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|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.
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
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:
- Value the vector 0s. (Hint: Based upon how the problem has been presented, the answer is trivial.)
- 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|0σ of using [1.11].
- 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]
- Calculate the portfolio’s 1-year 90% USDvalue-at-risk as
[1.27]
