Capital requirements of interest rate risk in the banking book : a solvency II-like approach

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ACTUARIAL

SCIENCE AND

M

ATHEMATICAL

F

INANCE

M

ASTER

’

S

T

HESIS

Capital requirements of Interest Rate Risk

in the Banking Book: A solvency II-like

approach

Author:

Sven VAN DEN BELD

Supervisor: Dr. Umut CAN

A thesis submitted in fulfillment of the requirements for the degree of Master of Science

in

Actuarial Science and Mathematical Finance

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ACTUARIAL SCIENCE AND MATHEMATICAL FINANCE

Abstract

University of Amsterdam

Actuarial Science and Mathematical Finance Master of Science

Capital requirements of Interest Rate Risk in the Banking Book: A solvency II-like approach

by Sven VAN DEN BELD

One of a bank’s key functions is maturity transformation: allowing the transfer from agents in surplus demanding short-term deposits to agents in deficit with long-term financing needs. From 2008 onwards, the global financial crisis revealed several in-efficiencies associated with maturity transformation. These risks are amplified by the current low interest rate environment created by central banks. In this context of historically low rates, policymakers have expressed concerns about interest rate risk in banks. However, despite its importance, banking regulation, until now, does not impose a minimum capital requirement for the identified interest rate risks in banks. On the other hand, Solvency II does provide such minimum capital requirement for insurance companies. This thesis investigates the differences and the possibilities to adopt elements from Solvency II into banking regulation. A subcomponent of bank-ing regulation, which closely resembles the capital framework for interest rate risk in Solvency II is the the "Delta-EVE" calculation. This thesis shows Delta-EVE calcu-lation can be enhanced with the following elements of the Solvency II framework: (1) harmonization of market-consistent valuation of the balance sheet, (2) restriction of critical technical aspects of measuring interest rate risk (e.g. the term structure, exclusion/inclusion of commercial margins, cashflows projections and treatment of behavorial assumption) and (3) Transparancy and increased flexibility of generation of stressed interest rate scenarios to calculate the valuet-at-risk.

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Introduction

1.1 Background

1.1.1 Interest Rate Risk in the banking industry

One of a bank’s key functions is maturity transformation: allowing the transfer from agents in surplus demanding short-term deposits to agents in deficit with long-term financing needs (Hicks, 1946 [1]). The value for the bank of transforming maturities and its associated vulnerabilities were already recognized in literature. However, research was limited to bank runs, deposit insurance and liquiduty (Diamond and Dybvig, 1983 [2]).

From 2008 onwards, the global financial crisis revealed several inefficiencies as-sociated with maturity transformation. During normal (non-crisis) periods banks can increase their porfitability by increasing the maturity mismatch on their balance sheet. Main drivers for increased returns are the additional risk premium charged to customers and the expected excess return by riding the curve. (Segura and Suarez, 2016 [3]. Therefore, banks have an incentive to increase the maturity mismatch be-tween assets and liabilities. As a consequence, exposing themselves to interest risk (Brunnermeier, 2013 [5]). In case interest rates changing, this maturity mismatch can have a significant impact on the banks earnings. Since short term liabilities reprice faster than assets, rising interest rates will have a decreasing effect on the net interest income of banks. Nanks have to pay a an increased interest rate on short rate liabili-ties, wheras the interest rate earned on long term assets remain unchanged. This can lead to a decline in net interest income (Memmel, 2011 [6]). Therefore, excessive ma-turity transformation exposes banks to tail risks that have the potential to threaten their profitability and, therefore also their solvability, even if if the credit position of the bank is highly-rated (van Rixtel, 2013 [7]).

These risks are amplified by the current low interest rate environment created by central banks. The global financial crisis and the subsequent recession have led to fundamental changes in the design and implementation of monetary policy. Many central banks reduced policy rates to near zero (or even negative) in 2009 and adopted less conventional policies in order to provide additional monetary stimulus. Claessens, 2014 [8]).

In this context of historically low rates, policymakers have expressed concerns about interest rate risk in banks (ESRB, 2016 [9]). However, despite its importance, banking regulation, until now, does not impose a minimum capital requirement (Pil-lar 1 requirement in the Basel capital framework) for the identified interest rate risks in banks (CRR, 2013 [10]).

Currently, Interest rate risk in the banking book (IRRBB) is part of Pillar 2 of the Basel capital frameworks (Supervisory Review Process) and is subject to the Basel

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Committees guidance set out in the 2004 Principles for the management and super-vision of interest rate risk, which lay out the Committees expectations for bank’s identification, measurement, monitoring and control of IRRBB as well as its supervi-sion (BIS, 2016 [12]). In 2015 the Basel comittee published a consultation document (BIS, 2015 [11]), where two options were for the regulatory treatments of IRRBB: a standardised Pillar 1 approach (Minimum Capital Requirements) and an enhanced Pillar 2 approach (which also included elements of Pillar 3 Disclosure). In the final standard published in 2016 (BIS, 2016 [12]) the committee concluded that a Pillar 1 approach to IRRBB was not feasible. In particular the complexities, involved in formulating a standardised measure of IRRBB which would be both sufficiently ac-curate and risk-sensitive, could not be capured to allow the approach to act as a means of setting regulatory capital requirements. The Committee concluded that the heterogeneous nature of IRRBB would be more appropriately captured in Pillar 2.

1.1.2 Interest Rate Risk in the insurance industry

Interest rate risk is one of the most significant risks of insurance companies. Since insurers are facing long term obligations, they invest over long time horizons and a large portion of their assets are fixed income investments, such as bonds or mortgage loans. Typically, the duration of assets and liabilities are not matched. Life insurers attain much longer durations on their liability side than on their asset side.In order to reduce interest rate risk, life insurers seek to match the maturities of their interest-bearing assets and liabilities. However, asset-liability matching is often imperfect, leaving insurers engaged in maturity transformation. The liabilities of life insurers, unlike those of banks, typically have maturity profiles that are longer and thus more sensitive to interest rate changes than those of investments. Life insurers therefore tend to benefit from rises in interest rates but lose if interest rates fall. (Mohlman, 2017 [14])

To hedge their interest rate risk, life insurers seek to match the maturities of their interest-bearing assets and liabilities. Nevertheless, hedging interest rate risk for insurance policies and annuities is not always straightforward. Many life insurers use hedging strategies such as cashflow or duration matching. These techniques generally do a good job of hedging interest rate risk when rates are relatively stable and near historical averages, as they were in the early part of the century. However, these strategies may not do as well when there is a large change in interest rates such as the sustained decrease in low interest rates that occurred after the 2008 financial crisis (Lubochinsky, 2015 [15]).

