IRRBB in a Low Interest Rate Environment

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IN

DEGREE PROJECT

MATHEMATICS,

SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2020

IRRBB in a Low Interest Rate

Environment

SIMON BERG

VICTOR ELFSTRÖM

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

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IRRBB in a Low Interest Rate

Environment

SIMON BERG

VICTOR ELFSTRÖM

Degree Projects in Financial Mathematics (30 ECTS credits) Master's Programme in Applied and Computational (120 credits) Mathematics KTH Royal Institute of Technology year 2020 Supervisors at zeb Nordics: Markus Ahlgren

Supervisor at KTH: Boualem Djehiche Examiner at KTH: Boualem Djehiche

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TRITA-SCI-GRU 2020:058 MAT-E 2020:021

Royal Institute of Technology

School of Engineering Sciences KTH SCI

SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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ii

Abstract

Financial institutions are exposed to several different types of risk. One of the risks that can have a significant impact is the interest rate risk in the bank book (IRRBB). In 2018, the European Banking Authority (EBA) released a regulation on IRRBB to ensure that institutions make adequate risk calculations. This article proposes an IRRBB model that follows EBA’s regulations. Among other things, this frame-work contains a deterministic stress test of the risk-free yield curve, in addition to this, two different types of stochastic stress tests of the yield curve were made. The results show that the deterministic stress tests give the highest risk, but that the out-comes are considered less likely to occur compared to the outout-comes generated by the stochastic models. It is also demonstrated that EBA’s proposal for a stress model could be better adapted to the low interest rate environment that we experience now. Furthermore, a discussion is held on the need for a more standardized framework to clarify, both for the institutions themselves and the supervisory authorities, the risks that institutes are exposed to.

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IRRBB i en Lågräntemiljö

Sammanfattning

Finansiella institutioner är exponerade mot flera olika typer av risker. En av de ris-ker som kan ha en stor påverkan är ränterisk i bankboken (IRRBB). 2018 släppte European Banking Authority (EBA) ett regelverk gällande IRRBB som ska se till att institutioner gör tillräckliga riskberäkningar. Detta papper föreslår en IRRBB modell som följer EBAs regelverk. Detta regelverk innehåller bland annat ett deter-ministiskt stresstest av den riskfria avkastningskurvan, utöver detta så gjordes två olika typer av stokastiska stresstest av avkastningskurvan. Resultatet visar att de deterministiska stresstesten ger högst riskutslag men att utfallen anses vara mindre sannolika att inträffa jämfört med utfallen som de stokastiska modellera generera-de. Det påvisas även att EBAs förslag på stressmodell skulle kunna anpassas bättre mot den lågräntemiljö som vi för tillfället befinner oss i. Vidare förs en diskus-sion gällande ett behov av ett mer standardiserat ramverk för att tydliggöra, både för institutioner själva och samt övervakande myndigheter, vilka risker institutioner utsätts för.

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Acknowledgements

We would express our gratitude to everyone at zeb, who has welcomed us and contributed their expertise. In particular, we want to thank Markus Ahlgren, our supervisor at zeb, for his advice and support. We would also like to thank Boualem Djehiche, our supervisor at KTH, for his feedback and guidance. Last but not least, we want to thank our families for their support. We are very grateful that you are always there for us.

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List of Figures

5.1 Cash flows in SEK, EUR and DKK converted to SEK. Note the scale difference for the different currencies. . . 36 5.2 Original yield curves for SEK, EUR and DKK. . . 36 5.3 Yield curves in SEK, EUR and DKK generated by the EBA model. 37 5.4 Describes the generated yield curves of the Cholesky and PCA

models . . . 40 5.5 Describes the maximum/minimum yield generated for each bucket

by the PCA and Cholesky model. . . 41 5.6 Describes the gap risk measure for both the PCA and Cholesky

model. . . 46

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List of Tables

4.1 Describes the different time buckets and the time period each bucket represent. . . 22 4.2 Describes the individual shock parameters for some currencies. . . 25 4.3 Describes scenario multipliers for each EBA scenario. . . 29 5.1 Gap risk measures given in million SEK for each of the six EBA

scenarios. . . 43 5.2 Describes gap risk measures given in million SEK for the three

models. . . 45 5.3 Describes the basis risk in each currency given in million SEK. . . 48 5.4 Behavioral risk measures given in million SEK for each of the six

EBA scenarios. . . 49 5.5 Describes behavioral risk measures given in million SEK for the

three models. . . 50

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Chapter 1 Introduction

In this chapter, a shorter background about interest rate risk in the banking book and the paper itself will be presented. Furthermore, this chapter will present the paper’s aim, research question, and outline.

1.1 Background

When talking about investments, people mainly want to analyze the expected earn-ings and the risk associated with an investment. When talking about investment strategies, it is common to use the words risk-averse, risk-neutral, and risk-seeking to explain how much risk an investor is willing to take in order to get a higher expected return on the investment. Financial institutions have different investment strategies, and their risk exposure is varying. However, regardless of an institution’s investment strategy, they need to do risk assessments regularly [1]. This in order to ensure that their business still can operate, even in case of unfavorable events. Financial institutions are exposed to many different kinds of risks. This paper will focus on the interest rate risk, which is one type of risk that can have a significant impact on companies. Interest rate risk is a subject that has been important for banks and institutions for decades [2]. One of the more common interest rate risks is the so-called Interest Rate Risk In The Banking Book (IRRBB).

“IRRBB refers to the current or prospective risk to the bank’s capital and earn-ings arising from adverse movements in interest rates that affect the bank’s banking book positions. When interest rates change, the present value and timing of future cash flows change. This, in turn, changes the underlying value of a bank’s assets, liabilities, and off-balance sheet items and hence its economic value. Changes in

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CHAPTER 1. INTRODUCTION 2

interest rates also affect a bank’s earnings by altering interest-rate-sensitive in-come and expenses, affecting its net interest inin-come (NII). Excessive IRRBB can pose a significant threat to a bank’s current capital base and/or future earnings if not managed appropriately” [3].

The quote above is the Basel Committee on Banking Supervision (BCBS) descrip-tion of IRRBB. The importance of IRRBB comes from the possibility of wrongly estimate an institution’s exposure to this risk. Inaccurate estimates of IRRBB can lead banks to reserve an amount of internal capital that does not reflect the bank’s actual riskiness. Both underestimation and overestimation potentially have negative consequences. Underestimation might jeopardize banks’ stability, whereas overes-timation might charge banks higher opportunity costs and, eventually, reduce their credit supply to the economy [2].

The banking sector plays a vital role in society, and therefore the role of regu-lators becomes essential in this sector. The purpose, briefly explained, with the regulations is to promote stability and efficiency in the financial system as well as to ensure effective consumer protection [4]. Different institutions are controlled by different supervisory authorities and are also acting under various regulations. The Swedish financial supervisory authority, which decides the rules that the Swedish banking sector needs to act under, is called Finansinspektionen (FI) [5]. FI recently decided that the Swedish banking sector is required to follow the European Banking Authority’s (EBA) new guidelines on the management of interest rate risk arising from non-trading book activities, i.e., the IRRBB. EBA’s guidelines on IRRBB are based on the Basel Committee on Banking Supervision updated standards on IR-RBB, which was published in 2016.

“Institutions should manage and mitigate risks arising from their IRRBB exposures

that affect both their earnings and economic value “ [1].

