Asset-Liability and Liquidity Management

By Pooya Farahvash

Asset-Liability and Liquidity Management distils the author’s extensive experience in the financial industry, and ALM in particular, into concise and comprehensive lessons. Each of the topics are covered with a focus on real-world applications, based on the author’s own experience in the industry.

The author is the Vice President of Treasury Modeling and Analytics at American Express. He is also an adjunct Professor at New York University, teaching a variety of analytical courses.

Learn from the best as Dr. Farahvash takes you through basic and advanced topics, including:

• The fundamentals of analytical finance
• Detailed explanations of financial valuation models for a variety of products
• The principle of economic value of equity and value-at-risk
• The principle of net interest income and earnings-at-risk
• Liquidity risk
• Funds transfer pricing

A detailed Appendix at the end of the book helps novice users with basic probability and statistics concepts used in financial analytics.

• Language: English
• Publisher: Wiley
• Release date: May 26, 2020
• ISBN: 9781119701910

Book preview

About the Author

Pooya Farahvash is vice president of Treasury Modeling and Analytics at American Express Company, overseeing the development of models used in ALM, liquidity risk management, stress testing, and deposit products. He previously worked at investment bank Jefferies in liquidity risk management and at CIT Group in asset-liability and capital management. His experience in the banking industry is focused in treasury department activities, specifically in the areas of interest rate risk, liquidity risk, asset-liability management, deposit modeling, and economic capital. Dr. Farahvash is also an adjunct instructor at New York University, teaching analytical courses. He received both his PhD degree in Industrial and Systems Engineering and MS degree in Statistics from Rutgers University, New Jersey. He currently lives in New York City.

Preface

In recent years, use of quantitative methods in asset-liability management (ALM) has increased significantly, particularly among medium- to large-size banks and insurance companies. This partly reflects the importance of effective balance sheet planning and managing related risks in achieving earnings and equity valuation targets. Traditionally and in the past, balance sheet management efforts were mainly focused on funding activities to ensure that the bank's assets are properly funded at the lowest cost possible. Lack of risk awareness, however, was always a major weakness in this approach and recent history has shown that poorly managed balance sheets can lead to catastrophic events for banks. In one view, the failures of several banks and investment banks during the financial crisis of 2007–2009 were partially due to ineffective balance sheet management practices. Newer banking strategies rely on ALM techniques that are based on accurate and precise calculations to evaluate the impact of various risk factors on earnings and value of the firm. These metrics are designed to assess the efficiency of the balance sheet management efforts while taking various risks, such as interest rate risk, into consideration.

This book presents the fundamentals of asset-liability management in banking. During my years of practice as an ALM analyst in various banks, I generally felt that there was a need for a book that provides a comprehensive view of ALM as it is exercised in practice. The goal of this book is to present the fundamentals and methodologies that are commonly used by banks in their ALM analysis. The book is written for professionals who are active in asset-liability management, financial risk management, and treasury analytics. This book can also be used as the main textbook for a graduate-level course in the aforementioned areas.

The main materials in the book are organized in three parts. The first part, consisting of Chapters 1 through 7, is focused on the interest rate concept and related topics, interest rate modeling methods, and valuation of financial instruments. Many ALM analyses require valuation of positions on the balance sheet of a bank, as well as valuation of off-balance-sheet exposures, such as derivative contracts. Materials in this part provide the fundamentals for valuation of common financial instruments, including fixed- and floating-rate loans, fixed-income securities such as bonds, equity securities, mortgage-backed and asset-backed securities, and callable or putable bonds. Valuations of common derivative products such as stock options, future options, interest rate swaps, interest rate forwards, interest rate caps and floors, and swaptions are also discussed. Since some topics reviewed in the interest rate models chapter require knowledge of valuation methods, that chapter is placed after the fundamentals of valuation are explained.

The second part of the book, consisting of Chapters 8, 9, and 10, is focused on two fundamental ALM metrics: economic value of equity and net interest income, and their related scenario analysis. The topics discussed in this part rely on the materials explained in Part One.

The third part of the book, consisting of Chapters 11 and 12, covers two topics that are closely related to ALM: liquidity risk and funds transfer pricing. Liquidity risk is the risk factor behind one of the gap measurements that the ALM process aims to optimize and funds transfer pricing is an internal allocation method of the net interest income. There are some practitioners who view liquidity risk management and funds transfer pricing as separate and independent topics from ALM. Recent trends, however, indicate that banks are moving toward a holistic view in managing the interest rate risk and the liquidity risk by combining the resources and required analysis of the two risk types. Particularly, there are many commonalities between data required for ALM and liquidity risk management. Funds transfer pricing, if done properly, internalizes the interest rate risk and liquidity risk among business units of a bank, and hence plays an important role in balance sheet management.

Asset-liability management studies are part of quantitative finance. In ALM, mathematical modeling and statistical concepts are mixed with high-level business decision making on how to run a bank. For the quantitative techniques discussed and used in this text, the general approach is to focus on applications and outcomes rather than providing deep discussions on supporting theories and proof of equations. For readers who are interested in theoretical background, each chapter provides a list of references for the origins of methods and further discussions. Since several subjects introduced in this book rely on statistical concepts, an appendix is added to cover the basic elements of probability and statistics in a concise form. These materials should help a reader who is not proficient in statistics to gain an understanding of the subjects that are needed in other parts of the book.

Methods discussed in this text when applied to the entire balance sheet of a bank require extensive computations. For the most part, examples provided are simple enough so the reader can follow and understand the topics. In practice, software packages are available that can perform the analysis explained here for balance sheets with a large number of positions. The book is not written with any particular software in mind, however, as the concepts discussed here are applicable to any ALM analysis, regardless of the software used.

