1.8.5 Transformation Procedures

1.8.5 Transformation Procedures

A transformation procedure—or transformation—characterizes a conditional distribution for and uses this characterization to value a desired value-at-risk metric or other PMMR. According to Holton (2004), risk has two components:

  1. exposure, and
  2. uncertainty.

A portfolio mapping incorporates both. The characterization of a conditional distribution of reflects our uncertainty. The mapping function θ reflects our exposure. The challenge for a transformation procedure is to combine both components to characterize a conditional distribution for . To intuitively understand what this entails, consider some simple examples.

A portfolio’s value depends upon a single normally distributed key factor . The mapping function θ is a linear polynomial. The situation is depicted in Exhibit 1.8. The graph on the left depicts the mapping function θ. Evenly spaced realizations for have been mapped into corresponding realizations for . The resulting realizations of are also evenly spaced, indicating that θ imparts no distortions. Since is conditionally normal, will also be conditionally normal, as illustrated in the graph on the right.

Exhibit 1.8: A linear mapping function θ is applied to a key factor 1R1. This is illustrated intuitively by mapping evenly spaced realizations for 1R1 through the mapping function. The output values for 1P are also evenly spaced, indicating that the mapping function causes no distortion. Since 1R1 is conditionally normal, so is 1P.

For our second example, consider a portfolio comprising a single call option with a conditionally normal key factor as its underlier. To avoid a need for additional key factors, treat applicable interest rates and implied volatilities as constant. In Exhibit 1.9, the left graph depicts the familiar “hockey stick” mapping function of a call option. Evenly spaced realizations for do not map into evenly spaced realizations for , so the mapping function causes distortions. Since is conditionally normal, the resulting distribution of is conditionally non-normal, as illustrated on the right.

Exhibit 1.9: A nonlinear mapping function θ is applied to a conditionally normal key factor 1R1. The result is a conditionally non-normal portfolio value 1P. This is illustrated intuitively by mapping evenly spaced realizations for 1R1 through the mapping function. The corresponding values for 1P are not evenly spaced, reflecting how the mapping function distorts the distribution of 1P.

Our third example considers a long-short options position applied to a short position in the underlier. The mapping function θ, which is illustrated in the left graph of Exhibit 1.10, causes realizations of to cluster in two regions. If the underlier is conditionally normal, will have the dramatically non-normal conditional distribution shown on the right.

Exhibit 1.10: A long-short options position can result in a bimodal distribution for the portfolio’s value 1P.

These are simple examples, especially since each entails a single key factor. Practical value-at-risk measures often entail 100 or more key factors. If a portfolio holds complex instruments such as exotic derivatives or mortgage-backed securities, a mapping function can be extremely complex. Such issues can make it difficult to design a practical transformation procedure.

In our examples of Section 1.6, we illustrated three types of transformations:

  1. linear transformations,
  2. quadratic transformations, and
  3. Monte Carlo transformations.

The first applies if a portfolio mapping function θ is a linear polynomial. The second applies if θ is a quadratic polynomial and is joint-normal. The third applies quite generally and is one example of a category of transformations called numerical transformations. We discuss transformation procedures in Chapter 10.


Exhibit 1.11 illustrates three portfolio mapping functions θ for portfolios whose values depend upon a single key factor . As we did in Exhibits 1.8, 1.9, and 1.10, sketch what each conditional PDF of might look like assuming is conditionally normal with its mean at the mid-point of each graph.

Exhibit 1.11: Portfolio mapping functions θ for Exercise 1:13



Describe real-world portfolios whose mapping functions might appear like those of the previous exercise.



1.8.4 Covariance Matrix Construction

1.8.4 Inference Procedures

In order to characterize a distribution for conditional on information available at time 0, we must characterize a conditional distribution for . We do so with an inference procedure. It is not always necessary to fully specify a distribution. We require only information sufficient to value our chosen value-at-risk metric or other PMMR.

Inference procedures take various forms. Leavens simply makes up a distribution suitable for his example. In practice, techniques of time series analysis are employed—in conjunction with financial theory—to obtain a covariance matrix for or some other reasonable characterization. We discuss inference procedures in Chapter 7.


