8.4 Example: Equities

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


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:






We specify key vector 1R using component vectors:


where the first component vector, 1RFX, is a vector of spot exchange rates, which settle in 2 days:


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:


More generally, expressed with component vectors,


Combining [8.30] with [8.34], we obtain our primary mapping:


Because exchange rates are multiplied by local-currency accumulated values, this defines 1P as a quadratic polynomial of 1R.


In our international equities example, the primary mapping [8.35] has a quadratic mapping function. It can be represented in matrix form as


using some symmetric matrix c. In this exercise, you will construct such a representation. To simplify the task, consider a reduced asset vector


Assume portfolio holdings


and use key factors