In some of the examples and illustrations throughout the book I occasionally use a LIBOR–swap curve for coupon calculation of floating-rate instruments or for discounting. The principles discussed, however, are applicable if any other interest rate, such as SOFR or OIS, was used instead. In some of the examples presented in the book, the reader may notice some minor differences between the results shown here and results if calculations are performed using a spreadsheet software. This is due to rounding errors that may occur at a calculation step and those errors generally make no difference in the final outcomes.

I would like to thank those individuals who commented on the manuscript, and those who were involved in the production process of the book.

Pooya Farahvash

New York

February 2020

Abbreviations

Introduction

A bank at its core is a financial intermediary institution that collects funds from those individuals or entities who do not have immediate use for them and lends to those who can use the capital to generate economic benefits. Depositors with excess cash can benefit from the interest earned on their deposits while borrowers can benefit from the borrowed funds for their personal needs, such as purchasing real properties, or business needs, such as investing in their small business ventures. As the facilitators of such fund transfers, banks earn the difference between the interest paid to the depositors and the interest earned from the borrowers. A bank with an asset-driven business model seeks to originate assets through lending activities and simultaneously pursue funding methods to fund those assets, whereas a bank with a liability-driven business model primarily focuses on collecting deposits and then attempts to lend or invest the proceeds from the deposits. While traditionally deposits are the main source of funds in the banking industry, nowadays banks use a variety of methods to raise funds, including the issuance of short-term and long-term notes, securitization, and collateralized borrowings. Use of funds is also evolved from traditional lending in the form of loans to individuals and businesses, in investment in securities, and even in speculation using derivatives. The net revenue a bank makes is the difference between the costs associated with its sources of funds and earnings from the instruments where available funds are invested and used.

A bank manages its sources and uses of funds by trying to match them based on different criteria. One such criterion is based on the principal cash flows. The status of a bank as a financial intermediary, which is often supported by the central bank of the country, allows it to have a lower cost of funds compared to other entities. In particular, the bank's short-term borrowings are usually significantly cheaper compared to long-term alternatives. This allows the bank to fund long-term assets that are more profitable by cheaper short-term liabilities. While economically this seems like a sound business model, it potentially increases the risk for banks of not being able to fulfill their obligations when they are due. When the return of the principal amount borrowed by the bank is due before the principal lent is returned, this may lead to the bank's failure, should it not have any alternative source to replace the needed funds. A prudent banking practice is to align or overlap the terms of asset and liability positions such that there are always available funds to cover short- to medium-term liability maturities. However, in practice this is hard to achieve for individual asset or liability positions. Except in rare cases in which a particular debt position is raised to fund a large asset portfolio or a particular investment project, individual asset positions, such as loans and investment in securities, are not funded by distinct liability positions. Banks raise funds in micro form through deposits or in bulk form by issuing bonds. This makes the principal matching between assets and liability difficult, if not impossible. Due to this, banks may attempt to match the principal cash flows on a portfolio level. But even this approach has its limitations, since non-maturing products such as credit card accounts or savings accounts do not have contractual maturity dates. To overcome this, existing balances of non-maturing products are assumed to follow some modeled runoff profiles that act as amortization schedules for them. This allows the bank to estimate principal cash flows related to these products and to create principal cash flow schedules at an aggregated level, for example, for the bank as a whole or at a subsidiary level. Such schedules provide an overview of amounts and timings of expected principal cash flows and help in the planning and coordination of asset originations and debt issuances. This approach, however, does not incorporate planned changes in the assets and the liabilities. For example, if the bank is planning to grow a certain asset portfolio or to issue new debt securities in the near future, they are not reflected in a static cash flow schedule. Particularly, expected changes in balances of non-maturing products due to macroeconomic factors are not included. A dynamic cash flow schedule incorporates planned and expected changes in the asset and liability portfolios. A more sophisticated version of such a schedule considers all principal and interest payments to create a comprehensive view of cash flows a bank can experience in a short- to medium-term time horizon in the future. Cash flow gap, sometimes referred to as maturity gap, is the net value of cash flows generated by assets and liabilities in a specific time period. Minimizing cash flow gap is one way to reduce the risk of adverse events due to the mismatch between asset and liability cash flows, particularly their principal flows.

A bank may manage its uses of funds based on the reliability and persistency of the sources of funds. Funding sources being unavailable when they are needed may lead to the bank's failure. To assess their readiness, banks often perform scenario analysis to evaluate the impact of unavailability of one or more funding sources on their cash flow schedules and ultimately on their balance sheets. This enables them to obtain a view of the potential liquidity gap they may face in the future.

Interest rate is another criterion used by banks for matching their sources and uses of funds. The margin between the interest rate a bank pays on its liabilities and the rate it earns on its assets is a defining factor in the net profit of a bank, and maintaining this margin is crucial for future earnings that grow along with the growth of the balance sheet. In one categorization view, asset and liability positions of a bank can be either fixed-rate or floating-rate. The rate of a fixed-rate instrument is constant and does not change throughout its term, whereas the rate of a floating-rate instrument can change periodically, based on its contractual setting. For example, a conventional mortgage loan is a fixed-rate instrument while a home equity line of credit (HELOC) is an example of a floating-rate product. The effective rates of floating-rate instruments are often set in conjunction with a market interest rate index, such as prime rate index or effective federal funds rate index. As these rate indexes change, so do the rates of the floating-rate positions. However, the change in rates may not be simultaneous, as floating-rate instruments often follow specific rate-setting schedules, for example, every three or six months. Therefore, once a market rate index is changed it may take a few months until the rate of a floating-rate instrument that is pegged to that index changes.