Risks arising from adverse interest rate movements can be analyzed in several ways, where Economic Value (EV) and Net Interest Income (NII) are commonly used ap-proaches. However, historically there has been no clear opinion about which per-spective an institution should mainly focus on when determining its risk exposure and hence its capital requirement. Both of the measures have advantages and dis-advantages, and neither has yet to prevail as the standard [6]. Swedish institutions have typically focused on EV, since FI’s old regulations mainly took this measure into account.

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banking sector to manage their interest rate risks that arise from both the changes in economic value and earnings [1]. More precisely, the EBA framework requires financial institutions to have at least one NII measure, and one EVE measures that combined should calculate the three risks Gap, Basis, and Option. By which meth-ods these measures should be calculated is not defined by the EBA. This creates great freedom for a risk manager to choose its own methods. However, do this cre-ate uncertainties in the reported calculations from the financial institutions? Which methods are, in fact, best adopted to calculate these risk measures under a low-interest environment? Furthermore, EBA has a few suggestions on how to calculate some features in an IRRBB model. One of these is how to stress the current yield curve, where the suggested method is highly standardized. How do these standard-ized stresses affect the IRRBB calculations? Is there any other method during the current circumstances that is better to use. These questions are some that this paper set out to answer.

1.2 Previous Research

Many previous studies have dealt with the topic of IRRBB. Several different ways of generating interest rate scenarios and different methods of calculating the changes in earnings and economic value have been proposed.

For instance, Fiori and Iannotti [7] evaluated risk exposure through a principal component Value-at-Risk method based on Monte Carlo simulations. They cre-ated different interest rates scenarios by doing a principal component analysis on historical yield curves. By shocking the principal components randomly and using a Monte Carlo simulation, they got theoretical yield curves scenarios. Kreinin et al. [8] used a similar approach, and they demonstrate that dimensionality reduction can result in substantial computational savings reduction. Combining PCA and Monte Carlo simulation is a useful way to revalue a portfolio fully under many different scenarios [9].

Abdymomunov and Gerlach [10] looked into six different methods based on in-dustry practices, academic studies, and banking supervisory requirements. One of these methods was based upon a PCA approach. Both the studies of Fiori and Ian-notti [7] and Abdymomunov and Gerlach [10] came to the conclusion that stochas-tic approaches can be used in favor of the determinisstochas-tic scenarios.

Cerrone et al. [2] created their own model based upon a Cholesky factorization of historical yields covariance matrix. They argue in their paper that the model is superior to other models due to that it is more consistent with actual riskiness

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than, for example, the standardized shocks presented in BCBS [11]. Furthermore, they argue that stochastic simulation techniques overcome the limits of the current deterministic scenarios since they allow estimating a bank’s equity sensitivity in a wider set of adverse scenarios and capturing interest rates dynamics over time [2]. Fiori and Iannotti [7], Cerrone et al. [2] and Abdymomunov and Gerlach [10] eval-uated their models towards a Value-at-Risk approach where they only measured the change of economic value of equity, i.e. EV EGap, which is not a sufficient model according to EBA [1]. However, at the time of their research, the new EBA framework was not yet presented. Therefore they compare their model to earlier frameworks such as BCBS [11] and BCBS [3]. These frameworks were less strict where, for example, in BCBS [11], the only required deterministic stresses were the parallel up and parallel down. In comparison, EBA [1] has four additional stresses in its framework, see Section 4.3.2. Furthermore, the old frameworks had more differing features that make these studies a bit out of date. All three of the earlier mentioned paper restricted the yield curves according to BCBS [11] and, BCBS [3], where no negative interest rates nor negative forward rates were allowed. Neg-ative interest rates are not only allowed in the new EBA framework, but it is also encouraged to exist. Furthermore, these articles are mainly focused on how interest rate scenarios could be generated, not on how the interest rate generators affect a complete IRRBB model.

As mentioned earlier, there exist many studies that investigate how to measure IR-RBB. For instance, Barbi [12] takes a closer look at the skewness behavior of the basis risk and how to hedge this risk in an appropriate way. Both Memmel [13] and Hibiki and Iwakuma [14] papers evaluate how the behavior of gap risk earnings and Economic value of equity measures relates to each other. Both evaluate these risk measures overtime, where Hibiki and Iwakuma [14] simulate data with the help of a Nelson-Siegel-Svensson, see Equation 3.1, and an AR(1) approach. While Memmel [13] looks at actual yield data over the period January 1980 to Decem-ber 2010. Memmel [13] concludes that changes in banks’ economic value and net interest income are highly correlated. Moreover, banks’ portfolio composition has a huge impact on the ratio of changes in net interest income relative to changes in Economic value [13]. However, to the best of our knowledge, there exist no articles that suggest a complete IRRBB model that captures all parts of the risk agreeing to EBA [1].

To conclude, as far as we know, this is the first paper that proposes a complete IRRBB model. In other words, a model that follows the EBA-framework and cap-tures gap, basis, and behavioral risk. Furthermore, this paper analyzes how

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differ-CHAPTER 1. INTRODUCTION 5

ent interest rate scenarios affect the EVE and NII for a given banking book. It also provides an insight into how modeling of IRRBB can look like and how different assumptions can be made.

1.3 Purpose

The new framework is creating new challenges for risk managers who need to adapt to these new guidelines. Especially under the historically low interest rate environ-ment that the financial world finds themself in. This paper aims to test this new framework by creating a method that follows the EBA standards to calculate a fi-nancial institution’s IRRBB. The model will be stressed with different techniques to generate yield curve scenarios. This in order to test how an EBA model behaves in a low interest environment and to investigate which method is better suited to stress this model under the current climate. Furthermore, a discussion will be included on which mathematical choices a risk manager has when creating their IRRBB model and which advantages some calculations have compared to others. This will show the significant variation each institutions’ IRRBB model will have in both resulting risk and behavior. This paper will contribute to the current research in the field of IRRBB by creating a better understanding of the effects of a low interest environ-ment on the calculations of IRRBB.

In order to perform this research, some limitations of the scope were needed. This paper will only focus on the actual calculations in an IRRBB model. This article will leave the more managerial questions unaddressed, for example, who should be in charge of the IRRBB management and how the information system connected to IRRBB should work.

1.4 Research Question

To fulfill the purpose of the article, the main question that this paper set out to answer is formulated as:

• How to manage IRRBB according to the new EBA-regulations in a low-interest environment?

In order to answer this research question, an IRRBB model that fulfills the new EBA standards will be created. This model, combined with the following sub-research questions, will be used to enrich the answer to the main question:

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• How do the individual choices of a risk manager affect the calculated risk? • How to stress test this model?

• What are the pros and cons of the different stress test?

• What effects do a low-interest environment have on the risk model? • What weaknesses do the new EBA-regulations have?

1.5 Outline of the Thesis

The layout of the report is formatted in a way to first give the reader an introduction and the necessary information about the topic. Then comes a more mathematical and theoretical base that has laid the foundation for the report’s calculation. Fi-nally, the report’s results are presented with associated analysis, discussion, and conclusions. In particular, chapter 2 will address the background on IRRBB, ex-plaining the three sub-categories which constitute the IRRBB, and the standard measures that are used to determine the risk. Furthermore, a summary of the signif-icant changes between previous frameworks and the current framework will be pre-sented. In Chapter 3, the mathematical framework is provided. It will address the essential mathematical methods and concepts considered in this paper, like Prin-cipal Component Analysis, Cholesky decomposition, and Bachelier option pricing model. Chapter 4 explains how the mathematical framework was used to construct the models that, in turn, generated the result. The result, together with analysis and discussion, are presented in chapter 5. Finally, conclusions and suggestions for further research are presented in Chapter 6.

