Risk Budgeting: Portfolio Problem Solving with Value-at-Risk

preface

This book describes the tools and techniques of value-at-risk and risk decomposition, which underlie risk budgeting. Most readers will never actually compute a value-at-risk (VaR) estimate. That is the role of risk measurement and portfolio management systems. Nonetheless, it is crucial that consumers of value-at-risk estimates and other risk measures understand what is inside the black box. This book attempts to teach enough so the reader can be a sophisticated consumer and user of risk information. It is hoped that some readers of the book will actually use risk information to do risk budgeting.

While it is not intended primarily for a student audience, the level of the book is that of good MBA students. That is, it presumes numeracy (including a bit of calculus), some knowledge of statistics, and some familiarity with the financial markets and institutions, including financial derivatives. This is about the right level for much of the practicing portfolio management community. The book presents sophisticated ideas but avoids the use of high-brow mathematics. The important ideas are presented in examples. That said, the book does contain some challenging material.

Every effort has been made to make the book self-contained. It starts with the basics of value-at-risk before moving on to risk decomposition, refinements of the basic techniques, and issues that arise with VaR and risk budgeting. The book is organized into five parts. Part I (Chapters 1–2) presents the concept of value-at-risk in the context of a simple equity portfolio and introduces some of the ways it can be used in risk decomposition and budgeting. Then, Part II (Chapters 3–9) describes the basic approaches to computing value-at-risk and creating scenarios for stress testing. Following this description of value-at-risk methodologies, Part III (Chapters 11–13) turns to using value-at-risk in risk budgeting and shows how risk decomposition can be used to understand and control the risks in portfolios. A few refinements of the basic approaches to computing value-at-risk are described in Part IV (Chapters 14–16). Recognizing that value-at-risk is not perfect, Part V (Chapters 17–19) describes some of its limitations, and Part VI (Chapter 20) concludes with a brief discussion of some issues that arise in risk budgeting. Clearly some readers will want to skip the first few chapters on the basic value-at-risk techniques. The notes to the chapters guide diligent readers toward much of the original (and sometimes mathematically challenging) work on value-at-risk.

It should also be said that the book does not address credit, operational, or other risks. It is about measuring market risk. Also, it stays away from software packages, partly because it is hoped that the shelf life of the book will be longer than the life cycle of computer software. I will be sorely disappointed if this turns out to be incorrect.

PART ONE

Introduction

CHAPTER 1

What Are Value-at-Risk and Risk Budgeting?

It is a truism that portfolio management is about risk and return. Although good returns are difficult to achieve and good risk-adjusted returns can be difficult to identify, the concept and importance of return requires no explanation. Larger returns are preferred to smaller ones. This is true at the level of the pension plan, at the level of each asset manager or portfolio used by or within the plan, and at the level of the individual assets. It follows from the fact that the contribution of an asset to the portfolio return is simply the asset’s weight in the portfolio.

Risk is more problematic. Risk is inherently a probabilistic or statistical concept, and there are various (and sometimes conflicting) notions and measures of risk. As a result, it can be difficult to measure the risk of a portfolio and determine how various investments and asset allocations affect that risk. Equally importantly, it can be difficult to express the risk in a way that permits it to be understood and controlled by audiences such as senior managers, boards of directors, pension plan trustees, investors, regulators, and others. It can even be difficult for sophisticated people such as traders and portfolio managers to measure and understand the risks of various instruments and portfolios and to communicate effectively about risk.

For years fund managers and plan sponsors have used a panoply of risk measures: betas and factor loadings for equity portfolios, various duration concepts for fixed income portfolios, historical standard deviations for all portfolios, and percentiles of solvency ratio distributions for long-term asset/liability analysis. Recently the fund management and plan sponsor communities have become interested in value-at-risk (VaR), a new approach that aggregates risks to compute a portfolio- or plan-level measure of risk. A key feature of VaR is that it is forward-looking, that is, it provides an estimate of the aggregate risk of the current portfolio over the next measurement period. The existence of a forward-looking aggregate measure of risk allows plan sponsors to decompose the aggregate risk into its various sources: how much of the risk is due to each asset class, each portfolio manager, or even each security? Alternatively, how much of the risk is due to each underlying risk factor? Once the contribution to aggregate risk of the asset classes, managers, and risk factors has been computed, one can then go on to the next step and use these risk measures in the asset allocation process and in monitoring the asset allocations and portfolio managers.

