Liquidity, banking and financial crises

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North-West University Mafikeng Campus Library

Liquidity, banking and financial

• crises

By

B. de Waal

(B

Sc· I\1Sc

)

20230257

Thesis

s

ubmitt

ed in p

a

rtia

l

fulfilment of

t

h

e

r

equirem

e

nt

s

for

the

d

egree

Phil

osophiae

Doctor in

Economics

a

t

the Mafikeng

C

a

mpus

of

the

)lorth-

W

est Univers

i

ty

(NWU-MC)

~larch

2013

Supe

rvisor

:

Prof

Mark

A P

eter

sen

(NWU-MC

)

Co-Supervisor:

Prof Janine

Mukuddem-Petersen

(N

WU-MC)

A

ssistant Sup

e

rvisor: Dr Thahir Bosch

(T

U

T

)

Call No \\""\ ~::l:

.

'

2015 -02- 0 2

Ace. No.:

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Liquidity, banking and financial crises

Acknowledgements

Firstly, I thank God for His grace in enabling me to complete this thesis.

I would like to acknowledge the emotional support that was provided by my husband, Heinrich and immediate family. Johan {father) and Wilma (mother).

I would like to express my gratitude towards my supervisor, Prof MA Petersen, co-supervisor, Prof Janine Mukuddem-Petersen, as well as my assistant supervisor, Dr Thahir Bosch, for their guidance and moral support.

Finally, I would like to thank the National Research Foundation (NRF) for providing me with funding for the duration of my studies.

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Liquidity, banking and financial crises ii

Preface

One of the contributions that was made by the NWU-MC to the activities of the stochastic analysis community has been the establishment of an active Finance, Modelling and Optimization Research Group (FMORG) that has an interest in institutional finance. In particular, FMORG has made con-tributions about modelling, optimization, regulation and risk management in insurance and banking. Students who have participated in projects in this programme under Prof Petersen's supervision, are the following:

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Liquidity, banking and financial crises iii

Level Student Graduation Title

MSc T Bosch May 2003 Controllability of HJI\IM Cum Laude Interest Rate Models MSc CH Fouche May 2006 Continuous-Time Stochastic

Cum Laude Modelling of Capital Adequacy Ratios for Banks ~1Sc .\IP 1\lulaudzi .\lay 2008 A Decision-making Problem

Cum Laude in the Banking Industry

PhD CH Fouche May 2008 Dynamic Modelling of Banking Activities

PhD F Gideon Sept. 2008 Optimal Provisioning for Deposit Withdrawals and Loan Losses in the Banking Industry

}viSe MC Senosi May 2009 Discrete Dynamics of Bank S2A3 Winner Credit and Capital and

for NWU their Cyclicality

PhD T Bosch May 2009 ~!anagement and Auditing of Bank Assets and Capital PhD BA Tau May 2009 Bank Loan Pricing and Profitability and Their

Connections .,vith Basel II and the Subprime 1\!ortgage Crisis PhD MP /1-lulaudzi May 2010 The Subprime Mortgage Crisis: Asset

Securitization and Interbank Lending .\!Sc B de Waal May 2011 Stochastic Optimization of Subprime Cum Laude Residential Mortgage Loan Funding

and its Risks

PhD MC Senosi May 2011 Discrete-Time Modelling of Subprime Mortgage Credit PhD S Thomas .\lay 2011 Residential Mongage Loan Securitization

and The Subprime Crisis

1\!Comm C Scheepers 2011 The Impact of the Global Financial Crisis Onwards on the South African Steel Industry .\!Comm G Mah 2012 Sovereign Debt

Onwards

MComm C Meniago 2012 An Econometric Analysis of the Impact of the Global Onwards Financial Crisis on HousPhold lndebteduess in South Africa l\!Comm .\1 .\lubanga 2012 Shocks to .\lortgages and

Onward~ Related Debt Obligation~

PhD CO Miruka 2012 Sove:eign Guarantees in South Africa Onwards

PhD I .\longale 2011 An Analysis of the Impact of the Global Onwards Financial Crisis on Savings in South Africa

PhD F Louw 2011 Monetary Policy Transmission .\lechanisms Onwards in South Africa

Postdoc J 1\!ukuddem-Petersen 2006-2009 Finance, Risk and Banking Postdoc T Bosch 2010 Finance. Risk and Banking Postdoc M Agaze Dessi 2011 Business Incubation Postdoc S Thomas 2011-2012 Subprime l\lortgage Models

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Liquidity, banking and fi.nancial crises iv

Declaration

I declare that the research that is presented in this thesis is my own unaided work, apart from the assistance that has already been acknowledged. It is being submitted in partial fulfilment of the requirements for the degree Philosophiae Doctor in Economics· at the Mafikeng Campus of the North-West University. It has not been submjtted before to any other University for any degree or examination.

I\obody. including Prof :\lark A Petersen, Prof Janine ~Iukuddem-Petersen and Dr Thahir Bosch, but myself is responsible for the final version of this thesis.

Signature ...

I1J;? ...

.... .

27 -03-2.0

i3

Date ...•...

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Liquidity, banking and fi_nancial crises v

Executive Summary

In recent times, the frequency of financial crises bas increased dramatically. In the last five years, for instance, the subprime mortgage and global financial crises have affected financial markets thr ough-out the world. Probleq1s with liquidity, mortgage funding, valuation and subprime mortgage design have been common to all of these crises and wUJ be studied in this thesis. The researcher's results on these problems are collected in articles [1] (see Chapter 2), [5] (see Chapter 3), [3] (see Chapter 4), [4] (see Chapter 5) and (2] (see Chapter 6) that are briefly described below.

Basel III attempts to raise the quality, consistency and transparency of the regulatory capital base in order to enhance the risk coverage of Basel II. In Chapter 2, the researchers explore the way in which these new Basel liquidity standards, as encapsulated in the liquid coverage ratio (LCR) and net stable funding ratio (NSFR), could be effectively implemented in mitigating liquidity problems by considering a solution paradigm for proposed Basel III liquidity regulation. It has been found that higher LCRs mitigate Liquidity risk and the ·sE'R limits an over-reliance on short-term wholesale funding. However, the overall impression of the researchers is that Basel III has not fully addressed the factors that were responsible for the crises and the fundamental problems of previous Basel regulation.

Chapter 3 asserts that the subprime mortgage crisis (S:MC) is an ongoing housing and financial crisis that was triggered by a marked increase in mortgage delinquencies and foreclosures in the US. Since it became apparent in 2007, it has had major adverse consequences for banks and financial markets around the globe. In the research, an originator's nonlinear stochastic optimal control problem that is related to choices regarding deposit inflow rates and marketable securities allocation is examined. Here, the primary aim is to minimize liquidity risk- more specifically, funding and credit crunch risk. In this regard, the researchers consider two reference processes, namely the deposit reference process and the residential mortgage loan reference process. This enables them to specify optimal deposit inflows as well as optimal marketable securities allocation by using actuarial cost methods to establish an ideal level of subprime mortgage extension. In their research, relationships are established in order to construct a stochastic continuous-time banking model to determine a solution for this optimal control problem which is driven by geometric Brownian motion.