Depending on the business model, a bank may originate both fixed-rate and floating-rate assets and fund them using a combination of fixed-rate and floating-rate liabilities. The mismatch between rate type of asset and liability positions can have an adverse impact on the net interest margin. Consider a bank with a balance sheet where the majority of asset positions are fixed-rate and the majority of liability positions are floating-rate. If the overall market interest rate increases, the interest income from assets would not materially change due to the fixed-rate nature of those positions, but interest expense from the floating-rate liabilities would increase, resulting in an overall decrease in the net interest margin. Conversely, a decline in overall market interest rate would result in an increase in the net interest margin. This volatility eventually impacts the overall net income of the bank.

One approach for maintaining a stable interest margin is to keep the composition of fixed- and floating-rate positions comparable and unchanged through time. This approach, however, is often impractical. Having comparable portfolios with respect to their rate types may not be in line with the business model of a bank. For example, a local or a regional bank that is focused on the mortgage market usually has a large portfolio of fixed-rate mortgages while such bank's main source of funding is often floating-rate savings products. Maintaining a position-level composition of assets and liabilities with respect to rate type is also prohibitive. For example, when a fixed-rate position matures it may not be feasible, or even economically justified, to replace it with another fixed-rate position. As an alternative, banks try to maintain the composition of their assets and liabilities at the portfolio level.

Even if the position-level or portfolio-level fixed- and floating-rate composition is maintained, the bank is still exposed to volatility in the net interest margin due to the timing of changes in the rates of floating-rate instruments. Rate resetting refers to the timing and periodicity that the rate of a floating-rate instrument changes. Consider a bank with a large floating-rate asset portfolio that resets on a monthly basis and a comparably large floating-rate debt note that resets quarterly. In an interest rate declining environment, the interest rate of the asset portfolio changes every month and decreases in line with decreases in the overall market interest rate, leading to a constant decline in the interest income from that portfolio. However, the interest rate of the debt note resets every three months. Hence the decline in the market interest rate does not impact the income from the asset and the expense of the liability at the same time, leading to periodical tightening of the net interest margin.

To assess the impact of changes in the interest rate on the net interest income, banks compare principals of assets and liabilities that are repriced during a period, where repricing here refers to changes in the effective rates. Earning gap is the net value of repriced principal of assets and liabilities during a specific period (e.g., one year). Minimizing the earning gap is one way to reduce the risk of adverse impact on the net interest margin due to changes in the interest rate level and timing of changes.

Aside from effect on earnings, changes in the interest rate can also impact a bank from a valuation point of view. A bank's management team, as agents of shareholders, are tasked with safeguarding the value of the firm against different risks, most notably the interest rate risk. Changes in the interest rate impact the value of fixed-rate and floating-rate instruments differently, where generally the value of a fixed-rate instrument is more sensitive to change in the interest rate compared to the value of a floating-rate instrument. The term of the instrument also influences the sensitivity of its value to the interest rate, where generally a long-term position is more sensitive compared to a short-term position. Duration of a financial instrument reflects the sensitivity of its value to a change in the interest rate level. To protect the value of the firm, banks compare durations of asset and liability portfolios and obtain a duration gap for the balance sheet. Minimizing the duration gap is one way to reduce the risk of adverse impact on the value of the firm due to changes in the interest rate.

Asset-liability management (ALM) is the process of optimizing the earning gap, the duration gap, the cash flow gap, and the liquidity gap of a bank in a holistic way and on a risk-aware basis. The close relationships between these notions require a comprehensive approach in managing their absolute impacts on the bank's earnings and the firm's value, as well as their interactions among themselves. A well-defined ALM process takes all aforementioned gaps into account in managing the balance sheet, as actions taken to optimize one gap may have an undesirable effect on another. Normally, the treasury department of a bank is in charge of ALM and in doing so, it coordinates its balance sheet management efforts with other business units that are in charge of asset originations or deposit collections.

ASSET-LIABILITY MANAGEMENT METRICS

ALM analysis is mainly focused on two metrics that have a close relationship to the gap measurements introduced above: economic value of equity and net interest income.

The economic value of equity (EVE) of a bank is the difference between the economic value of its assets and the economic value of its liabilities, calculated at a specific point in time. One of the main responsibilities of a bank's management team is to preserve the economic value of equity, protect it against various risks, and increase it through business activities. ALM studies that are focused on the EVE aim to identify and analyze scenarios where changes in risk factors such as interest rate would lead to a decrease in the EVE of the bank. Once those scenarios are identified, the bank management team sets limits against the potential decreases in the EVE associated with each scenario and tasks the treasury department, as the entity in charge of the balance sheet management, to coordinate asset originations and funding activities to adhere to those limits.

The net interest income (NII) of a bank is the difference between the interest income from the assets and the interest expense from the liabilities during a specific time period. For traditional banks with significant lending businesses, the net interest income is usually the main component of total earnings. ALM studies that are focused on the NII aim to identify and analyze scenarios where changes in risk factors, specifically the interest rate, would lead to a decrease in the NII of the bank. Similar to the EVE, for identified scenarios the bank has specific limits for a potential decrease in the NII and manages its balance sheet to adhere to those limits. Earnings-based ALM studies are often focused on net interest income but more comprehensive analysis may include other incomes, such as fee-based revenues, to provide a more comprehensive view on scenarios leading to a potential decrease in total earnings.