The process of decomposing the aggregate risk of a portfolio into its constituents, using these risk measures to allocate assets, setting limits in terms of these measures, and then using the limits to monitor the asset allocations and portfolio managers is known as risk allocation or risk budgeting. This book is about value-at-risk, its use in measuring and identifying the risks of investment portfolios, and its use in risk budgeting. But to write that the book is about value-at-risk and risk budgeting is not helpful without some knowledge of these tools. This leads to the obvious question: What are value-at-risk and risk budgeting?

VALUE-AT-RISK

Value-at-risk is a simple, summary, statistical measure of possible portfolio losses due to market risk. Once one crosses the hurdle of using a statistical measure, the concept of value-at-risk is straightforward. The notion is that losses greater than the value-at-risk are suffered only with a specified small probability. In particular, associated with each VaR measure are a probability α, or a confidence level 1 − α, and a holding period, or time horizon, h. The 1 − α confidence value-at-risk is simply the loss that will be exceeded with a probability of only α percent over a holding period of length h; equivalently, the loss will be less than the VaR with probability 1 − α. For example, if h is one day, the confidence level is 95% so that α = 0.05 or 5%, and the value-at-risk is one million dollars, then over a one-day holding period the loss on the portfolio will exceed one million dollars with a probability of only 5%. Thus, value-at-risk is a particular way of summarizing and describing the magnitude of the likely losses on a portfolio.

Crucially, value-at-risk is a simple, summary measure. This makes it useful for measuring and comparing the market risks of different portfolios, for comparing the risk of the same portfolio at different times, and for communicating these risks to colleagues, senior managers, directors, trustees, and others. Value-at-risk is a measure of possible portfolio losses, rather than the possible losses on individual instruments, because usually it is portfolio losses that we care most about. Subject to the simplifying assumptions used in its calculation, value-at-risk aggregates the risks in a portfolio into a single number suitable for communicating with plan sponsors, directors and trustees, regulators, and investors. Finally, value-at-risk is a statistical measure due to the nature of risk. Any meaningful aggregate risk measure is inherently statistical.

VaR’s simple, summary nature is also its most important limitation—clearly information is lost when an entire portfolio is boiled down to a single number, its value-at-risk. This limitation has led to the development of methodologies for decomposing value-at-risk to determine the contributions of the various asset classes, portfolios, and securities to the value-at-risk. The ability to decompose value-at-risk into its determinants makes it useful for managing portfolios, rather than simply monitoring them.

The concept of value-at-risk and the methodologies for computing it were developed by the large derivatives dealers (mostly commercial and investment banks) during the late 1980s, and VaR is currently used by virtually all commercial and investment banks. The phrase value-at-risk first came into wide usage following its appearance in the Group of Thirty report released in July 1993 (Group of Thirty 1993) and the release of the first version of RiskMetrics in October 1994 (Morgan Guaranty Trust Company 1994). Since 1993, the numbers of users of and uses for value-at-risk have increased dramatically, and the technique has gone through significant refinement.

The derivatives dealers who developed value-at-risk faced the problem that their derivatives portfolios and other trading books had grown to the point that the market risks inherent in them were of significant concern. How could these risks be measured, described, and reported to senior management and the board of directors? The positions were so numerous that they could not easily be listed and described. Even if this could be done, it would be helpful only if senior management and the board understood all of the positions and instruments, and the risks of each. This is not a realistic expectation, as some derivative instruments are complex. Of course, the risks could be measured by the portfolio’s sensitivities, that is, how much the value of the portfolio changes when various underlying market rates or prices change, and the option deltas and gammas, but a detailed discussion of these would likely only bore the senior managers and directors. Even if these concepts could be explained in English, exposures to different types of market risk (for example, equity, interest rate, and exchange rate risk) cannot meaningfully be aggregated without a statistical framework. Value-at-risk offered a way to do this, and therefore helped to overcome the problems in measuring and communicating risk information.

WHY USE VALUE-AT-RISK IN PORTFOLIO MANAGEMENT?

Similar issues of measuring and describing risk pervade the investment management industry. It is common for portfolios to include large numbers of securities and other financial instruments. This alone creates demand for tools to summarize and aggregate their risks. In addition, while most investment managers avoid complex derivative instruments with risks that are difficult to measure, some investment managers do use them, and some use complicated trading strategies. As a result, for many portfolios the risks may not be transparent even to the portfolio manager, let alone to the people to whom the manager reports.

Moreover, pension plans and other financial institutions often use multiple outside portfolio managers. To understand the risks of the total portfolio, the management, trustees, or board of directors ultimately responsible for an investment portfolio must first aggregate the risks across managers. Thus, although developed by derivatives dealers in a different context, value-at-risk is valuable in portfolio management applications because it aggregates risks across assets, risk factors, portfolios, and asset classes. In fact, a 1998 survey of pensions, endowments, and foundations reported that 23% of large institutional investors used value-at-risk.

