Quantitative Financial Risk Management

Michael B. Miller

• Publisher: Wiley
• Year: 2018
• Print ISBN: 9781119522201
• eBook ISBN: 9781119522263
• Edition: 1
• Topic: Business
• Subtopic: Finance
• Index: Business

A mathematical guide to measuring and managing financial risk.

Our modern economy depends on financial markets. Yet financial markets continue to grow in size and complexity. As a result, the management of financial risk has never been more important. Quantitative Financial Risk Management introduces students and risk professionals to financial risk management with an emphasis on financial models and mathematical techniques. Each chapter provides numerous sample problems and end of chapter questions. The book provides clear examples of how these models are used in practice and encourages readers to think about the limits and appropriate use of financial models.

Topics include:

• Value at risk
• Stress testing
• Credit risk
• Liquidity risk
• Factor analysis
• Expected shortfall
• Copulas
• Extreme value theory
• Risk model backtesting
• Bayesian analysis
• ... and much more

1
OVERVIEW OF FINANCIAL RISK MANAGEMENT

Imagine you are a chef at a restaurant. You've just finished preparing eggs benedict for a customer. The eggs are cooked perfectly, the hollandaise sauce has just the right mix of ingredients, and it all sits perfectly on the plate. The presentation is perfect! You're so proud of the way this has turned out that you decide to deliver the dish to the customer yourself. You place the plate in front of the customer, and she replies, “This looks great, but I ordered a filet mignon, and you forgot my drink.”

Arguably, the greatest strength of modern financial risk management is that it is highly objective. It takes a scientific approach, using math and statistics to measure and evaluate financial products and portfolios. While these mathematical tools can be very powerful, they are simply that—tools. If we make unwarranted assumptions, apply models incorrectly, or present results poorly—or if our findings are ignored—then the most elegant mathematical models in the world will not help us. The eggs might be perfect, but that's irrelevant if the customer ordered a steak.

This is not a new idea, Vitruvius, a famous Roman architect wrote, “Neque enim ingenium sine disciplina aut disciplina sine ingenio perfectum artificem potest efficere”, which roughly translates to “Neither genius without knowledge, nor knowledge without genius, will make a perfect artist.” Applying this to risk management, we might say, “Neither math without knowledge of financial markets, nor knowledge of financial markets without math, will make a perfect risk manager.”

Before we get to the math and statistics, then, we should take a step back and look at risk management more broadly. Before delving into the models, we explore the following questions: What is risk management? What is the proper role for a risk manager within a financial organization? What do risk managers actually do on a day‐to‐day basis?

We end this chapter with a brief history of risk management. As you will see, risk management has made many positive contributions to finance, but it is far from being a solved problem.

WHAT IS RISK?

Before we can begin to describe what financial risk managers do, we need to understand what financial risk is. In finance, risk arises from uncertainty surrounding future profits or returns. There are many ways to define risk, and we may change the definition slightly, depending on the task at hand.

In everyday speech, the word risk is associated with the possibility of negative outcomes. For something to be risky, the final outcome must be uncertain and there must be some possibility that the final outcome will have negative consequences. While this may seem obvious, some popular risk measures treat positive and negative outcomes equally, while others focus only negative outcomes. For this reason, in order to avoid any ambiguity when dealing specifically with negative outcomes, risk managers will often talk about downside risk.

Risk is often defined relative to expectations. If we have one investment with a 50/50 chance of earning $0 or $200, and a second investment with a 50/50 chance of earning $400 or $600, are both equally risky? The first investment earns $100 on average, and the second $500, but both have a 50/50 chance of being $100 above or below this expected value. Because the deviations from expectations are equal, many risk managers would consider the two investments to be equally risky. By this logic, the second investment is more attractive because it has a higher expected return, not because it is less risky.

It is also important to note that risk is about possible deviations from expectations. If we expect an investment to make $1 and it does make $1, the investment was not necessarily risk free. If there were any possibility that the outcome could have been something other than $1, then the investment was risky.

Absolute, Relative, and Conditional Risk

There may be no better way to understand the limits of financial risk management—why and where it may fail or succeed—than to understand the difference between absolute, relative, and conditional risk.

Financial risk managers are often asked to assign probabilities to various financial outcomes. What is the probability that a bond will default? What is the probability that an equity index will decline by more than 10% over the course of a year? These types of predictions, where risk managers are asked to assess the total or absolute risk of an investment, are incredibly difficult to make. As we will see, assessing the accuracy of these types of predictions, even over the course of many years, can be extremely difficult.

It is often much easier to determine relative risk than to measure risk in isolation. Bond ratings are a good example. Bond ratings can be used to assess absolute risk, but they are on much surer footing when used to assess relative risk. The number of defaults in a bond portfolio might be much higher or lower next year depending on the state of the economy and financial markets. No matter what happens, though, a portfolio consisting of a large number of AAA‐rated bonds will almost certainly have fewer defaults than a portfolio consisting of a large number of C‐rated bonds. Similarly, it is much easier to say that emerging market equities are riskier than U.S. equities, or that one hedge fund is riskier than another hedge fund.

