## 1.7.5 Example: Australian Equities (Linear Remapping)

###### 1.7.5 Example: Australian Equities (Linear Remapping)

As an alternative solution, let’s approximate θ with a linear polynomiale  based upon the gradient3 of θ. We must choose a point at which to take the gradient. A reasonable choice is 0E(), which is the expected value of , conditional on information available at time 0. Let’s assume 0E() = . We define

[1.40]

Our approximation = () of the portfolio mapping is an example of a portfolio remapping. We obtain = (.20.080, 10.800, 1.150, 0.3892) from Exhibit 1.5 and evaluate

[1.41]

[1.42]

Our remapping [1.40] is

[1.43]

[1.44]

[1.45]

The portfolio remapping is represented schematically as

[1.46]

The upper part of the schematic is precisely schematic [1.34] indicating the original portfolio mapping . The lower part indicates the remapping . In such schematics, vertical arrows indicate approximations. approximates .

Because [1.45] is a linear polynomial, we can apply [1.11] to obtain the conditional standard deviation 1| of :

[1.47]

Assume is conditionally normal with this conditional standard deviation 1| and conditional mean 1| = . The .05-quantile of a normal distribution occurs 1.645 standard deviations below its mean, so

[1.48]

and the .95-quantile of portfolio loss is

[1.49]

The portfolio’s 1-day 95% GBPvalue-at-risk is approximately GBP 5,876. This result compares favorably with our previous result of GBP 5,925, which we obtained with the Monte Carlo method.

## 1.7.4 Example: Australian Equities (Monte Carlo Transformation)

###### 1.7.4 Example: Australian Equities (Monte Carlo Transformation)

We discuss the Monte Carlo method formally in Chapter 5. For now, an intuitive treatment will suffice. We assume is joint-normal with conditional mean vector  1|0μ  = and conditional covariance matrix 1|0Σ given by [1.35] Based upon these assumptions, we “randomly” generate 10,000 realizations, , , … , , of . We set

[1.36]

for each k, constructing 10,000 realizations, , , … , , of . Results are indicated in Exhibit 1.6.

Realizations of are summarized with a histogram in Exhibit 1.7. We may approximate any parameter of with the corresponding sample parameter of the realizations.

Exhibit 1.06: Results of the Monte Carlo analysis
Exhibit 1.07: Histogram of realizations for the portfolio’s value at time 1

The sample .05-quantile of our realizations is GBP 191,614. We use this as an approximation of the .05-quantile, (.05), of . The .95-quantile of portfolio loss is:

[1.37]

[1.38]

[1.39]

The portfolio’s 1-day 95% GBPvalue-at-risk is approximately GBP 5925.

## 1.7.3 Example: Australian Equities

###### 1.7.3 Example: Australian Equities

Our next example is ostensibly similar to the last. As we work through it, a number of issues will arise as a consequence of exchange rate risk. These will motivate different approaches for a solution.

Suppose today is March 9, 2000. A British trader holds a portfolio of Australian stocks. We wish to calculate the portfolio’s 1-day 95% GBPvalue-at-risk. The portfolio’s current value is GBP 0.198MM. Let represent its value tomorrow. Define the random vector

[1.28]

accumulated values reflect price changes, dividends, and changes in the GBP/AUD exchange rate since time 0. The portfolio’s holdings are:

• 10,000 shares of National Australia Bank,
• 30,000 shares of Westpac Banking Corp.,
• –15,000 shares of Goodman Fielder (short position),

which we represent with a row vector

[1.29]

The portfolio’s future value is a linear polynomial of :

[1.30]

We face a minor problem. In the last example, we used historical data to construct a covariance matrix for . In the present example, components of are denominated in GBP, but any historical data for Australian stocks will be denominated in AUD. We solve the problem with a change of variables :

[1.31]

where

[1.32]

Composing ω with φ we obtain a function that relates to :

[1.33]

This is a quadratic polynomial—the exchange rate combines multiplicatively with the accumulated values , , . It is our portfolio mapping, and we represent it schematically as

[1.34]

Exhibit 1.5 provides historical data for .

