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  • V. Illustrations of EAD Calculations

    • A. Illustration 1

      Consider a netting set with three interest rates derivatives: two fixed versus floating interest rate swaps and one purchased physically settled European swaption. The table below summarizes the relevant contractual terms of the three derivatives. All notional amounts and market values in the table are given in USD. We also know that this netting set is not subject to a margin agreement and there is no exchange of collateral (independent amount/initial margin) at inception.

      Trade #NatureResidual maturityBase currencyNotional (thousands)Pay Leg (*)Receive Leg (*)Market value (thousands)
      1Interest rate swap10 yearsUSD10,000FixedFloating30
      2Interest rate swap4 yearsUSD10,000FloatingFixed-20
      3European swaption1 into 10 yearsEUR5,000FloatingFixed50

      (*) For the swaption, the legs are those of the underlying swap.

      The EAD for un-margined netting sets is given by:

      EAD = 1.4 * (RC + PFE)
       

      • 1. Replacement Cost Calculation

        The replacement cost is calculated at the netting set level as a simple algebraic sum (floored at zero) of the derivatives’ market values at the reference date. Thus, using the market values indicated in the table (expressed in thousands):

        RC = max {V - C; 0} = max {30 - 20 + 50; 0} = 60
         

        Since V-C is positive (equal to V of 60,000), the value of the multiplier is 1, as explained in the Standards.

      • 2. Potential Future Exposure Calculation

        All the transactions in the netting set belong to the interest rate asset class. So the Add-on for interest rate class must be calculated as well as the multiplier since

        PFE = multiplier × Add-onagg
         

        For the calculation of the interest rate add-on, the three trades must be assigned to a hedging set (based on the currency) and to a maturity category (based on the end date of the transaction). In this example, the netting set is comprised of two hedging sets, since the trades refer to interest rates denominated in two different currencies (USD and EUR). Within hedging set “USD”, Trade 1 falls into the third maturity category (>5 years) and Trade 2 falls into the second maturity category (1-5 years). Trade 3 falls into the third maturity category (>5 years) of hedging set “EUR”.

        S and E represent the start date and end date, respectively, of the time period referenced by the interest rate transactions.

        Trade #Hedging setTime bucketNotional (thousands)SE
        1USD310,000010
        2USD210,00004
        3EUR35,000111

         

        The following table illustrates the steps typically followed for the add-on calculation:

        StepsActivities
        1. Calculate Effective NotionalCalculate supervisory duration
        Calculate trade-level adjusted notional as trade notional (in domestic currency) × supervisory duration
        Effective notional for each maturity category = Σ(trade-level adjusted notional × supervisory delta × maturity factor), with full offsetting for each of the three maturity categories, in each hedging set (that is, same currency)
        2. Apply Supervisory FactorsAdd-on for each maturity category in a hedging set (that is, same currency) = Effective Notional Amount for maturity category × interest rate supervisory factor
        3. Apply Supervisory CorrelationsAdd-on for each hedging set = Σ(Add-ons for maturity categories), aggregating across maturity categories for a hedging set. One hedging set for each currency.
        4. AggregateSimple summation of the add-ons for the different hedging sets
           Calculate Effective Notional Amount

        The adjusted notional of each trade is calculated by multiplying the notional amount by the calculated supervisory duration SD as defined in the Standards.

        d = Trade Notional × SD = Trade Notional × (exp(-0.05×S) – exp(-0.05 × E)) / 0.05

        TradeNotional AmountTime BucketSESupervisory Duration SDAdjusted Notional d
        Trade 110,000,00030107.86938680678,693,868.06
        Trade 210,000,0002043.62538493836,253,849.38
        Trade 35,000,00031117.48559228237,427,961.41
           Calculate Maturity Category Effective Notional

        A supervisory delta is assigned to each trade in accordance with the Standards. In particular:

        1. Trade 1 is long in the primary risk factor (the reference floating rate) and is not an option so the supervisory delta is equal to 1.
        2. Trade 2 is short in the primary risk factor and is not an option; thus, the supervisory delta is equal to -1.
        3. Trade 3 is an option to enter into an interest rate swap that is short in the primary risk factor and therefore is treated as a purchased put option. As such, the supervisory delta is determined by applying the relevant formula using 50% as the supervisory option volatility and 1 (year) as the option exercise date. Assume that the underlying price (the appropriate forward swap rate) is 6% and the strike price (the swaption’s fixed rate) is 5%.