Additionally, as interest rates remain low, policies sold in the past with relatively high guaranteed minimum yearly rate of return are becoming very expensive to fund. This is particularly true in case of traditional life policies, which often en-tail a minimum guaranteed rate of return over a very long period of time. As a consequence, profit margins for life insurers are decreasing and as expected future returns are bound to fall, also the attractiveness of such saving and retirement prod-ucts might decrease going forward.

Therefore, insurance companies have drawn considerable attention in recent years, due to the persistent low yield environment. Supervisory authorities are particularly concerned about the financial stability of the life insurance industry. For instance EIOPA in its Financial Stability Reports pays special attention to the interest rate risk across European life insurers (EIOPA, 2015 [16]; EIOPA 2016 [17]).

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1.1. Background 9 In contrary to the Basel capital framework for banks, the capital framework for insurance companies, Solvency II, imposes a risk-based capital requirement (pillar I) for interest risk. Under Solvency II, the capital requirement is defined as the 99.5% Value-at-Risk of the change in economic capital over one year. Most insurers deter-mine the capital requirement with a standard formula. The standard formula applies multiplicative stress factors to the current yield curve to determine an upward and a downward shift of interest rates, and insurers need to recalculate their capital in these scenarios. Alternatively, insurance companies are given the option to develop an internal model to calculate the capital requirement. These models are subject to approval by a compentent authority, the supervisor.

1.1.3 Research Problem

Interest rate risk is the sensitivity of an financial institution to adverse movements in interest rates. It refers to the current or prospective risk to the institutions capital and earnings arising from adverse movements in interest rates that affect the insti-tutions positions (BCBS, 2016 [12]). When interest rates change, the present value and timing of future cash flows change. This in turn changes the underlying value of a insitutions assets ans liabilities, hence its economic value. This risk is inherent to the financial business where the maturity of assets and liabilities or not matched, e.g. banks and insurance companies. For both, bank’s and insurance companies, interest rate risk can form a serious threat for its profitability and shareholder value. Despite the common sensisitivty to interest rate risks, regulation on interest rate risk management for insurance companies (Solvency II) and banks (Basel III), and in par-ticular the regulatory minimal capital requirements on interest risks, are not aligned. This misalignment could influence the allocation of interest rate risk among differ-ent types of financial institutions and ultimately, could influence the global financial stability due to mismanagement of this risk type.

1.1.4 Research Objective

The objective of this thesis is to investigate whether components of the Solvency II framework with respect to interest rate risk could be applied on banks. In particular, this thesis focus will focus on the differences in regulatory capital frameworks. We will analyze the differences in the regulatory framework, We will provide a imple-mentation of the solvency II capital framework for interest risk and apply this frame-work on a stylized banking balance sheet. Furthermore, we will test this frameframe-work against multiple interest rate scenarios using a stochastic interest rate model, based on Principal Componenta Analysis. Summarizing, the main research question of this thesis is:

• What elements of the regulatory capital framework on interest rate risk for

insurance companies (Solvency II) can be potentially adopted by banking reg-ulation (Basel III)?

which will be answered by adressing the subquestions:

• What are the differences in the regulatory frameworks on interest rate risk for

banks vis-a-vis insurance companies? Literature study

• How can the Solvency II capital framework on interest rate risk be applied on

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• What will be the impact on future capital projections for banks, when applying

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Chapter 2

Literature Research

2.1 Interest Rate Risk in the Banking Book

The assets and liabilities on a bank’s balance sheet have different maturities, which makes the balance sheet sensitive to changes of interest on the money and capital markets. This risk specifically applies to the banking book, since these balance sheet items are supposed to be hold to maturity. This is the so-called interest rate risk in the banking book (IRRBB). More, specifically the BCBS has defined IRRBB as "the

current or prospective risk to the banks capital and earnings arising from adverse movements in interest rates that affect the banks banking book positions".

The level of interest rate risk a bank is exposed to is significantly influenced by the degree of maturity transformation between the longer term fixed interest peri-ods on the asset side versus the shorter term fixed interest rates of funding on the liability side (Memmel, 2011 [6]), i.e. a retail bank, which focuses on selling long term mortgages funded with retail sight deposits with a fixed term.

There are basically two methods to measure and controll IRRBB : at present value of future cashflows or at looking at the perdiocal flow of interest rates (BCBS, 2016 [12]). When we look from a present value point of view, risks are measured as the sensitivity of the economic value of the total banking book to changes in the the yield curve, i.e. an increase of the yield curve, will reduce the net present value of assets and liabilities, and consequently also the economic value of interest rate exposure. From a periodic point of view, we don’t look at the total cashflows, but focus on the interest rate earned and paid on the short term, i.e. an increase of the yield curve will directly impact the earnings on new assets, whereas the interest paid on sight deposits will increase more slowly and will increase the earnings of the bank from a short term perspective. Credit intitutions should be able to measure their IRRBB exposure from both point of views in different interest rate risk scenario in order to allow them to manage the underlying risks (BCBS, 2016 [12]). As financial intermediaries, banks encounter interest rate risk in several ways. Four sources of interest rate risk can be discerned:

1. Gaps in maturity and/or repricing between assets and liabilities (repricing risk); 2. (Asymmetric) changes of the interest rates at different maturities (yield curve

risk);

3. Imperfect hediging due to differences in underlying curves used to price dif-ferent financial instruments (basis risk);

4. Embedded options in the financial instruments such as prepayment options, withdrawal options or caps and floors (option risk).

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In the following subsection we will go into more detail about the different build-ing blocks of interest rate risk, like the subcomponents of which interest rates are build up, banking versus trading book, regulatory framework and risk measures to monitor and manage interest rate risk.

2.1.1 Components of interest rates

Banks requires a compensation for the different types of risks they take upon when lending sums of money. Therefore, they charge the client several risk premiums. The final interest rates are the market price of the sum of risks, that banks charge for lending money to clients or have to pay to fund themselves, build up by several risk premiums. Theoretically, we can distinguish five different components of which an interest rate is build up (BCBS, 2016 [12]):

1. The risk-free rate: this is the fundamental building block for an interest rate. This component assumes no risk or uncertainty, simply reflecting differences in timing: the preference to spend now/pay back later versus lend now/collect later;

2. A market duration spread: an instrument will be more sensitive to interest rate fluctuations the longer the time to maturity is. To be compensated for the uncertainty of both cash flows and the prevailing interest rate environment, banks charge a premium or spread over the risk-free rate to cover for this type of risk;

3. A market liquidity spread: some investments are highly liquid, meaning they are easily sold for cash without a premium to be paid (e.g. Dutch government bonds). Other securities are less liquid, and there may be a haircut if a bank needs to sell these types of instruments quickly. A bank will charge a premium a to be compensated for this potential loss;

4. A general market credit spread: This risk premium is charged by banks for a given credit quality, which can be seen as the market price of credit risk and depends on the risk appetite of the market;

5. Idiosyncratic credit spread: this reflects the specific credit risk attached to the individual borrower and the specifics of the underlying instrument.

