###### 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,
- 500 tons of lead,
- 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|0}σ and ^{1|0}**Σ** 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|0}**Σ** to obtain ^{1|0}σ. Exhibit 1.4 indicates historical metals price data.

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|0}σ of USD 0.217MM.

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

We have no expression for . All we have is a conditional standard deviation ^{1|0}σ for . We need additional assumptions or information. A simple solution is to assume that is conditionally normal with conditional mean ^{1|0}μ = = 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

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|0}**Σ** for :

[1.25]

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

- Value the vector
^{0}. (Hint: Based upon how the problem has been presented, the answer is trivial.)*s* - Using the formula , value .
- Specify a portfolio mapping that defines as a linear polynomial of .
- Draw a schematic for your portfolio mapping.
- Determine the conditional standard deviation
^{1|0}σ of using [1.11]. - Assume is normally distributed with conditional mean
^{1|0}μ = and conditional standard deviation obtained in part (e). Calculate the .10-quantile of with the formula[1.26]

- Calculate the portfolio’s 1-year 90% USDvalue-at-risk as
[1.27]