs.

5.2 Varied Transaction Limits

Solving the optimisation problem with a transaction limit of 10 lowers the cost by roughly 20 per cent of the optimal cost reduction (0.54 / 9.32). The majority of the cost reduction consists of lowered opportunity cost. Ten transactions are clearly not enough to reach a solution which is even close to being optimal.

When the number of allowed transactions is small the majority of the cost

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reduction comes from reduced opportunity cost. This does not mean that re-ducing counterparty risk and triparty cost is not important. As the number of transactions increase the reduction of portfolio and triparty cost becomes significant.

Reducing counterparty risk cost is done by replacing high haircut assets by low haircut assets. High haircut assets often consist of equities (stocks).

When allocating equity to a portfolio the collateral value is often not con-centrated in a single stock. Equity is allocated in blocks of multiple stocks. This means that to reduce the counterparty risk cost in a significant way it is necessary to bring home multiple different stocks, each homebringing being a transaction. With a low number of transactions, say 10, the counterparty risk cost cannot be reduced since not enough high haircut assets can be brought home.

5.3 Repeated Optimisation

The repeated optimisation experiment, where a 10 transaction limit problem is solved several times in a row, achieves a reduction in cost of 5.07 per cent of the initial allocation. This may be compared to the optimisation with no transaction limit, which reduces the cost by 9.32 per cent.

After two iterations the cost is reduced by 3.08 per cent. This is a similar reduction to the optimisation with a transaction limit of 20, which reduces the cost by 3.24 per cent. The cost reductions after 3, 4, 5, 6 and 7 iterations come close to, but fall short of, the optimisations with transaction limits of 30, 40, 50, 60 and 70.

The third iteration has a large triparty cost reduction when compared to the other iterations. Similarly the sixth iteration has an especially large coun-terparty risk cost reduction.

The cost reduction per iteration increases steadily and the execution time increases drastically to the point where the eighth iteration takes 50 hours.

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CHAPTER 5. DISCUSSION 41

to exchange assets between portfolios and other portfolios rather than only between portfolios and the banks inventory to achieve optimal results. With a larger inventory in the bank the single swap solution could come closer to the optimal solution.

The 3.98 per cent discrepancy between the single swap iteration and the optimal solution is the value the bank can save through exchanging assets be-tween portfolios and other portfolios, rather than only bebe-tween portfolios and bank inventory.

An interesting detail is that it is mainly counterparty risk and triparty costs which improve with an increased number of swap iterations. The opportunity and counterparty risk costs reductions have reached their peak numbers after three swap iterations, but two additional iterations are required to reach the peak triparty cost reduction.

5.5 Optimising Only the Largest Portfolios

Over 90 per cent of the cost reduction can be achieved when running the op-timisation on the top 27.59 per cent of portfolios. This is an expected result since a portfolio which does not have a large amount of assets allocated to it will not be able to decrease the allocation cost significantly. The total alloca-tion cost is concentrated in the largest portfolios.

As the number of portfolios increase, so does the reduction in counter-party risk cost. Larger portfolios are generally low risk, which leads to small potential of reducing counterparty risk.

5.6 Transaction Threshold

As the threshold of how profitable a transaction must be to be performed in-creases the total cost reduction of the hometaking subproblem dein-creases. This is as expected. A larger threshold means more transactions will be too small to perform.

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5.7 Cost Component Distribution

It is of interest to examine how to reduce the different allocation cost compo-nents, namely opportunity cost, counterparty risk and triparty costs.

Reducing counterparty risk cost mainly consists of bringing home excess posted collateral. This is a task which is handled by the subproblem which brings home assets from portfolios without posting anything from the bank. There is a not insignificant portion of counterparty risk cost to be reduced in the optimisation phase also, but it is nowhere near as large as what is accom-plished in the homebringing subproblem.

Opportunity cost reductions are the largest and in a sense also the easiest reductions to achieve. Reducing the opportunity cost requires fewer transac-tions than the reduction of counterparty risk and triparty costs.

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Chapter 6 Conclusions

6.1 Usability of Optimisation for Collateral

Man-agement

The improved optimisation model can find new transaction suggestions which minimise counterparty risk as well as opportunity and triparty costs. Due to problems in retrieving all required input data the optimisation model has only been tested on historical data from last year 2019. To ensure that the model works in practice one would need to run tests on the bank’s current collateral allocation and have the suggested transactions manually evaluated. When all input data gathering is working as required the bank will be able to perform a more hands on evaluation of the quality of the suggested transactions.