Derivatives dealers typically express the value-at-risk as a dollar amount, while in investment management value-at-risk may be expressed as a percentage of the value of the portfolio. Given this, it is clear that value-at-risk is closely related to portfolio standard deviation, a concept that has been used by quantitative portfolio managers since they first existed. In fact, if we assume that portfolio returns are normally distributed (an assumption made in some VaR methodologies), value-at-risk is proportional to the difference between the expected change in the value of a portfolio and the portfolio’s standard deviation. In investment management contexts, value-at-risk is often expressed relative to the return on a benchmark, making it similar to the standard deviation of the tracking error. What then is new or different about value-at-risk?

Crucially, value-at-risk is a forward-looking measure of risk, based on current portfolio holdings. In contrast, standard deviations of returns and tracking errors are typically computed using historical fund returns and contain useful risk information only if one assumes both consistency on the part of the portfolio managers and stability in the market environment. Because value-at-risk is a forward-looking measure, it can be used to identify violations of risk limits, unwanted risks, and managers who deviate from their historical styles before any negative outcomes occur.

Second, value-at-risk is equally applicable to equities, bonds, commodities, and derivatives and can be used to aggregate the risk across different asset classes and to compare the market risks of different asset classes and portfolios. Since a plan’s liabilities often can be viewed as negative or short positions in fixed-income instruments, value-at-risk can be used to measure the risk of a plan’s net asset/liability position. Because it aggregates risk across risk factors, portfolios, and asset classes, it enables a portfolio manager or plan sponsor to determine the extent to which different risk factors, portfolios, and asset classes contribute to the total risk.

Third, the focus of value-at-risk is on the tails of the distribution. In particular, value-at-risk typically is computed for a confidence level of 95%, 99%, or even greater. Thus, it is a measure of downside risk and can be used with skewed and asymmetric distributions of returns.

Fourth, the popularity of value-at-risk among derivatives dealers has led to a development and refinement of methods for estimating the probability distribution of changes in portfolio value or returns. These methodologies are a major contribution to the development of value-at-risk, and much of this book is devoted to describing them.

Finally, and perhaps most importantly, the development of the concept of value-at-risk, and even the name itself, has eased the communication of information about risk. Phrases such as portfolio standard deviation and other statistical concepts are perceived as the language of nerds and geeks and are decidedly not the language of a typical pension plan trustee or company director. In contrast, value and risk are undeniably business words, and at is simply a preposition. This difference in terminology overcomes barriers to discussing risk and greatly facilitates the communication of information about it.

RISK BUDGETING

The concept of risk budgeting is not nearly as well defined as value-at-risk. In fact, it has been accused of being only a buzzword. Not surprisingly, it is also controversial. That it is a controversial buzzword is one thing upon which almost everyone can agree. But risk budgeting is more than a buzzword.

Narrowly defined, risk budgeting is a process of measuring and decomposing risk, using the measures in asset-allocation decisions, assigning portfolio managers risk budgets defined in terms of these measures, and using these risk budgets in monitoring the asset allocations and portfolio managers. A prerequisite for risk budgeting is risk decomposition, which involves

identifying the various sources of risk, or risk factors, such as equity returns, interest rates, and exchange rates;

measuring each factor’s, manager’s, and asset class’s contribution to the total risk;

comparing the ex post realized outcomes to the ex ante risk; and

identifying the risks that were taken intentionally, and those taken inadvertently.

This risk decomposition allows a plan sponsor to have a better understanding of the risks being assumed and how they have changed, and to have more informed conversations with the portfolio managers. In the event that there are problems, it allows the sponsor to identify unwanted risks and managers who deviate from their historical styles before any negative outcomes occur.

If this risk decomposition is combined with an explicit set of risk allocations to factors, managers, or asset classes, it is called risk allocation or risk budgeting. The risk budgeting process itself consists of

setting limits, or risk budgets, on the quantity of risk due to each asset class, manager, or factor;

establishing asset allocations based on the risk budgets;

comparing the risk budgets to the measures of the risk due to each factor on an ongoing basis; and

adjusting the asset allocations to keep the risks within the budgeted limits.

Risk decomposition is crucial to risk budgeting, because the aggregate value-at-risk of the pension plan or other organization is far removed from the portfolio managers. At the risk of stating the obvious, the portfolio managers have control only over their own portfolios. For them, meaningful risk budgets are expressed in terms of their contributions to portfolio risk.