1.8.3 Portfolio Mappings

1.8.3 Portfolio Mappings

In mathematics, a mapping is a function. The words are synonyms. In the context of value-at-risk, we reserve the word “mapping” for functions relating specific risk vectors to one another. If and are risk vectors, a mapping is a functional relationship:


We call φ the mapping function.

A portfolio mapping is a mapping that defines a portfolio’s value as a function of some risk vector :


Portfolio mappings play a simple but inevitable role in value-at-risk measures. Let’s focus on two of our earlier examples: Leavens’ PMMR and our Australian equities value-at-risk measure. To quantify a portfolio’s market risk, we must calculate the value of some function—value-at-risk metric or other PMMR—of and the conditional distribution of . We interpret as the portfolio’s market value at time 1, but this is not a definition. Mathematically, there are two ways we may define the random variable :

  1. we can directly specify a conditional distribution for ;
  2. we can define as a function of some random vector.

The first approach is hardly feasible. Portfolios and financial markets tend to be complicated, so it is difficult to directly specify a conditional distribution for . Inevitably, we define using the second approach—which leads to portfolio mappings. Both the Leavens and Australian equities value-at-risk measures define as a function of some asset vector :


We interpret as a vector of accumulated values, but this is not a definition. To complete our definition of , we must mathematically define . As with , there are two ways to define :

  1. we can directly specify a conditional distribution for ;
  2. we can define as a function of some other random vector.

At this point, Leavens uses the first approach. He specifies a conditional distribution for and uses this to infer a binomial distribution for . We schematically represent Leavens’ portfolio mapping as


The Australian equities value-at-risk measures don’t stop there. Rather than directly specify a joint distribution for , they define as a mapping of another random vector . We schematically represent the resulting portfolio mapping as


No matter how many mappings are composed, ultimately must be defined as a function of some random vector for which we directly characterize a joint distribution. That random vector is the key vector . We denote the mapping function that relates to its key vector with θ. Accordingly, the notation


recurs frequently in this text. An exception is if asset values are used as key factors. In this case, the relationship is


and plays the dual role of asset vector and key vector.

Here we have described not only portfolio mappings, but also a general procedure for constructing them. Portfolio mappings constructed in this manner—starting with asset vector and holdings ω, and perhaps mapping to some key vector —are called primary mappings. The name distinguishes them from portfolio mappings constructed as remappings. All portfolio mappings stem from primary mappings. They either are left in that form, or are approximated using one or more remappings. We discuss primary mappings in Chapter 8.


1.8.2 Holdings

1.8.2 Holdings

When we design a value-at-risk measure, we must decide what physical or financial assets to represent with mathematical assets (, ). We might measure equity positions in shares or round lots. In Exercise 1.12 we measured them as the number of USD held in a given stock at time 0. Positions in cocoa might be measured in pounds, bags, or tons. The choice of units is largely arbitrary, but it must be explicit if we are to define portfolio holdings.

A portfolio’s holdings is a row vector ω indicating the number ωi of units held by the portfolio in each asset.


1.8 Value-at-Risk Measures

1.8 Value-at-Risk Measures

In the previous section, we described several market risk measures, most of them value-at-risk measures. Despite a disparity in modeling techniques, our treatment was standardized. Certain concepts recurred. You are now familiar with notation such as ω, , θ, , , and 1|0Σ.

We have many value-at-risk measures to consider. Before long, we will stop describing entire value-at-risk measures and start describing stand-alone components of value-at-risk measures—much as auto enthusiasts might discuss types of brakes or fuel injectors without having a particular automobile in mind. In this sense, our discussions will have a “building block” quality. We don’t want every value-at-risk measure to be a unique monolith standing on its own. Instead, we will treat them as modular. Avoiding the top-down approach of discussing Toyotas, Fords, and Mercedes, we will take a bottom-up approach, discussing fuel injectors, suspension systems, and brakes. To this end, we must identify the essential components that make up any value-at-risk measure—indeed, any PMMR measure. In doing so, we will lay out a framework for much of this book.

1.8.1 Risk Factors

A risk factor is any random variable whose value will be realized during the interval (0,1] and will affect the market value of a portfolio at time 1. A risk vector is a random vector of risk factors. If a risk vector reflects a future value of some time series, we may speak of its current value 0q or historical values 0q, –1q, –2q, –3q

One particular risk factor and two risk vectors play important roles in value-at-risk measures. We give them special names and notation. These are:

  1. the portfolio’s future value ,
  2. the asset vector , and
  3. the key vector .