        The trade-level supervisory delta is therefore:

        TradeDeltanstrument Type
        Trade 11inear, long (forward and swap)
        Trade 2-1inear, short (forward and swap)
        Trade 31purchased put option

         

        The Maturity Factor MF is 1 for all the trades since they are un-margined and have remaining maturities in excess of one year.

        Based on the maturity categories, the Effective Notional D for the USE and EUR hedging sets at the level of the maturity categories are as shown in the table below:

        Hedging SetTime BucketAdjusted NotionalSupervisory DeltaMaturity FactorMaturity category-level Effective Notional D
        HS 1 (USD)378,693,8681178,693,868
        236,253,849-11-36,253,849
        HS 2 (EUR)337,427,961-0.271-10,105,550

        In particular:

        Hedging set USD, time bucket 3: D = 1 * 78,693,868 * 1 = 78,693,868

        Hedging set USD, time bucket 2: D = -1 * 36,253,849 * 1 = -36,253,849

        Hedging set EUR, time bucket 3: D = -0.27 * 37,427,961 * 1 = -10,105,550

           Apply Supervisory Factor

        The add-on must be calculated for each hedging set.

        For the USD hedging set there is partial offset between the two USD trades:

        Effective notional(IR) USD = [D22 + D32 + 1.4 x D2 x D3]1/2

           = [(-36,253,849)2 + 78,693,8682 + 1.4 × (-36,253,849) × 78,693,868]1/2

           = 59,269,963

        For the Hedging set EUR there is only one trade (and one maturity category):

          Effective notional(IR)EUR = 10,105,550

        In summary:

        Hedging setTime BucketMaturity category-level Effective Notional Dj,kHedging Set level Effective Notional Dj,k (IR)
        HS 1 (USD)378,693,86859,269,963
        (Partial offset)
        2-36,253,849
        HS 2 (EUR)3-10,105,55010,105,549.58

         

        Aggregation of the calculated add-ons across different hedging sets:

        Effective Notional(IR) = 59,269,963 + 10,105,550 = 69,375,513(No offset between hedging sets)

         

        The asset class is interest rates; thus the applicable Supervisory factor is 0.50%. As a result:

         Add-on = SF × Effective Notional = 0.005 × 69,375,513 = 346,878

           Supervisory Correlation Parameters

        Correlation is not applicable to the interest rate asset class, and there is no other asset class in the netting set in this example.

           Add-on Aggregation

        For this netting set, the interest rate add-on is also the aggregate add-on because there are no trades assigned to other asset classes. Thus, the aggregate add-on = 346,878

           Multiplier

        The multiplier is given by:

        multiplier = min { 1; Floor+(1-Floor) × exp [(V-C) /(2 ×(1-Floor)×Add-onagg)]}

           = min {1; 0.05 + 0.95 × exp [60,000 / (2 × 0.95 × 346,878]}

             =1

           Final Calculation of PFE

        In this case the multiplier is equal to one, so the PFE is the same as the aggregate Add-On:

        PFE = multiplier × Add-onagg = 1 × 346,878 = 346,878
         

      • 3. EAD Calculation

        The exposure EAD to be risk weighted for counterparty credit risk capital requirements purposes is therefore

        EAD = 1.4 * (RC + PFE) = 1.4 x (60,000 + 346,878) = 569,629
         

    • B. Illustration 2

      Consider a netting set with three credit derivatives: one long single-name CDS written on Firm A (rated AA), one short single-name CDS written on Firm B (rated BBB), and one long CDS index (investment grade). All notional amounts and market values are denominated in USD. This netting set is not subject to a margin agreement and there is no exchange of collateral (independent amount/initial margin) at inception. The table below summarizes the relevant contractual terms of the three derivatives.

      Trade #NatureReference entity / index nameRating reference entityResidual maturityBase currencyNotional (thousands)PositionMarket value (thousands)
      1Single-name CDSFirm AAA3 yearsUSD10,000Protection buyer20
      2Single-name CDSFirm BBBB6 yearsEUR10,000Protection seller-40
      3CDS indexCD X.IGInvestment grade5 yearsUSD10,000Protection buyer0

       

      According to the Standards, the EAD for un-margined netting sets is given by:

      EAD = 1.4 * (RC + PFE)
       

      • 1. Replacement Cost Calculation

        The replacement cost is calculated at the netting set level as a simple algebraic sum (floored at zero) of the derivatives’ market values at the reference date. Thus, using the market values indicated in the table (expressed in thousands):

        RC = max {V - C; 0} = max {20 - 40 + 0; 0} = 0
         

        Since V-C is negative (i.e. -20,000), the multiplier will be activated (i.e. it will be less than 1). Before calculating its value, the aggregate add-on needs to be determined.