In Chapter 4, an important aspect of the subprime mortgage crisis is discussed, namely subprime mortgage design in both a theoretical-and numerical-quantitative framework. Such design utilizes real financial market interest rates, securitization structuring and mortgage pricing to explain the economic mechanism behind the recent crisis. In particular, the researchers model mortgages that are able to fully amortize, voluntarily prepay (involving prepayment and possibly refinancing) or default. It is found that mortgage refinancing is curtailed by high loan-to-value ratios due to house price depreciation, while low loan-to-,·alue ratios increase mortgagor house equity. Furthermore. an optimal originator valuation problem under mortgage origination is solved. In this case, optimal mortgage value and rates, as well as profit, are computed. The paper supports the view that the

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Liquidity, banking and financial crises vi

subprime mortgage crisis was partially caused by the intricacy of design of subprime mortgages that led to information problems (asymmetry. contagion, inefficiency and loss), valuation opaqueness and ineffective risk mitigation.

The researchers provide a stochastic model for a global liquidity standard that is proposed by Basel III banking regulation in Chapter 5. In particular, the focus is on the modelling of the liquidity coverage ratio (LCR) that is defined by the stochastic dyhamics of the quotient of high-quality liquid assets (HQL.As) to net cash outflows (:\COs). This model enables the researchers to solve a nonlinear optimal stochastic LCR problem with quadratic cost where LCRs are used as one of the metrics in ratio analysis to measure bank liqnirlity. The main novelty is the introduction of an LCR reference process with respect to which optimal liquidity provisioning and HQLA allocation are characterized. The researchers also introduce the notion of an adjustment to the rate of liquidity provisioning per unit of the bank"s KCOs for deficit that, under c~rtain circumstances, can be related to a bank bailout rate in a theoretical-quantitative framework. Finally, numerical-quantitative results concerning LCRs and their connections with HQLAs and :\COs are provided to supplement the previous analysis.

:\umerical results, im·olving new Basel III liquidity regulation, are obtained in Chapter 6. More specifically, the net stable funding ratio is computed in accordance with the prescripts of the proposed banking rules. ln this regard, the researchers investigate the effects of shareholder cash-flow rights on the aforementioned funding ratio and a non-Basel III liquidity coverage ratio for certain developing countries during the period Ql:2005 to Q-1:2009. The study finds that the funding ratio appears to have satisfied Basel III minimum liquidity standard.s during this period and that more concentrated cash-flow rights result in improved liquidity.

Keywords: Bank Bailouts, Basel III, Credit Risk, Deposits, Global Financial Crisis, House Equity, Liquid Assets, Liquidity Coverage Ratio, Liquidity Coverage Ratio Reference Process, Liquidity Risk, Loan-to-Value Ratio, :\larketable Securities, ~fortgagcs, 11ortgage Funding, :\lortgage Rate, :\et Cash Outf!O\V. r'\et Stable funding Ratio. Prepayment, Procyclicality. Refinancing, Subprime Mortgage Crisis, Subprime Mortgage Insurance.

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Liquidity, banking and financial crises vii

Samevatting

In die afgelope tyd het die frekwensie van finansiEHe krisisse dramaties gestyg. In die laaste vyf jaar, byvoorbeeld, het die subprima-verband en wereldwye finansiele krisisse finansiele markte regoor die wereld geraak. Problerne met likiditeit, verbandbefondsing, waardasie en subprima-verbandontwerp was algemeen aan al hierdie krisisse en sal in hierdie proefskrif bestudeer word. Die navorser se resultate op hierdie probleme word in artikels [1] (sien Hoofstuk 2), [5] (sien Hoofstuk 3), [3] (sien Hoofstuk 4), [4J (sien Hoofstuk 5) en [2J (sien Hoofstuk 6) bespreek en word kortliks hieronder beskryf.

Basel III poog om die kwaliteit, konsekwentheid en deursigtigheid ,·an die regulerende kapitaalba-sis te verbeter ten einde die risikodekking van Basel II te verbeter. In Hoofstuk 2 ondersoek die navorsers hoe hierdie nuwe Basel-likiditeitstandaarde, soos saamgevat deur die likiditeitdekkingsver-houding en netto stabiele befondsingsverhouding. effektief ge·irnplementeer kan word in die tempering van likiditeitsprobleme deur die oorweging van 'n oplossingparadigma vir voorgestelde Basel III-Iikiditeitregulasie. Daar is gevind dat boer likiditeitdekkingsverhoudings likiditeitsrisiko versag en dat die netto stabiele befondsingsverhouding 'n oorafl1anklikheid op korttermyn-groothandelbefondsing beperk. Die algehele indruk van die navorsers is egter dat die Basel III nie die faktore wat verant-woordelik was vir die krisisse en die fundamentele problerne met vorige Basel-regulasie ten voile aanspreek nie.

Die bydrae in Hoofstuk 3 beweer dat die subprima-verbandkrisis 'n deurlopende behuisings- en finansiele krisis is wat vcroorsaak is deur '11 rnerkbare toename in agren;taHigc vt:rbande en die oproeping van verbande in die VSA. adat dit in 2007 duidelik geword het, hct dit belangrike nadelige ge,·olge \"ir die banke en finansiele markte regoor die wereld gehad, nadat dit duidelik geword het in 2007. In die navorsi11g word 'n bank se nie-lineere stogastiese optirnale beheerprobleem ondersoek met betrekking tot keuses rakende deposito-invloei en bemarkbare sekuriteite-allokasie. Die primere doe! is om likiditeitsrisiko, meer spesifiek befondsings- en kredietkrisisrisiko, tot die minimum te beperk. In hierdie verbnnd oorweeg die navorsers twee verwysingsprosesse, naamlik die deposito- en die residensiele verbandlening-verwysingsproses. Dit stel hulle in staat om optimale deposito-invloei, asook optirnale bemarkbare sekuriteite-allokasie, te bepaal deur \"aO aktuariele kostemetodes gebruik te maak om 'n ideate vlak van subprima-verbanduitreiking vas te stel. In bulle navorsing word verhoudings vasgestel om 'n stogastiese bankmodel in kontinue-tyd te bou ten einde 'n oplossing te bepaal vir hierdie optimale beheerprobleem wat deur 'n geometriese Brownse beweging gedryf word.

