Cutting edge: Regulation

Revisiting SA-CCR

In this article, Mourad Berrahoui, Othmane Islah and Chris Kenyon propose a comparison for SA-CCR, termed Revisited SA-CCR, which reconstructs SA-CCR in a self-consistent and appropriately risk-sensitive way by cashflow decomposition in a three-factor Gaussian market model identified from SA-CCR itself. This framework can guide banks in resolving ambiguities in their SA-CCR interpretation

he recent request for comments from the Office of the Comptroller dynamics. Table C shows EEPE versus moneyness and direction as of Decem- of the Currency (OCC 2018) offers an opportunity to address ber 5, 2018. The EEPE was calculated using a one-factor Hull-White model T observations on the Basel standardised approach for measuring identified from SA-CCR (mean reversion 0.05, flat volatility 0.0189). Note counterparty credit risk exposures, SA-CCR (BCBS 2014b, 2018a), and may that although the supervisory factor isServices 50bp, this translates directly into a inform other jurisdictions (BCBS 2018b). Our major observations include Hull-White volatility of 189bp. a lack of self-consistency for linear trades; a lack of appropriate risk sen- Thus, SA-CCR does not have appropriate risk sensitivity because (i) eco- sitivity (zero positions can have material add-ons; moneyness is ignored); nomically zero positions can have material add-ons and (ii) moneyness is and dependence on economically equivalent confirmations. Other signifi- ignored, which can exclude material risk differences. cant observations we address include the ambiguity of risk assignment (ie, the  Dependence on economically equivalent confirmations. Trade-level requirement for a single primary risk factor) and a lack of clear extensibility. add-ons in SA-CCR are defined as: Our observations about the SA-CCR’s properties, and the point sugges- trade (a) (a) AddOn = ıi SF d MFi tions of other authors (BCBS 2013), highlight principles by which we could i i i (a) assess SA-CCR, namely: appropriate risk sensitivity (the same exposure for SF captures risk-factorMedia volatility for asset class a, ı is 1 for direction (a˙) the same economics; positive exposure for non-zero risk), transparency, con- and MF is the maturity factor. The adjusted notional d incorporates a sistency and extensibility. The scope of this article is limited to non-option duration measure. interest rate products, for which these issues are most significant. Our objec- We can construct zero add-ons for any IRS with maturity T > 1 year, tive is to provide a comparison of the current SA-CCR with a Revisited SA- where T and 2T are in the same maturity bucket, as follows:  a receiver (rate R) amortising swap with notional 3(e0:05T /(e0:05T CCR (RSA-CCR) that is consistent with the existing structure and principles 1))N between today and T , and then notional (e0:05T /(e0:05T 1))N of SA-CCR but free of the observations discussed above. This RSA-CCR can guide banks in resolving ambiguities in their SA-CCR interpretation during between T and 2T (N is an arbitrary positive number);  a payer (rate R) swap with maturity 2T and notional 2(e0:05T /(e0:05T implementation. Longer proofs, additional details and derivations can be found in theDigital as- 1))N ; accepted version of this paper and the technical appendix (Berrahoui et al  a forward start (starting at T with maturity 2T ) receiver (rate R) swap with notional (e0:05T /(e0:05T 1))N ; 2019), both available on SSRN.  a payer swap (rate R) with maturity T and notional (1/(e0:05T 1))N . (IR) Observations on SA-CCR The adjusted notional (ie, time average) d for the first two swaps is the same: 2(e0:05T /(e0:05T 1))N . They are in opposite directions, so their Our observations on SA-CCR are for linear instruments, which we illustrate net add-on is zero. The maturity factor (MF) is the same for the thirdand using interest rate swaps (IRSs). We demonstrate these can be material for fourth trades: 1 (as T > 1). This is also true of the supervisory factor (SF(IR) simple situations, and then we describe RSA-CCR. in the regulation), which is 0:005 for both trades. The sum of direction and  Lack of self-consistency for linear trades. Economically equivalent adjusted delta is: positions have different capital requirements.Infopro For example, a vanilla IRSor 0:05T 0:05T 0:05 2T a set of forward rate agreements (FRAs) with the same strike have different X (IR) e e e  ıi d = N add-ons (see table A). Thus, SA-CCR is not self-consistent. i e0:05T 1 0:05 i=3;4  Lack of appropriate risk sensitivity. We illustrate two aspects of this 0:05T 1 1 e here: that economically zero positions can have material add-ons, and that N e0:05T 1 0:05 moneyness is ignored. The latter is probably more important; for example, = 0 moneyness produces systematic risk for existing positions when the general level of rates changes. So, the net add-on for the third and fourth trades is also 0. Now the net To illustrate the first case, consider an IRS that is economically hedged economics over all four trades is 0 for T to 2T , but it is a net receiver swap with a set of FRAs. There is a non-zero add-on despite there being zerorisk from 0 to T with flat notional N (e0:05T 1)/(e0:05T 1) = N . (see table B). Thus, SA-CCR is dependent on economically equivalent confirmations, Non-at-the-money (non-ATM) swaps have the same regulatory add-ons and this is not an objective of the standard (unlike leverage ratio capital). but can have wildly different effective expected positive exposures (EEPEs), even whenCopyright using very simple dynamics (see table C). RSA-CCR: cashflow and model-based SA-CCR SA-CCR ignores the moneyness of linear derivatives, although non- To address our observations about SA-CCR we propose a cashflow- and ATM swaps can have wildly different EEPEs even when using very simple model-based comparison for SA-CCR, which we call Revisited SA-CCR

