# 8.4 Example: Equities

Measure value-at-risk as 1-week 95% USDvalue-at-risk. Measure basis days as actual days. Assume 2^{nd}-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 ^{1}*P* = θ(^{1}** R**) of the form

[8.26]

For each stock traded on one of the exchanges, define an asset value ^{1}*S _{i}* 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

^{1}

**and the holdings**

*S***ω**into sub-vectors by country:

[8.27]

[8.28]

Then

[8.29]

[8.30]

We specify key vector ^{1}** R** using component vectors:

[8.31]

where the first component vector, ^{1}*R** ^{FX}*, 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 2^{nd}-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 ^{1}** R** to

^{1}

**with a simple currency conversion. For Bangchak Petroleum:**

*S*[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 ^{1}*P* as a quadratic polynomial of ^{1}** R**.

###### 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]