Both the EVE and the NII are suitable metrics for quantification of the interest rate risk. Changes in the interest rate impact the EVE and NII and expected future interest rate levels are used in the calculation of both. In the EVE analysis, the expected future interest rates are used to estimate future cash flows needed for valuation of the positions on the balance sheet and in the NII analysis they are used to estimate accrued interests in future periods, for example, during quarterly time buckets in the next two years. Interest rate also impacts the EVE through the discounting process where present values of expected future cash flows are obtained using discount factors that depend on the interest rate.

The EVE and the NII metrics are closely related, although the EVE has a broader scope. The value of a firm depends on its earnings and for a bank it specifically depends on the net interest income. Therefore the firm's earnings are reflected in the value of its equity. The main differences between the two metrics are in the time horizon of the analysis and the treatment of changes in the balance sheet during that period:

The NII analysis is usually focused on a short-to-medium time horizon, for example, one to three years from the analysis date. However, since the EVE metric is based on the economic value of the balance sheet positions, the analysis time horizon is effectively extended to the time of the last expected future cash flow.

The EVE is based on values of existing positions on the balance sheet at a specific point in time and expected changes in the balance sheet in the future are usually not included in the valuation process. In the NII analysis and depending on the selected treatment, the balance sheet may be assumed to be in a runoff mode, where positions matured during the analysis time horizon are not replaced, or it may be assumed to be constant with no changes during the horizon, or a dynamic balance sheet may be assumed when planned changes in the portfolios are reflected in the interest income and the interest expense calculation.

While the EVE and the NII are appropriate metrics for measuring the interest rate risk at the balance sheet level, neither of them is suitable for quantification of the liquidity risk. Banks often assess their exposures to liquidity risk by performing liquidity stress test analysis. In such analysis, several hypothetical stress scenarios are assumed and the impact of each case on (i) cash flows from the assets and liabilities, (ii) funding sources, and (iii) available cash and cash-equivalent assets, are estimated. Using this approach, banks can identify the potential scenarios that lead to a liquidity event and subsequently create contingency plans accordingly.

Funds transfer pricing (FTP) is one of the areas that are closely related to the ALM but often overlooked in practice. FTP is an internal allocation method of the net interest income among business units of banks on a risk-aware basis. Implementation of the FTP system is often based on the ALM platform, since both systems require position-level data as well as market data.

ALM RISK FACTORS

Interest rate is by far the most important risk factor considered in the ALM analysis. Depending on its balance sheet, a bank may be exposed to different types of interest rate risk. Yield curve risk is generally referred to as the potential for an adverse impact on NII or EVE of the bank due to inconsistent changes in the interest rates with different terms. If compositions of fixed- and floating-rate instruments on the balance sheet are uneven, or when the timing of floating-rate instrument rate settings are not aligned, the bank may be exposed to interest rate gap risk. When floating-rate assets and liabilities of a bank are based on different market rate indexes (e.g., assets are based on the prime index and liabilities are based on LIBOR or SOFR), uneven movement of those indexes may have an adverse impact on the EVE and the NII, exposing the bank to interest rate basis risk.

When certain conditions are met, some financial contracts allow one side to terminate the contract prior to its maturity or to significantly alter it. This is usually referred to as the counterparty having an option on the contract. In turn, this can materially change the expected cash flows of such instruments, exposing the banks to option risk. Particularly, when the borrower of a loan has the option to return the principal amount earlier than scheduled, it may force the bank to re-lend the returned fund in unfavorable conditions (e.g., lending in a low interest rate environment), impacting the EVE and the NII adversely. This is referred to as prepayment risk. Prepayment risk is notably present for mortgage loans when borrowers' refinancing activities can change the expected cash flows significantly. Option and prepayment risks usually have a close relationship with the level of the interest rate, and for that reason they are often characterized as interest rate risk.

Interest rate risk is not the only risk factor included in the ALM analysis. For a bank with a multi-currency balance sheet, exchange rate risk can be a significant source of uncertainty in the earnings and value of the bank. Functional currency of a bank usually is the currency of the country where the bank is incorporated and conducts most of its business. When a bank transfers funds from the functional currency to another currency, the income earned is eventually expected to be returned to the functional currency. The exchange rate at each conversion step can impact the earnings amount and the net income of the bank. The economic value of equities of multi-currency banks also depends on the exchange rates and their volatilities. From a balance sheet management point of view, and depending on the amount of foreign currency exposures, the exchange rate can be a significant source of risk; this makes it essential for banks to include the currency exchange rate risk factor in their ALM analysis based on both EVE and NII metrics.

Changes in securities prices can impact the EVE of banks with significant investment or trading portfolios. While a change in the market interest rate is the main driver behind the change in the securities prices, other factors such as changes in supply and demand forces, the release of new information regarding a security or its issuer, or even the spread of misinformation in the market can lead to unexpected price changes. The risk for a potential adverse impact from this on the EVE of a bank is generally referred to as price risk. Practitioners often use market risk term to refer to the interest rate risk, the exchange rate risk, and the price risk, collectively.

Credit risk in a financial contract generally refers to the risk of counterparty default. For example, for a commercial loan, credit risk arises from the possibility that the borrower will fail to make required payments at scheduled times. For a bond, since the price depends on the credit rating, credit risk may refer to the potential of an adverse impact on the price of the security due to the change in its rating. Credit risk factor is usually included in the ALM analysis by considering the impact on the position's cash flows, both in the NII and the EVE analysis. An alternative approach in incorporating credit risk in the EVE analysis is to discount cash flows based on a risk-adjusted interest rate curve instead of the risk-free rate.