      • 2. Potential Future Exposure Calculation

        The following table illustrates the steps typically followed for the add-on calculation:

        StepsActivities
        1. Calculate Effective NotionalCalculate supervisory duration
        Calculate trade-level adjusted notional = trade notional (in domestic currency) × supervisory duration
        Calculate trade-level effective notional amount = trade-level adjusted notional × supervisory delta × maturity factor
        Calculate effective notional amount for each entity by summing the trade-level effective notional amounts for all trades referencing the same entity (either a single entity or an index) with full offsetting
        2. Apply Supervisory FactorsAdd-on for each entity in a hedging set = Entity-level Effective Notional Amount × Supervisory Factor, which depends on entity’s credit rating (or investment/speculative for index entities)
        3. Apply Supervisory CorrelationsEntity-level add-ons are divided into systematic and idiosyncratic components weighted by the correlation factor
        4. AggregateAggregation of entity-level add-ons with full offsetting in the systematic component and no offsetting in the idiosyncratic component

         

           Effective Notional Amount

        The adjusted notional of each trade is calculated by multiplying the notional amount with the calculated supervisory duration SD specified in the Standards.

        d= Trade Notional × SD = Trade Notional × {exp(-0.05×S) – exp(-0.05 × E)} / 0.05

        TradeNotional AmountSESupervisory Duration SDAdjusted Notional d
        Trade 110,000,000032.78584047127,858,405
        Trade 210,000,000065.18363558651,836,356
        Trade 310,000,000054.42398433944,239,843

         

        The appropriate supervisory delta must be assigned to each trade: in particular, since Trade 1 and Trade 3 are long in the primary risk factor (CDS spread), their delta is 1; in contrast, the supervisory delta for Trade 2 is -1.

        TradeDeltaInstrument Type
        Trade 11linear, long (forward and swap)
        Trade 2-1linear, short (forward and swap)
        Trade 31linear, long (forward and swap)

         

        Thus, the entity-level effective notional is equal to the adjusted notional times the supervisory delta times the maturity factor (where the maturity factor is 1 for all three derivatives).

        1

         

         

        TradeAdjusted NotionalSupervisory DeltaMaturity FactorEntity Level Effective Notional
        Trade 127,858,4051127,858,405
        Trade 251,836,356-11-51,836,356
        Trade 344,239,8431144,239,843

         

           Supervisory Factor

         

        The add-on must now be calculated for each entity. Note that all derivatives refer to different entities (single names/indices). A supervisory factor is assigned to each single-name entity based on the rating of the reference entity, as specified in Table 1 in the relevant Standards. This means assigning a supervisory factor of 0.38% for AA-rated firms (Trade 1) and 0.54% for BBB-rated firms (for Trade 2). For CDS indices (Trade 3), the supervisory factor is assigned according to whether the index is investment or speculative grade; in this example, its value is 0.38% since the index is investment grade.

         

        Asset ClassSubclassρSF
        Credit, Single NameAA50%0.38%
        Credit, Single NameBBB50%0.54%
        Credit, IndexIG80%0.38%

         

         

         

        Thus, the entity level add-ons are as follows:

         

        Add-on(Entity) = SF × Effective Notional
         

         

        TradeEffective NotionalSupervisory factor SFAdd-on (Entity)
        Trade 127,858,4050.38%105,862
        Trade 2-51,836,3560.54%-279,916
        Trade 344,239,8430.38%168,111

         

        Supervisory Correlation Parameters

         

        The add-on calculation separates the entity level add-ons into systematic and idiosyncratic components, which are combined through weighting by the correlation factor. The correlation parameter ρ is equal to 0.5 for the single-name entities (Trade 1-Firm A and Trade 2-Firm B) and 0.8 for the index (Trade 3-CDX.IG) in accordance with the requirements of the Standards.

