8.4 Example: Equities
Measure value-at-risk as 1-week 95% USDvalue-at-risk. Measure basis days as actual days. Assume 2nd-day valuation. A US fund manager runs a portfolio of Pacific Basin equities. The fund does not hedge foreign exchange exposures. Holdings are in actively traded stocks on the following exchanges:
- Jakarta Stock Exchange,
- Philippine Stock Exchange,
- Stock Exchange of Hong Kong,
- Stock Exchange of Singapore,
- Stock Exchange of Thailand,
- Taiwan Stock Exchange.
Let’s construct a primary mapping 1P = θ(1R) of the form
[8.26]
For each stock traded on one of the exchanges, define an asset value 1Si to represent the USD accumulated value of a single share. Accumulated value reflects the stock’s price, dividends, stock splits, and the USD exchange rate versus the stock’s local currency. For expositional convenience, we segment 1S and the holdings ω into sub-vectors by country:
[8.27]
[8.28]
Then
[8.29]
[8.30]
We specify key vector 1R using component vectors:
[8.31]
where the first component vector, 1RFX, is a vector of spot exchange rates, which settle in 2 days:
[8.32]
The remaining component vectors indicate, for each stock, accumulated value in local currency based on 2nd-day valuation. If represents the USD accumulated value of a share of Bangchak Petroleum,
represents the THB accumulated value of that same share. We map 1R to 1S with a simple currency conversion. For Bangchak Petroleum:
[8.33]
More generally, expressed with component vectors,
[8.34]
Combining [8.30] with [8.34], we obtain our primary mapping:
[8.35]
Because exchange rates are multiplied by local-currency accumulated values, this defines 1P as a quadratic polynomial of 1R.
Exercises
In our international equities example, the primary mapping [8.35] has a quadratic mapping function. It can be represented in matrix form as
[8.36]
using some symmetric matrix c. In this exercise, you will construct such a representation. To simplify the task, consider a reduced asset vector
[8.37]

Assume portfolio holdings
[8.38]
and use key factors
[8.39]