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Chapter 2 Theoretical Background

This chapter will provide a more thorough background description of the topic as a whole as well as a description of concepts that are important for understanding the necessary calculations.

2.1 IRRBB

Interest rate risk in the banking book may, for practitioners, be a self-explanatory phrase. It refers to the risk which arises from adverse movements in interest rates that affect the banking book positions [15]. Where banking book positions refer to all interest-rate sensitive instruments from non-trading activities, i.e., assets, liabili-ties, and off-balance-sheet items in the non-trading book, excluding assets deducted from CET1 capital [1]. To understand the difference between the banking book and the trading book, one must understand the difference in the assets held by these ac-counts. The clear distinction between these accounting books is that the banking book instruments are, in general, intended to be held to maturity. In contrast, the assets in the trading book are, in general, held on a shorter horizon. For example, trading book assets might be bought or sold to facilitate trading activities for clients or to profit from spreads between the bid and ask prices.

Interest rate changes affect both a financial institution’s economic value and its earning capability. It is these risks that constitute interest rate risk in the bank book. An institution’s IRRBB may sound simple to determine, but a banking book exists of multiple positions in a wide range of asset classes, which makes the calcu-lations more complicated. Therefore, to investigate this risk, one must understand the drivers of it. There are different names and definitions of these drivers, but since this paper aims to propose a model that fulfills the EBA-standards, its definitions

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CHAPTER 2. THEORETICAL BACKGROUND 8

will mainly be used in this paper. EBA [1] divides the drivers into the three major groups Gap risk, Basis risk, and Option risk. To avoid confusion between option risk and the automated option risk, we will, in this paper, refer to option risk as be-havioral risk, which is a commonly adopted name of this risk. These three drivers depend directly on the level and shape of the yield curve, which forces financial institutions to do significant stress tests on the yield curve. The stress testing is es-sential to get information about how different economic environments would affect the IRRBB for an institution. Therefore, the effectiveness of a stress test depends directly on the yield curve scenarios that have been used in the stress test [10].

2.1.1 Gap Risk

The first major driver of risk defined by EBA [1] is called gap risk and is defined as : "Risk resulting from the term structure of interest rate sensitive instruments that

arises from differences in the timing of their rate changes, covering changes to the term structure of interest rates occurring consistently across the yield curve (par-allel risk) or differentially by period (non-par(par-allel risk)." [1]

This driver of risk is essentially one measure of two different risks that are closely related to each other, namely yield curve risk and repricing risk [11].

Repricing risk is the primary and most discussed form of interest rate risk, and it arises mainly from timing differences in the maturity and repricing of financial institutions assets, liabilities, and OBS position. While such repricing mismatches are fundamental to the business of banking, they can expose a bank’s income and underlying economic value to unanticipated fluctuations as interest rates vary. For instance, a bank that funded a long-term fixed-rate loan with a short-term deposit could face a decline in both the future income arising from the position and its underlying value if interest rates increase. These declines occur because the cash flows from the loan are fixed over its lifetime, while the interest paid on the funding is varying [11].

Repricing mismatches can also expose a bank to risks in case of changes in the slope and shape of the yield curve (Yield Curve Risk). This risk occurs when unanticipated shifts of the yield curve have adverse effects on a financial institu-tion’s income or underlying economic value. Consider a portfolio with only two instruments consisting of a long position in 10-year government bonds and a short position in 5-year government notes. The economic value of this portfolio could decline sharply if the yield curve steepens (the longer-horizon yield drastically

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in-CHAPTER 2. THEORETICAL BACKGROUND 9

creases while medium-horizon yield stays the same). The reason for this is that the value of the long position would, in this scenario, drastically decrease while the short position would essentially remain the same.

2.1.2 Basis Risk

The second major driver of IRRBB is commonly called basis risk. This risk refers to the impact of an imperfect correlation between indices underlying asset and li-ability rates, with otherwise similar repricing frequency. For example, 6-month PLR-linked loans can be funded by 6-month LIBOR-linked deposits, but there is no guarantee that a 1% change in LIBOR will imply a 1% change in PLR. If the spread between index rates suddenly changes, it might lead to a decrease in earn-ings and net worth [16]. In other words, basis risk captures the possible loss from the imperfect correlation between different interest rates, which assets and liabili-ties are priced upon [12]. It is important to capture this risk since it is a common practice to ignore which yield curve instruments are based upon when doing gap risk calculations. Thus, one can see the basis calculations as a complement to the gap risk calculations.

2.1.3 Behavioral Risk

The third major driver of IRRBB is called behavioral risk, which EBA call option risk. There is no consensus about a definition of behavioral risk [17]. However, in this paper, behavioral risk will be parted into two different categories in accor-dance with EBA [1]. The first part of the risk arises from options (embedded and explicit), where the institution or its customer can alter the level and timing of their cash flows. Hence, this risk arises from interest-rate sensitive instruments where the holder will almost certainly exercise the option if it is in their financial interest to do so (embedded or explicit automatic options) [1]. This part of the behavioral risk will, in this paper, be referred to as option risk, which is not to be confused with EBA’s definition. They use this term when they are talking about what this paper refers to as behavioral risk, i.e., both risk that occurs from financial options and customer behavior.

The second part is commonly called prepayment risk, withdrawal risk, refinanc-ing risk, or optionality risk. This risk occurs when bankrefinanc-ing books contain numer-ous implicit options such as prepayment options on mortgages, borrowing options, early withdrawal options, interest rate options, etc. As these options may be exer-cised in response to market interest rate changes, they induce (non-linear) interest rate risk [18]. In other words, this risk occurs because different interest rate swings affect customer behavior. For example, people tend to pay back more on their fixed

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interest rate loans during periods of falling interest rates and less during periods of rising interest rates [3]. One of the main problems of estimating this risk is that one needs to predict the behavior of the customers, which are extremely hard in some businesses, like in retail products [18]. Consider the case when a customer has a fixed-rate mortgage loan at 5%. This customer has strong incentives to refinance his loan when the interest rate drops, and there is, for example, a similar fixed-rate loan at 3%. This phenomenon creates uncertainties in future cash flow for institu-tions, which may affect the NII or/and EVE significantly. In this paper, this part of the behavioral risk will be referred to as refinancing risk.

2.2 The Legal Environment of IRRBB

The Basel Committee on Banking Supervision is the primary global standard-setter for the prudential regulation of banks and provides a forum for regular cooperation on banking supervisory matters. Its 45 members comprise central banks and bank supervisors from 28 jurisdictions [4]. The committee publishes standards for bank-ing supervision, but each country has its authority who sets the regulations. It is up to each authority to take action to implement these regulations through their na-tional systems. In Sweden, it is the Swedish Financial Supervisory Authority (FI) who is the government agency responsible for financial regulation. FI is, in turn, governed by EU laws. However, it is of great importance to keep these regulations updated so that they hold for the current environment.

In 2016, BCBS [3] issued new standards for the interest rate risk in the banking book. These new standards are a revised version of the committees "Principles for the Management and Supervision of Interest Rate Risk" from 2004. Much has hap-pened since 2004. Several countries are experiencing a low-interest rate climate, which they have not experienced before. However, BCBS’s revised version con-tains a lot of changes and updates. This section will mainly focus on the changes that have an impact on the modeling and calculation of the IRRBB, and less on the management/governance changes.