Exhibit 1.5: Two months of historical data for the GBP/AUD exchange rate and AUD prices for the indicated stocks. None of the stocks had ex-dividend dates during the period indicated. Source: Federal Reserve Bank of Chicago and Dow Jones.

Using time-series methods described in Chapter 4, we construct a conditional covariance matrix for :

[1.35]

Now we face another problem. We have a portfolio mapping = θ() that expresses as a quadratic polynomial of , and we have a conditional covariance matrix 1|0Σ for . This is similar to the previous example where we had a portfolio mapping that expressed as a linear polynomial of , and we had a covariance matrix 1|0Σ for . Critically, in the previous example, our portfolio mapping was linear. Now it is quadratic. In the previous example, we could apply [1.11] to obtain the conditional standard deviation of . Now we cannot.

Nonlinear portfolio mappings pose a recurring challenge for measuring value-at-risk. There are various solutions, including:

• apply the Monte Carlo method to approximate the desired quantile;
• approximate the quadratic polynomial θ with a linear polynomiale and then apply [1.11] as before;
• assume is conditionally joint-normal and apply probabilistic techniques appropriate for quadratic polynomials of joint-normal random vectors.

Each is a standard solution used frequently in value-at-risk measures. Each has advantages and disadvantages. We will study them all in later chapters. For now, we briefly describe how each is used to calculate value-at-risk for this Australian equities example.

## 1.7.2 Example: Industrial Metals

###### 1.7.2 Example: Industrial Metals

Suppose today’s date is June 30, 2000. A US metals merchant has a portfolio of unsold physical positions in several industrial metals. We wish to calculate the portfolio’s 1-week 90% USDvalue-at-risk. Measure time t in weeks. Specify the random vector

[1.12]

where accumulated values are in USD and reflect the value of a ton of metal accumulated from time 0 to time 1. Accumulated value might reflect price changes, cost of financing, warehousing, and insurance. For simplicity, we consider only price changes in this example.

Current values in USD/ton for the respective metals are

[1.13]

The portfolio’s holdings are:

• 1000 tons of aluminum,
• 2000 tons of copper,
• 250 tons of nickel,
• 1000 tons of tin, and
• 100 tons of zinc,

which we represent with a row vector:

[1.14]

The portfolio’s current value is

[1.15]

Its future value is random:

[1.16]

We call this relationship a portfolio mapping. We represent it schematically as

[1.17]

Let 1|0σ and 1|0Σ be the standard deviation of and the covariance matrix of , both conditional on information available at time 0. Let’s apply [1.11]. By [1.16], is a linear polynomial of , so:2

[1.18]

We know ω. We need 1|0Σ to obtain 1|0σ. Exhibit 1.4 indicates historical metals price data.

Exhibit 1.4: Thirty weekly historical prices for the indicated metals. All prices are in USD per ton. Source: London Metals Exchange (LME).

Applying time-series methods described in Chapter 4, we construct

[1.19]

Substituting [1.14] and [1.19] into [1.18], we conclude that has conditional standard deviation 1|0σ of USD 0.217MM.

Let denote the cumulative distribution function (CDF) of portfolio loss 1L conditional on information available at time 0. Its inverse  provides quantiles of 1L. Our value-at-risk metric is 1-week 90% USDvalue-at-risk, so we seek the .90-quantile, 1|, of portfolio loss 1L.

We have no expression for . All we have is a conditional standard deviation 1|0σ for . We need additional assumptions or information. A simple solution is to assume that is conditionally normal with conditional mean 1|0μ = = 13.011MM. Since a normal distribution is fully specified by a mean and standard deviation, we have now specified a conditional CDF, , for .

The .90-quantile of portfolio loss is

[1.20]

A property of normal distributions is that, as described in Section 3.10, a .10-quantile occurs 1.282 standard deviations below the mean, so

[1.21]

Substituting [1.21] into [1.20]:

[1.22]

The portfolio’s 1-week 90% USDvalue-at-risk is USD 0.278MM. Note that dropped out of the calculations entirely, so we did not actually need to calculate its value in [1.15].