In theory all rates of balance sheet items could be decomposed into these rate components. In practice, decomposing interest rates into their component parts is very challenging, because not all components are traded on a liquid market. Hence, the market price of each risk is not easily identifiable. In particular non-market traded assets, like bank loans or mortgages, for which even the market price of the total spread is often not frequently traded, it is very challenging to decompose the interest rates. Therefore, banks often, and also regulation, simplifies the decompo-sition of of interest rates for risk management purposes into two parts (BCBS, 2016 [12]):

• The funding rate, which is equivalent to the reference rate plus a funding

mar-gin. The funding rate is the blended internal cost of funding the loan. It con-sists of a reference rate , which is an externally set benchmark rate, such as Eu-ribor , and a bank-specific funding margin to reflect the premium the market

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2.1. Interest Rate Risk in the Banking Book 13 charge to fund the bank. Both the funding rate and the reference rate incor-porate other components of the theoretically decomposition of interest rates, such as liquidity, duration spread and market credit spread. However, the al-location between these components may fluctuate over time;

• The credit margin (or commercial margin): these are market specific risk

pre-miums which banks charge to componsated for all other remaining risks, with-out specifying specifically for which risk type a premium is charged.

Changes to the market liquidity spreads and market credit spreads are combined within the definition of CSRBB, but out of scope for our thesis. Figure2.1provides a visual representation on the decomposition of interest rates into its theoretical com-ponents. It shows that changes to the risk-free rate, market duration spread, ref-erence rate and funding margin are within scope of IRRBB, wheras the remaining components are covered by other parts of regulation (e.g. Credit Spread Risk in the Banking Book (CSRBB).

FIGURE2.1: Schematic overview of the components of interest rate

risk (source: BCBS [12]

2.1.2 Banking Book versus Trading Book

As the name suggest IRRBB only focuses on interest rate risk in the banking book, incidating that a distinction should be made between interest rate risk resulting from this banking book and other "books". Books held by banks can be distinguished in two types: banking book and trading book. All banking activities should, for capital and risk management purposes, be divded over these two books.

In accounting, a banking book refers to registers of accounts covering both assets and liabilities, which are intended to be held to maturity. These product are not int-eded to be traded. This allows banks to account these assets at "acquisitions costs" or at "book value", and not as opposed to traded instruments "mark-to-market". There-fore, these instruments are not subject to day-to-day changes in market value, i.e. changes of interest rate yields do not directly influence the accounting value of these instrument, so will not have an impact on the income statement. On the other hand,

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instruments designated to the trading book, are held with the intention to be traded. These assets are valued "mark-to-market", which means that the accounting value is equivalent to the market value or fair value of the assets: "The price that would be

received to sell an asset or paid to transfer a liability in an orderly transaction between mar-ket participants at the measurement date". These instruments are subject to day-to-day

changes in value, i.e. changes of interest rate yields directly influence the accounting value of these instrument, which will directly flow into the income statement of the bank.

Banking book instruments are generally intended to be held to maturity. Changes in market value are therefore not necessarily reflected in profit and loss accounts (BCBS, 2015). Instruments held in the trading book are often not meant to be held to maturity, and changes in the fair value impact profit and loss accounts. Before and during the crisis banks could designate instruments with observable market prices to the trading book by claiming the intention to trade these assets. However, during the crisis many of these positions became illiquid and volatile. In order to avoid the impact of changes in valuation of these instruments on profit and loss accounts, many banks transferred instruments to the banking book (BCBS, 2015). After the crisis the BCBS started a fundamental review of the trading book, where different capital charges for the same types of products in the trading and banking book were addressed. In addition, more strict boundaries were set to prevent transferring trad-ing instruments to the banktrad-ing book and vice versa (BCBS, 2015; BCBS, 2016b). Both trading and banking book products are subject to interest rate risk. In this thesis, the focus will be on interest rate risk on banking book products, leaving the trading book out of scope.

Summarizing, financial instruments, despite having the same economic cashflow profile, will be treated differently accountingwise, depending on their designation to a specific accounting book. This misalignment of treatment is also observed in capital regulation (BCBS, 2013 [?]) and is particularly the case for interest rate risk. Where interest rate risk in the trading book should be capitalized with a minimal capital requirement (Pillar I), interest rate risk in the banking book should be capi-talised within the "Supervisory Review Process" (Pillar II), resulting in different cap-ital charges.

2.1.3 IRRBB regulation

In December 2010, the Basel Committee on Banking Supervision (BCBS) published its reforms on capital and liquidity rules to address problems, which arose during the financial crisis. This new set is of rules is better known as Basel III. Basel III strengthens the Basel II regulatory framework. Whereas Basel II focused on the asset side of the balance sheet, Basel III als put emphasis on the liabilities, i.e. capital and liquidity. The new framework will (a) impose higher capital ratios, including a new ratios focusing on common equity, (b) increase capital charges for many activities, particularly involving counterparty risk and (c) narrow the scope of what constitutes Tier 1 (T1) and Tier 2 (T2) capital.

The three pillar system in the Basel III accords is intended to introduce stan-dards for capital, risk management and supervision. The level of required capital depends on the risk profile and business profile of the individual banks. The first pillar addresses the regulatory capital and identifies the three major risk compo-nents, i.e. credit, market and operational risk. The second pillar sets out the process by which a bank should review its overall capital adequacy and the process under which the supervisors evaluate how well financial institutions are assessing their

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2.1. Interest Rate Risk in the Banking Book 15 risks and take appropriate actions in response to the assessments. The third pillar sets out the disclosure requirements for banks to publish certain details of their risks, capital and risk management, with the aim of strengthening market discipline. This is intended to improve effective risk management by allowing for comparison of the performance across sectors through these disclosure requirements.

Due to the high relevance of the interest rate risk in the banking book, especially in the context of a continued low interest rate phase, there are a number of regulatory requirements for the measurement and management of interest rate risks defined by international and national supervisory authorities. In June 2015 the BCBS pub-lished a guidance document, which included a proposal for a a two-pillar approach for banks to assure appropriate capital to cover potential losses from exposures to changes in interest rates and to "limit potential capital arbitrage by switching expo-sures from the trading book and the banking book, and vice versa":

• Pillar 1: Standardized Minimum Capital Requirement (MCR) based on two

metrics, the Economic Value of Equity (EVE) and Net Interest Income (NII) measured under six interest rate scenarios

• Pillar 2: A set of 12 principles to guide banks (9 principles) and supervisors (3

principals) nin measuring risk and assessing capital adequacy internally and via supervisory assesment

In September the financial institutions heavenly critized the proposal of introducing a Pillar 1 minimal capital requirement for interest rate risk in the banking book. The feedback was basically driven by one argument, which argues that a one-size-fits-all standardized approach is not feasibile given the diversity of bank’s business model, balance sheets and behaviour of bank’s customers. Ultimately, in 2016 the Basel Committee on Banking Supervision (BCBS) published standards (2016, [12]), which only contained the strenghtening of pillar 2 requirements for the measurement and management of this type of risk type, still leaving the gap unfilled for having a har-monized approach to capitalize IRRBB in pillar 1. As can be seen in [?] the pillar I framework lacks a block for interest rate risk, whereas Solvency specifically adresses interest rate risk in their framework (see figure2.4, which will be discussed later).