To understand the new standards and how it affects financial institutions, it is a good start to get a view of how the standards from 2004 looked like. First of all, the old standards contained principles on how to manage the interest rate risk (IRR), i.e., the interest rate risk for both the banking book and the trading book. The new standards are only focusing on the interest rate risk in the banking book. Moreover, the previous regulations contained only two stress scenarios, one positive and one negative parallel shift of the yield curve. It was considered important to add several possible scenarios to capture potential risks. Hence, the new framework contains

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six stress scenarios. However, BCBS decided to update the standards and came up with two options for the regulatory treatments of IRRBB. Pillar 1, which is a stan-dardized approach that calculates the Minimum Capital Requirements, and pilar 2, a less standardized approach which instead contains principals to guideline banks to measuring risk and assessing capital adequacy internally. BCBS concluded that Pilar 2 was more appropriate to capture the heterogeneous nature of IRRBB. This was also the proposal the industry preferred since the feasibility of Pilar 1 was questionable. This since all banks have different products, plans, strategy, etc. and therefore, there is no model which fit all banks.

The European Banking Authority (EBA) is one of the EU’s three joint financial reg-ulators. On 19 July 2018, EBA published the new guidelines on the Management of IRRBB. This update was the first step towards the implementation of BSBC’s new standards [1]. In Sweden, FI decided to follow these standards, and this has had a quite significant impact on Swedish financial institutions since the guidelines from EBA differ a bit from the old regulations provided by FI. The old standards from FI required institutions to calculate the changes in EVE under different stress tests, which mostly capture the gap risk [5]. Furthermore, institutions were supposed to fill out a questionnaire about their exposure to basis and options risk, i.e., they did not have to do any calculations about it.

The updated standards require institutions to use at least one economic value mea-sure and one earnings-based meamea-sure. Such requirements did not exist before. Usually, only one of those measures was used. Nevertheless, it is still not com-pletely clear how these calculations should be applied. Institutions still have a lot of freedom in how the IRRBB should be measured, as long as they stay inside the boundaries of the framework.

2.3 Measures of IRRBB

"The Committee acknowledges the importance of managing IRRBB through both economic value and earnings-based measures. If a bank solely minimises its eco-nomic value risk by matching the repricing of its assets with liabilities beyond the short term, it could run the risk of earnings volatility." [3]

There is no single method for measuring IRRBB. Still, it is common to divide the used methods into two different types of measures, namely earning-based measures and economic-value measures. Interest rate changes can affect both a bank’s earn-ings and economic value, therefore this categorization. The two categories differ from each other and provide different perspectives on IRRBB.

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Earnings measures look at changes in expected future profitability within a given time horizon resulting from interest rate movements. Changes in the interest rate will affect the size of cash flows, and it is the sensibility against these changes that earnings measures look at. The time horizon for earnings-based measures is usually the nearest 1-5 years, i.e., it captures the short term risks [1]. However, the most common earnings-based measure is ∆N II, which basically compares cash flows under different interest rate scenarios. How sensitive institutions are to ∆N II depends on their specific assets and liabilities, where fixed or floating rates and repricing dates have a significant impact.

In contrast to earnings measures, economic value measures look at the net present value of the interest rate sensitive instruments over their remaining lifetime, i.e., un-til all positions have expired. One of the most common economic value measures is the Economic Value of Equity. It is calculated by taking the present value of all expected asset cash flows and subtract it with the present value of the expected cash flows on liabilities, where equity is excluded from the cash flows. To calculate risk with this measure, it is a common practice to compare the present scenario to all other created scenarios and look at the change in EVE, i.e., ∆EV E

"For measuring and monitoring of IRRBB, institutions should use at least one earnings-based measure and at least one economic value measurement method that, in combination, capture all components of IRRBB." [1]

It is important for institutions to use both types of measures since small changes in earnings measures dose not imply small changes in the economic value. Banks can reduce their risk measured by earnings measures at the expense of EV. There-fore, according to the new guidelines from EBA, institutions need to capture all of the components of IRRBB (gap, basis, and behavioral risk) by using at least one earnings-based measure and at least one economic value measurement method.

"In addition to the existing and prospective exposure to IRRBB, when implementing the guidelines, institutions should also consider their general level of sophistication and internal approaches to risk management to make sure that their approaches, processes and systems for the management of IRRBB are coherent with their gen-eral approach to risk management and their specific approaches, processes and systems implemented for the purpose of the management of other risks." [1]

EBA [1] has made a sophistication matrix containing four different categories. The different categories reflect different sizes, structures, the nature scope, and

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complexity of activities of institutions. Category 1 is the one with the most so-phisticated institutions, and category 4 is the one for less soso-phisticated institutions. The guidelines for measuring IRRBB differ a bit between the categories. Supervi-sors have different expectations on metric and modeling depending on the level of sophistication an institution is placed in. This paper will mainly be based on the guidelines for institutions in categories 3 and 4.

2.4 Balance Sheet

When calculating the IRRBB, an institution’s current balance sheet is used. But since the calculations are done over a long time horizon, one must make assump-tions about how the balance sheet will change over time. Basically, it is three dif-ferent ways to do this; assume a run-off balance sheet, a dynamic balance sheet, or a constant balance sheet. A run-off balance sheet is a balance sheet where existing positions are not replaced when matured. A dynamic balance sheet is a balance sheet where future business expectations, adjusted for the relevant scenario, are in-cluded. A constant balance sheet is a balance sheet where maturing or repricing cash flows are replaced with new cash flows that have identical features with regard to the amount, repricing period, and spread components. Different calculations as-sume different types of balance sheets. Hence, in Chapter 4, the asas-sumed portfolio type will be mentioned before each associated risk calculation is defined.

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Chapter 3 Mathematical Framework

This chapter presents the most important and more complex mathematical concepts used in this article. Here, the theories are presented in a comprehensive way, and the next chapter describes how these theories have been applied in this study.

3.1 Nelson-Siegel-Svensson Parameter

Estima-tion

Central banks and other market participants widely use the Nelson-Siegel-Svensson (NSS) equation as a model for the term structure of interest rates [19]. The NSS equation models the yield curve at a point in time as follows [20]:

ˆ Y (t) = β1+ β2 1 − e−tλ1 t/λ1 + β3 1 − e−tλ1 t/λ1 − e −t λ1 +β4 1 − e−tλ2 t/λ2 − e−tλ2 (3.1)

WhereY (t) denotes the zero rate for maturity t. To calibrate the model we needˆ to estimate six parameters: β1,β2,β3, β4and λ1, λ2. For N observed yields with different maturities t1, . . . , tN , we have N equations. These parameters can be estimated by minimizing the difference between the model rates ˆY , and observed rates YM where the superscript stands for ‘market’ [20]. An optimization prob-lem can be stated and solved by a least squared method minimizing the quadratic difference between the approximated yields and market yields by calibrating the β and λ values in accordance to:

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CHAPTER 3. MATHEMATICAL FRAMEWORK 15

min

λ,β(

X

( ˆY − YM)2) (3.2) To avoid potential challenges in numerical optimization, Diebold and Li [21] in-troduces a method of fixing the λ1and λ2 parameters and then estimate the rest of the parameters using a least squares algorithm. Based upon the earlier research of Diebold and Li [21], Muvingim and Kwinjo [22] tested which values of λ1and λ2that would generate the best-fitted yield curve. Their research showed that a λ1

value of 0.714 and λ2value of 10 gives the most appropriate curves. With these values set, the optimization problem can be solved with the ordinary least square estimation:

YM_t = Xt· ˆβ + t (3.3) Where YMt expresses the interest rates at maturity date t, tis the random error term, Xtis the vector of prediction variables that explain the dependent variable yield. The vector Xtcan be estimated given maturity t and λ. Using the equation for Nelson-Siegel-Svensson model, see Equation 3.1, the following vector for Xt

can be obtained: Xt=               1 1 − e −t1 λ1 t1/λ1 1 − e −t1 λ1 t1/λ1 − e −t1 λ1 1 − e −t1 λ2 t1/λ2 − e −t1 λ2 1 1 − e −t2 λ1 t2/λ1 1 − e −t2 λ1 t2/λ1 − e −t2 λ1 1 − e −t2 λ2 t2/λ2 − e −t2 λ2 .. . ... ... ... 1 1 − e −tN λ1 tN/λ1 1 − e −tN λ1 tN/λ1 − e−tλ1 1 − e −tN λ2 tN/λ2 − e −tN λ2               (3.4)

Whereβ is a column vector consisting of the four βˆ iconstants:

ˆ β =     β1 β2 β3 β4     (3.5)

The optimal parameters ofβ can then be estimate by the equation:ˆ ˆ

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Finally the fitted yield valueY (t) for maturity t can be obtained by inserting theˆ estimated values of βiand λiinto Equation 3.1:

3.2 Principal Component Analysis

Principal component analysis (PCA) is a method that can be used to reduce the di-mension of a data set and still retain most of the information. This is done by a linear orthogonal transformation of a d-dimensional data set onto a k-dimensional linear space (where k < d), such that the variance of the projected data is maximized. The dimension of the transformed data are orthogonal, i.e., they are independent and are mutually uncorrelated [23]. This technique can briefly be described in the following steps;

i. Normalize and center the data to have mean zero (Let us call this data for X) ii. Determine the corresponding covariance (or correlation) matrix of the data X and extract eigenvectors and eigenvalues from the covariance matrix or correlation matrix. Where Singular Vector Decomposition also can be used.

iii. Construct a projection matrix W consisting of the k selected eigenvectors that correspond to the k largest eigenvalues.

iv. Use the projection matrix W to transformed the observed d-dimensional data to the new k-dimensional feature subspace. A deeper theoretical explanation of the steps is presented below.

Consider a data set with d features X1, X2, ..., Xd. Each of the dimensions found by PCA is a linear combination of the d features.The first principal component of a set of features X1, X2, ..., Xdis the normalized linear combination of the features that has the largest variance. The normalized linear combination of the features can be expressed as:

Z1= φ11X1+ φ21X2+ ... + φd1Xd (3.7)

Moreover, normalized can be expressed asPpj=1φ 2

j1= 1. The elements φ11, ..., φd1

can be seen as the direction of the first principal component, which, together, make up the principal component direction vector φ1= φ11, φ21, ..., φTd1[23].

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Now, consider the n · d data set X, where X has been normalized and centered to have mean zero. To calculate the first principal component, the linear combination of the sample feature values is considered:

zi1= φ11xi1+ φ21xi2+ ... + φp1xip (3.8)

for i = 1,...,n that has largest sample variance, subject to the constraintPpj=1φ 2 j1=

1. This means, that the first principal component vectors solves the following opti-mization problem max {φ11,...,φp1}    1 n n X i=1   p X j=1 (φj1xij)   2   subject to p X j=1 φ2_j1= 1 _(3.9)

We refer to Z1as the first principal component with realized values z11, ..., zn1. The optimization problem can be solved by different methods, e.g., by finding the eigenvectors and eigenvalues from the covariance matrix or correlation matrix of X. The obtained eigenvectors and eigenvalues can be seen as the core of a PCA. The eigenvectors are the principal components, which define the directions of the new feature space. The eigenvalues represents the magnitude of the principal com-ponents [23].

The second principal component is the linear combination of X1, X2, ..., Xdthat has maximal variance among all linear combinations that are uncorrelated with Z1

[23]. The second principal component scores z12, z22, ..., zn2take the form: zi2= φ12xi1+ φ22xi2+ ... + φd2xid (3.10)

Where φ2is the second principal component direction vector with elements φ12, φ22, ..., φd2. It turns out that constraining Z2to be uncorrelated with Z1is equivalent to

con-straining the direction φ2to be orthogonal to the direction φ1, and so on [23]. Principal components Z can be expressed as

Z = XΦ (3.11)

Where X is the normalized observed data with zero mean and Φ is the matrix with the eigenvectors (principal components) of the sample covariance matrix of X, i.e., the columns in the matrix φ are the principal component direction vectors

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φ1, φ2, ..., φk[23]. We refer to the matrix φ as the projection matrix.

The inverted PCA representation can be used to reproduce the correlation struc-ture of the original risk factors

X = ZW−1 _(3.12)

Where W−1is the inverse of the projection matrix W .

3.3 Cholesky Factorization

Every positive definite matrix ARn×ncan be factored as

A = LLT (3.13)

where L is a lower triangular matrix with real and positive diagonal entries, and LT denotes the conjugate transpose of L [24]. The elements of L can be determined for k = 1, 2...n and for i = 1, 2...k − 1 in accordance with Datta [25] as:

Lk,k= v u u tAk,k− k−1 X j=1 L2 k,j Lk,i= 1 Lii (Ak,i− i−1 X j=1 Li,jLk,j) (3.14)

The Cholesky decomposition is a commonly used mathematical tool in a Monte Carlo simulation for systems with multiple correlated variables. The covariance matrix is decomposed to give the lower-triangular L. Applying L to a vector of uncorrelated samples u produces a sample vector Lu with the covariance proper-ties of the system being modeled [24].

3.4 Forward Rate

The forward rate is a projection of future interest rates between the period t to T at time s, where the continuously compounded forward rate are given by:

F (s; t, T ) = −ln(p(s, T ) − ln(p(s, t))

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Where p(t, T ) is the price of a non-defaultable zero-coupon bond with maturity at time T. This is a financial contract that, with certainty, pays its owner 1 at time T and are given by the formula:

p(s, T ) = e−r(s,T )·(T −s) _(3.16) Where r(s, T ) is the interest rate between s and T at time s, if s = 0 we approxi-mate in this paper that the (continuously compounded) interest rate is equal to the yield.

3.5 Bachelier Pricing Model

In 1990, the French mathematician Louis Bachelier presented the normal model, which is now one of many methods that can be used to price interest rate options [26]. For simplicity, we will refer to the model as the Bachelier model. Unlike the Black-76 Model, which probably is the most widely used option pricing method, Bachelier’s model allows negative interest rates. However, it is known that caps and floors are the interest rate options that are most likely to occur in the banking book [3]. Caps and Floors are a series of caplets and floors. Hence, we start to calculate them. Bachelier’s pricing formula of caplets follows:

Caplet = F (s; t, T ) − K · N (d) + σ ·√T − t · n(d) _(3.17)

Where k is the strike rate, T is the option maturity date, N is the cumulative normal distribution function, n is the normal density function, and d is given by:

d =F (s; t, T ) − K

σ ·√T (3.18)

As mention before, caplets are used to price caps, which are instruments containing a series of caplets. Hence, the pricing formula for a cap is given by:

Cap =

N

X

i

capleti· τti−1,ti· DF (ti) (3.19) Each caplet is weighted by the year-fraction τti−1,tiand the discount factor DF (Ti).

Bachelier’s pricing formula of floorlets follows:

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Just like a cap, a floor can be priced as a discounted sum of floorlets:

F loor =

N

X

i

F loorleti· τti−1,ti· DF (ti) (3.21)

It can be seen that the valuation of caps and floors are quite similar. A cap contains a series of caplets. Each caplet is a potential cash flow and can be seen as a call option on a floating rate index level. Whereas, a floor contains a series of floorlets. A floorlet can be seen as a put option on a floating rate index level [26].