However, risk budgeting is more than a list of steps or procedures. Defined more broadly, risk budgeting is a way of thinking about investment and portfolio management. For this reason, to find a definition that attracts broad agreement is difficult, and perhaps impossible. The world view that underlies risk budgeting takes for granted reliance upon probabilistic or statistical measures of risk and the use of modern risk- and portfolio-management tools to manage risk. Thinking about the asset-allocation problem in terms of risk allocations rather than traditional asset allocations is a natural outgrowth of this world view.

From a logical perspective, there is no special relation between value-at-risk and risk budgeting. Risk budgeting requires a measure of portfolio risk, and value-at-risk is one candidate. It is a natural candidate, in that: (i) it is a measure of downside risk, and thus useful when the distribution of portfolio returns is asymmetric; and (ii) when returns are normally distributed, it is equivalent to a forward-looking estimate of portfolio standard deviation. However, the risk budgeting process could be implemented using any of a number of risk measures. For example, it could be implemented using either a forward-looking estimate of portfolio standard deviation or a scenario-based measure of the type advocated by Artzner, et al. (1997, 1999) and described in chapter 19. In fact, it is widely recommended that value-at-risk measures be used in combination with stress testing (procedures to estimate the losses that might be incurred in extreme or stress scenarios).

In practice, however, value-at-risk and risk budgeting are intimately related. Because risk budgeting involves the quantification, aggregation, and decomposition of risk, the availability of a well-recognized aggregate measure of portfolio risk is a prerequisite for its use and acceptance. In this sense, risk budgeting is an outgrowth of value-at-risk. But for the popularity and widespread acceptance of value-at-risk, you would likely not be hearing and reading about risk budgeting today. Nonetheless, value-at-risk has some well known limitations, and it may be that some other risk measure eventually supplants value-at-risk in the risk budgeting process.

DOES RISK BUDGETING USING VaR MAKE SENSE?

To those who share its underlying world view, the process of risk budgeting outlined above is perfectly natural—how else would one think about asset allocation? Of course, one can think about asset allocation in the traditional way, in terms of the fractions of the portfolio invested in each asset class. But seen through the lens of risk budgeting, the traditional approach is just an approximation to the process described above, where portfolio weights proxy for risk measures. An advantage of risk budgeting over this traditional view of asset allocation is that it makes explicit the risks being taken and recognizes that they change over time. In addition, risk budgeting provides a natural way to think about nontraditional asset classes, such as hedge funds and the highly levered strategies often pursued by them. In contrast to traditional asset classes, the dollar investment in a highly leveraged strategy often says little about the quantity of risk being taken, and the label hedge fund does not reveal the nature of the risks.

A significant part of the controversy stems from the broader definition of risk budgeting as the natural outgrowth of a way of thinking about investment and portfolio management. This is not about the precise definition of risk budgeting (i.e., whether the preceding list of the steps that define the risk budgeting process is better or worse than another) or whether risk budgeting is cost effective. Much of the controversy seems to stem from the fact that not all plan sponsors and portfolio managers share the same underlying paradigm. This is not just the source of the controversy; the difference in world views is much of the controversy. It is difficult to imagine that it will ever be resolved.

However, some of the disagreement about risk budgeting is eminently practical and can be addressed by a book. The computation of value-at-risk, and the processes of risk decomposition and risk budgeting, involve considerable trouble and expense. Given the imperfections of and errors in quantitative measures such as value-at-risk, reasonable people who share the view of portfolio management underlying risk budgeting may nonetheless conclude that it is not cost effective, that is, that the additional information about and understanding of portfolio risk provided by the risk budgeting process are not worth the cost that must be incurred. It is likely that the practical argument against risk budgeting will become less compelling over time, as increases in the extent of risk-management education and knowledge and the evolution of risk-measurement systems both increase the benefits and reduce the costs of the risk budgeting process. Regardless, to make an informed judgment about the benefits, limitations, and cost-effectiveness of value-at-risk and risk budgeting requires an understanding of them. One of the goals of this book is to provide enough information about value-at-risk methodologies and risk budgeting to enable readers to understand them and make informed choices about them.

NOTES

The development of value-at-risk is generally attributed to J.P. Morgan (e.g., see Guldimann 2000). To my knowledge, the first publication in which the phrase appeared was the widely circulated Group of Thirty report (Group of Thirty 1993). It was subsequently popularized by the RiskMetrics system originally developed by J.P. Morgan (Morgan Guaranty Trust Company 1994).

