IMA ES FX Conversions
The calculations for modellable risk-factors (IMCC) start with the expected shortfall in the specifications. In Atoti FRTB, we start with P&L vectors indexed by scenario, which can be aggregated before calculating the expected shortfall. Moreover, Atoti FRTB allows the P&L vectors to be specified in any currency and will perform FX conversions to the reporting currency.
In the case when the FX rates themselves are shocked as part of the scenario, then an FX conversion of the P&L value using a base FX rate is not sufficient. For example, the FX risk-class with liquidity horizon of 10 or 20 days will involve shocks to the FX rates. See table 2 in [MAR33.12].
To handle this, instead of applying the FX conversion to the P&L value, we will support using shocked and base PV values, then apply shocked and base FX rates for the FX conversion.
Definitions
| Element | Description |
|---|---|
| $P\&L^{scenario}_{(CCY)}$ | The P&L value for the scenario ($scenario$), expressed in the specified currency ($CCY$). |
| $PV^{base}_{(CCY)}$ | The PV value for the base scenario, expressed in the specified currency ($CCY$). |
| $PV^{scenario}_{(CCY)}$ | The PV value for the scenario ($scenario$), expressed in the specified currency ($CCY$). |
| $FX^{base}_{CCY1/CCY2}$ | The spot FX rate for the $CCY1/CCY2$ currency pair in the base scenario. |
| $FX^{scenario}_{CCY1/CCY2}$ | The spot FX rate for the $CCY1/CCY2$ currency pair in the scenario ($scenario$). This is the multiplication factor that will convert from $CCY2$ to $CCY1$ in the scenario ($scenario$). |
Formulas
In the instrument’s payout currency, the P&L value is:
$$ P\&L^{scenario}_{(payout\ ccy)} = PV^{scenario}_{(payout\ ccy)} - PV^{base}_{(payout\ ccy)} $$
The IMA ES calculations require the scenario P&L values in the reporting currency ($P\&L^{scenario}_{(reporting\ ccy)}$). We calculate this by converting the PV values to the reporting currency. To do this we use both the scenario FX rate and the base FX rate.
$$ \begin{align} P\&L^{scenario}_{(reporting\ ccy)} & = PV^{scenario}_{(reporting\ ccy)} - PV^{base}_{(reporting\ ccy)} \\\\ & \\\\ & = PV^{scenario}_{(payout\ ccy)} \times FX^{scenario}_{reporting/payout} - PV^{base}_{(payout\ ccy)} \times FX^{base}_{reporting/payout} \end{align} $$
Implementation
We calculate the ES values (ES (Basic)) from the vectors of P&L values. These vectors are indexed by scenario.
In the input file Expected Shortfall PL Trade we load both PV vector (also index by scenario) and a base PV value. The vector of P&L values is obtained by subtracting the base PV value from each entry of the PV vector. If currency conversion is required, then scenario FX rate vectors are used. Additionally, the usual spot FX rates are used as the base FX rate.
Scenario FX Rates Vectors
The vectors of scenario FX rates are loaded from the IMA ES Scenario FX Rates files. These vectors are also indexed by scenario; they share the index of the PV vectors.
When looking at the scenario FX rates between any two currencies, the same approach is used as for the base FX rate. In particular, if a direct rate cannot be found, then cross rates (with the common currency) or inverse rates are used. This is the same apprach as described in FxRate Lookup, with the vector-valued calculations evaluated by performing the operations on individual entries.
In addition to the currencies and as-of date, these vectors are also looked up by data set, risk-class, and liquidity horizon. This includes looking up cross and inverse rates.
It is only expected that the FX or “allin” risk-classes with liquidity horizons of 10 or 20 days will need vectors of scenario FX rates. Other combinations of risk-class and liquidity horizon are supported, but these other combinations represent scenarios with unshocked FX rates.
Backwards Compatibility
The functionality described here is optional.
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If the base PV = 0, then the PV vector becomes the P&L vector (in the instrument’s payout currency)
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If no scenario FX rate vector is available to provide per-scenario FX rates, then the base FX rate is used to convert the scenario PV (or P&L) values.
Because only the FX and “allin” risk-classes with liquidity horizons of 10 or 20 days use shocked FX rates, scenario FX rate vectors do not need to be provided for other combinations of risk-class and liquidity horizon. For these other combinations, the base FX rate is sufficient.