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Chapter 4 Model

In this chapter, we introduce notations and calculation that has been used to build the models in this study. Several methods have been used in order to determine different risks in different scenarios. The primary model is a model that follows EBA’s guidelines on how to measure IRRBB. We have referred to this model as the IRRBB model. In addition to the IRRBB model, three different models that generate interest rates scenarios have been built. We will, in this paper, refer to these models as the EBA model, Cholesky model, and PCA model.

4.1 Mapping of the Cash Flow Data

One essential part of the IRRBB calculation is cash flows. The timing, size, and of course, the type of cash flows will have significant effects on the IRRBB. Which cash flows should be used, and how should they be modeled? This section will answer how these questions were handled in this paper.

All future notional repricing cash flows arising from interest rate sensitive assets, li-abilities, and off-balance sheet items were aggregated and mapped onto 19 different time buckets (indexed numerically by k). A notional repricing cash flow includes any repayment of principal, any repricing of principal and any interest payment on a tranche of principal that has not yet been repaid or repriced [3]. Cash Flows under interest rate shock scenario i and currency c is denoted as CFi,c(k). Each bucket

is represented by a time period, and cash flows were mapped into the different time buckets according to their repricing date. All cash flows in each time bucket were assumed to take place at the midpoint of the buckets time interval. The 19 time buckets, with corresponding time interval and midpoint, can be seen in Table 4.1. The time bucket approach is commonly used and suggested by EBA [1]. An

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CHAPTER 4. MODEL 22

native approach is to measure each cash flow to its exact time, which would give a more precise result but is hard in practice.

Table 4.1: Describes the different time buckets and the time period each bucket represent.

Time bucket intervals (M: months; Y: years) Short-term rates Overnight (0.0028Y) O/N< tC F 1M (0.0417Y) 1M< tCF 3M (0.1667Y) 3M< tC F 6M (0.375Y) 6M< tC F 9M (0.625Y) 9M< tC F 1Y (0.875Y) 1Y< tC F 1.5Y (1.25Y) 1.5Y< tC F 2Y (1.75Y) Medium-term rates 2Y< tC F 3Y (2.5Y) 3Y< tC F 4Y (3.5Y) 4Y < tC F 5Y (4.5Y) 5Y< tC F 6Y (5.5Y) 6Y< tC F 7Y (6.5Y) Long term rates 7Y< tC F 8Y (7.5Y) 8Y < tC F 9Y (8.5Y) 9Y < tC F 10Y(9.5Y) 10Y<tC F 15Y(12.5Y) 15Y< tC F 20Y(17.5Y) t C F > 20Y(25Y)

The type of a particular position decides how its cash flow should be slotted. For fixed-rate positions, which generate cash flows that are certain till the point of contractual maturity, should all coupon cash flows, and periodic or final princi-pal be mapped into the time bucket closest to the contractual maturity. Examples of such positions are fixed-rate loans without embedded prepayment options, term deposits without redemption risk, and other amortizing products such as mortgage loans.

Floating rate instruments are managed in a slightly different way since they gener-ate cash flows that are not predictable past the next repricing dgener-ate other than that the present value would be reset to par. These instruments are assumed to reprice fully at the first reset date. Hence, the entire principal amount is slotted into the bucket in which that date falls, with no additional slotting of notional repricing cash flows to latter time buckets.

4.2 Yield Curve Construction

Before any consideration of IRRBB could be made, an interest rate curve, more commonly known as yield curve, needed to be built. A yield curve is a plot of yields of bonds having equal credit quality but different maturity dates. This will be the foundation of the entire model, especially for the gap risk calculations, since these calculations assume that for each currency, all assets and liabilities are priced upon the same yield curve.

In order to generate a yield curve, bond prices or equivalent instruments need to be extracted from the market. In this paper, the yield curves for each currency are based upon the market price of interest rate swaps, which is recommended by

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CHAPTER 4. MODEL 23

BCBS [27]. However, one is not done there. Remember that the EBA framework suggests an IRRBB model that is based upon a time bucket approach. Since the maturity dates of the market’s bonds do not precisely match the time bucket’s mid-points, a bootstrapping method was needed to generate an approximation of the yield for each bucket.

In the model proposed in this paper, the Nielson-Siegel-Svensson (NSS) method was used. NSS is a commonly used method to generate yield curves, and it can, for instance, also be found in Hibiki and Iwakuma [14]. Remember from Section 3.1 that the NSS equation models the yield curve at a point in time as follows [20]:

ˆ Y (t) = β1+ β2 1 − e−tλ1 t/λ1 + β3 1 − e−tλ1 t/λ1 − e −t λ1 +β4 1 − e−tλ2 t/λ2 − e −t λ2 (4.1)

WhereY (t) denotes the estimated zero rate for maturity t. The parameters βˆ 1,β2,β3, β4 and λ1, λ2were calibrated with a least square method see Section 3.1. Once these parameters have been estimated, one can use this equation to generate yields for any maturity. In this study, yields for each specific scenario were created for each time bucket, see Table 4.1, based upon each simulated market data point. Re-member that in this paper, yields with a specific maturity approximate the interest rate for instruments with the same maturity.

4.3 Interest Rate Models

Hypothetical interest rate scenarios can be generated with many different methods. In this paper, the different methods that are presented can be divided into two differ-ent main approaches, namely deterministic scenarios and stochastic scenarios. In the deterministic approach, key scenarios are hypothetically assumed and created by predefined stresses. In the stochastic approach, a large number of hypothetical scenarios are randomly generated [10]. In this section, the deterministic method that EBA [1] proposes will be introduced first. This paper has also implemented two different stochastic approaches, which will also be presented in this section.

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CHAPTER 4. MODEL 24

4.3.1 Interest Rate Restrictions

Before presenting the different models for generating the yield curve, it is of utmost importance to be aware of the limitations EBA has on acceptable yield curves. Earlier frameworks, such as BCBS [11], suggested that no negative interest rates should be allowed. Since then, negative interest rates have become more prevalent in many countries. Hence, the EBA has given a lower allowed limit which can be described by the function:

YF loor(t) =      −1%¸ _{for t ∈ [0, 1]} −1%¸ + 0.05%¸ · (t − 1) for t ∈ (1, 20] 0%¸ for t ∈ [21, 25] (4.2)

In the yield curve generation models presented in this paper, this lower limit was handle in two ways. For the deterministic EBA method, yields that fall below the interest rate boundary were replaced with the value of the floor. For example, if the 1-year yield in one scenario was equal to -2 %, this value was replaced and set to −1%. For the stochastic models yield curves that did not fulfill the requirement, set by EBA [1], these curves were rejected and replaced by a new unrelated stochastic simulation.

4.3.2 EBA Interest Rate Model

To evaluate how interest rate sensitive a financial institution is, EBA [1] suggests that predefined stress tests should stress the yield curve for each currency c. The six suggested interest rate shocks suggested by the EBA framework are:

(i) Parallel shock up (ii) Parallel shock down (iii) Short rates shock up (iv) Short rates shock down

(v) Steepener shock (short rates down and long rates up) (vi) Flattener shock (short rates up and long rates down)

In order to provide these stressed yield curves, EBA suggests a simple model where a constantR¯x,care generated from a predefined method see EBA [1] for calcula-tions, where x is the type of stress, and c is for which currency the stress is applied on, see Table 4.2 for currency depended constants.