The use of the phrase 1 − α percent confidence VaR to mean the loss that is exceeded with a probability of α percent over a holding period of length h is a misuse of the terminology confidence or confidence level. A better terminology would be to refer to the α or 1 − α quantile VaR, because value-at-risk is the α quantile of the distribution of portfolio profits (or returns), or, equivalently, the 1 − α quantile of the loss distribution. However, the misuse of the terminology confidence in the context of value-at-risk is well established, and this book will not try to fight it.

Since 1995, the Basel Committee on Banking Supervision and the International Organization of Securities Commissions have been examining the risk-management procedures and disclosures of leading banks and securities firms in the industrialized world. The latest surveys (Basel Committee on Banking Supervision and the International Organization of Securities Commissions 1999 and Basel Committee on Banking Supervision 2001) indicated that virtually all banks and securities firms covered by the survey used value-at-risk techniques to measure market risk. The finding that 23% of institutional investors use value-at-risk is from the 1998 Survey of Derivative and Risk Management Practices by U.S. Institutional Investors conducted by New York University, CIBC World Markets, and KPMG (Levich, Hayt, and Ripston 1999; Hayt and Levich 1999).

The nature of the controversy about risk budgeting is described by Cass (2000), who describes the debate at the Risk 2000 Congress in June 2000. Cass quotes Harris Lirtzman of the New York City Retirement Systems as saying: There is almost a theological divide in this discussion among public plan sponsors—VaR versus non-VaR, risk budgeting versus asset allocation.

CHAPTER 2

Value-at-Risk of a Simple Equity Portfolio

To introduce the concept of value-at-risk, consider a simple example of a portfolio exposed to changes in the U.S. and U.K. stock market indexes. The portfolio consists of $110 million invested in a well-diversified portfolio of large-capitalization U.S. equities, together with positions in U.S. (S&P 500) and U.K. (FT-SE 100) index futures contracts. The portfolio of U.S. equities is well diversified, and its returns are highly correlated with the returns on the S&P 500 index. For simplicity, it is assumed that the returns on the portfolio are perfectly correlated with changes in the S&P 500 index. To gain exposure to the U.K. market, the portfolio manager has established a long position of 500 FT-SE 100 index futures contracts traded on the London International Financial Futures Exchange (LIFFE). Through the standard cost-of-carry formula for the futures price (see the notes to this chapter) and using the multiplier of £10, a one-point change in the FT-SE 100 index results in a £10.131 change in the position value. The current value of the FT-SE 100 is 5862.3, so the index futures position is equivalent to an investment of £29.696 million in the portfolio that underlies the index. At the current exchange rate of 1.6271 $/£, this is equivalent to an investment of $48.319 million in the portfolio underlying the index.

To reduce his exposure to the U.S. market, the portfolio manager has shorted 200 of the S&P 500 index futures contract traded on the Chicago Mercantile Exchange (CME). The current level of the S&P index is 1097.6, and the contract has a multiplier of 250, so, through the cost-of-carry formula, a one-point change in the index results in a $253.48 change in the position value, implying that this position is equivalent to a short position of $55.643 million in the portfolio that underlies the S&P 500 index. Combined with the $110 million invested in the cash market, the combined stock and futures position is equivalent to an investment of $54.357 million in the index portfolio.

It has been estimated that the standard deviation of monthly rates of return on the portfolio underlying the S&P 500 index is σ1 = 0.061 (6.1%), the standard deviation of monthly rates of return on the portfolio underlying the FT-SE 100 index is σ2 = 0.065 (6.5%), and the correlation between the monthly rates of return is estimated to be ρ = 0.55. The expected rates of change in the S&P 500 and FT-SE 100 indexes are estimated to be μ1 = 0.01 (1%) and μ2 = 0.0125 (1.25%) per month, respectively. In addition, the portfolio of U.S. stocks pays dividends at the rate of 1.4% per year, or 1.4/12 = 0.1167% per month.

STANDARD VALUE-AT-RISK

To compute the value-at-risk, we need to pick a holding period and a confidence level 1 − α. We choose the holding period to be one month and somewhat arbitrarily pick a confidence level of 1 − α = 95%, or α = 5%. Given these choices and the information above, it is easy to compute the value-at-risk if one assumes that the returns on the S&P 500 and FT-SE 100 are normally distributed. If they are, then the portfolio return is also normally distributed and the expected change and variance of the value of the portfolio can be calculated using standard mathematical results about the distributions of sums of normal random variables. Then, because the normal distribution is completely determined by the expected value and variance, we know the distribution of profit or loss over the month.

For example, suppose that the distribution of possible profits and losses on a portfolio can