In Hoofstuk 4 word 'n belangrike aspek van die subprima-verbandkrisis, naamlik subprima-verbandontwerp, in beide 'n teoretiese en numeriese kwantitatiewe raamwerk bespreek. Sodanige ontwerp maak ge -bruik van reele finansiele markrentekoerse. sekuritering, strukturering en verbandprysing om die ekonomiese rneganisme agter die onla.ngse krisis te verduidelik. Die navorsers modelleer in die

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Liquidity, banking and financial crises viii

besonder verbande wat in staa.t is om ten volle te amortiseer, vrywillig voora.f te beta.al (voorafbe-taling en moontlik herfinansiering) of wan te betaal. Daar word gevind dat die verbandherfinan-siering ingekort word deur hoe lening-tot-waarde-verhoudings as gevolg van 'n afname in huispryse, terwyl lae lening-tot-waa.rde-verhoudings huisverbandaandele verhoog. Verder word 'n optimale bankwaardasieprobleem in verband met verbanduitbreiding opgelos. In hierdie geval word opti-male verbandwaarde en rentekoerse, asook wins, bereken. Die studie ondersteun die siening dat die subprima-verbandkrisis gedeeltelik ,·eroorsaak is deo.r die kompleksiteit van die ontwerp van subprima-verbande wat gelei het tot inligtingsprobleme (asimmetrie, besmetting, ondoeltreffend-heid en verlies). waardasie-ondeursigtigheid en ondoeltreffende risikoversagtings.

Die navorsers ondersoek 'n stogastiese model vir 'n globale likiditeitstandaard soos deur die Basel III-bankregulasie in Hoofstuk 5 voorgestel. In die besonder is die fokus op die modellering van die likiditeitdekkingsverhouding wat deur die stogastiese .dinarnika van die kwosient van hoe-gehalte likiede bates tot netto kontantuitvloei gedefinieer word. Hierdie model stel die navorsers in staat om 'n nie-lineere optimale stogastiese likiditeitdekkingsverhouding-probleem met kwadratiese koste op te los, waar die likiditeitdekkingsverhouding een van die statistieke is wat gebruik word in ver-houdingsanalise om banklikiditeit te meet. Die vernaamste nuutheid is die bekendstelling van 'n likiditeitdekkingsverhouding-verwysingsproses met betrekking tot optimale likiditeitvoorsiening en hoe-gehalte likiede batetoekenning. Die navorsers stel ook die moontlikheid bekend van 'n aan-passing aan die tempo van likiditeitvoorsiening per eenheid van die bank se netto kontantuitvloei vir die tekortkoming wat onder sekere omstandighede verwant is aan 'n bankuitbetalingskoers in 'n teoreties-kwantitatiewe raamwerk.

In Hoofstuk 6 ,·erkry ons numeriese resultate. insluitend resultate ,·ir die nuwe Basel III-likiditeitregulasie. Mecr spesifick, die nctto stabicle befondsingsverhouding word in ooreenstemming met die voorskrifte van die ,·oorgestelde bankregulasies bereken. In hierdie verband ondersoek die navorsers die gevolge van aandeelilouer-kontantvloeiregtc op die bogenoemde befondsingsverhouding en 'n nie-Basel III-likiditeitdekkingsverhouding ,·ir sekere ontwikkelende Iande gedurende die tydpcrk Kl:2005 tot K4:2009. Die stuclie bevind dat die befondsingsverhouding gedurende hirdie tydperk aan Basel III-minimum likiditeitstandaa.rde blyk te voldoen. Daar word afgelei dat meer gekonsentreerde kon-tantvloeiregte verbeterde likiditeit veroorsaak.

Sleutelwoorde: Bankuitbetalings, Basel III. Bemarkbare Sekuriteite, Deposito's. Globale Finansiele

Krisis, Herfinansiering, Huisaandeel, Kredietkrisis, Likiditeitdekkingsvcrhouding, Likiditeitdekkingsverhouding-verwysingsproses, Likiditeitsrisiko. Likiede Bates. :\etto Kontantuitvloei, Netto Stabiele

Befonds-ingsverhouding, Pro-siklies, Subprima-verbandkrisis, Subprima-verbandversekering, Verbandbe fonds-ing, Vcrbande, Verbandrentekoers, Verband-tot-Waarde Verhoudfonds-ing, Voorafbetaling.

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Liquidity, banking and financial crises ix

Glossary

Bank bailouts are financial assistance or rescue from the government, stakeholder or other entities to prevent a financial institution from bankruptcy and total liquidation.

Basel III is a comprehensive set of rules and measures for banking regulation, supervision and risk management. The Basel Committee on Banking SuperYision (BCBS) developed an international framework for risk measurements, standards and monitoring. In response to the global financial crisis, banks should always have a 30-day liquidity cover for emergency situations and meet certain capital requirements in order to minimize systemic risk that is caused by financial shocks.

Cost of mortgages is the interest cost that a bank must pay for the use of funds to originate mortgages.

Credit risk involves a bank's risk of loss from a mortgagor or special purpose vehicle that does not make scheduled payments and its securitization equi\·alent. This risk category generally includes counterparty risk that, in the researchers' case, is the risk that a subprime agent does not fulfil its obligations on a bond, credit derivative or insurance contract.

Deposits refer to the amount of money that is placed into a bank by a depositor and that gains interest in return. This deposited amount is a liability that is owed by the bank. In this study. deposits include both demand and time deposits. Demand deposits are the larger part of an orig-inator's money supply which are payable immediately on request, while time deposits are money deposits which can only be withdrawn after a preset fi..xed time period.

Global financial crisis refers to the worldwide economic crisis which became apparent in 2007. The global financial crisis was caused by, amongst other things, the downturn in the US housing mar-ket, risky lending and borrowing practices, inaccurate credit ratings, as well as excessive individual and corporate debt levels.

House equity refers to the current market value of the house less the outstanding mortgage pay-ments on this property.

Liquid assets refer to assets that can be converted into cash quickly with a minimal impact on the price of the assets. This means that there should be enough buyers to absorb the assets that are sold on the market. In this study, the stock of high-quality liquid assets is constituted by cash, central bauk reserves, marketable securities and government/central bank debt that has been issued. Liquidity coverage ratio {LCR) refers to the stock of high-quality liquid assets in regard to the net cash outflow over a 30-day period. This ratio measures the banking system·s liquidity position that allows the assessment of a bank's capacity to ensure the coverage of some of its more immediate liabilities and that also identifies the amount of high-quality liquid assets and institution holds that can be used to offset the net cash outflows that it would encounter under a short-term stress scenario. Liquidity coverage ratio (LCR) reference process refers to a structure of optimal control laws with respect to optimal liquidity provisioning and high-quality liquid asset (HQLA) allocation.

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Liquidity, banking and financial crises X

Liquidity risk refers to the risk of an asset that cannot be converted quickly enough to prevent a financial loss. Liquidity risk arises from situations in which a banking agent who is interested in selling (buying) residential mortgage products cannot do it, because nobody in the market wants to buy (sell) those residential mortgage products.