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CHAPTER 4. MODEL 25

Table 4.2: Describes the individual shock parameters for some currencies. Currency SEK USD GBP DKK EUR JPY

¯ Rparallel,c 200 200 250 200 200 100 ¯ Rshort,c 300 300 300 250 250 100 ¯ Rlong,c 150 150 150 150 100 100

Parallel shock for currency c:

A constant parallel shock up or down across all time buckets a certain amount of basis points based upon in which currency is to be stressed [1].

∆Rparallel,c(tk) = ± ¯Rparallel,c (4.3)

Short rate shock for currency c:

Shock up or down that is greatest at the shortest tenor and diminishes toward zero at the tenor of the longest point on the term structure [1].

∆Rshort,c(tk) = ± ¯Rshort,c· e −tk

4 _(4.4)

Rotational steepener shock for currency c:

Involving rotations to the term structure of the interest rates, whereby both the long (up) and short (down) rates are shocked and the shift in interest rates at each tenor midpoint is obtained by applying the following formula [1]:

∆Rsteepener,c(tk) = −0.65 · | ¯Rshort,c· e −tk

4 _{| + 0.8 · | ¯}_R_long,c_{· 1 − e} −tk

4 | (4.5)

Rotation flattener shock for currency c:

Involving rotations to the term structure of the interest rates, whereby both the long (down) and short (up) rates are shocked and the shift in interest rates at each tenor midpoint is obtained by applying the following formula [1]:

∆Rf lattener,c(tk) = 0.8 · | ¯Rshort,c· e −tk

4 _{| − 0.6 · | ¯}_R_long,c_{· 1 − e} −tk

4 | (4.6)

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CHAPTER 4. MODEL 26

yi,c(tk) = y0,c(tk) + Ri,c(tk) (4.7)

4.3.3 PCA Interest Rate Model

One common yield scenario-generating method is the Principal Component Analy-sis (PCA) [10], which is described in Section 3.2. Several studies show that the first three principal components factors can describe most variations in the yield curve. These factors represent the shift, tilt and twist of the yield curve [10] [7]. This pa-per is also using the first three principal components. Furthermore, the model can be described in the following way;

i) Semiannual historical data of interest rate changes were collected. The PCA was implied on the data set, and the first three principal components were extracted. ii) The PCA based Monte Carlo simulation was performed by drawing indepen-dent random shocks from each principal component distribution. We know from Section 3.2 that the yield curve can be expressed by inverting the PCA representa-tion, i.e.:

X = Z · W−1 (4.8)

Here, the coefficients W−1give the sensitivity of interest rate changes along the yield curve with respect to each Z.

iii) By shocking the right hand side of the equation, new theoretical yield curve changes was obtained. The shock vector U that represents the simulated change in each scenario can be expressed as:

U = W−1√Λη = η1 p λ1W1+ η2 p λ2W2+ ... + ηk p λkWk (4.9)

Where Λ is the diagonal matrix of the corresponding eigenvalues, and η = η1, η2, ..., ηk

are the vectors of independent shocks. Usually, scenarios based on PCs are sim-ulated through vectors of standard normal shocks ηj ∼ N (0, Ik), which also has

been used in this case [7].

iv) Finally, the simulated change for each scenario was applied upon the current yield curve to generate the i:th scenario yi, i.e.:

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CHAPTER 4. MODEL 27

This procedure was repeated K times until 1000 generated scenarios that fulfills the EBA standards of a yield curve were created.

4.3.4 Cholesky Interest Rate Model

Cerrone et al. [2] used a Monte-Carlo approach in their research in order to gener-ate simulgener-ated scenarios. They argue that their model was superior to other methods suggested by other papers. However, in their simulation, they used some constrains on the yield curve simulation that is no longer up to date. For instance, they did not allow negative yields nor negative forward rates. In newer standards like EBA [1], these constraints should not be applied. Furthermore, Cerrone et al. [2] only tested their generated scenarios on an EVE gap risk model. To check if their conclusions are still in effect, we tested their method on a more up-to-date IRRBB model with current standards. The model can be explained in seven numbers of steps: i) Estimate the means µi of the semiannual changes in the key rates and their variance-covariance matrix Ω.

ii) Generate a random number ui(i = 1, . . . 14) ranging from zero to one at each

node of our key rates term structure.

iii) Convert each uiinto a value zi(i = 1, . . . 14) distributed according to a

stan-dard normal. In symbols:

zi= F−1(ui) (4.11)

Where F−1is the inverse of the distribution function of a standard normal. iv) Use the algorithm of Cholesky 3.3 in order to decompose the matrix Ω in two matrices L and LT such that:

Ω = L · LT (4.12)

Remember from section 3.3 that L is a lower-triangular matrix containing the variance-covariance matrix relationship between the key rates.

v) Calculate the vector x, whose elements are the simulated semiannual changes in the key rates, through the following formula:

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CHAPTER 4. MODEL 28

x = L · z + µ (4.13)

Where z is the vector of the values calculated in step iii) and µ is the vector of the 14 means of the distributions of the key rates annual changes calculated in step i). x represents a simulated change of the yield curve.

vi) The simulate change vector x is then applied to the current yield curve to get the simulated scenario i

yi= y0+ x (4.14)

vii) The steps ii)-vi) were then repeated until we reached a number of 1000 scenar-ios [2].

4.4 Refinancing Risk

After both stressed and non-stressed yield curves had been generated, the second part of behavioral risk was modeled, namely refinancing risk. This since the gap risk is highly dependent on each CFi,c(k) size, which will differ for each yield and

currency for the instruments that are behavioral sensitive.

Several factors are important determinants of financial institutions estimations of how different interest rate scenarios will affect the average prepayment and with-drawal speed. A financial institution must assess the expected average prepayment and withdrawal speed under each scenario [27]. It is also essential for a risk man-ager to have a deep understanding of the balance sheet in order to identify which products are sensitive to refinancing risk. This since it is far from every product that the customer has the embedded option to prepay or withdrawal capital. To calculate how the different cash flows change during different stress tests, a function for the behavioral sensitive instruments based on BCBS [27] original idea with smaller alterations was created.

First, the conditional prepayment rate, CP Rpi,c, was calculated with the formula

CP Rp_i,c= min(1, γi· CP R p

0,c) (4.15)

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CHAPTER 4. MODEL 29

sensitive instrument, and γidenotes the multiplier applied for scenario i. To sim-plify the model, assumptions for all prepayments sensitive instruments were made. The original CP Rp0,cwas set, in accordance with BCBS [28] recommendations, to

0.01. For the EBA yield curves scenarios, the γiwere defined in accordance with BCBS [28] and can bee seen in Table 4.3.

Table 4.3: Describes scenario multipliers for each EBA scenario. Scenario number (i) Interest rate shock scenarios γi(scenario multiplier)

1 Parallel up 0.8

2 Parallel down 1.2

3 Short rate up 0.8

4 Short rate down 1.2

5 Steepener 0.8

6 Flattener 1.2

For the Monte-Carlo simulations, a similar approach was used to include con-sistency between the models as far as possible. It was assumed that γiwas equal to 0.8 if the short term rates were increasing and that γiwas equal to 1.2 if the short term rates were decreasing.

When the CPRs for each scenario had been calculated, the new scenario-depended cash flows were calculated. It was assumed that all prepayments were made in each time bucket and went into effect instantly. In accordance with the formula:

CF_i,cp (k) = CF_i,cs (k) · CP Rp_i,c (4.16) CFs

i,cdenotes the originally scheduled interest rate and principal payment. These

new scenario sensitive cash flows were used in the gap risk calculation, see section 4.5. Hence, this option risk is not calculated in an isolated fashion as the other risks are.