The nine principles which apply for banks, included in the BCBS standards, are:

• Principal 1: Principle 1: IRRBB is an important risk for all banks that must be

specifically identified, measured, monitored and controlled. In addition, banks should monitor and assess CSRBB.

• Principle 2: The governing body of each bank is responsible for oversight of

the IRRBB management framework, and the banks risk appetite for IRRBB. Monitoring and management of IRRBB may be delegated by the governing body to senior management, expert individuals or an asset and liability man-agement committee (henceforth, its delegates). Banks must have an adequate IRRBB management framework, involving regular independent reviews and evaluations of the effectiveness of the system.

• Principle 3: The banks risk appetite for IRRBB should be articulated in terms

of the risk to both economic value and earnings. Banks must implement pol-icy limits that target maintaining IRRBB exposures consistent with their risk appetite.

• Principle 4: Measurement of IRRBB should be based on outcomes of both

eco-nomic value and earnings-based measures, arising from a wide and appropri-ate range of interest rappropri-ate shock and stress scenarios.

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FIGURE2.2: The Pillar 1 Basel III capital framework. The framework

lacks a specific capital requirement for IRRBB, as apposed to Solvency II capital framework, which do include a capital requirement for

in-terest rate risk (see figure2.4).

• Principle 5: In measuring IRRBB, key behavioural and modelling assumptions

should be fully understood, conceptually sound and documented. Such as-sumptions should be rigorously tested and aligned with the banks business strategies.

• Principle 6: Measurement systems and models used for IRRBB should be

based on accurate data, and subject to appropriate documentation, testing and controls to give assurance on the accuracy of calculations. Models used to mea-sure IRRBB should be comprehensive and covered by governance processes for model risk management, including a validation function that is indepen-dent of the development process.

• Principle 7: Measurement outcomes of IRRBB and hedging strategies should

be reported to the governing body or its delegates on a regular basis, at rele-vant levels of aggregation (by consolidation level and currency).

• Principle 8: Information on the level of IRRBB exposure and practices for

mea-suring and controlling IRRBB must be disclosed to the public on a regular ba-sis.

• Principle 9: Capital adequacy for IRRBB must be specifically considered as

part of the Internal Capital Adequacy Assessment Process (ICAAP) approved by the governing body, in line with the banks risk appetite on IRRBB.

In this thesis, we will particularly focus on principle 4, which includes guidelines how to measure IRRBB exposure and how you should include incorporrate a range of interest rate scenario’s into the risk management framework.

2.1.4 Interest rate risk measures

For measuring interest-rate risk banks use a variety of methods. The level of so-phistication and complexity of individual methods varies. Nawalka and Soto (1999 [30]) provides an overview of risk managmenet tools to measure interest rate risk.

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2.1. Interest Rate Risk in the Banking Book 17 The most widely used tecnniques are gap analysis, the duration of equity, economic value approach and earnings at risk approach. In the following sections we will provide an overview of these measures.

Gap analysis

Gap analysis is based on the classification of interest rate sensitive cashflows of as-sets and liabilities into maturity/repricing buckets. The netted differences between assets and liabilities in the different buckets indicate the gap positions. The gap po-sitions indicate the cashflow released and the cashflow that is outstanding during a particular time period. Using gap analysis, the earnings sensitivity of the banking book to interest rate movements can be derived, i.e. when the value of interest-sensitive liabilities exceeds that of interest-interest-sensitive assets and the interst rate rises, the net interest income is adversely affected. This method is developed over 50 years ago, when the banking business was characterized by stable interest rates, the banks balance sheet comprised of simple instruments with a fixed interest rate and the golden balance rule applied in banks, wich means that people should only bor-row money to invest and not to fund current spending. The current banking and macro-economic environment differs significantly from the period when gap analy-sis was founded. Interest rates exhibit a high degree of volatility and are currently extremely low, banks issue more products with variable interest rates, instruments include many embedded options (e.g. prepayment, caps and floors) and govern-ment and corporates and retail clients do not only borrow for investgovern-ment purposes, but also include short term financing of spending (Licak, 2004 [31]). However, Gap analysis is still very popular in banks and is often used mainly due to its simplicity, but because of it simplicity it also has some weaknesses (DNB, 2005 [32]):

• it is a static model that does not take account of the interest sensitivity of retail

products and/or of changes in savings or payment behaviour as a result of interest rate movements;

• yield curve and/or basis risk cannot be analysed properly using gap analysis; • no account is taken of the changes in interest rate spreads that may occur as a

result of a change in the general level of interest rates;

• it is based on the assumption that all positions within a particular maturity

segment mature or are repriced simultaneously.

Therefore, additional risk measures are needed in conjunction with gap analysis to monitor the interst rate risk approriately.

Duration Gap

Another widely used risk measure for interest rate risk is the duration gap, which is based on the duration of balance sheet items. The main buidling block for cal-culating the duration is the net present value of cashflow (NPV). The NPV of an instrument is given by a function, which depends on cashflows of that particular in-strument and the interest rate. Through the first derivation of the inin-strument’s NPV and division by NPV itself, we get the equation for calculating price volatility. If we subsequentltly replace the first derivation by differentials, we get a definition of the modified duration, the value of which is defined as the share of weighted financial

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flows (weighted by the time from the moment of valuation until maturity) and the current value of the financial flows:

ModD(y)≡ −1 V · ∂V ∂y =− ∂ ln(V) ∂y (2.1) with ∂V ∂y =− n

∑

i=1 ti·CFi·e−y·ti (2.2) where

y = is the continiously compounded interest rate;

i = indexes the cash flows;

ti = is the time in years until the i-th payment will be received;

V = he present value of all cash payments from the asset.