What is the probability that the S&P 500 will be down more than 10% next year? What is the probability that a particular U.S. large‐cap equity mutual fund will be down more than 8% next year? Both are very difficult questions. What is the probability that this same mutual fund will be down more than 8%, if the S&P 500 is down more than 10%? This last question is actually much easier to answer. What's more, these types of conditional risk forecasts immediately suggest ways to hedge and otherwise mitigate risk.

Given the difficulty of measuring absolute risk, risk managers are likely to be more successful if they limit themselves to relative and conditional forecasts, when possible. Likewise, when there is any ambiguity about how a risk measure can be interpreted —as with bond ratings— encouraging a relative or conditional interpretation is likely to be in a risk manager's best interest.

Intrinsic and Extrinsic Risk

Some financial professionals talk about risk versus uncertainty. A better approach might be to contrast intrinsic risk and extrinsic risk.

When evaluating financial instruments, there are some risks that we consider to be intrinsic. No matter how much we know about the financial instrument we are evaluating, there is nothing we can do to reduce this intrinsic risk (other than reducing the size of our investment).

In other circumstances risk is due only to our own ignorance. In theory, this extrinsic risk can be eliminated by gathering additional information through research and analysis.

As an example, an investor in a hedge fund may be subject to both extrinsic and intrinsic risk. A hedge fund investor will typically not know the exact holdings of a hedge fund in which they are invested. Not knowing what securities are in a fund is extrinsic risk. For various reasons, the hedge fund manager may not want to reveal the fund's holdings, but, at least in theory, this extrinsic risk could be eliminated by revealing the fund's holdings to the investor. At the same time, even if the investor did know what securities were in the fund, the returns of the fund would still not be fully predictable because the returns of the securities in the fund's portfolio are inherently uncertain. This inherent uncertainty of the security returns represents intrinsic risk and it cannot be eliminated, no matter how much information is provided to the investor.

Interestingly, a risk manager could reduce a hedge fund investor's extrinsic risk by explaining the hedge fund's risk guidelines. The risk guidelines could help the investor gain a better understanding of what might be in the fund's portfolio, without revealing the portfolio's precise composition.

Differentiating between these two fundamental types of risk is important in financial risk management. In practice, financial risk management is as much about reducing extrinsic risk as it is about managing intrinsic risk.

Risk and Standard Deviation

At the start of this chapter, we said that risk could be defined in terms of possible deviations from expectations. This definition is very close to the definition of standard deviation in statistics. The variance of a random variable is the expected value of squared deviations from the mean, and standard deviation is just the square root of variance. This is indeed very close to our definition of risk, and in finance risk is often equated with standard deviation.

While the two definitions are similar, they are not exactly the same. Standard deviation only describes what we expect the deviations will look like on average. Two random variables can have the same standard deviation, but very different return profiles. As we will see, risk managers need to consider other aspects of the distribution of expected deviations, not just its standard deviation.

WHAT IS FINANCIAL RISK MANAGEMENT?

In finance and in this book, we often talk about risk management, when it is understood that we are talking about financial risk management. Risk managers are found in a number of fields outside of finance, including engineering, manufacturing, and medicine.

When civil engineers are designing a levee to hold back flood waters, their risk analysis will likely include a forecast of the distribution of peak water levels. An engineer will often describe the probability that water levels will exceed the height of the levee in terms similar to those used by financial risk managers to describe the probability that losses in a portfolio will exceed a certain threshold. In manufacturing, engineers will use risk management to assess the frequency of manufacturing defects. Motorola popularized the term Six Sigma to describe its goal to establish a manufacturing process where manufacturing defects were kept below 3.4 defects per million. (Confusingly the goal corresponds to 4.5 standard deviations for a normal distribution, not 6 standard deviations, but that's another story.) Similarly, financial risk managers will talk about big market moves as being three‐sigma events or six‐sigma events. Other areas of risk management can be valuable sources of techniques and terminology for financial risk management.

Within this broader field of risk management, though, how do we determine what is and is not financial risk management? One approach would be to define risk in terms of organizations, to say that f...

Table of contents

• COVER
• TABLE OF CONTENTS
• PREFACE
• ABOUT THE AUTHOR
• 1 OVERVIEW OF FINANCIAL RISK MANAGEMENT
• 2 MARKET RISK: STANDARD DEVIATION
• 3 MARKET RISK: VALUE AT RISK
• 4 MARKET RISK: EXPECTED SHORTFALL, AND EXTREME VALUE THEORY
• 5 MARKET RISK: PORTFOLIOS AND CORRELATION
• 6 MARKET RISK: BEYOND CORRELATION
• 7 MARKET RISK: RISK ATTRIBUTION
• 8 CREDIT RISK
• 9 LIQUIDITY RISK
• 10 BAYESIAN ANALYSIS
• 11 BEHAVIORAL ECONOMICS AND RISK
• APPENDIX A MAXIMUM LIKELIHOOD ESTIMATION
• APPENDIX B COPULAS
• ANSWERS TO END-OF-CHAPTER QUESTIONS
• REFERENCES
• INDEX
• END USER LICENSE AGREEMENT