4.5 Gap Risk

Gap risk is the most comprehensive risk and has, in this study, been calculated in terms of EVE and NII. Both of these risk measures will be presented separately below.

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CHAPTER 4. MODEL 30

4.5.1 EVE

Under each scenario i, all notional repricing cash flows are slotted into the respec-tive time bucket k (1, 2, . . . , 19). Remember that for each scenario, the cash flows differ due to refinancing risk. Within a given time bucket k, all positive and nega-tive notional repricing cash flows are netted to form a single cash flow. Hence, the offsetting part of each cash flow within the time bucket k was removed from the calculation. By following this process across all the time buckets, a set of netted notional repricing cash flows denoted as CFi,cp (k) was obtained [1]. Netted cash

flows in each time bucket k were weighted by a continuously compounded discount factor computed as:

DFi,c(tk) = e−ri,c(tk)·tk (4.17)

Which leads to that for each scenario i and currency c the estimated EV E is de-scribed as: EV Ei,c= 19 X k=1 CF_i,cp (k) · DFi,c(tk) (4.18)

Finally, the changes in EVE in currency c under the interest rate scenario i, denoted as ∆EV Ei,cGAP, is obtained by subtracting the original EV E, denoted as EV E0,c, from the EV E for scenario i.

∆EV EGAP_i = N X c=1 EV Ei,c− EV E0,c (4.19)

4.5.2 NII

The risk of loss in earnings in this model will be measured with the use of net in-terest income. More precisely explained, the changes in NII depending on shifts in the yield curve. Even though earlier framework such as BCBS [3] only required risk managers to measure their NII risk for parallel shifts of the yield curve, EBA [1] has implemented stricter rules for this measure. EBA [1] requires risk managers to evaluate the NII risk fully, i.e., evaluate the change of NII for all six stress tests. In this section, a method to measure this risk in accordance with the demands of EBA [1] is presented.

First and foremost, a more precise explanation of NII and which instruments are affected by this measure is necessary. NII risk is measured in the relative change

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CHAPTER 4. MODEL 31

of NII in accordance with 4.22. Since NII gap is a measure of the difference in interest earnings, the net worth of asset and liabilities are not of interest for these calculations but instead the interest payments of these instruments. This implies that fixed interest rate instruments are not of interest since these interest incomes will offset each other when the interest rate changes. Hence, the only relevant part of fixed interest rate instruments is the change of prepayment rate, since this change the amount of interest payment received.

However, instruments affected by a floating interest rate are of high importance in these calculations. Floating interest rate instruments do not change continuously. Instead, the interest rate for these changes occurs periodically at the instruments’ specific repricing date. The frequency of how often an instrument’s repricing date occurs will, in this paper, be referred to as repricing frequency, which typically takes place every month, every two months, etc. The effect of this is that a floating interest rate instrument can be handled as a fixed interest rate instrument between each of the repricing dates for that instrument. Furthermore, for these calculations, it is important to remember that NII is calculated with a so-called constant balance sheet, which means that maturing assets are instantly replaced with an instrument with identical features as the instrument matures. Another important aspect to re-member for these calculations is that the EBA approach requires risk managers to slot institutions’ notional cash flows into time buckets in accordance with each cash flows repricing date. This means that the interest rate income during the period t to t + R, where R denotes the repricing frequency, will be assumed to act under the period tkto Tj. Where tk is the midpoint of the bucket closest to t and Tj is the midpoint of the bucket closest to t + R.

The first step to calculate the NII gap risk is to aggregate all notional cash flows with the same repricing bucket, repricing frequency, and currency.

The second step is to evaluate the interest earnings of the notional cash flow with repricing frequency R under scenario i, between tk and Tj, this is done with the formula:

N IIi,c,k= CFi,c,k,· F (0; tk, Tj) (4.20)

Where i refers to stress scenario, c currency, k is the index of the bucket closes to t, and j is the index of time bucket closest to t + R. The effective interest rate under the period tk to Tjis given by the forward rate described in section 3.4:

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CHAPTER 4. MODEL 32

the notional cash flow was added to the aggregated notional cash flow with the same repricing frequency in the time bucket of Tj. This procedure was then repeated for all time buckets and repricing frequencies until the NII for the next three years had been calculated.

To calculate the total interest earnings for scenario i, the NII for all repricing fre-quencies was summed with the formula:

N IIi,c= C X c=1 19 X k=1 N IIi,c,k (4.21)

The final step to receive the resulting NII exposure was to calculate the difference between the interest earnings in scenario i and the original scenario.

∆N II_i,cGAP = N IIi,c− N II0,c (4.22)

4.6 Basis Risk

The next necessary step to measure a financial institution’s IRRBB is to calculate the basis risk in the banking book. Remember that basis risk refers to the impact of an imperfect correlation between indices underlying asset and liability rates. in other words, the possible impact of relative changes in interest rates on interest rate sensitive instruments that have similar tenors but are priced using different interest rate indices [1]. In this paper, an approach suggested by BCBS [28] was used to determine the basis risk. In this method, basis risk is computed by first aggregating the net notional repricing cash flows for a specific time period and for each refer-ence rate. In this paper, the used time period is three years. After that, the net cash flows exposed to basis risk between each pair of reference rates are computed as the minimum of the associated aggregated net notional repricing cash flows. This procedure is repeated for all combinations of pairs of reference rates in currency c. This can be expressed by the following equation:

∆N II_crb= RR X Y =2 Y X X=1 ((min(max(CF_0,c,rr∗ _X; 0); |min(CF_0,c,rr∗ _Y; 0)|) +min(|min(CF0,c,rr∗ X; 0)|; max(CF ∗ 0,c,rrY; 0))) · H C rrX,rrY) (4.23)

Where CF0,c,rr∗ is the aggregated notional cash flows during the chosen time

pe-riod for the specific reference rate rr. RR is the number of reference rates which the bank is exposed to in currency C. Hrrx,rryC is the historical basis risk shock

IRRBB in a Low Interest Rate Environment

IN

DEGREE PROJECT

MATHEMATICS,

SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2020

IRRBB in a Low Interest Rate

Environment

SIMON BERG

VICTOR ELFSTRÖM

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

IRRBB in a Low Interest Rate

Environment

SIMON BERG

VICTOR ELFSTRÖM

Abstract

IRRBB i en Lågräntemiljö

Sammanfattning

Acknowledgements

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Background

1.2

Previous Research

1.3

Purpose

1.4

Research Question

1.5

Outline of the Thesis

Chapter 2

Theoretical Background

2.1

IRRBB

2.1.1

Gap Risk

2.1.2

Basis Risk

2.1.3

Behavioral Risk

2.2

The Legal Environment of IRRBB

2.3

Measures of IRRBB

2.4

Balance Sheet

Chapter 3

Mathematical Framework

3.1

Nelson-Siegel-Svensson Parameter

Estima-tion

3.2

Principal Component Analysis

3.3

Cholesky Factorization

3.4

Forward Rate

3.5

Bachelier Pricing Model

Chapter 4

Model

4.1

Mapping of the Cash Flow Data

4.2

Yield Curve Construction

4.3

Interest Rate Models

4.3.1

Interest Rate Restrictions

4.3.2

EBA Interest Rate Model

4.3.3

PCA Interest Rate Model

4.3.4