However in financial markets, yields are usually not expressed continiously, but instruments are periodically compounded instead. Then the expresion to calculate

V will be: V(yk) = n

∑

i=1 PVi = n

∑

i=1 CFi (1+yk/k)k·ti (2.3) and taking the differentials yields

∂V ∂yk =− 1 (1+yk/k)· n

∑

i=1 ti· CFi (1+yk/k)k·ti =−∑ n i=1V(tyik)· CFi (1+yk/k)k·ti ·V(yk) (1+yk/k) (2.4) where n

∑

i=1 ti V(yk)· CFi (1+yk/k)k·ti = MacaulayDuration (2.5) which gives the formula for the modified duration in case of periodally com-pounded interest rates:

ModD =− 1 V(yk)· ∂V ∂yk = MacD (1+yk/k) (2.6) Modified duration shows the change in price of a financial instrument corre-sponding to marginal and parallel shifts of the yield curve. It is a relative measure-ment, which is expressed as a percentage. The price value of a basis point (PVO1) measurement is equivalent measure, but expresses an absolute measurement de-rived from duration, which shows in monetary units the change in price resulting from a one basis point (0.01%) shift in the yield curve.

The modified duration or PVO1 measures can also be applied on an aggregated basis to portfolios or the total banking book. The result of this calculation expresses the economic value impact on the economic value of the positions concerned re-sulting from the specified change in interest rates. Finally, if we take the volume weighted difference between the duration of the assets and liabilites we can derive the duration gap:

DurationGap=DurationAssets−VLiabilities

VAssets ∗DurationLiabilities

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2.1. Interest Rate Risk in the Banking Book 19 One benefit of the duration method is that it analyses the economic value impact of a particular change in interest rates relating to a particular class of assets and/or liabilities and/or the balance sheet in a simple way, but also has some weaknesses (DNB, 2005 [32]):

• it only applies to parallel shifts of the yield curve and it cannot be used to

measure basis or yield curve risk;

• it only applies to marginal shifts of the yield curve. Relatively large

move-ments in interest rates, and therefore convexity, cannot be measured accu-rately;

• it takes no account of the change in expected cash flows resulting from interest

rate movements;

• it is a static method in the sense that it shows a snapshot in time of the risk

based upon the current composition of the portfolio or balance sheet.

In order to overcome some of the disadvantages of modified duration described above banks can also calculate the effective duration of assets, liabilities and equity. Effective duration does not only takes account of changes in the interest rates, but also considers the change of the expected cash flows due to changes in the interest yield curve, which is very important when calculating the interest rate sensitivity of instruments that contain embedded options, e.g. mortgages which contain prepay-ment options.

Earnings at Risk

Earnings at risk (EaR) is a scenario-based method that analyses the interest rate risk in the banking book in terms of earnings. It measures the amount that the net interest income, or earnings of a bank, will be impacted, resulting from interest rate curve changes, which can be either a gradual or an instantaneous shocks of the yield curve. Compared to the gap analysis and duration gap, which are based on a point-in-time view of the interest rate risk where cashflows are fixed, the EaR method is more dynamic. It evaluates the risk exposure of the banking book over a particular time horizon (1 to 2 years), taking account dynamically projected changes in cashflows, maturities, repricing and the total size of the balance sheet.

As done for the gap analysis, as a first step, all of the relevant assets and liabil-ities are allocated to maturity/repricing buckets, followed by calculation of the Net Interest Income (NII) in the projected scenario Rsversus the base case scenario R0:

EaR=N I I(Rs)−N I IR0 (2.8)

Due to the dynamic character of the EaR approach, a crucial role is played by assumptions regarding instruments with embedded options, such as prepayment and/or savings behaviour. Under each scenario, the interest rate depending be-haviour of customers should be captured, adding a layer of complexity to the cal-culations. However, the BCSB prescribes a standardised way to calculate the regu-latory EaR. The BCBS-method assumes a static balance sheet, meaning EaR should be computed assuming a constant balance sheet, where maturing or repricing cash flows are replaced by new cash flows with identical features with regard to the amount, repricing period and spread components. Additionally, the BCBS only con-siders the two parallel shocks (up and down) of yield curve, and presribes that EaR

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should be disclosed as the difference in future interest income over a rolling 12-month period.

The disadvantages of the EaR method are (DNB, 2005 [32]):

• it only analyses the short-term earnings effect (accrued interest) resulting from

interest rate fluctuations and not the long-term economic value effects (capital gains/losses);

• it can be complex and non-transparent as a result of the underlying

assump-tions. However the BCBS prescribes a harmonised approach, making results more comparable

Economic Value of Equity

The Economic Value of Equity (EVE) is a valuation technique where all net cash flows, i.e. all asset cash flows subtraced by all liability cash flows, are discounted with an interest yield curve, resulting in the net present value (NPV) of a bank’s bal-ance sheet’s cash flows. Conceptually, it can be interpreted as the amount of future earnings capacity residing in the banks balance sheet. In contrast to the earnings perspective, that has a short term focus on IRRBB, the economic value perspective has a longer time horizon. The economic value perspective takes into account all future cashflows and takes into account the time value of money.

The EVE-at-risk measures the change in the NPV of equity resulting from dif-ferent interest rate scenario’s. As in the case of the EaR, the value of equity under alternative interest rate scenarios is compared with the value under a base scenario. The base interest rate scenario is the NPV of the assets less the liabilities under the current interest rate conditions. After revaluating the EVE under alternative inter-est scenario, a particular scenario (e.g. maximum loss scenario or 99.5% in case of Monte Carlo simulation):

EVEj = n

∑

i=1 CFj(ti) (1+yjk/k)k·ti (2.9)

∆EVEm = EVE0−EVEm (2.10)

where j = 0 is the current interest rate yield curve, CFj(ti)is the net cash flow of instruments that mature in the time bucket ti, yjkthe interest rate between 0 and ti in a specifif interest scenario j.

The calculation of the regulatory EVE-at-risk is prescribed by the BCBS (2016, [12]). It prescribes the following requirements:

• Banks should exclude their own equity from the computation of the exposure

level.

• Banks should include all cash flows from all interest rate sensitive assets and

liabilities in the banking book in the computation of their exposure. Banks should disclose whether they have excluded or included commercial margins and other spread components in their cash flows.

• Cashflows should be discounted using either a risk-free rate or a risk-free rate

including commercial margins and other spread components (only if the bank has included commercial margins and other spread components in its cash flows). Banks should disclose whether they have discounted their cash flows

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2.1. Interest Rate Risk in the Banking Book 21 using a risk-free rate or a risk-free rate including commercial margins and other spread components.

• EVE-at-risk should be computed with the assumption of a run-off balance

sheet, where existing banking book positions amortise and are not replaced by any new business.

Moreover, BCBS defines six interest scenario’s that a banks should use to calcu-late their EVE-at-risk. The EVE-at-risk is equal to the maximum amount of EVE lost in one of the six scenario’s:

EVE@Risk0= max

i∈{1,2,...,6}△EVEi (2.11)

The six interest rate scenarios are described in the "Standards on IRRBB" by the BCBS (2016, [12]).

1. Parallel up shock: A parallel shock on the yield curve of +200 basis points. 2. Parallel down shock: A parallel shock on the yield curve of -200 basis points. 3. Steepener: A shock where the yield curve is rotated to get a steeper version of

the base curve.