###### Exercises
1.12

This exercise is based upon an equity example in Harry Markowitz’s (1959) book Portfolio Selection. Suppose today is January 1, 1955. Measure time t in years and define:

[1.23]

Each accumulated value represents the value at time 1 of an investment worth 1 USD at time 0 in the indicated stock. Accumulated values include price changes and dividends. Consider a portfolio with holdings

[1.24]

Based upon data provided by Markowitz, we construct a conditional covariance matrix 1|0Σ for :

[1.25]

Calculate the portfolio’s 1-year 90% USDvalue-at-risk according to the following steps:

1. Value the vector 0s. (Hint: Based upon how the problem has been presented, the answer is trivial.)
2. Using the formula , value .
3. Specify a portfolio mapping that defines as a linear polynomial of .
4. Draw a schematic for your portfolio mapping.
5. Determine the conditional standard deviation 1|0σ of using [1.11].
6. Assume is normally distributed with conditional mean 1|0μ = and conditional standard deviation obtained in part (e). Calculate the .10-quantile of with the formula

[1.26]

7. Calculate the portfolio’s 1-year 90% USDvalue-at-risk as

[1.27]

# 1.7 Examples

Let’s consider some examples of risk measures. These will introduce basic concepts and standard notation. They will also illustrate a framework for thinking about value-at-risk measures (and, more generally, measures of PMMRs), which we shall formalize in Section 1.8.

###### 1.7.1 Example: The Leavens PMMR

Value-at-risk metrics first emerged in finance during the 1980s, but they were preceded by various other PMMRs, including Markowitz’s (1952) variance of simple return. Even earlier, Leavens (1945) published a paper describing the benefits of diversification. He accompanied his explanations with a simple numerical example:

Measure time t in appropriate units. Let time t = 0 be the current time. Leavens considers a portfolio of 10 bonds over some horizon [0, 1]. Each bond will either mature at time 1 for USD 1000 or default and be worthless. Events of default are assumed independent. The portfolio’s market value at time 1 is given by the sum of the individual bonds’ accumulated values at time 1:

[1.6]

Let’s express this relationship in matrix notation. Let be a random vector with components . Let ω be a row vector whose components are the portfolio’s holdings in each bond. Since the portfolio holds one of each, ω has a particularly simple form:

[1.7]

With this matrix notation, [1.6] becomes the product:

[1.8]

Let 1|0ϕi denote the probability function, conditional on information available at time 0, of the ith bond’s value at time 1:

[1.9]

Measured in USD 1000s; the portfolio’s value has a binomial distribution with parameters n = 10 and p = 0.9. The probability function is graphed in Exhibit 1.3.

Exhibit 1.3: The market value (measured in USD 1000s) of Leavens’ bond portfolio has a binomial distribution with parameters 10 and 0.9.

Writing for a non-technical audience, Leavens does not explicitly identify a risk metric, but he speaks repeatedly of the “spread between probable losses and gains.” He seems to have the standard deviation of portfolio market value in mind. For the portfolio in his example, that PMMR has the value USD 948.69.

Our next two examples are more technical. Many readers will find them simple. Other readers—those whose mathematical background is not as strong—may find them more challenging. A note for each group:

• For the first group, the examples may tell you things you already know, but in a new way. They introduce notation and a framework for thinking about value-at-risk that will be employed throughout the text. At points, explanations may appear more involved than the immediate problem requires. Embrace this complexity. The framework we start to develop in the examples will be invaluable in later chapters when we consider more complicated value-at-risk measures.
• For the second group, you do not need to master the examples on a first reading. Don’t think of them as a main course. They are not even an appetizer. We are taking you back into the kitchen to sample a recipe or two. Don’t linger. Taste and move on. In Chapters 2 through 5, we will step back and explain the mathematics used in the examples—and used in value-at-risk measures generally. A purpose of the examples is to provide practical motivation for those upcoming discussions.

There is a useful formula that we will use in the next two examples. We introduce it here for use in those examples but will cover it again in more detail in Section 3.5.