Liquidity risk is the risk of a bank not being able to fulfill its financial obligation at the required time without incurring excessive costs or suffering extreme losses. Unexpected large cash outflows, inability to roll a maturing debt note, and run on demand deposits are some of the circumstances that can lead to a liquidity event. While realization of adverse events due to market risk or credit risk results in financial losses, a liquidity event can be catastrophic to a bank, leading to bank failure and eventual bankruptcy. There is a close relationship between ALM and liquidity risk management and often a single group within the treasury department of a bank is responsible for both. Since ALM systems can produce position-level expected cash flows required for the EVE and the NII analysis of different scenarios, the same cash flows can be used in liquidity risk analysis.

ORGANIZATION OF THIS BOOK

The topics discussed in this book are organized in several chapters and one appendix. A brief overview of each chapter is presented below.

Chapter 1: Interest Rate

The concept of interest rate is at the core of any financial analysis and understanding notations used to represent the interest rate and its market conventions is essential for the topics discussed throughout this book. In this chapter we start our discussion by presenting the interest rate concept, different types of interest rates, and various ways they are quoted and used in financial analysis. In doing so we explain the present and future values of cash flows and discuss how they are related to the interest rate. We also introduce the important concept of compounding and explain the difference between simple interest and compounded interest. Cash flows of financial instruments are derived based on their contractual characteristics. For example, a borrower of a mortgage loan usually makes monthly payments but the contract of a commercial loan may be structured such that the borrowing company pays every three months. There are different market conventions that define these characteristics and understanding them is crucial for cash flow analysis. In this chapter we introduce a few of these market conventions, including payment and accrual periods, day count conventions, and business day adjustments. Quoted interest rates in the market have different terms (e.g., 1-month rate or 1-year rate) and may reflect periodic coupon payments. In this chapter we discuss a technique known as bootstrapping to put rates with different terms and coupon payments on a similar footing. Rates obtained from this process are known as spot rates or zero-coupon rates and play an important role in quantitative finance. In this chapter we also discuss the important concepts of forward rate, implied forward rate, and future rate. These are all indications of interest rates in the future, but derived using different methods and used for different purposes.

An interest rate swap contract is a derivative product widely used in finance for hedging against movement of interest rate. Swap contracts usually have a long term (e.g., 2 years to 10 years). For this reason, interest rates used in swap contracts are often used to extend short-term rates, and together they create an interest rate curve. In this chapter we introduce interest rate swap contracts, discuss their valuation, and explain how a swap rate is derived and is associated with short-term rates. Components of interest rates, namely their risk and term structures, and a few methods for interpolation of rates are also explained in this chapter.

Chapter 2: Valuation: Fundamentals of Fixed-Income and Non-Maturing Products

In this chapter, the discounted cash flow method as the fundamental technique for valuation of financial instruments is introduced. We start our discussion by explaining the most common amortization methods that are used for derivation of principal cash flows. One broad categorization of financial instruments is based on the type of their coupon rate. A fixed-rate instrument has a constant coupon rate that does not change during its term, but the coupon rate of a floating-rate instrument changes based on the value of an interest rate index. In this chapter we discuss the valuation of fixed-rate and floating-rate instruments and in doing so we introduce the important concepts of yield, duration, and convexity. In particular, we explain how duration and convexity can be used to approximately estimate the change in value of a financial instrument due to the change in the interest rate. We then use this method to explain how the value of a fixed-income financial instrument, such as a bond, can be immunized against the change in the interest rate.

Another broad categorization of financial instrument is based on their terms. A maturing product has a contractual term and a specific maturity date. A bond or a mortgage loan are examples of maturing financial instruments. A non-maturing product does not have a contractual maturity date. A savings account or a credit card account are examples of non-maturing products. In this chapter we review a few treatments commonly used in practice to estimate cash flows of non-maturing products that are needed for valuation purposes. In the last topic of this chapter we discuss the issue of prepayment and how it impacts the cash flows of a financial instrument and its valuation.

Chapter 3: Equity Valuation

This chapter covers three equity valuation methods: the dividend discount model, the discounted free cash flow method, and the comparative approach. The dividend discount model and the discounted free cash flow method are both based on discounting expected future cash flows, where in the former they are dividend payments and in the latter they are cash flows generated by the firm that are available to the equity holders. In the comparative valuation approach a measurement of the stock price, often in the form of a ratio, is compared with the firm's peers to estimate if the market price of equity is overvalued or undervalued.

Chapter 4: Option Valuation

An option contract provides its holder the right, but not the obligation, to take an action, such as purchasing an equity security or returning a bond to the issuer, within specific settings defined at the contract. Execution of an option can materially change the cash flows of the contract and therefore the contract value should reflect this. The value of an option contract depends on the value of other products or financial indexes, generally referred to as the underlying. In this chapter we introduce the fundamentals of option valuation by first focusing on stock options. Specifically, three option valuation techniques are discussed here: binomial trees, the Black-Scholes-Merton model, and Monte Carlo simulation. The binomial tree method is a discrete-time approach that defines the change in the value of the underlying in a tree-like structure. Black-Scholes-Merton is a continuous-time model that presents evolution of the value of the underlying through time. Monte Carlo simulation is based on creating a large number of simulated paths of the underlying value during a specific time horizon and using them to obtain the expected value of the option.

