###### 8.3.3 Example

Consider a value-at-risk metric of 5-day 95% USDvalue-at-risk. Measure basis days as actual days. Time 0 is May 29, 2001. Based upon the calendar of Exhibit 8.12, time 1 is June 5, 2002. A value-at-risk measure is applied to a foreign exchange portfolio comprising:

- a forward contract to exchange USD 10MM for CAD 15.47MM on June 8, 2001, and
- a short European-exercise call option on CAD 30MM struck at 1.55 CAD/USD and expiring on May 31, 2001.

Our example is contrived, but it will illustrate some important concepts. In particular the short option expires prior to the end of the value-at-risk horizon. This will be our first example of a portfolio mapping that must account for an intra-horizon event.

We adopt 2^{nd}-day valuation and specify asset values:

[8.14]

Holdings are **ω** = (15,470,000 –30,000,000), so we define

[8.15]

The option expires in τ(2) = 2 basis days. If it is exercised, CAD will be delivered against USD 2 trading days later, which is at τ(4) = 6 basis days. The value-at-risk horizon ends at τ(5) = 7 basis days. Its value date is at τ(7) = 9 basis days. The forward contract matures at τ(8) = 10 basis days. These events are plotted on a time line in Exhibit 8.13.

^{nd}-day.

Because of the option’s expiration, the portfolio value ^{1}*P* depends upon market conditions on both May 31 and June 5—at τ(2) = 2 and τ(5) = 7 basis days. Specify risk factors as

[8.16]

The first risk factor will determine the value of the option at expiration. Because the option settles 2 trading days later, that value is realized at basis day 6. The second risk factor will accumulate that value to the value date, basis day 9. The last three risk factors will determine the value of the outstanding forward at the end of the value-at-risk horizon, again for a value date at 9 basis days.

Note that we do not use fractional time superscripts to indicate when values of risk factors will be realized. All five risk factors have time superscript 1, which is intended only to indicate that each is realized at some point in the time interval (0,1]. Our convention is that a time superscript *t* indicates that a value * ^{t}Q_{i} *is realized at some point between times

*t*– 1 and

*t*. Even over a 1-day horizon, risk factors are often realized nonsynchronously because of markets being in different time zones, etc. Nothing is gained by trying to identify in our notation the precise times at which each is realized.

To specify a mapping ^{1}* S* = φ(

^{1}

*), we must specify two component formulas φ*

**Q**_{1}and φ

_{2}for valuing the forward and option, respectively. Both must reflect value based upon information available at the end of the value-at-risk horizon—actual day 7—but for value date 2 trading days later—actual day 9. Starting with φ

_{1}, the forward comprises two cash flows. At actual day 10, for each CAD we receive, we will pay .6464 USD. Discounting USD .6464 back to actual day 9 yields a USD market value of .6464

^{1}

*Q*

_{4}. Discounting a CAD back to actual day 9 yields a CAD market value of

^{1}

*Q*

_{5}. To convert this to a USD market value, we divide by the CAD/USD exchange rate

^{1}

*Q*

_{3}. Conveniently,

^{1}

*Q*

_{3}is a spot exchange rate for actual day 7, which means it settles on the value date, which is actual day 9. This isn’t coincidence—we chose to use 2

^{nd}-day valuation for a reason! The value of the forward per CAD is

[8.17]

Turning now to φ_{2}, the option expires at actual day 2. If exercised, it settles as a spot foreign exchange transaction at actual day 6 for value (1/1.55 – 1/^{1}*Q*_{1}) per CAD. We will know at actual day 2 that this value is to be realized, so financing from actual day 6 to actual day 9 can be arranged at that time. Accumulated to actual day 9, the option’s value per CAD is

[8.18]

Together, [8.15], [8.17], and [8.18] define a portfolio mapping. We could let ^{1}** Q** be our key vector and be done. However, the accumulation/discount factors

^{1}

*Q*

_{2},

^{1}

*Q*

_{4}, and

^{1}

*Q*

_{5}make odd key factors, even for a contrived example such as this. To tidy things up, let’s express

^{1}

**in terms of key factors that are quoted in the market:**

*Q*[8.19]

Using techniques similar to those of the previous section, we construct a mapping:

[8.20]

This completes our primary portfolio mapping:

[8.21]

This example is useful for two reasons. First, it illustrates the generality of our framework for modeling value-at-risk. We have been developing value-at-risk measures as single-period models. They characterize at time 0 a conditional distribution of a portfolio’s value at time 1. It might seem that intra-horizon events—such as the expiration of an option, resetting of a floating interest rate or knocking out of a barrier option between times 0 and 1—might force us to employ multi-period models. Our example illustrates that this is not necessary. Key factors ^{1}*R _{i}* can represent quantities that will be realized at any point in the interval (0,1]. Timing issues do not prevent us from specifying a portfolio mapping

^{1}

*P*= θ(

^{1}

**). They will not prevent us from characterizing a joint distribution for**

*R*^{1}

**or from applying a suitable transformation procedure.**

*R*Second, the example illustrates how intra-horizon events can complicate mappings, if they require the modeling of additional key factors and the construction of specialized model libraries. Such complexities are avoided if we use a 1-day value-at-risk horizon—which is a compelling reason to do so!

###### Exercises

Measure value-at-risk as 1-day 95% USD value-at-risk and employ cash valuation. A portfolio comprises 130 troy ounces of gold, which will be received in one trading day. The gold has already been paid for, so there is no offsetting cash flow. Construct a primary mapping for this portfolio based upon a single key factor:

[8.22]

Repeat the previous exercise, but construct a primary mapping using two key factors:

[8.23]

Measure value-at-risk as 1-day 95% USD value-at-risk and employ cash valuation. NYMEX gold futures have a notional amount of 100 troy ounces. Today’s date is March 2, 2001. A portfolio comprises 12 April contracts. The April contract’s current settlement price is 263.50 USD/ounce. Construct a primary mapping for the portfolio based upon a single key factor:

[8.24]

Measure value-at-risk as 1-day 95% USD value-at-risk and employ cash valuation. Today’s date is March 2, 2001. A gold portfolio comprises:

- 200 ounces of gold to be received in one trading day;
- a long position of 6 April contracts; and
- a short position of 2 June contracts.

Construct a primary mapping for the portfolio based upon three key factors:

[8.25]

Measure value-at-risk as 5-day 95% USD value-at-risk. Today’s date is September 20, 1999. A portfolio comprises an OTC up-and-in barrier call option on 10,000 ounces of gold. The option expires on September 24, 1999. It has strike price 275 and barrier 300. If, at any time up to and including the expiration date, the price of gold (London afternoon fixing) reaches USD 300/ounce, the option will convert to a European call struck at 275. Otherwise, it will expire worthless. If exercised, the option settles for cash (as opposed to physical delivery) 2 trading days after expiration.

Employ 2^{nd}-day valuation. Construct a primary portfolio mapping as follows:

- Construct a time line for the problem similar to that of Exhibit 8.13.
- Identify variables—quoted in the markets—to model as key factors.
- Specify a primary mapping in terms of these.