Let X be a random vector with covariance matrix Σ. Define random variable Y as a linear polynomial

[1.10]

of X, where b is an n-dimensional row vector and a is a scalar. The variance of Y is given by

[1.11]

where a prime ′ indicates transposition. Formula [1.11] is a quintessential formula for describing how correlated risks combine, but there is a caveat. It only applies if Y is a linear polynomial of X.

###### Exercises
1.10

Using a spreadsheet, extend Leavens’ analysis to a bond portfolio that holds 20 bonds.

1. Graph the resulting probability function for .
2. Value Leavens’ “spread between probable losses and gains” PMMR for the portfolio?

1.11

Using only the information provided in the example, which of the following PMMR’s could we evaluate for Leavens’ bond portfolio:

1. 95% quantile of loss;
2. variance of portfolio value;
3. standard deviation of simple return.

# 1.5 Risk Limits

In a context where risk taking is authorized, risk limits are bounds placed on that risk taking.

Suppose a pension fund hires an outside investment manager to invest some of its assets in intermediate corporate bonds. The fund wants the manager to take risk on its behalf, but it has a specific form of risk in mind. It doesn’t want the manager investing in equities, precious metals, or cocoa futures. It communicates its intentions with contractually binding investment guidelines. These specify acceptable investments. They also place bounds on risk taking, such as requirements that:

• the portfolio’s duration always be less than 7 years;
• all bonds have a credit rating of BBB or better.

The first is an example of a market risk limit; the second of a credit risk limit

A risk limit has three components:

1. a risk metric,
2. a risk measure that supports the risk metric, and
3. a bound—a value for the risk metric that is not to be breached.

At any point in time, a limit’s utilization is the actual amount of risk being taken, as quantified by the risk measure. Any instance where utilization breaches the risk limit is called a limit violation.

A bank’s corporate lending department is authorized to lend to a specific counterparty subject to a credit exposure limit of GBP 10MM. For this purpose, the bank measures credit exposure as the sum amount of outstanding loans and loan commitments to the counterparty. The lending department lends the counterparty GBP 8MM, causing its utilization of the limit to be GBP 8MM. Since the limit is GBP 10MM, the lending department has remaining authority to lend up to GBP 2MM to the counterparty.

A metals trading firm authorizes a trader to take gold price risk subject to a 2000 troy ounce delta limit. Using a specified measure of delta, his portfolio’s delta is calculated at 4:30 PM each trading day. Utilization is calculated as the absolute value of the portfolio’s delta.

###### 1.5.1  Market Risk Limits

For monitoring market risk, many organizations segment portfolios in some manner. They may do so by trader and trading desk. Commodities trading firms may do so by delivery point and geographic region. A hierarchy of market risk limits is typically specified to parallel such segmentation, with each segment of the portfolio having its own limits. Limits generally increase in size as you move up the hierarchy—from traders to desks to the overall portfolio, or from individual delivery points to geographic regions to the overall portfolio.

Exhibit 1.1 illustrates how a hierarchy of market risk limits might be implemented for a trading unit. A risk metric is selected, and risk limits are specified based upon this. Each limit is depicted with a cylinder. The height of the cylinder corresponds to the size of the limit. The trading unit has three trading desks, each with its own limit. There are also limits for individual traders, but only those for trading desk A are shown. The extent to which each cylinder is shaded green corresponds to the utilization of that limit. Trader A3 is utilizing almost all his limit. Trader A4 is utilizing little of hers.

Exhibit 1.1: A hierarchy of market risk limits is illustrated for a hypothetical trading unit. A risk metric—value-at-risk, delta, etc.—is chosen. Risk limits are specified for the portfolio and sub-portfolios based upon this. The limits are depicted with cylinders. The height of each cylinder corresponds to the size of the limit. The degree to which it is shaded black indicates current utilization of the limit. Fractions next to each cylinder indicate utilization and limit size. Units are not indicated here, as these will depend upon the particular risk metric used. Individual traders have limits, but only those for traders on desk A are indicated in the exhibit.

For such a hierarchy of risk limits to work, an organization must have a suitable risk measure to calculate utilization of each risk limit on an ongoing basis. Below, we describe three types of market risk limits, culminating withvalue-at-risk limits.