Volatility of the underlying value plays an important role in option valuation. In this chapter we discuss derivation of the volatility and introduce two methods to model the volatility when it is not constant through time: generalized autoregressive conditional heteroscedasticity (GARCH) and exponentially weighted moving average (EWMA).

In the last section of this chapter we introduce another type of option when the underlying is a futures contract. A futures contract is an agreement between two parties to buy and sell an asset at some time in the future within specific settings defined at the contract initiation. Futures options are options written on futures contracts based on prices of commodities or financial instruments. Here we first introduce futures contracts and then discuss the Black model for valuation of futures options.

Chapter 5: Interest Rate Models

An interest rate model is a mathematical formulation that characterizes the movement of interest rates over a period of time. Interest rate models are used for simulation of the interest rate, which in turn can be used for valuation of financial instruments or simulation of interest earnings in a specific scenario. Interest rate models are also useful for valuation of complex derivative products. In this chapter we introduce a few interest rate models that are commonly used in the ALM analysis. To do so, we first explain the concepts of short rate and instantaneous forward rate and discuss how the spot rate and the short rate are associated. We then present four interest rate models: Vasicek, Hull-White, Ho-Lee, and Black-Karasinski. These four models have a lot in common while some specific features distinguish them from each other. The Hull-White model can be considered an extension of the Vasicek and the Ho-Lee is a special case of the Hull-White model. Short rates produced by these three models can be positive or negative but a short rate produced by the Black-Karasinski model is always positive.

An interest rate tree is a discrete-time representation of a continuous-time interest rate model. In this chapter we introduce construction of a trinomial tree based on the Hull-White model and then extend our discussion to construction of an interest rate tree based on the Black-Karasinski model. These interest rate trees, also known as lattice models, can be used for valuation of complex financial securities, where cash flows and other characteristics of the instruments at a specific time depend on the level of the interest rate at that time.

Parameters of an interest rate model are often estimated using values of liquid market-traded financial products. This process is referred to as model calibration. Interest rate cap and floor and swaption contracts are usually used in calibration of interest rate models. In this chapter we introduce these derivative products and then demonstrate how they are used in the calibration process.

Chapter 6: Valuation of Bonds with Embedded Options

Callable and putable bonds have contractually embedded options that allow the investor (in the case of the putable bond) or the issuer (in the case of the callable bond) to recall or redeem the note. In this chapter we explain how the interest rate tree, discussed in the previous chapter, can be used for valuation of callable and putable bonds. In doing so, we also introduce the important concept of the option-adjusted spread.

Chapter 7: Valuation of Mortgage-Backed and Asset-Backed Securities

For many traditional banks, mortgage loans comprise a large portion of their asset originations. To free up the funding required for these assets, banks often package the mortgage loans into new securities through a process known as securitization. These mortgage-backed securities are then sold to investors who usually could not directly invest in the mortgage sector. Valuation of the mortgage-backed securities is a challenging task as many factors, such as interest rate level, prepayment, and potential default of the borrowers, are involved in the process. In this chapter we first introduce the basic mathematics of mortgage loans and then discuss how the Monte Carlo simulation method can be used to value the mortgage-backed securities. Asset-backed securities are another class of financial products that are created through the securitization process. The most common asset-backed securities are based on pools of auto loans, credit card receivables, student loans, and home equity loans. In the second part of this chapter we briefly introduce these four types of securities and explain their valuation method using Monte Carlo simulation.

Chapter 8: Economic Value of Equity

Economic value of equity (EVE) is the difference between the economic value of the assets of a bank and the economic value of its liabilities at a specific point in time. Management of EVE and its relationship with the structure of assets and liabilities of the bank is an integral part of an effective balance sheet management strategy and consequently it is crucial for the bank to regularly assess the potential change in the EVE in different market conditions. In this chapter we first introduce the concept and calculation of the economic value of equity and then present the duration gap as a basic approach in analyzing the potential change in the EVE when the interest rate level changes. When the discounted cash flow method is used in valuation of asset or liability positions for the EVE analysis, the yield curve used for discount factor generation may need to be adjusted for the riskiness of positions being valued. Here we explain how risk-adjusted yield curves are created and used.

Beyond the basic duration gap method, scenario analysis is the main approach for EVE studies. In this chapter we explain EVE scenario analysis based on the interest rate and the exchange rate risk factors. The last section of this chapter presents an overview of the Basel committee guidelines on managing interest rate risk in the banking book as well as procedures for scenario generation and EVE analysis.

Chapter 9: Net Interest Income

Net interest income (NII) is the difference between the interest income and the interest expense during a specific time period. For traditional banks the net interest income is the main component of their total earnings. Sensitivity of the net interest income to the interest rate level highly depends on the type and rate-setting behavior of positions on the balance sheet. In this chapter we introduce calculation of the net interest income and specifically explain two approaches for calculation of the interest income or expense for a floating-rate instrument. Since the net interest income is evaluated during a time horizon in the future it is important to decide whether expected balance sheet changes in that period are to be incorporated in the NII analysis. In this regard there are three treatments usually considered in practice. In the first treatment the balance sheet is assumed in a runoff mode where matured, amortized, or runoff positions are not replaced while in the second treatment the balance sheet is assumed constant throughout the analysis horizon and matured positions are replaced such that making this assumption holds. The third treatment is the most comprehensive approach where forecasted changes in the balance sheet are considered in the analysis and hence a dynamic view of the NII is obtained.

The earning gap method is a first step in sensitivity analysis of the NII in different interest rate environments. In this chapter we introduce the earning gap and then move to more comprehensive scenario analysis where impacts on the NII of parallel or non-parallel shocks to the current interest rate curves are studied. The scenario analyses discussed in this chapter are focused on the interest rate and the exchange rate risk factors. The last section of this chapter provides an overview of the Basel committee guidelines for the NII analysis.