Loan-to-value ratio refers to the ratio of a mortgage with respect to the value of the house. Marketable securities represent claims on originator claims that are guaranteed by sovereigns, central banks, non-central government public sector entities, the Bank for International Settlements (BIS), the International :\Ionetary Fund (Il\lF), the European Commission or multilateral develop-ment bo.nks. These claims are assigned a 0% risk-weight. 11nrler the Basel II standardized approach. Also. deep repo markets should exist for these securities and they are not issued by banks or other financial service entities.

Mortgages refer to subprime mortgage loans in this s~udy. A subprime mortgage is a mortgage that is extended to mortgagors who do not qualify for market interest rates because of their poor credit history. The term subprime refers to mortgagors who are less likely to repay mortgages and who do not qualify for prime interest rates; high interest rates are therefor charged. A subprime mortgage is worse from an originator's view because it is in the riskiest category of mortgages. with high default rate::;. In general, a subprime mortgage loan is subprime if

1. the mortgagor has a poor credit history;

2. it is extended by an originator who specializes in high-cost subprime mortgage loans;

3. it is part of a reference subprime mortgage loan portfolio which is traded on secondary markets: or

4. it is issued to a mortgagor with a prime credit history. but is a subprime-only contract type. for example a 2/28 hybrid mortgage.

Mortgage funding refers to funds that are generated from marketable securities investments and deposit inflows to fund mortgage extension.

Mortgage rate is the rate of interest on a mortgage loan.

Net cash outflow refers to the total expected cash outflow minus total expected cash inflow for the ensuing 30 calendar days. A negative net cash outflow results in debt. Cash outflows are constituted by retail deposits, unsecured wholesale funding, secured funding and additional liabilities. Cash inflows are made up of receivable amounts from retail and wholesale counterparties, receivables in respect of repo and reverse repo transactions that are backed by illiquid assets and securities lending/borrowing transactions where illiquid assets are borrowed, as well as other cash inflows. Net stable funding ratio (NSFR) refers to a long-term ratio that measures how much stable funding a bank has to hold in order to endure a year-long liquidity crisis.

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Liquidity, banking and financial crises XI

Procyclicality refers to a positive correlation between an economic indicator and the economy. In this case, the researchers study the relati·onship between credit ratings, as well as mortgage losses and subprime mortgage insurance premium rates that are procyclical.

Refinancing: A refinancing mortgage undergoes a revision of its payment schedule or involves the replacement of an older mortgage by a new mortgage, offering better terms.

Subprime mortgage crisis refers to the ongoing housing and financial crisis during 2007 and 2008 that has been characterized by major losses due to the exposure to subprime financial products. This crisis was triggered by the downturn in the US housing market, inaccurate credit ratings, excessive individual and corporate debt llevP.Is, marked increase in mortgage delinquencies as well as foreclosures in the US. It has had major adverse consequences for banks and financial markets around the globe.

Subprime mortgage insurance is insurance on credit default swaps and other subprime mortgage products that are sold by monoline insurers. In case of a default. the monoline insurer will pay out the value of the loss that has been incurred. The problem was that insurers assigned contracts to other insurers to mitigate default risk to other parties, as seen during the subprime mortgage crisis.

A~I- Adjustable-rate mortgage ASF- Available stable funding

Abbreviations

BCBS - Basel Committee on Banking Supervision BIS - Bank for International Settlements

CDO -Collateralized debt obligation CDS - Credit default swap

CoVAR- Covariance

DSGE-Dynamic stochastic general equiilibrium ECAI -External Credit Assessment Institution GDP - Gross domestic product

HJBE- Hamilton-Jacobi-Bellman equatilon HQLA - High-quality liquid asset

IMF -International Monetary FUnd LlA- Level 1 asset

L2A - Level 2 asset

LCR - Liquidity coverage ratio

LIBOR- London interbank offered rate LTVR- Loan-to-value ratio

NCO - Net cash outflow

l\SFR - )let stable funding ratio

~1BS- Residential mortgage-backed security RSF - Required stable fundlng

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SCFR-Shareholder cash-flow rights SMC -Subprime mortgage crisis SMI - Subprime mortgage insurance Tl K -Tier 1 capital

T2K -Tier 2 capital VaR- Value at risk

VECM - Vector error correction model

B -Marketable securities

B* - Optimal marketable securities

B

-

Risky marketable securities B- Borrowings

b - Borel-measurable function

Basic notations

bi - Total earnings of the i-th marketable securities class br - Cost function

C - Insurer payments Yia the protection leg C -Credit rating

C(S(C))- Subprime mortgage insurance protection leg payments c- Deposit inflow rate

c~ - Optimal deposit inflow rate

c8 - Costs for holding marketable securities cf3 -Cost of borrO\Ying

cD - :\larginal cost of deposits cdw - Deadweight cost of total capital

d

-

Cost of liquidation

c;Mw - Average weighted cost of loans cP - Prepayment cost

D-Deposits

D (Ch 5) -Differential operator

oa -

Additional deposits Dr - Deposit reference process d -Number of dividends

D.F - Depreciation of fixed assets 6 -Stochastic discount factor E- Equity

E - Conditional expectation Ec - Common equity EP - Preferred equity

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er -

Retained earnings

e (Ch 3) - Volatility of stipulated levels of subprime mortgages

e (C

h 5) - Outflows per monetary unit of the bank's net cash outflows F -Fixed assets

:F -Right continuous filtration

F(·) -Cumulative distribution of the sho.:k to mortgag~s

f

-

Probability distribution

f

I - Fractions of mortgages that refinance

JM -

Fraction of the face value of an originator's subprime mortgages j5 - Fractions of mortgages that default

f(u) - Probability density function Q -Class of admissible control laws g - Control law

g• - Optimal control law

r-

Subprime mortgage mass "'- Factor

H - House prices

h -Investment return on bank's high-quality liquid assets

i -Increase of net cash outflow before outflows per monetary unit of net cash outflow J (Ch 4) - Originator's performance criterion

J (Ch 5) - Cost function J• -Optimal cost function [( - Bank capital

k- Constant

"' - Weighting factor L - Loan-to-value ratio

L" - Optimal loan-to-value ratio l (Ch 3) - Mortgage origination rate l (Ch 4) -Lagrangian multiplier l (Ch 5) -Liquidity coverage ratio

lr - Liquidity coverage ratio reference process A - 1\Iarket value of all loans

AI -Subprime mortgages

M• - Optimal subprime mortgages Ar - ~Iortgage reference process

uu-

Unfunded subprime mortgages N -Net cash flow

n- Number of originator shares 0- Subordinate debt

w8 - Risk weights related to the originator's risky marketable securities w(C) - Risk weights related to the originator's mortgages at face value

