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 .
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.