###### 1.5.2  Stop-Loss Limits

A stop-loss limit indicates an amount of money that a portfolio’s single-period market loss should not exceed. Various periods may be used, and sometimes multiple stop-loss limits are specified for different periods. A trader might be given the following stop-loss limits:

• 1-day EUR 0.5MM;
• 1-week EUR 1.0MM;
• 1-month EUR 3.0MM:

A limit violation occurs whenever a portfolio’s single-period market loss exceeds a stop-loss limit. In such an event, a trader is usually required to hedge material exposures—hence the name stop-loss limit. Stop-loss limits have shortcomings:

• Single-period market loss is a retrospective measure of risk. It only indicates risk after financial consequences of that risk have been realized.
• Single-period loss provides an inconsistent indication of risk. If a portfolio suffers a large loss over a given period, this is a clear indication of risk. If the portfolio does not suffer a large loss, this does not indicate an absence of risk!
• Traders cannot control the specific losses they incur, so it is difficult to hold them accountable for isolated stop-loss limit violations.

Despite their shortcomings, stop-loss limits are simple and convenient to use. Non-specialists easily understand stop-loss limits. A single proxy for risk—experienced loss—can be applied consistently across different types of exposures and with different trading strategies. Calculating utilization is as simple (or difficult, in some cases) as marking a portfolio to market. For these reasons, stop-loss limits are widely implemented by trading organizations.

###### 1.5.3  Exposure Limits

Exposure limits are limits based upon an exposure risk metric. For limiting market risk, common metrics include: duration, convexity, delta, gamma, and vega. Crude exposure limits may also be based upon notional amounts. These are called notional limits. Many exposure metrics can take on positive or negative values, so utilization may be defined as the absolute value of exposure.

Exposure limits address many of the shortcomings of stop-loss limits. They are prospective, indicating risk before its financial consequences are realized. Also, exposure metrics provide a reasonably consistent indication of risk. For the most part, traders can be held accountable for exposure limit violations because they largely control their portfolio’s exposures. There are rare exceptions. A sudden market rise may cause a positive-gamma portfolio’s delta to increase, resulting in an unintended delta limit violation. For the most part, utilization of exposure limits is easy to calculate. There may be analytic formulas for certain exposure metrics. At worst, a portfolio must be valued under multiple market scenarios with some form of interpolation applied to assess exposure.

Exposure limits pose a number of problems:

• At higher levels of portfolio aggregation, exposure limits can multiply. While a trader who transacts in 30 stocks might require 30 delta limits, the entire trading floor he works on might transact in 3,000 stocks, requiring 3,000 delta limits.
• Different exposure limits may be required to address dissimilar exposures or different trading strategies. For example, delta might need to be supplemented with duration or rho to address yield curve risk. It might need to be supplemented with gamma or vega to address options risk.
• Custom exposure limits may be required to address specialized trading strategies such as cross-hedging, spread trading or pairs trading that reduce risk by taking offsetting positions in correlated assets. In such contexts, any delta limits must be large to accommodate each of the offsetting positions. Being so large, they cannot ensure reasonable hedging consistent with the intended trading strategy.
• With the exception of notional limits, non-specialists do not easily understand exposure limits. For example, it is difficult to know what might be a reasonable delta limit for an electricity trading desk if you don’t have both a technical understanding of what delta means and practical familiarity with the typical size of market fluctuations in the electricity market.
###### 1.5.4  Value-at-Risk Limits

Value-at-risk is used for a variety of tasks, but supporting risk limits is its quintessential purpose. When risk limits are measured in terms of value-at-risk, they are called value-at-risk limits. These combine many of the advantages of exposure limits and stop-loss limits.

Like exposure metrics, value-at-risk metrics are prospective. They indicate risk before its economic consequences are realized. Also like exposure metrics, value-at-risk metrics provide a reasonably consistent indication of risk. Finally, as long as utilization is calculated for traders in a timely and ongoing manner, it is reasonable to hold those traders accountable forvalue-at-risk limit violations. As with exposure limits, there are rare exceptions. Consider a trader with a negative gamma position. While she is responsible for hedging the position on an ongoing basis, it is possible that a sudden move in the underlier will cause an unanticipated spike in value-at-risk.