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P -Distribution function

P -Probability pr - Cost of treasuries

pr - Provisions against deposit withdrawals

Pv -Percentage of subprime mortgages to be originated

p - Density function

pi(C) - Subprime mortgage insurance premium rate

II -Profit

IIP - Present value of future profits from additional loans, based on current loans 1i (Ch 3) - Marketable securities allocation

1i (Ch 4) - Probability

1i (Ch 5) -Investment strategy

1f - Marketable securities allocation strategy

1f• -Optimal marketable securities allocation strategy 1rk· - Optimal high-quality liquid asset allocation q -Riccati equation

R -Recovery amount R- Return on reserves ra - Rate of actualization

r8 - Rate of return from marketable securities rB -Rate of return from risky marketable securities r8 _- _Borrowe_{r r}_ate

r0 - Rate of return from deposits rd - Discounted rate of interest

re - Rate of outflows per monetary unit of the bank's net cash outflows rf - Forecasting rate of net cash outflow

rh - Rate of return on high-quality liquid assets ri - Rate of net cash outflows increase before outflows rL - London interbank borrowing rate

rA - Loan rate

r-'

1 _- _:._I_{ortgage rate}

rM • - Optimal mortgage rate r0 - Subordinate debt rate rP - Penalty rate

r111 - l\Iarket-based step-up rate rR- Recovery rate

rr - Rate of return from treasuries r£ -Teaser rates

rY - Rate of asset returns p - Risk premium

S - Mortgage losses

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ue - Volatility in the change per net cash outflow unit uh - Volatility in the rate of high-quality liquid assets returns u' -Volatility in the net cash outflow increase before outflow

u11 - A random exogenous shock to the demand for loans uM - Random shock to mortgages

EY -Matrix of high-quality liquid asset returns T- Riskless assets (treasuries)

to - Beginning of the period

t 1 - End of the period

u - Unanticipated withdrawals

u 1 - ='iormal rate of liquidity provisioning

u2 _- _{Adjustment to the rate of liquidity provisioning} u2• - Optimal bailout rate

u3 - Liquidity provisioning rate

V - Objecti\·e function

V - Value function

v· -

Optimal objective function v -Arbitrary instant of time

(} - Risk premium

~V -Standard Bro,,·nian motion x (Ch 4)- Deposit withdrawals x (Ch 5) -Liquidity coverage ratio

x1

- Dynamics of high-quality liquid asset

x2 - Dynamics of net cash outflow

xr -Liquidity coverage ratio reference process { - Market price of risk

y - Return per high-quality liquid asset unit

yk - High-quality liquid asset returns in the k-th asset class per unit of the k-th class

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f

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Figure 1: An overview of Basel III and liquidity

Figure 2: Subprime mortgage funding and liquidity risk

Figure 3: Optimal originator valuation and the global financial crisis Figure 4: Liquidity coverage ratios

Figure 5: A note on Basel Ill and liquidity Figure 2.1: Basel III liquidity model framework

Figure 3.1: Diagrammatic overview of mortgage funding

(17)

Figure 3.2: Dynamics of mortgage funding parameters

Figure 4.1: Evolution of house prices and the loan-to-Yalue ratio Figure 5.1: Risky high-quality liquid asset allocation vs. liquidity ratio Figure 5.2: Extra liquidity contributions vs. liquidity coverage ratio Figure 5.3: Trajectories of the liquidity coverage ratio reference function

Figure 5.4: Simulated liquidity coverage ratio and extra liquidity contribution rate

Figure 5.5: Simulated liquidity coverage ratio and riskier high-quality liquid asset allocation Figure 5.6: Simulated liquidity coverage ratio, using control Jaws

Figure 5.6.1.2: Liquidity coverage ratio trajectory for high-quality liquid assets

Figure 6.1: !\fodel framework for Basel III liquidity regulation

Li

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es

Table 1: Postgraduate and postdoctoral supervision

Table 2.1: Basel III on the short position

Table 2.2: Summary table of the net stable funding ratio

Table 2.3: Survey of quantitative methods for studying Basel III liquidity Table 4.1: Choices of subprime mortgage origination parameters

Table 4.2: Computed subprime mortgage origination parameters

Table 5.1: Liquidity parameter values

Table 5.2: \"alues of q and m for different ,-alues of r0 _and_lr Table 5.3: Choices of liquidity coverage ratio parameters Table 5.4: Bank level liquidity (in $ billions)

Table 5.5: Bond level liquidity

Table 6.1: Descriptive statistics for Hquidity Table 6.2: Liquidity correlation analysis Table 6.3: Liquidity regression analysis

Table G.4: Financial crisis regression analysis

(18)

Co-authors and contributions

Co-authors and the researcher's contributions:

1.

Type:

Article 1

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An overview of

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III

and liquidity

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Quantitative finance

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,

2012

My

co

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3. Type:

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:

Optimal originator valuation

and the globa

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financial

cris

i

s

Main

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Petersen MA

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in

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(19)

Liquidity, banking and financial crises xviii

Main

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Petersen MA

Authors:

P

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sen :YIA,

D

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B

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5 Title:

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III

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De

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Liquidity, banking and financial crises xix

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(22)

Co-author

permission

l

ette

r

s

Ms. Bernadine de Waal PhD Economics Student Department of Economics

North-West Umversny (Mafikeng Campus)

Dear Prof Mark Petersen

Intention: Request for permission

t.a;lti "h£i": t..~:rJt~il "'\t"'.~!..Y1 (!..~(It( !C.): ~~J, 1·=·•,;; U'T>!Ri""'UT MAf1K£SG CAMPUS

Faculty of Commerce and Adm1n1st:-at1on Tel 081 363 0906

E-Ma1l: 2023025i@nwu.ac.za

2012·10-08

Request for permission to include the following papers, of wh1::h you were a co-author, 111 my doctoral theSIS to be submitted in partial fulfillment of the reqwernents for the degre~ Phiiosophiae Doctor 111 Economics at the Mafikeng Campus of the North West Univers1ty.

'An Overview of Basel Ill and Liquidity', Bulletin of Econom1c Research. "A note on Basel Ill and liquidity". Applied Economic Letters

'Optimal O•iginator Valuat1on and the Global F1nanc1al Crisis·. Optimal Control Applications and Methods.

'LiqUldny C.O'Jerage rahos'. Malh-:!mat:cal Ftna'1ce

·subprtme mottgage tund•ng ana liqUid;ty risk , Quantitative Fman::e Please do no! heSitate to contact me Jf you have any quenes

Yours Sincerely,

Bernad1ne de Waal (Student Number: 20230257)

Prof Mark A Petersen (Supervisor)

(23)

Ms Bernadine de Waal

PhD Economics Student Department of Econom1cs

North-West University (Mafikeng Campus}

Dear Prof. Jamne Mukuddem-Petersen

Intention: Request for pemnission

Faculty of Commerce and Adm1mstration

Te:: 081 363 0906

E-Mail 20230257@nwu.ac.za

2012-10-08

Request for perm1ss1on to include the following paper of wh1ch you were a co-author. in my

doctoral thes1s to be submitted in parttal fulftllment of the requirements for the degree

Philosophiae Doctor in Economics at the Mafikeng Campus of the North West University.

