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FX calculation theory

The article presents the theory on how to calculate the PNL of a deal that is in position between two foreign currencies.

Sample deal

Let’s take a deal with the following attributes at t0 :

Deal Currency RiskClass RiskFactor Value
Deal1 CC1 MtM [1] 1000
Deal1 CC2 FX Delta CC2 600

[1] MtM means Mark to Market, it is the current price of the product on the market. It means that the pricing of the product under market conditions will produce the MtM.

We have a deal that is evaluated at 1000 CC1 and with a sensitivity to CC2 of 600 CC1.

Market values

FX rates t0 t1
CC0/CC1 1.2 1.25
CC0/CC2 10 9.8
CC2/CC1 0.12 0.127551

To calculate the 3rd currency exchange rate:

Here is the exchange rate conversion, the exchange rate name is the unit. So, multiplying n CC1 by CC0/CC1 gives CC0.

1 CC1 = 1.2 CC0

Conversion to Cash equivalent

To price this product, we will use a proxy that has the same characteristics on the market. In a perfect market this proxy must have the same price as the input product. We will use a basket of two currencies : CC1 and CC2.

Fx MtM Delta Total
CC1 1000 -600 400
CC2 72 72

The cash from delta are calculated as follows:

Let’s check that it is correct at t0

For the MtM

Currency Cash Cash in CC0 MtM
CC1 400 480 1000
CC2 72 720
Total in CC0
1200 1200

For the Delta

Currency Cash Cash in CC1 Cash in CC1 with CC2 + 1% Delta
CC1 400 400 400
CC2 72 600 606
Total in CC1
1000 1006 6 / 1% = 600

PNL explain calculation

Let’s calculate the PNL explain from the cash equivalent at t1:

Currency Cash Cash in CC0 Cash in CC1 Cash in CC2
CC1 400 500 400 51.02
CC2 72 705.6 564.48 72
Total = MtM at t1
1205.6 964.48 123.02
MtM at t0
1200 1000 120
PNL
5.6 -35.52 3.02
Variation of MtM
0.47% -3.55% 2.52%

VaR calculation

note

Simply multiplying the PNL vector by the Spot will not produce the right PNL vector in the domestic currency.

The currency rate is a stochastic variable that has to be taken into account in the VaR calculation.

For each scenario of the PNL vector, a specific exchange rate has to be used. This exchange rate must be consistent with the corresponding scenario.

This will give us :

with the shift defined as

VaR calculation and risk class

If the Risk Class axis is selected we must split the VaR between the FX risk class and the underlying risk class.

So the VaR will be split as follows :

You can see the formula in Excel: FxCalculation.xlsx