4. Flattener: A shock where the yield curve is rotated to get a flattener version of the base curve.

5. Short rate up: Shock up that is the greatest at the shortest tenor point 6. Short rate down: Shock down that is the greatest at the shortest tenor point.

FIGURE 2.3: The six prescribed shocks for EUR, USD, and USG (source: Moody’s [35])

The maximum delta EVE is subject to an outlier test. The outlier test applied by supervisors should at least include comparison of the banks maximum∆EVE under the six prescribes scenarios with 15% of its Tier 1 capital. Supervisors are required to use this information to idenitfy potential outlier banks for more intensive super-vision. Whenever a supervisor deems that a bank does not hold capital compared to its exposure to IRRBB, it should take remedial actions, requiring the bank to either increase capital and/or reduce its exposure to interest rate risk.

One major disadvantage of the EVE method, especially compared to the EaR appraoch is (DNB, 2005 [32]):

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• it is a static method in the sense that it shows a snapshot in time of the risk

based upon the current portfolio or balance sheet composition. It cannot make allowance for the market valuation of future (forecast) growth in existing or new business activities;

2.2 Interest Rate Risk for insurance companies

After being pushed back a couple of times Solvency II, a new regulatory frame-work for insurance companies, came into effect on 1 January 2016. The objective of this new framework is to harmonize the European insurance market and enhance consumer protection. One of the most important components of Solvency II is the new set of rules to calculate the solvency captial requirements (SCR), which from 2016 onwards will be risk based. The SCR should correspond to the one-year 99.5% Value-at-Risk of the basic own funds (= equivalent to a bank’s equity position) of an insurance company (Solvency II directive, 2009 [33]).

In general, Solvency II allows for two different approaches to determine the rel-evant capital requirements to compensate for a potential loss of basic own funds. Firstly, the insurance company can calculate the SCR based on a predefined set of formulas and risk parameters: the so-called standard formula approach or alterna-tively insurance companies can opt in to develop their own internal model to calcu-late the capital requirements. The model should be able to accurately estimate the the value-at-risk for a confidence level of 99.5%.

The SCR under the standard formula approach is derived by aggregating capital requirements specified in several sub-modules, as is illustrated in figure2.4, corre-sponding to the different risk types an insurance company potentially can have on their balance sheet. One of the sub-modules corresponds to interest rate risk. In the standard formula, Solvency II will prescribe a set of stress scenarios in order to derive the SCR for interest rate risk.

FIGURE2.4: The aggregation of risk types within the the solvency II

capital framework. Interest rate risk is categorized under the market risk module

In the following section we will explain in more detail the several elements of the SCR-framework for Interest rate risk, which can be split up in the standardized

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2.2. Interest Rate Risk for insurance companies 23 formula for interest rate risk and internal modelling approach, but first we will elob-orate more on the valuation framework, which serves as the basis for the capital calculation.

2.2.1 Market-consistent Valuation

In order to understand the dynamics of interest rate risk on the balance, it is impor-tant to understand the valuation of the balance sheet items. Whereas balance sheet item in the banking book are valued at acquisition costs or book value, the valua-tion of the assets and liabilities, as set out in Solvency II Directive ([33]), requires an economic, market-consistent approach. As apposed to the book value approach, changes in interest yield curves directly impact the valuation of assets and liabilities on a insurance company’s balance sheet. Compared with banks, this is a major dif-ference. Insurance regulation does not follow the accounting balance sheet, which could be based on book value or acquisition costs, but prescribes a market-consistent balance sheet.

Both the assets and liabilities are affected by the term structure of interest rates. However, the yield curve that needs to be used in the valuation of liabilities is differ-ent from the yield curve used in the valuation of assets. Insurance companies have very long-term liabilties. In particular life insurance companies can have expected cashflows on their balance sheet beyond 50 years. The issue with the valuation of these cashflows is the absence of liquid market instruments, which can be used to construct a mark-to-market yield curve. Therefore the valuation curve consists of two part: a mark-to-market part and a mark-to-model part. The splitting point be-tween the to parts is called the last-liquid-point (LLP): the last available maturity point for which liquid market data is available. For maturities up to the last liquid point, the curve is equal to the Euribor Swap curve. For maturities beyond this last liquid point, the yield curve is extrapolated using the Smith-Wilson extrapolation technique (Beers, 2012 [36]). The inputs for the Smith-Wilson extrapolation are the yield curve prior to the last liquid point, the unconditional ultimate forward rate (UFR) and the speed of transition, where for the euro, the Last Liquid Point (LLP) was set by EIOPA at the 20-year swap. The UFR is defined as the rate all term struc-tures of interest rates will eventually converge to and is assumed to be constant over time. A macro-economic assessment has been applied in order to derive its value for different currencies. For the Eurozone, the UFR is initially set equal to 4.2%, which is the sum of the expected annual inflation rate (2%) and the expected an-nual short-term return on risk-free bonds (2.2%). However, EIOPA reconsidered the methodology to determine the UFR in 2017. In line with the new methodology, and reflecting the significant changes in the long-term expectations of interest rates in recent years, the calculated value of the UFR for the euro was 3.65%. However an-nual changes will not be higher than 15 basis points. In a first step of the phasing-in the current UFR of 4.2% will therefore be lowered in January 2018 to 4.05% (EIOPA, 2017 [34]). Fortunately, this thesis is not impacted by the mark-to-model part of the valuation curve, because All cashflows from the hypothetical bank in our thesis, are before the LLP.

2.2.2 Solvency Capital requirement

The Solvency Capital Requirement (SCR) is a the regulatory minimum amount of Own funds that an insurance company should hold. In addition to the SCR capital requirement, a minimum capital requirement (MCR) must also be calculated. This

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figure represents the threshold below which a national regulatory agency would intervene. The MCR is intended to achieve a level of 85 percent probability of ade-quacy over a one-year period.

For regulatory purposes, the SCR and MCR figures should be regarded as "soft" and "hard" floors, respectively. That is, a tiered intervention process applies once the capital holding of the (re)insurance company falls below the SCR, with intervention becoming progressively more intense as the capital holdings approach the MCR. The Solvency II Directive provides regional regulators with a number of options to address breaches of the MCR, including the complete withdrawal of authorization from selling new policies and forced closure of the company.

2.2.3 Standard Formula Interest Rate Risk

All assets and liabilities of which the valuation is sensitive to movements of the term structure of interest rates and consequently in scope of the capital calculation for interest rate risk. The standard formula for interest rate risk prescribes two stress scenarios. These scenarios are defined by a downward and upward stress of the term structure of interest rates.