As with stop-loss limits, non-specialists may intuitively understandvalue-at-risk limits. If a portfolio has 1-day 90% USDvalue-at-risk of 7.5MM, a non-specialist can understand that such a portfolio will lose less than USD 7.5MM an average of 9 days out of 10. As with stop-loss limits, a single limit can suffice at each level of portfolio aggregation—at the position level, trader level, trading desk level, sub-portfolio level and portfolio level. Andvalue-at-risk limits are uniformly applicable to all sources of market risk and all trading strategies. Of course, for value-at-risk, such generality is theoretical. The ability of a particular value-at-risk measure to address the market risk associated with specific instruments or trading strategies depends on the generality and sophistication of that particular value-at-risk measure.

This brings us to the drawbacks ofvalue-at-risk limits:

• Depending on the level of generality and/or sophistication required of value-at-risk measures, they can be difficult to implement. This book is a testament to (and hopefully a palliative for) the potentially complicated analytics value-at-risk measures require.
• Utilization of somevalue-at-risk limits may be computationally expensive to calculate. While value-at-risk can be calculated in real time or near-real time for many portfolios, it may take minutes or hours to calculate for others.
• While most risk metrics entail some model risk or potential for manipulation, the complexity of value-at-risk measures make them particularly vulnerable.

The last point was illustrated in the aftermath of JPMorgan’s 2012 “London Whale” trading scandal. It came to light that bank employees had manipulated portfolio valuations, undermining stop-loss limits. They had also replaced a value-at-risk measure with a rudimentary spreadsheet, which further understated risk.

###### 1.5.5  Summary Comparison

Exhibit 1.2 summarizes the strengths and weakness of stop-loss, exposure, andvalue-at-risk limits.

Exhibit 1.2: Comparison of stop-loss, exposure, andvalue-at-risk limits.

# 1.4 Value-at-Risk

Suppose an investment fund indicates that, based on the composition of its portfolio and on current market conditions, there is a 90% probability it will either make a profit or otherwise not lose more than USD 2.3MM over the next trading day. This is an example of a value-at-risk (VaR) measurement. For a given time period and probability, a value-at-risk measure purports to indicate an amount of money such that there is that probability of the portfolio not losing more than that amount of money over that time period. Stated another way, value-at-risk purports to indicate a quantile of the probability distribution for a portfolio’s loss over a specified time period.

To specify a value-at-risk metric, we must identify three things:

1. The period of time over which a possible loss will be calculated—1 day, 2 weeks, 1 month, etc. This is called the value-at-risk horizon. In our example, the value-at-risk horizon is one trading day.
2. A quantile of that possible loss. In the example, the portfolio’s value-at-risk is expressed as a .90 quantile of loss.
3. The currency in which the possible loss is denominated. This is called the base currency. In our example, the base currency is USD.

In this book, we measure time in units equal to the length of the value-at-risk horizon, which always starts at time 0 and ends at time 1. We adopt the following convention for naming value-at-risk metrics: the metric’s name is given as the horizon, quantile, and currency, in that order, followed by “VaR” or “value-at-risk”. If the horizon is expressed in days without qualification, these are understood to be trading days. The quantile q is generally indicated as a percentage. Based on this convention, the value-at-risk metric of the investment fund in our example above is one-day 90% USD value-at-risk. If a British bank calculates value-at-risk as the 0.99 quantile of loss over ten trading days, as required under the Basel Accords, this would be called 10-day 99% GBPvalue-at-risk.

###### 1.4.1 Probabilistic Metrics of Market Risk (PMMRs)

Value-at-risk is one example of a category of risk metrics that we might call probabilistic metrics of market risk (PMMRs). While this book focuses on value-at-risk, we shall see that the computations one performs to calculate value-at-risk are mostly identical to those you would perform to calculate any PMMR. In this section, we formalize the notion of value-at-risk metrics by first formalizing PMMRs. This will provide a general perspective for understanding value-at-risk in a context of other familiar market risk metrics.