"An Overview of Basel Ill and Liquidly". Bulletin of Economic Research.

"A note on Basel Ill and llquidtty·. Applied Economtc Letters.

·optimal Origtnator Valuat1on and the Global Financ1al Cnsis". Opumal Control Applications and Methods.

"Liquidity coverage rat1os·. Mathematical F1nan~

··subprime mortgage fundtng and liquidity nsk", Quantitative Finance.

Please do not hestlate to contact me if you have any quenes.

Yours sincerely,

Bernadine de Waal (Student Number: 20230257)

Prof. Janine Mukuddem-Petersen (Co-Supervisor)

(24)

Liquidity, banking and financial crises x:xiii

Ms. Bernadine de Waal

PhD Economics Student

Department of Economics

North-West University (Mafikeng Campus)

Dear Dr. Soby Thomas

Intention: Request for permission

I

NORTH·WEST UtiiV(RSJTY YUNIBESITI YA. BO~OUE·BOPHIRIMA NOOROWES·UIIIVERSITEIT

MAFIKEHG CM\PUS

Faculty of Commerce and Administration

Tel: 081 363 0906

E-Mail: 20230257@nwu.ac.za

2012-10-08

Request for permission to include the f9llowing paper, of which you were a co-author, in my

.doctoral thesis to be submitted in partial fulfillment of the requirements for the degree Philosophiae Doctor in Economics at the Mafikeng Campus of the North West University.

"Optimal Originator Valuation and the Global Financial Crisis", Optimal Control Applications

and Methods.

Please do not hesitate to contact me if you have any queries.

Yours sincerely,

Bernadine de Waal (Student Number: 20230257)

(25)

Ms. Berredine de Weal PhD Economics Student Depe<1ment c1 Eeooomic:$

NorUI·WC$1 UnNOrsi!Y (Ma'ikllng CampuS}

Dsar Dr. MmbOr\tSent Mulauc:zl

~acuity of Coolme<ce end Administration TE!E. 081 363 Oi06

E-Mall: 2023025 7 (t ITi'W'..: .ac.za

2012·10.08

Req~t for pe~Kin lo include the lo~ p.apor. Qf 10tvch you wn:r~ Jl co·J\uthoc, In my

doctoral the-sis 10 be submlnoo in p&!1)al tutf:Jmet~ of the requuemoots tor 1he d~ree

Philosophiao Doctor i"l Economir..s a1 the Mafi-:8119 Campus of ltle North Wasl Uni~·orsity.

"Sub~ rnortg~ve lun6-lg and liquidty nsk•, Quan1ilalrn Finance. Please do not he-sic ate ~::l cor11act me d you have any qu~

\'ours sincerely.

~nadrle ~ Waal (Srudonl Numoor: 20230257)

Dr. Mmh~eru Mul21l11l.•

(26)

Ms. Bernadme de Waal PhD Economics Student

Department of Economics

North-West University (Mafikeng Campus)

Dear Ms. Lungile Hlatshwayo

Intention: Request for permission

~OI!Ttl wm UrtiVERSITY YUN·Sul"l YA B:lkOtiE·SCPII!RIW ~001\0'.'1£~ UlltV~f\Si.CiT

MAFIKENG CAMPUS

Faculty of Commerce and Admintstration

Tel: 081 363 0906

E-Mail: 20230257@nwu.ac.za

2012-10-08

Request for permission to include the following papers. of wnich you were a co-author. in my doctoral thes1s to be submit1ed m partial fulfillment of the requ1rements for the degree

Philosoph1ae Doctor in Economics at the Mafikeng Campus of the North West University. "An Overview of Basel Ill and Liquidity'', Bulletin of Economic Research.

"A note on Basel Ill and liquidity", Applied Economic letiers

"liqwdity coverage ratios", Mathematical Finance

Please do not hesitate to contact me If you have any 4uerit::s.

Yours sincerely,

Bernadtne de Waal (Student Number: 20230257)

Ms. lungtle Hlatshwayo

(27)

Liquidity, banking and financial crises xxvi

Bibliography

[lJ De Waal B., Petersen M.A., Hlatshwayo L ... P., ~Iukuddem-Petersen J. (2012). An Overview of Basel III and Liquidity. Bulletin of Economic Research. Submitted.

[2J De Waal B., Petersen .M.A., Hlatshwayo L.N.P., Mukuddem-Petersen J. (2012). A note on Basel III and liquidity. Applied Economic Letten doi: 10.1080/13504851.2012.744130. Accepted. [3J 1Iulaudzi :VLP., Petersen t\I.A., Mukuddem-Petersen J ., De ''-'aal B. (2012). Optimal

origina-tor valuation and the global financial crisis. Optimal Control Applications and Methods. doi: 10.1002foca.2022. Appeared electronically in 2012.

[4J Petersen .\I.A., De Waal B .. t\Iukuddem-Petersen J., Hlatshwayo L.)l'.P. (2012). Liquidity cov-erage ratios. Mathematical Finance. Submitted.

[5] Petersen M.A., De Waal B., Mukuddem-Petersen J., 'Mulaudzi t\l.P. (2012). Subprime mortgage funding and liquidity risk. Quantitative Finance. doi: 10.10 0/146976 .2011.637076. Appeared electronically in 2012.

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2.3.3 Benefits that are related to financial markets and institutions 2.3.3.1 Benefits that are related to financial markets . . 2.3.3.2 Benefits that are related to financial institutions 2.3.4 Benefits that are related to liquidity and other risks 2.3.5 Benefits that are related to the broader economy 2.4 Potential challenges of Basel III liquidity regulation . . .

2.4.1 Challenges that are related to liquidity standards . 2.4.2 Challenges that are related to crisis prevention . .

2.5

2.4.3 Challenges that are related to financial markets and institutions 2.4.3.1 Challenges that are related Lo financial markets . . 2.4.3.2 Challenges that are related to financial institutions 2.4.4 Challenges that are related to liquidity and other risks . 2.4.5 Challenges that are related to the broader economy Conclusions and future directions . . . .

3 LIQUIDITY AND MORTGAGE FUNDING

3.1 Introduction . . . . 3.1.1 ~lain questions and article outline

3 .1.1.1 ~lain questions .