The capital requirement is calculated by revaluating all assets and liabilities us-ing the yield discount curves derived from the two stress scenarios. The scenario that results in the maximum loss of own funds is used to calculate the SCR for inter-est rate risk and is set equal to the total loss in that specific scenario:

SCRIRR =max(Lup, Ldown, 0) (2.12)

where Lupand Ldownrepresent the losses of own funds in respectively, the up-ward and the downup-ward scenario. The underlying scenario rup_k and rdown_k are rela-tively stressed levels of rkwith factors sup_k and sdown_k :

rup_k =max([1+s_kup]rk, rk+1%) (2.13)

rdown_k =max([1+sdown_k ]rk, rk) (2.14) In order to achieve an European-wide agreement on a harmonized approach that should fit all insurance companies, a number of simplifications have been applied to define the stress scenarios for the interest rate risk module in the standard formula (Beers, 2012 [36]).

1. The stadard formule only prescribe two near parallel scenarios. Although these scenario’s are derived from a principal component analysis, which ex-plains 99.98% of the variablility of historical term structures, the number of scenarios extracted is limited to two scenario’s, which do not capture specific movements other than parallel moves, e.g. tils or butterfly effects.

2. The stress is also applied after the LLP. This contradict the theoretical founda-tion of the UFR, which states that longer term rates always, so also in stress scenarios, converge to the UFR. However, the standard formula assumes that the UFR might change in stress scenarios.

Detailed information on the calibration of the risk factors and application of the formula can be found in appendixA.1.

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2.3. Interest rate modelling 25

2.2.4 Internal model Interest Rate Risk

Under the draft directive companies may calculate their SCR using the standard for-mula or their own partial or full internal model. The standard forfor-mula is designed and calibrated to value the risks in accordance with a 99.5% probability of remain-ing solvent over a one year time horizon. An internal model aims to calculate the similar risk at the same percentile (99,5%), but applies assumptions and/or mod-els other than those set out in the standard formula. The quantification of the risk amount always uses the economic balance sheet as the starting point with the assets and liabilities valued using a market consistent approach. Allthough many compa-nies currently use internal models for product development and financial reporting, not all insurance companies already apply market-consistent valuation for account-ing purposes. Compared this with banks, insurance regulation does not follow the accounting balance sheet, which could be based on book value of acquisition costs, but prescribes a market-consistent balance sheet.

Internal models may be used only if they have been explicitly approved by the supervisory authority. Solvency II imposes strict requirements on internal models. These requirements relate to (1) suitability of the the models to assess company’s risk profile, (2) conceptual soundness of their basic assumptions, (3) their statistical quality, and (4) the system of governance and use of the models. These aspects are all considered when internal models are assessed. Model assessment therefore is a complex and lengthy process. However, on the other hand an internal model also comes with significant benefits. An internal model should enable improved insight into the company risk profile and capital requirements. This should result in substantial benefits to the management, governance and strategic decision making of the company and therefore in a more efficient use of the companys capital.

Under a full internal model approach, all risks are evaluated using the firms economic capital framework. However, not all insurers choose to implement a full internal model. There are a wide variety of possible approaches to a partial internal model. The model may be partial in the sense that the internal model is only adopted for specific risk modules, e.g. a Monte Carlo simulation-based approach may only be applied for the market risks, including interest rate risk, while underwriting risks quantified using the Standard Formula approach. Alternatively, the partial may re-fer to when the internal model only applies to specified lines of business.

In this thesis, we specifically focus on an internal model for interest rate risk, leaving all other internal models, including aggregation of risk modules out of scope.

2.3 Interest rate modelling

An interest rate model is a model that describes the evolution of a zero curve through time. In literature there are many different interest rate models available, which can be roughly divided in three categories of models: equilibrium or fundamental mod-els (e.g. Vasicek and CIR-model), no-arbitrage modmod-els (e.g. Hull-White model and Two-factor Vasicek model) and empirical or statistical models (e.g. Nelson-Siegel-Svensson and PCA-VAR models). All models, have their pro’s and con’s, where some models are focused on modelling the short rate others tend to model the dy-namics of the whole interest rate term structure. In the section we will describe the mostly used models. However, an exhaustive characterization of all available inter-est models is beyond the scope of this thesis.

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2.3.1 Equilibrium models

Equilibrium models assume, as the the name suggests, that the market is equilib-rium. It describes the economy based on the utiltiy fuction of a investor and sub-sequently derive the term structure of interest rates, the risk premium and other asset prices endogenously. Strictly theoretically, an equilibrium model should be arbritage-free, else the economy would not be in equilibrium. However, in practice this models cannot fit the market curves, and do provide opportunities for arbritage. Two examples of one-factor equilibrium models are the Vasicek (1977, [20]) and Cox, Ingersoll and Ross (1985, [21]) models. One-factor models, the short-rate process can be presented in the following general form

drt=κ(Θ−rt)dt+σrβdZt (2.15)

whereκ ,Θ ,σ and β are positive constants and dZtis an increment of a standard brownian motion over time interval dt . In these models, the interest rate, r , is reversed to a level Θ at rate κ . The special cases, where β = 0 or β = 0.5 are respectively the Vasicek model and the Cox, Ingersoll and Ross (CIR) model.

The Vasicek model is a method which is widely used in the market, easy to un-derstand and exerts reversion to a meanθ. A drawback of the Vasicek model is that the volatility of the short rate is constant, which is not supported empirically and model is that it permits negative interest rates. The latter is emprically observed in current interest rate environment. However, the Vasicek model does not include some kind of floor, which means it can be infinitely negative in theory. The Vasicek model assumes that for the short rate rt follows an so-called Ornstein-Uhlenbeck process:

drt =κ(Θ−rt)dt+σdZt (2.16)

Where

dr is the change in interest rate;

κ measures the speed of mean reversion;

Θ is the steady state mean, to which the process tends to revert in the long run;

σ is a measure of the process volatility;

dZt The standard Brownian motion, so dz∽ N(0, dt)

The Cox, Ingersoll and Ross (CIR) equilibrium model is widely used alternative to the Vasicek model. It always produces nonnegative interest rates, which at current interest rate levels can be seen as a disadvantage. Both the Vasicek and CIR model includes a mean-reversion component.However, the volatility of the short rate is assumed to be proportional to the level of interest rates through √rt. The model allows for more variability at higher levels of interest rates and less variability at lower levels. The CIR model assumes that the short rate r follows a process of the type:

drt =κ(Θ−rt)dt+σ√rtdZt (2.17)

Modeling the interest rate evolution through the short rate has some advantages, mostly the large flexibility one has in choosing the related dynamics (Brigo 2001 [37]). However, these models have also some clear drawbacks. For example, as mentioned, they lack the possibility to fit the market observed initial interest term structure of discount factors and are therefore not arbritage-free. Additionally, these

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2.3. Interest rate modelling 27 models have the undesirable property that returns for all maturities are perfectly correlated, and thus these models do not accurately reflect the potential change in interest rate dynamics (e.g. tilt and butterfly effects), which are observed empirically time (Kuo, 2006 [38]).