Suppose a portfolio were to remain untraded for a certain period—say from the current time 0 to some future time 1. The portfolio’s market value at the start of the period is known. Its market value at the end of the period is unknown. It is a random variable. As a random variable, we may assign it a probability distribution conditional upon information available at time 0. We might quantify the portfolio’s market risk with some real-valued parameter of that conditional distribution.

Formally, we define a PMMR as a real-valued function of:

• the distribution of conditional on information available at time 0; and
• the portfolio’s current value .

Standard deviation of , conditional on information available at time 0, is an example:

[1.1]

Volatility, defined as the standard deviation of a portfolio’s simple return 1Z, is a PMMR:

[1.2]

If we define portfolio loss as

[1.3]

then the conditional standard deviation of 1L is a PMMR:

[1.4]

###### 1.4.2 Value-at-Risk as a PMMR

Let and denote cumulative distribution functions (CDFs) of 1P and 1L, conditional on information available at time 0. The preceding superscripts 1|0 are a convention to alert you that the distributions are “for random variables at time 1 but conditional on information available at time 0.”

If these conditional CDFs are strictly increasing, their inverses and exist and provide quantiles of 1P and 1L. As we have already indicated, value-at-risk metrics represent a q-quantile of loss 1L, and this satisfies the definition of PMMR:

[1.5]

Recall that risk measures are categorized according to the metrics they support. Having defined value-at-risk metrics, we define value-at-risk as the category of risk measures that are intended to support value-at-risk metrics. If a risk measure is intended to support a metric that is a value-at-risk metric, then the measure is a value-at-risk measure. If we apply a value-at-risk measure to a portfolio, the value obtained is called a value-at-risk measurement or, less precisely, the portfolio’s value-at-risk.

To use a value-at-risk measure, we must implement it. We must secure necessary inputs, code software, and install the software on computers and related hardware. The result is a value-at-risk implementation.

###### Exercises
1.7

Which of the following represent PMMRs?

1. conditional variance of a portfolio’s USD market value 1 week from today;
2. conditional standard deviation of a portfolio’s JPY net cash flow over the next month.
3. beta, as defined by Sharpe’s (1964) Capital Asset Pricing Model, conditional on information available at time 0.
4. expected tail loss (ETL), which is defined as the expected value of a portfolio’s loss over a specified horizon, assuming the loss exceeds the portfolio’s value-at-risk for that same horizon.1

1.8

Using the naming convention indicated in the text, name the following value-at-risk metric: conditional 0.99 quantile of a portfolio’s loss, measured in GBP, over the next trading day.

1.9

As part of specifying a value-at-risk metric, we must indicate a base currency. This makes sense because value-at-risk indicates an amount of money that might be lost. It is measured in units of currency. But what about other PMMRs? Consider, for example, a 1-day standard deviation of simple return. A portfolio’s return is a unitless quantity; so is its conditional standard deviation of return. Must we specify a base currency for this PMMR?

# 1.3 Market Risk

Business activities entail a variety of risks. For convenience, we distinguish between different categories of risk: market risk, credit risk, liquidity risk, etc. Although such categorization is convenient, it is only informal. Usage and definitions vary. Boundaries between categories are blurred. A loss due to widening credit spreads may reasonably be called a market loss or a credit loss, so market risk and credit risk overlap. Liquidity risk compounds other risks, such as market risk and credit risk. It cannot be divorced from the risks it compounds. A convenient distinction for us to make is that between market risk and business risk.

Market risk is exposure to the uncertain market value of a portfolio. Suppose a trader holds a portfolio of commodity forwards. She knows what its market value is today, but she is uncertain as to its market value a week from today. She faces market risk.

Business risk is exposure to uncertainty in economic value that cannot be marked-to-market. The distinction between market risk and business risk parallels the distinction between market-value accounting and book-value accounting. Suppose a New England electricity wholesaler is long a forward contract for on-peak electricity delivered over the next 12 months. There is an active forward market for such electricity, so the contract can be marked to market daily. Daily profits and losses on the contract reflect market risk. Suppose the firm also owns a power plant with an expected useful life of 30 years. Power plants change hands infrequently, and electricity forward curves don’t exist out to 30 years. The plant cannot be marked to market on a regular basis. In the absence of market values, market risk is not a meaningful notion. Uncertainty in the economic value of the power plant represents business risk.