3.1.1.2 Outline of the article 3.2 Stochastic model for subprime originators

3.2.1 Subprime mortgages and marketable securities 3.2.1.1 Subprime mortgages .

3.2.1.2 Marketable securities 3.2.2 Deposits and equity

3.2.2.1 Deposits 3.2.2.2 Equity .

3.2.3 Mortgage and deposit reference processes

xxvili 30 31 32 32 32 33 34 34 34 35 35 36 36 37 38 38 39 41 43 44 50 52 54 54 54 55 55 55 56 5 58 59 59

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3.2.3.1 3.2.3.2

Description of mortgage and deposit reference processes . Relationship between mortgage and deposit reference processes . 3.3 Optimal liquidity risk management . . . . . . . .

3.3.1 Stochastic dynamics of marketable securities 3.3.2 Optimal originator liquidity risk management 3.3.3 Spread method of mortgage funding . . . . . 3.3.4 Optimal originator liquidity risk management .

3.3.4.1 The main optimal originator liquidity management results 3.3.4.2 Optimal allocation strategies for marketable securities . 3.3.5 Numerical examples involving mortgage funding

3.4 Conclusions and future directions . . . . . . . , . . . 4 SUBPRIME MORTGAGE DESIGN

4.1 Introduction . . . .

4.2

4.1.1 Literature review of subprime mortgage design 4.1.2 Preliminaries about subprime mortgage origination .

4.1.2.1 The balance sheet . . . . 4.1.2.2 Credit ratings for subprime mortgages . 4.1.2.3 Subprime mortgage insurance . . .

4.1.2.-l The economy. economic agents and equilibrium . 4.1.3 \lain questions . . .

Subprime mortgage design . 4.2.1 Subprime mortgage rates 4.2.2 Subprime mortgages . ..

4.2.3 Subprime mortgage options and LTVRs 4.3 Optimal subprime mortgage design . . . .

4.3.1 Risk and profit under subprime mortgage origination . 4.3.1.1 A motivating e..'<ample . . .

4.3.1.2 Retained earnings under subprime mortgage origination . 4.3.1.3 l\lodel for profit under subprime mortgage origination 4.3.2 Originator valuation under subprime mortgages . . . . . . . .

4.3.2.1 Net cash flow under subprime mortgage origination 4.3.2.2 Optimal originator valuation under subprime mortgages . 4.3.3 Optimal originator valuation and LTVRs . . . . . . . . . . . . . .

xxix 60 60 62 62 63 64 65 65 72 73 74 77 79 80 81 81 83 83 83 4 4 4 85 87 87 91 92 !:13 93 94 100

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4.4 Originator valuation example . . . . . . . .

4.4.1 Choices of subprime mortgage origination parameters 4.4.2 Computation of subprime mortgage origination parameters 4.5 Conclusions . . . . . . . . . . . . . . . . . .

5 LIQUIDITY MODELLING 5.1 Introduction . . . . 5.2 A liquidity co,·erage ratio model

5.2.1 Description of the liquidity coverage ratio model 5.2.2 Description of the simplified LCR model .

5.3 Optimal bank liquidity coverage ratios . 5.3.1 The optimal bank LCR problem

5.3.2 Optimal bank LCRs in the simplified case 5.3.3 Discussion of the cost function and control laws .

5.3.3.1 Discussion of the cost function 5.3.3.2 Discussion of the control laws. 5.4 Numerical results involving liquidity . . . .

5.4.1 Values of q and m for different values of r0 _and_lr

XXX 103 103 103 105 108 llO ll2 ll2 ll7 ll8 119 120 129 129 130 132 133

5.4.2 Trajectories of the LCR reference function, xr, and of m for lr

=

1.05 13--1 5.4.3 Simulated LCR, Xt, and extra liquidity contribution rate; lr = 1.05. r0 ₌_0.01 _1:35 5.4.~ Simulated LCR. x1• and risky HQLA allocation; lr = 1.05, r0 = 0.01 136 5.4.5 Simulated LCR, Xt. using control laws: lr = 1.05. r0 = 0.01 136 5.5 Conclusions and future directions . . .

5.6 AppendLx: Kumerical results for LCRs 5.6.1 Appendix A: LCR simulation .

5.6.1.1 Appendi.x A1: LCR simulation parameters 5.6.1.2 Appendi.x A2: LCR dynamics . . . . 5.6.1.3 Appendix A3: Features of the LCR trajectory 5.6.2 Appendix B: Bank leYelliquidity data

5.6.3 Appendix C: Bond level liquidity data

6 LIQUIDITY AND BASEL III 6.1 Introduction . . . . 6.2 Data and methodology .

137 139 139 139 139 1~0 1~1 141 143 145 146

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6.2.1 Data .

6.2.1.1 Descriptive statistics. 6.2.2 l\Iethodology .

6.3 Results and discussion

6.3.1 Liquidity correlation analysis

6.3.2 Liquidity regression analysis .

6.3.3 Financial crisis regression analysis

6.4 Conclusions and future directions . . .

7 CONCLUSIONS AND FUTURE RESEARCH

7.1 Discussions and concluding remarks .

7.2 Recommendations and future research

7.3 Bibliography . . . .. . 1 146 146 147 148 148 148 149 150 152 152 154 155

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Chapter

1 INTRODUCTION

As calamitous as the subprime blowup seems. it is only the beginning. The credit bubble spawned abuses throughout the system. Subprime lending just happened to be the most egregious of the lot, and thus the first to ha\'e the cockroaches scurrying out in plain view. The housing market will collapse. ~ew-home construction will collapse. Consumer pocketbooks will be pinc~ed. The consumer spending binge will be o,·er. The US economy will enter a recession.

- Eric Sprott (Sprott Asset Management), 2007.

These days America is looking like the Bernie l\ladoff of economies: For many years it was held in respect, even awe, but it tum~ out lu ltavt! ut:t:u a fraud all aluug.

- Prof Paul Krugman (200 :\obel ).Iemorial Prize Laureate in Economic Sciences.

Princeton U ni,·ersity, t:S). 2009.

During the global financial crisis, banks were under SC\'ere pressure to maintain adequate liquidity. In general, empirical evidence shows that banks with sufficient liquidity can meet their payment obligations, while banks with low liquidity are not able to do that. The global financial crisis highlighted the fact that liquidity risk can proliferate quickly with funding sources dissipating and with concerns about asset valuation, capital adequacy realizing and subprime mortgage design which im·olves subprime mortgage e>.:tension. prepayment and refinancing, credit risk mitigation, subprime mortgage insurance. house prices. house equity and loan-to-value ratio, as well as procyclicality. Such

subprime design utilizes real financial market interest rates. securitization structuring and mortgage pricing to explain the economic mechanism behind the recent crisis. In this regard, important issues ha\'e to be accountant for, such as moral hazard in expanding mortgage portfolios, incomplete information among market players about their counterparties, myopia in decision making in the subprime mortgage market and monetary policy incentives that are boosting the growth of the subprime market.