Chapter 10: Equity and Earnings-at-Risk

EVE and NII scenario analyses focus on a limited number of scenarios. This chapter extends the topics discussed earlier by considering the cases in which a large number of simulated scenarios are generated to study the potential extreme changes in the EVE and the NII. We start this chapter by introducing the value-at-risk (VaR) method. VaR is a popular technique widely used for risk management purposes. We explain three general approaches in obtaining the VaR of a position or a portfolio: historical sampling, Monte Carlo simulation, and the variance-covariance method. We then use the VaR concept in the EVE and the NII analysis.

Equity-at-risk is the near-maximum potential decrease in the economic value of equity due to changes in risk factors and earnings-at-risk is the near-maximum potential decrease in estimated earnings due to changes in risk factors. Both metrics are defined for a specific time horizon and for a given probability confidence level. The first step in the equity-at-risk and the earnings-at-risk analyses is to generate a large number of simulated scenarios for the risk factors considered in the analysis. One method for scenario generation is to sample the historical values of a risk factor and use them to generate simulated paths of the risk factor in the future. The second method is based on the Monte Carlo simulation where a quantitative model, often in the form of a stochastic process, is developed to represent the behavior of a risk factor through time and then the model is used to generate simulated paths of the factor in the future. In this chapter we discuss both methods and introduce a few general models for simulating risk factors such as exchange rates or prices of stocks and commodities. We then follow by explaining how the equity-at-risk and the earnings-at-risk analyses are performed by focusing on the interest rate and the exchange rate risk factors. In this chapter we also introduce the delta-gamma approximation method, which is commonly used in the value-at-risk analysis to reduce the computational burden of a valuation process.

Chapter 11: Liquidity Risk

After the financial crisis of 2007–2009, the importance of liquidity risk management increased considerably and many banks implemented various tools to measure, monitor, and control this risk. We start this chapter by introducing some of the common funding channels and discuss the potential of each channel in producing or contributing to a liquidity event. Particularly, liquidity events related to repurchase agreement (repo) contracts were some of the most impactful during the financial crisis period. In this chapter we introduce repo contracts and discuss the circumstances in which they can cause a liquidity stress event. We then follow by discussing a few methods to manage the liquidity risk of the repo contracts.

Cash flow gap analysis is one of the main tools in liquidity risk measurement. Cash flow gap schedules are often created for at least three scenarios. In a business-as-usual case, the cash inflows and outflows during a specific period based on the normal course of business are estimated and compared. In an idiosyncratic stress case, a stress event particular to the firm is assumed and its impacts on cash flows and their timings are evaluated. In a market-wide stress case, it is assumed that the stress event has impacted the entire market, hence cash flows are reevaluated based on the assumption that the firm's counterparties are also under stress. Multi-currency cash flow gap analysis is applicable to a firm with a multi-currency balance sheet. In this chapter we explain these cash flow gap schedules using several examples.

The last section of this chapter provides an overview of two liquidity risk monitoring tools proposed by the Basel committee. Regulatory entities in many countries have already required the banks in their jurisdictions to implement these tools. They are the liquidity coverage ratio and the net stable funding ratio. Here we introduce both ratios and provide their calculation details using examples.

Chapter 12: Funds Transfer Pricing

Analysis and measurement of net interest income is usually performed at the balance sheet level of a bank or its standalone entities such as its major subsidiaries. In doing so, when sensitivity of the net interest income to risk factors is analyzed the assumption is that the entity as a whole attempts to minimize the risk. Past experiences have shown that this may not be the case for large banks where individual business units aim to maximize their own earnings irrespective of impacts on the total bank's risk level. The association of compensation and year-end rewards to individual business unit performance is one of the main factors in encouraging this behavior. This often has a negative consequence for the bank when business units' risk-taking activities to maximize their own performance measures expose the bank to a level of risk that is beyond its designated tolerance limits. Funds transfer pricing (FTP) is an internal allocation method of distributing the interest margin and the net interest income among relevant units within the bank and to internalize the interest rate and liquidity risks to the business units that are sources of risk-generating activities. There is a close relationship between the net interest income measurements at the aggregated bank level and funds transfer pricing as a distribution method of the net interest income to business units.

In this chapter we first introduce the concept of funds transfer pricing and its benefits. We then explain two FTP techniques commonly used in practice: the pool method and the match maturity method. Particularly, we demonstrate FTP rate assignment based on the matched maturity method for fixed-rate instruments, for floating-rate instruments, and for non-maturing products. In the last section of this chapter characteristics of a good FTP system are discussed.

Appendix: Elements of Probability and Statistics

In the appendix we review a few important topics in probability and statistics that help in understanding the topics discussed in the main body of the book.

CHAPTER 1

Interest Rate

Finance charges, fees, and principal gains are the most common earnings channels in the finance industry. The most traditional revenue source for banks and many non-bank financial institutions is the finance charges associated with lending activities. Commercial and personal loans, student loans, lines of credit, mortgages, and many other financial instruments create revenue for the entities that underwrite and sell these products. Many banks provide fee-based services to their customers as well and charge them accordingly. Various checking, savings, retirement, and investment accounts offered by financial institutions include monthly fees that may vary by account activities and outstanding balances. Investment banks often offer a range of fee-based advisory services, including equity and debt issuances, mergers and acquisitions, and restructuring. Trading, clearing, and bookkeeping services offered by brokerage houses are also fee-based. Financial institutions with trading desks have significant, although volatile, revenue through gains on principal of traded securities. Financial institutions always attempt to diversify their earnings and utilize more than one of these channels. For example, broker-dealer firms rely on fee-based earnings for services offered through their brokerage divisions, as well as principal gain revenues from their trading activities on their own security inventory.