         

        Add-on(Credit) = [ [ ∑k ρk CR × Add-on (Entityk) ]2 + ∑k (1- (ρk CR)2) × (Add-on (Entityk))2]1/2

         

         

         

        TradeρAdd-on(Entityk)ρ × Add-on(Entityk)(1 – ρ2)(1 – ρ2) × (Add-on(Entityk))2
        Trade 150%105,86252,9310.758,405,062,425
        Trade 250%-279,916-139,9580.7558,764,860,350
        Trade 380 %168,111134,4890.3610,174,120,000
        Systematic Component47,462Idiosyncratic Component77,344,042,776
        Full offsettingNo offsetting

         

           Add-on Aggregation

         

        For this netting set, the interest rate add-on is also the aggregate add-on because there are no trades assigned to other asset classes. Thus, the aggregate add-on = 346,878

         

        Aggregation of entity-level add-ons with full offsetting in the systematic component and no offsetting benefit in the idiosyncratic component.

         

        Systematic Component47,462
        Idiosyncratic Component77,344,042,776

         

         

         

           Thus,

         

        Add-on = [ (47,462)2 + 77,344,042,776 ]1/2 = 282,129

         

           Multiplier

         

           The multiplier is given by

         

        multiplier = min {1; Floor+(1-Floor) × exp [(V-C)/(2×(1-Floor)×Add-onagg)]}

         

           = min {1; 0.05 + 0.95 × exp [-20,000 / (2 × 0.95 × 282,129)]}

         

              =0.96521

         

           Final Calculation of PFE

         

        PFE = multiplier × Add-onagg = 0.96521 × 282,129= 272,313
         

         

      • 3. EAD Calculation

        The exposure that would be risk-weighted for the purpose of counterparty credit risk capital requirements is therefore:

        EAD = 1.4 * (RC + PFE) = 1.4 x (0 + 272,313) = 381,238
         

    • C. Illustration 3

      Consider a netting set with three commodity forward contracts. All notional amounts and market values are denominated in USD. This netting set is not subject to a margin agreement and there is no exchange of collateral (independent amount/initial margin) at inception. The table below summarizes the relevant contractual terms of the three commodity derivatives.

      Trade #NatureUnderlyingPositionDirectionResidual maturityNotional (thousands)Market value (thousands)
      1Forward(WTI)
      Crude Oil
      Protection BuyerLong9 months10,000-50
      2Forward(Brent)
      Crude Oil
      Protection SellerShort2 years20,000-30
      3ForwardSilverProtection BuyerLong5 years10,000100

       

      • 1. Replacement Cost Calculation

        The replacement cost is calculated at the netting set level as a simple algebraic sum (floored at zero) of the derivatives’ market values at the reference date, provided that value is positive. Thus, using the market values indicated in the table (expressed in thousands):

        RC = max {V - C; 0} = max {100 - 30 - 50; 0} = 20
         

        The replacement cost is positive and there is no exchange of collateral (so the bank has not received excess collateral), which means the multiplier will be equal to 1.

      • 2. Potential Future Exposure Calculation

        The following table illustrates the steps typically followed for the add-on calculation, for each of the four commodity hedging sets with non-zero exposure:

        StepsActivities
        1. Calculate Effective NotionalCalculate trade-level adjusted notional = current price × number of units referenced by derivative
        Calculate trade-level effective notional amount = trade-level adjusted notional × supervisory delta × maturity factor
        Calculate effective notional for each commodity-type = Σ(trade-level effective notional) for trades referencing the same commodity type, with full offsetting in commodity type
        2. Apply Supervisory FactorsAdd-on for each commodity type in a hedging set = Commodity-type Effective Notional Amount × Supervisory Factor
        3. Apply Supervisory CorrelationsCommodity-type add-ons are divided into systematic and idiosyncratic components weighted by the correlation factor
        4. AggregateAggregation of commodity-type add-ons with full offsetting in the systematic component and no offsetting in the idiosyncratic component
        Simple summation of absolute values of add-ons across the four hedging sets
           Effective Notional Amount

        Trade-level Adjusted Notional calculation for each commodity derivative trade:

        di(COM) = current price per unit × number of units in the trade
         

        TradeCurrent price per unit (unit is barrel for oil; ounces for silver)Number of units in the tradeAdjusted Notional
        Trade 1100100 barrels10,000
        Trade 2100200 barrels20,000
        Trade 320500 ounces10,000

         

        The appropriate supervisory delta must be assigned to each trade:

        TradeDeltaInstrument Type
        Trade 11linear, long (forward & swap)
        Trade 2-1linear, short (forward & swap)
        Trade 31linear, long (forward & swap)

         

        Since the remaining maturity of Trade 1 is less than a year, at nine months (approximately 187 business days), and the trade is un-margined, its maturity factor is scaled down by the square root of 187/250 in accordance with the requirements of the Standards. On the other hand, the maturity factor is 1 for Trade 2 and for Trade 3, since the remaining maturity of those two trades is greater than one year and they are un-margined.