During this period, large amounts of deposits flowed into the uS banking system which funded 2

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Liquidity, banking and financial crises 3

mortgages in a low interest rate environment. Consequently, credit was easy to obtain, thus boosting housing and credit markets. Originators passed on mortgages via securitization to shift credit risk to investors in risky marketable securities such as residential mortgage-backed securities (RMBSs) and collateralized debt obligations (CDOs). As a consequence, mortgage innovation rose dramatically. The proceeds of mortgage origination were invested in marketable securities for higher yields than

earnings on treasuries. In turn, the funds from deposits and marketable securities were then used for funding new subprime mortgages. The sharp rise in delinquencies and foreclosures of subprime mortgages had a major adverse impact on the liquidity of financial institutions and markets around the globe.

The three recent financial crises. namely the subprime mortgage crises. global financial crisis and sovereign debt crisis, were also characterized by too-big-to-fail institutions that suffered from a lack of high-quality liquid assets (HQLAs). The realization of such liquidity risk led to credit crunches and had deleterious effects on financial markets globally.

This situation underscores the important relationship between funding risk (invoh·ing the raising of

funds to bankroll asset holdings) and market liquidity (involving the efficient conversion of assets

into liquid funds at a given price). During these financial crises, banks experienced difficulties due to lapses in principles of liquidity risk management. In response to this, the Basel Committee on Banking Supervision (BCBS) published banking regulations (see 14]) in 2008 - commonly known

as ''Sound Principles" - that provided detailed guidance on risk management and supervision of

funding liquidity risk. Subsequently,

I

3 J

proposed via Basel III regulation that banks should always

have liquidity cover for stress scenarios. Although pre-Basel III regulation establishes procedures for assessing credit, market and operational risk, it does not provide effective protocols for managing liquidity and systemic risks. The drafting of Basel III represents an effort to address the latter (see,

for instance, 12].

ll

]

and

l

-1

;}.

The net stable funding ratio (~SFR) is mandated by proposed Basel III regulation as a measure of bank capital stability (see. for instance, [3J and H~).

In response to this, among other things, the Basel Committee on Banking Supervision (BCBS) is proposing that banks should always have a 30-day liquidity cover for stress scenarios. As far as

Basel III liquidity proposals are concerned, the BCBS is suggesting a liquidity coverage ratio (LCR)

that is defined as

LCR -_ High-quality liquid assets (HQLAs) >1. 30-day net cash outflows (NCOs)

-These ratios measure the banking system's liquidity position that allows the assessment of a bank's capacity to ensure the coverage of some of its more immediate liabilities with highly a'-ailable assets

and also identifies the amount of unencumbered, high-quality liquid assets and institution holds that can be used to offset the net cash outflows it would encounter under a short-term stress scenario,

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1.1 M

e

thod

s

According to Protter [6], stochastic calculus is a branch of mathematics that operates on

stochas-tic processes. It allows a consistent theory of integration to be defined for integrals of stochastic

processes with respect to stochastic processes. It is used to model systems that behave randomly.

A stochastic process, or sometimes random process, is

a

collection of random variables; it is often

used to represent the evolution of some random value or system oYer time. This is the probabilistic

counterpart to a deterministic process (or deterministic system). Instead of describing a process

which can only evolve in one way (as in the case, for example. of solutions of an ordinary differential

equation), in a stochastic or random process, there is some indeterminacy; even if the initial condi-tion (or starting point) is known. there are several (often infinitely many) directions in which the process may evolve.

Time can be discrete, for example t = 1, 2, 3, ... , or continuous,

t;:::

0. Stochastic calculus deals with functions of time t, 0 $ t $ T. Calculus is more suited for continuous time processes. At any time

t, the observation is described by a random variable which is denoted by Xe or X(t). A stochastic

process {X(t)} is frequently denoted by X or, with a slight abuse of notation, also by X(t) (see al5o

[5] for more information on stochastic calculus).

A function g is called continuous at the point t

=

to if the increment of g over small interYals is

small:

Clg(t)

=

g(t) - g(to) _,

o

as Cit- to -+

o.

If g is continuous at every point of its domain of definition, it is simply called continuous. g is called differentiable at the point t

=

t0 if. at that point,

lim D.g(t) =C.

6t-O Dot

This constant C is denoted by g'(t0 ). If g is differentiable at every point of its domain, it is called

differentiable.

In 1 2 . botanist R. Brown described the Brownian motion of a pollen particle that is suspended

in fluid. It was observed that a particle mo\·ed in an irregular, random fashion. Albert Einstein,

in 1905, argued that the movement is due to bombardment of the particle by the molecules of the

fluid, and obtained the equations for Brownian motion. In 1900, L Bachelier used the Brownian

motion as a model for movement of stock prices in his mathematical theory of speculation. The

mathematical foundation for Brownian motion as a stochastic process was laid by N Wiener in 1931, and this process is also called the \Viener process. The Brownian motion process B(t) serves as a

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basic model for the cumulative effect of pure noise. If B(t) denotes the position of a particle at time t, then the displacement B(t) - B(O) is the effect of the purely random bombardment by the molecules of the fluid, or the effect of noise over time

t

.

The Brownian motion {B(t)} is a stochastic process with the following properties:

1. (Independence of increments) B(t) - B(s), for t

>

s is independent of the past, that is of Bl).,

0 :::;

u:::; s, or of F., the() field generated by B(u), u:::; s.

2. (i'\ormal increments) B(t) - B(s) has a normal distribution with mean 0 and variance t - s. This implies (taking s

=

0) that B(t) - B(O) has N(O, t) distribution.

3. (Continuity of paths) B(t), t :2:: 0 are continuous functions of date t.

Stochastic differential equations: If x(t) is a differentiable function that is defined for t > 0, u(x, t) is a function of x and t and the following relation is satisfied for all t, 0:::; t :::; T

~~t)

=

x'(t)

=

u(x(t), t) and x(O) = Xo,

then x(t) is a solution of the ordinary differential equation with the initial condition xo. UsualJy the

requirement that x'(t) is continuous is added. The above equation can be written in other forms:

dx(t) - u(x(t), t)dt

and (by continuity of x'(t))

x(t) = x(O)

+

lot

u(x(s), s)ds.

1.2 R

esearc

h

qu

es

tions and outlin

e

of the th

es

i

s

In this section, the main research questions are stated and an outline of the thesis is provided.

1.2.

1 R

esearc

h

q

u

es

ti

o

n

s

The main aim of this research was to develop optimal banking models and to consider their rela

-tionships with financial crises. In particular, this modelling aims to improve the understanding of

these subprime activities. An additional objective is to use the principles of stochastic control and

other methods to implement optimal preventative strategies that offset the deleterious effects of this subprime design and risk mitigation.