An important challenge in using optimisation for collateral management is that of avoiding precedence complications in the suggested transactions. Ex-periments show that reducing opportunity cost is often a simple task without complex transaction chains. Reducing counterparty risk and triparty costs however require many transactions and the experiment with precedence con-straints show that especially triparty cost reduction needs many portfolio to portfolio transactions to be fully optimised.

6.2 Further Research

6.2.1 Further Testing

The current optimisation model and solver is ready to be implemented in a semi-automatic process, where all transactions are evaluated and approved or

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rejected by a human. A process where no human input is needed will re-quire extensive testing since the consequences could be large if something goes wrong.

6.2.2 Improving Execution Speed

Decreasing the execution speed of the program would be a big improvement and add a lot of value for the bank. The execution speed becomes an issue when running multiple iterations, as described in section 4.5, when solving the problem with large transaction limits and when solving the precedence constrained problem with transaction limits.

Alternative formulations, simplifications and clever heuristics could be a way forward. A way of improving the performance without any actual changes could be to use a commercial solver, rather than the open source one.

6.2.3 Additional Constraints

A way of making the optimisation even more true-to-life would be to model the fact that the bank is able to acquire assets from the market if the inventory is insufficient. To model this it is necessary to know exactly which assets can be easily acquired and in which quantities.

This is partly handled in the current model through artificial assets, as de-scribed in section 3.2.3. The flaws of the current way to handle artificial assets is that it is assumed that all assets can be acquired in infinite quantities. The reason for this choice is that the current input data does not include a list of which assets can be assumed to be readily sourced if the current inventory is lacking.

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Bibliography

[1] Finansinspektionen. “Capital requirements for Swedish banks”. In: (2014). [2] Alexander Schrijver. Theory of linear and integer programming. John

Wiley & Sons, 1998.

[3] Gerard Sierksma and Yori Zwols. Linear and integer optimization:

the-ory and practice. CRC Press, 2015.

[4] Thomas Schiebe et al. “Techniques for Post-Trade Collateral Optimiza-tion”. In: (2016).

[5] Johanna Bylund. Collateral Optimization. 2017.

[6] PG Reynolds. “Computational studies of collateral optimisation prob-lems”. PhD thesis. North-West University, 2018.

[7] Konstantinos Papalamprou, Efthymios P. Pournaras, and Styliani Tycha-laki. “A Mathematical Programming Approach for the Optimal Collat-eral Allocation Problem”. In: Operations Research Proceedings 2018. Ed. by Bernard Fortz and Martine Labbé. Cham: Springer International Publishing, 2019, pp. 209–215. isbn: 978-3-030-18500-8.

[8] Magnus Orrsveden and Emil Tarukoski. “Optimization of Collateral al-location for Securities Lending : An Integer Linear Programming Ap-proach”. In: 2019.

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Mixed Integer Linear Programming for Allocation of Collateral within Securities Lending

Mixed Integer Linear Programming

for Allocation of Collateral within

Securities Lending

MARTIN WASS

Mixed Integer Linear

Programming for Allocation of

Collateral within Securities

Lending

MARTIN WASS

Abstract

Sammanfattning

Acknowledgements

Contents

Chapter 1

Introduction

1.1

Research Questions

Chapter 2

Background

2.1

Collateral

2.1.1

Types of Collateral

2.1.2

Value of Collateral Assets

2.1.3

Cost of Posting Collateral

2.1.4

Exposure and Margin Calls

2.1.5

Concentration Limits

2.2

Securities Lending and Borrowing

2.2.1

Incentive of Lending Securities

2.2.2

Incentive of Borrowing Securities

2.3

Collateral Management

2.3.1

Triparty Collateral Management

2.4

Thresholding Transactions

2.5

Mixed Integer Linear Programming

2.5.1

Optimisation Solver

2.5.2

Branch and Bound

2.5.3

Cutting Planes

2.5.4

Branch and Cut

2.5.5

Big M Method

2.6

Previous Research

Chapter 3

Methods

3.1

Optimisation Problem Formulation

3.1.1

Formulating the Optimisation Problem

3.1.2

Objective Function

3.1.3

Constraints

3.1.4

Sets, Variables and Constants

3.1.5

Complete Problem

3.2

Subproblems and Optimisation Flow

3.2.1

Problems with the Optimisation Model

3.2.2

Subproblem 1 - Homebringing

3.2.3

Subproblem 2 - Posting