Finance charge as one of the main sources of revenue in finance is based on the well-known economic concept of the time value of money, signifying that a sum of money at the present time is worth more than an equal sum in the future, due to its potential earning power. What associates the value of present-time money to future-time money is known as the interest rate.

In this chapter we introduce the interest rate and its measurement methods. Concepts of compounding frequency, day count convention, and business day adjustment rules that are closely related to interest rates are explained as well. We introduce the risk-free rate and its proxies as the fundamental component of interest rate. Common interest rate measurements, including Treasury yield, LIBOR, federal funds, prime, swap, and OIS rates, are introduced and their relationships are discussed. We then continue our discussion of interest rates by providing details on construction of various interest rate curves and explain the relationship between spot and forward rates that is critical for many financial analysis. We finish this chapter by introducing some of the interest rate shocks that are commonly used in asset-liability management analysis.

Understanding of the interest rate concept is crucial for comprehension of other finance topics; hence materials presented in this chapter are fundamental to the topics discussed in the remainder of this book.

INTEREST RATE, FUTURE VALUE, AND COMPOUNDING

The interest rate is a method of measuring the cost of borrowing or the return on lending. Consider investor A, who invests in a bank savings account that promises to pay 8% annual interest. Assume the investor put $100 in that account and keeps it there for one full year. At the end of the year he earns 8% of $100, which is $8 interest in addition to his original invested amount of $100, for a total of $108. This can be presented as:

(1.1) equation

where 0.08 in the above formula is the interest rate for the duration of the investment. Here the original $100 is principal and the $8 is interest. Assume investor A decides to reinvest the original $100 principal plus the $8 interest from the first year in the same savings account and keep it there for another year, and assume the account interest rate for the second year is 8% annual interest. At the beginning of the second year the invested amount is $108; therefore during the second year the investor earns interest. Total investment in that savings account at the end of the second year is:

equation

The same result is achieved by considering that investor A invested $100 in the savings account that pays 8% annually for two years (two periods); therefore:

equation

The $110.64 balance at the end of the second year is known as future value. It has two components: the original principal of $100 and the earned interest of $16.64. Since in each year the annual interest rate is 8% the earned interest for two years based on the original principal of $100 is $16, calculated as:

equation

The extra $0.64 is the interest earned in the second year over the reinvested interest from the first year:

equation

The concept of reinvesting interest earned in previous period(s) and earning interest on both principal and reinvested interest is known as compounding interest. Compounding frequency defines the periodicity of interest calculation and its reinvestment. In contrast to compounding interest, simple interest earns interest on principal only.

A bank quote of 8% annual interest is incomplete without stating the compounding frequency. Compounding frequency indicates how often the interest is calculated and reinvested. In the above example, the bank quote should be 8% annual interest, compounded annually, meaning that the quoted interest rate of 8% is an annual rate and the compounded frequency is also annual. To illustrate the concept of compounding better, now assume that there is a different bank offering a savings account with a quoted interest rate of 8% annual interest, compounded semi-annually. The annual rate is 8% as before, but this product has a semi-annual compounding frequency. This means that the interest is calculated and reinvested every six months. Suppose investor B invests $100 in the savings account with semi-annual compounding frequency for one year. In the first six months he earns on his original principal of $100 and his total investment at the end of six months is:

equation

The $4 interest earned in the first half of the year is reinvested and for the second half of the year the 4% interest is earned on $104; therefore, at the end of the first year total investment value is:

(1.2) equation

Comparing Equations (1.1) and (1.2), at the end of the first year investor B earned $0.16 more than investor A by investing in a savings account with semi-annual compounding frequency. The extra $0.16 is the interest earned in the second half of the year over the reinvested interest earned in the first half. Dividing by 2 in the Equation (1.2) is needed since the quoted interest rate is annual and to convert it to the interest for six months we need to divide it by 2. In this case, every half-year is known as the period and there are two periods in each year. The 4% here is known as the periodic interest rate.

If investor B keeps the $108.16 balance at the end of the first year in the same savings account for one more year and the quoted interest rate for the second year is 8% annual interest, compounded semi-annually, the total investment (principal and interest) at the middle of the second year is (rounded to two decimals):

equation

and total investment at the end of the second year is:

equation

An alternative way of obtaining the same result is to note that investor B invested $100 for four periods of a half-year each while the interest earned on each half-year period is . So:

equation

Generally, if principal (P) is invested for years, where the annual interest rate is and the number of periods in a year (compounding frequency) is m, the total future value (FV) at the end of years is:

(1.3) equation

In the above equation, is annual interest rate in decimal, is measured in years, and is the number of periods. For semi-annual compounding frequency, is 2. Other compounding frequencies that are commonly used are quarterly, monthly, and daily. values for these compounding frequencies are presented in Table 1.1.

For example, an investment of $100 for two years ( ) in a savings account that pays 8% ( ) annually with quarterly compounding ( ) has a future value of:

equation

A compounding frequency that is extensively used in quantitative finance is continuous compounding. Continuous compounding is achieved when in Equation (1.3) an infinite value of is considered. It can be shown that in the limiting case of the future value of the investment is obtained as:

(1.4) equation

TABLE 1.1 Compounding frequency