2.3.2 No-arbritage models

In contrast to equilibrium models, no-arbritage models assumes that they lack, as the name suggest, arbritage opportunities. It describes the stochastic behaviour of one or many interest rates and the market price of risks and derives the volatility of the interest rates around the, in the market observed, interest rate term structure. In other terms, there is no risk-free financial strategy with zero cost that should give with absolute certainity a positive return. The Ho and Lee (1986, [23]) model was the first term structure model to solve these problems. The model can be calibrated to market observed yield curves, which can also be done with the Hull-White model, but this model also includes a mean reversing property. Another family of mod-els, which are arbritage free, are represented by the HeathJarrowMorton (HJM, 1990 [24]) framework. HJM models are different from the previously described short-rate models. The HJM-type models are capable to model the full dynamics of the complete interest rate term structure, while the short-rate models only model the dynamics of the rt, the short rate.

In 2986 Ho-Lee developed an innovative no-arbitrage model of interest-rate evo-lution. This model have become popular for two reasons:

• It has a built-in ability to price exactly a given vector of bond-prices;

• It resembles the binomial approach to value non-lineair instruments like

op-tions, simplifiying pricing of such instruments.

As the model assumes that dZ is normal distributed, generating a symmetric dis-tribution of rates aroundθt,allowing for negative rates. The model can be calibrated to market observed interest rates Thetat:

drt =θtdt+σdZt (2.18)

The Hull and White model is an extension of the Vasicek model that can be cali-brated to the initial term structure. It assumes that movement of short rates are nor-mally distributed, and that the short rates are subject to mean reversion. Volatility is likely to be low when short rates are near zero, which is reflected in a larger mean reversion in the model. Additionally, the Hull-White model prices instruments as a function of the entire term structure, rather than at a single point like the Ho Lee model. It estimates the future interest rates, rather than observable market rates:

drt =κ(θt−rt)dt+σtdZt (2.19) where θt = ∂ f (0, t) ∂t +α f(0, t) + σ2 2α(1−e −αt₎ _(2.20) and f(0, t) = ∂lnP(0, t) ∂t (2.21)

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In short rate models, of which we explained a few above, we specify the dy-namics of the short rates. The HeathJarrowMorton model (HJM) takes a different approach, it specifies the evolution of the entire forward curve directly. As a conse-quence the initial forward curve enteres as an exogenous parameters. The drift of the forward rate dynamics will be fully determined by the volatility of the forward rate dynamics. The instantaneous forward interest rate with maturity T is assumed to satisfy the stochastic differential equation:

d f(t, T) =α(t, T)dt+σ(t, T)dZt (2.22) where

α(t, T) is the drift process

σ(t, T) is the volatility process ;

f(t, T) is the exogeneous initial forward curve;

2.3.3 Empirical models

The final set of interest rate models can be categorised as empirical models. These type of models do not necessarily that require risk neutrality or the no-arbitrage condition, but aim to model the relationship between emprirical macro-economic variables and interest rate term stuctures. A well described example in litereature is the the dynamic version of the Nelson and Siegel (1987), which was translated to a dynamic model of time series of term structures by Diebold and Li (2006). Fi-nally, Christensen, Diebold and Rudebusch (2009), showed that empircal models of Diebold and Li can be extended to be made arbitrage-free and make them equivalent to arbritage free models, like those within the HJM framework.

The Nelson Siegel Model, extended by Svensson (1994) to add an additional hump, captures the term structure with a parametric function with three param-eters. Nelson and Siegel show that with their model they are capable to capture the historically observed shapes of the yield curve. However, the Nelson Siegel ap-proach restricts the factor weights by imposing a parametric structure given by the structural form of the model. This limits the variety of potential shapes that can be modelled. The functional form uses Laguerre functions which consist of the prod-uct between a polynomial and an exponential decay term, resulting in the following formula to model spot rates:

yt(τ) =β0+β1 [1−exp(−τ/λ_t)] τ/λt +β₂ ( [1−exp(−τ/λ_t)] τt/λt − exp(−τ/λ_t) ) (2.23) hereβ1,β2,β3, lambdatare time-varying parameters. yt(τ)denotes the yield at observed me t with maturity meτ.

Svensson (1994) added a "second hump" term. The extended version of the Nel-son Siegel model is called the NelNel-sonSiegelSvensNel-son model. The additional term is: β3 ( [1−exp(−τ/λ₂)] τ/λ2 − exp(−τ/λ₂) ) (2.24) By transforming the Nelson-Siegel model into a dynamic framework, Diebold and Li (2006) proposes a dynamic version of the Nelson Siegel model, in which the

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2.3. Interest rate modelling 29 coefficientsβ1,β2,β3can be transformed in three latent dynamic variables Lt, Stand

Ct, which represent level, slope and curvature respectively of which the dynamics can be modelled using empirical data:

yt(τ) =Lt+St [1−exp(−τ/λt)] τ/λt +Ct ( [1−exp(−τ/λt)] τt/λt − exp(−τ/λt) ) (2.25) This three-factor Diebold and Li model can be used to construct projections of future term structure. They propose a two-step estimation procedure not only per-froms well constructing in-sample term structures of the yield curve, but also is able to construct out-of-sample priojection of interest rate forecasts.

2.3.4 Model selection

Despite the wide-availability of interest models, we will investigate an empirical, non-parametric empirical approach to generate scenarios, based on Principal Com-ponent Analysis, which is closley related to the Diebold and Li empirical model. This methodology is chosen, because it aims to duplicate the calibration methodology of the standard formula within Solvency II. However, Solvency II only generates two specific interest rate scenario, where we aim to utilize the model to generate a distri-bution of out-of-sample interest rate term structures. Our PCA approach estimates the components and the related contributions jointly. This approach only imposes one restriction, which requires that components are orthogonal, while imposting no restrictions on contributions of the components. This is different from the approach as proposed by Diebold and Li, in which the components may be correlated, but the weigths are restricted. These restrictions limit the performance in a low interest rate environent, which requires the modelling of negative interest rates (Inui, 2015 [39]).

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Capital requirements of interest rate risk in the banking book : a solvency II-like approach

ACTUARIAL

SCIENCE AND

M

ATHEMATICAL

F

INANCE

M

’

T

Capital requirements of Interest Rate Risk

in the Banking Book: A solvency II-like

approach

Abstract

Contents

Chapter 1

Introduction

1.1

Background

Chapter 2

Literature Research

2.1

Interest Rate Risk in the Banking Book

∑

∑

∑

∑

∑

∑

2.2

Interest Rate Risk for insurance companies

2.3

Interest rate modelling