The portfolio’s future value represents the market value at time 1 of the portfolio for which value-at-risk is to be measured. The portfolio is assumed fixed in the sense that it will not be traded during the period [0,1] and no assets will be added or withdrawn. This does not preclude traders or portfolio managers from trading! It simply means that a value-at-risk measure quantifies the market risk of a portfolio based upon its composition at time 0. The value-at-risk measure can recognize changes in the portfolio’s composition during the period [0,1] due to planned events such as options expiring, dividends being paid, or scheduled payments being made on a swap. We are interested in the portfolio’s current value because value-at-risk metrics depends upon it. We generally do not consider or attempt to define prior historical portfolio values.

Asset vector has asset values as components. These represent accumulated values of specific assets that may make up a portfolio. Realizations may be negative, so our definition recognizes no accounting distinction between assets and liabilities. Accumulated value is denominated in the base currency employed by the value-at-risk metric. It may reflect such variables as capital gains, dividends, coupons, margin payments, reinvestment income, storage costs, insurance, financing, changes in exchange rates, leasing income, etc.

Mathematically, we define a portfolio as a pair (, ) where the constant is the portfolio’s current value, and the random variable is the portfolio’s future value. Similarly, we mathematically define an asset as a pair (, ), where is the asset’s current value, and is the asset’s future value.

We have considerable leeway in how we select what financial instruments to represent with assets. This may affect value-at-risk results. Consider an investor who borrows EUR 100,000 and invests it in Hoechst stock. We might model the portfolio three different ways:

  1. as comprising holdings in two assets whose values 1S1 and 1S2 represent the accumulated values of the stock and the financing, respectively;
  2. as comprising a single asset whose value 1S1 represents the accumulated value of the stock less the accumulated value of its financing;
  3. as comprising a single asset whose value 1S1 represents the accumulated value of the stock.

The first two representations are financially equivalent. One approach (probably the first) will be computationally easier to work with, but both will result in the same value-at-risk. The third representation is different. It excludes financing from the portfolio. With the third approach, the random variable represents something different than it does with the first two approaches.

As we shall see, every value-at-risk measure must directly characterize a conditional probability distribution for some vector of risk factors, such as prices, interest rates, spreads, or implied volatilities. Those risk factors are called key factors. They are the components of the key vector . Occasionally, we use asset values as key factors. This was the case in our examples of Leavens’ PMMR and the value-at-risk measure for industrial metals. We explore the role of key factors in more detail shortly.


1.7.6 Example: Australian Equities (Quadratic Transformation)

1.7.6 Example: Australian Equities (Quadratic Transformation)

For a third approach to calculating value-at-risk for our Australian equities portfolio, assume that is conditionally joint-normal with conditional mean vector 1|0μ = and covariance matrix 1|0Σ obtained previously. Our original portfolio mapping defines as a quadratic polynomial of a conditionally joint-normal random vector . As we will discuss in Section 3.13, any real-valued quadratic polynomial of a joint-normal random vector can be expressed as a linear polynomial of independent normal and chi-squared random variables. In this case, the expression takes the form


where X1 and X2 are independent chi-squared random variables, each with 1 degree of freedom and respective non-centrality parameters 674.2 and 14,195.5 This is not an approximation. The representation is exact.

There are various ways to extract a quantile of portfolio loss from a representation such as [1.50]. Two approaches that we shall discuss in Section 3.17 are:

  1. approximate the desired quantile using the Cornish-Fisher (1937) expansion,
  2. invert the characteristic function of using numerical integration,

Applying the first approach to our Australian equities portfolio yields an approximate 1-day 95% GBPvalue-at-risk of GBP 5854.


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


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



Our remapping [1.40] is




The portfolio remapping is represented schematically as


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 :


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


and the .95-quantile of portfolio loss is


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


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:




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


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


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


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 :




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


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


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 :


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


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


The portfolio’s holdings are:

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

which we represent with a row vector:


The portfolio’s current value is


Its future value is random:


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


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


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


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


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


Substituting [1.21] into [1.20]:


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].


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:


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


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


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


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