Most risk metrics apply to a specific category of risks. There are market risk metrics, credit risk metrics, etc. We do not categorize risk measures according to the specific operations those measures entail. We characterize them according to the risk metrics they are intended to support. Gamma—as used by options traders—is a metric of market risk. There are various operations by which we might calculate gamma. We might:

• use a closed form solution related to the Black-Scholes formula;
• value the portfolio at three different underlier values and interpolate a quadratic polynomial; etc.

Each method defines a risk measure. We categorize them all as measures of gamma, not based upon the specific operations that define them, but simply because they all are intended to support gamma as a risk metric.

###### Exercises
1.6

Describe two different risk measures, both of which are intended to support duration as a risk metric.

# 1.2 Risk Measures

In the context of risk measurement, we distinguish between:

• a risk measure, which is the operation that assigns a value to a risk, and
• a risk metric, which is the attribute of risk that is being measured.

Just as duration and size are attributes of a meeting that might be measured, volatility and credit exposure are attributes of bond risk that might be measured. Volatility and credit exposure are risk metrics. Other examples of risk metrics are delta, beta and duration. Any procedure for calculating these is a risk measure. For any risk metric, there may be multiple risk measures. There are, for example, different ways that the delta of a portfolio might be calculated. Each represents a different risk measure for the single risk metric called delta.

According to Holton (2004), risk has two components:

1. exposure, and
2. uncertainty.

If we swim in shark-infested waters, we are exposed to bodily injury or death from a shark attack. We are uncertain because we don’t know if we will be attacked. Being both exposed and uncertain, we face risk.

Risk metrics typically take one of three forms:

• those that quantify exposure;
• those that quantify uncertainty;
• those that quantify exposure and uncertainty in some combined manner.

Probability of rain is a risk metric that only quantifies uncertainty. It does not address our exposure to rain, which depends upon whether or not we have outdoor plans.

Credit exposure is a risk metric that only quantifies exposure. It indicates how much money we might lose if a counterparty were to default. It says nothing about our uncertainty as to whether or not the counterparty will default.

Risk metrics that quantify uncertainty—either alone or in combination with exposure—are usually probabilistic. Many summarize risk with a parameter of some probability distribution. Standard deviation of tomorrow’s spot price of copper is a risk metric that quantifies uncertainty. It does so with a standard deviation. Average highway deaths per passenger-mile is a risk metric that quantifies uncertainty and exposure. We may interpret it as reflecting the mean of a probability distribution.

###### Exercises
1.2

Give an example of a situation that entails uncertainty but not exposure, and hence no risk.

1.3

Give an example of a situation that entails exposure but not uncertainty, and hence no risk.

1.4

In our example of the deaths per passenger-mile risk metric, for what random variable’s probability distribution may we interpret it as reflecting a mean?

1.5

Give three examples of risk metrics that quantify financial risks:

1. one that quantifies exposure;
2. one that quantifies uncertainty; and
3. one that quantifies uncertainty combined with exposure.

# 1.1 Measures

Measures are widely used in science and in every-day activities. While it is common to speak of measuring things, we actually measure attributes of things. For example, we don’t measure a meeting, but we may measure the duration of a meeting or the size of a meeting. Duration and size are attributes.

A measure is an operationally defined procedure for assigning values. An attribute is that which is being measured—the object of the measurement.

A highway patrolman points a Doppler radar at an approaching automobile. The radar transmits microwaves, which are reflected off the auto and return to the radar. By comparing the wavelength of the transmitted microwaves to that of the reflected microwaves, the radar generates a number, which it displays. This entire process is a measure. An interpretation of that number—speed in miles/hour—is an attribute.

There are measures of length, temperature, mass, time, speed, strength, aptitude, etc. Assigned values are usually numbers, but can be elements of any ordered set. Shoe widths are sometimes assigned values from the ordered set {A, B, C, D, E}.

Let’s consider our first exercise.

###### Exercises
1.1

Describe a measure and corresponding attribute that might be used in weather forecasting.