        The trade-level effective notional is equal to the adjusted notional times the supervisory delta times the maturity factor. The basic difference between the WTI and Brent forward contracts effectively is ignored since they belong to the same commodity type, namely “Crude Oil” within the “Energy” hedging set, thus allowing for full offsetting. (In contrast, if one of the two forward contracts were on a different commodity type within the “Energy” hedging set, such as natural gas, with the other on crude oil, then only partial offsetting would have been allowed between the two trades.) Therefore, Trade 1 and Trade 2 can be aggregated into a single effective notional, taking into account each trade’s supervisory delta and maturity factor.

        1

         

        Hedging SetCommodity TypeTradeAdjusted NotionalSupervisory DeltaMaturity FactorEffective Notional
        EnergyCrude OilTrade 110,0001 187250=0.865 10,000 x 1 x 0.865 + 20,000x(-1)x1 =-11,350 (full off-setting within the ‘Crude Oil’ commodity type)
        EnergyCrude OilTrade 220,000-11
        MetalsSilverTrade 310,0001110,000
           Supervisory Factor

        For each commodity-type in a hedging set, the effective notional amount must be multiplied by the correct Supervisory Factor (SF). As described in the Standards, the Supervisory Factor for both the Crude Oil commodity type in the Energy hedging set and the Silver commodity type in the Metals hedging set is SF=18%.

        Thus, the add-on by hedging set and commodity type is as follows:

        Add-on(Typekj) = SFTypek(Com) × Effective NotionalTypek(Com)
         

        Hedging SetCommodity TypeEffective NotionalSupervisory Factor SFAdd-on by HS and Commodity type
        EnergyCrude Oil-11,35018%-2,043
        MetalsSilver10,00018%1,800
           Supervisory Correlation Parameters

        The commodity-type add-ons in a hedging set are decomposed into systematic and idiosyncratic components. The commodity subclass correlations parameters are as stated in the Standards, in this case 40% for commodities.

        Thus, the hedging set level add-ons are calculated for each commodity hedging set:

        Add-on(COM) = [( Σk ρj(COM) × Add-on (Typekj) )2 + Σk (1- (ρj(COM) )2) × (Add-on (Type j))2]k1/2

        Hedging SetCommodity TypeρAdd-on(Typek)Systematic Component (ρ × Add-on(Typek))2(1 – ρ2)Idiosyncratic Component (1 – ρ2) x (Add-on(Typek)) 2Add-onj (Only one commodity type in each HS) 
        EnergyCrude Oil40%-2,043(-817)20.840.84 × (-2,043)22,043 
        MetalsSilver40%1,800(720)20.840.84 × (1,800)21,800 

         

        However, in this example, since only one commodity type within the “Energy” hedging set is populated (i.e. all other commodity types within that hedging set have a zero add-on), the resulting add-on for the hedging set is the same as the add-on for the commodity type. This calculation shows that when there is only one commodity type within a commodity hedging set, the hedging-set add-on is equal to the absolute value of the commodity-type add-on. (The same comment applies to the commodity type “Silver” in the “Metals” hedging set.)

           Add-on Aggregation

        Aggregation of commodity-type add-ons uses full offsetting in the systematic component and no offsetting benefit in the idiosyncratic component in each hedging set. As noted earlier, in this example there is only one commodity type per hedging set, which means no offsetting benefits. Computing the simple summation of absolute values of add-ons for the hedging sets:

        Add-on = Σj Add-onj

        Add-On = 2,043 + 1,800 = 3,843

           Multiplier

        The multiplier is given by

        multiplier = min {1; Floor+(1-Floor) × exp [(V-C)/(2×(1-Floor)×Add-onagg)]}

           = min {1; 0.05 + 0.95 × exp [20 / (2 × 0.95 × 3,843)]}

              = 1, since V-C is positive.

           Final Calculation of PFE

        PFE = multiplier × Add-onagg = 1 × 3,843 = 3,843
         

      • 3. EAD Calculation

        The exposure EAD to be risk weighted for counterparty credit risk capital requirements purposes is therefore

        EAD = 1.4 * (RC + PFE) = 1.4 x (20 + 3,843) = 5,408