risk.net 1 Cutting edge: Regulation

A. Equivalent derivatives positions with different add-ons For an ATM swap with start date Ts, end date Te and notional N , we show Instrument SA-CCR add-on in theorem 1 that the volatility of the present value of the swap (assuming a ATM swap 3,934,693 flat discount curve with zero rate) is: FRA replication 3,433,691  Split at 3Y 3,654,794 HW aTs aTe ATM(0) = N (e e ) ATM swap is 10Y USD, 100M notional. Split means a 3Y swap and a separate forward starting swap a (starting at 3Y for 7Y), with the same fixed rate as the ATM 10Y swap Given the relation (1) between theoretical add-on and trade volatility, we B. Material add-on for zero economic position obtain a model-based add-on for the ATM swap, identical to the SA-CCR Instrument SA-CCR add-on add-on: ATM net of FRA replication 1,646,936 aT aT 2HW e s e e Position is an ATM swap 10Y USD, 100M notional hedged by a strip of FRAs AddOntrade = N
3p2 a e aTs e aTe C. EEPE versus moneyness and direction calculated using a one-factor Hull-White = N SFIR model identified from SA-CCR; mean reversion 0.05, flat volatility 0.0189
Servicesa Instrument Pay Receive Thus, SA-CCR and BCBS (2014a) implicitly identify an appropriate model. ATM+500bp 5,778,184 3,898,079 ATM+100bp 4,271,792 3,707,917 Given that SA-CCR and BCBS (2014a) can be interpreted as using a one- ATM 3,903,699 3,668,881 factor Hull-White model, we extend this to a three-factor model to take into ATM 100bp 3,539,968 3,634,569 account the correlation structure of the different maturity zones in a later ATM 500bp 2,166,837 3,577,668 section. Correlations between add-ons are mapped to correlations between Fwd start ATM 1y–10y +500bp 4,712,043 4,891,493 zero-coupon bonds. Fwd start ATM 1y–10y +100bp 3,932,090 3,931,231  Fwd start ATM 1y–10y 3,737,350 3,691,418 Identity of SA-CCR and a one-factor Hull-White for a single matu- Fwd start ATM 1y–10y 100bp 3,542,763 3,451,756 rity bucket. Here,Media we provide details to establish a formal correspon- Fwd start ATM 1y–10y 500bp 2,766,729 2,495,425 dence of ATM swap add-ons between SA-CCR and a one-factor Hull-White Under SA-CCR, the top five trades have the same add-ons, as do the bottom five. ATM swapis10Y model. This correspondence being established, the regulatory parameters cor- USD. All instruments have a USD100M notional responding to model parameters are as shown in the previous section. This process also shows how to extend SA-CCR consistently to non-ATM trades (RSA-CCR). We pick the model by identifying it from the SA-CCR stan- in order to take moneyness into account. dard itself using the link between trade level add-ons and the volatility of A one-factor Hull-White model (Hull & White 1990) has short rate the present value of the trade in BCBS (2014a). Basing RSA-CCR on cash- (r(t)) dynamics, given by (2) above, so the dynamics of a zero-coupon bond flows automatically solves the lack of self-consistency, lack of appropriate risk P (t; T ) price are: sensitivity, dependence on economically equivalent confirmations, and accu-Digital  a(T t) rate risk assignment. Using a model provides clear extensibility, and provides dP (t; T ) = rt P (t; T ) dt (1 e )P (t; T ) dWt (3) a parameter transparency.  Model identification. Similarly to BCBS (2014a), we can define the Consider a payer, ie, pay-fixed rate R, notional N and swap price V , at t: theoretical add-on at horizon T of a trade with a value process Vt as the V (t) = V (t) V (t) following average expected positive exposure: Float Fixed e e X X Z T = N P (t; Ti )ıi L(t; Ti 1;Ti ) N P (t; Ti )ıi R 1 + AddOnV (T ) = E[(V (t) V (0)) ] dt i=s+1 i=s+1 T 0 We show the identity between SA-CCR and a one-factor Hull-White model Assuming dV (t) =  dW (t), we obtain the trade-level theoretical add-on in theorem 1 below. at horizon T : Infopro r 2 T Theorem 1 With zero bond dynamics given by (3), if V is the value process AddOnV (T ) = 3  (1) 2 of a forward starting payer swap, then the instantaneous volatility of V for t 6 Ts Recall the SA-CCR trade add-ons for interest rate trades: can be decomposed into three contributions:

0:05S 0:05E e e V (t) = ATM(t) + Float(t) + Fixed(t) AddOntrade = ıN SFIR MF
0:05 where: Taking a payer swap of longer than one year means MF = 1 and ı = 1. The e IR  supervisory factor SF = 0:005 for interest rates, so we define: X a(Ti 1 t) a(Ti t) ATM(t) = N P (t; Ti 1) (e e ) a i=s+1 3 IR  := p2SF e HW 2  X a(Ti 1 t) Float(t) = N P (t; Ti ) (1 e )ıi L(t; Ti 1;Ti ) a Consider a one-factor Hull-White model with short rate r(t) and mean i=s+1 e reversion parameterCopyrighta = 0:05:  X a(Ti t)  (t) = N P (t; Ti ) (1 e )ıi R Fixed a drt = a(t rt ) dt +  dWt (2) i=s+1 HW

2 risk.net May 2019 Cutting edge: Regulation

D. Correlation structure of SA-CCR interest rate add-ons by instrument maturity T ,  Cashflow decomposition for linear products. We first define the where 1 = 0.7 and 2 = 0.3 scope, concentrating on common linear product cashflows (which we call 6 6 T < 1 1 T < 5 5 T elementary cashflows), in definition 1 and then provide theorem 2,which T < 1 1   1 2 gives the add-ons for each of the different cashflow types. In a later sec- 1 6 T < 5 1 1 1 5 6 T 2 1 1 tion, we give an example derivation of the entries in table E for a float- ing cashflow (with a deterministic tenor basis). Full derivations, including a stochastic basis, are in the accompanying technical appendix (Berrahoui et al If at time t the swap is ATM, taking the standard weight-freezing assumption 2019). In the penultimate section, we compare the performance of SA-CCR (see proof), we have: and RSA-CCR.

e From the examples here and in Berrahoui et al (2019), the add-ons for  X a(Ti 1 t) a(Ti t) other (less common) cashflows in linear products can be derived in asimilar  (t) = N P (t; Ti ) (e e ) =  (t) V a ATM i=s+1 manner. Services T Moreover, if the yield curve is flat and equal to zero, then the instantaneous Definition 1 An elementary cashflow with payment date and pro- jected value CF(T ) (viewed from 0) is the projected value of one of the volatility of an ATM swap in one maturity bucket is given by the SA-CCR following six possibilities: regulatory formula, ie:  a fixed cashflow, so CF(T ) = N ;   a(Ts t) a(Te t) a standard interest rate floating cashflow, with notional N , money market ATM(t) = N (e e ) a index L(t; t + ı()), tenor  and fixing date 0 < Tf 6 T defined as: See the technical appendix (Berrahoui et al 2019) for a proof. ı NL(Tf ;Tf + ı ) The swap volatility can be decomposed into two contributions: Media  volatility of the floating rate index itself, ATM(t); and  a standard constant maturity swap (CMS) cashflow with notional N and  volatility of the present value of the cashflows, Float(t) + Fixed(t). swap rate S (Ts;Te) fixing at 0 < Ts 6 T , with underlying money market Since, for SA-CCR, only exposures below one year are relevant, we freeze tenor , maturity at Te and swap tenor ı(Ts;Te), defined as: the volatility V (t) and use its initial value V (0) so that: ı NS (Ts;Te) e  X aTi 1 aTi V (t) N P (0;Ti 1) (e e )  a standard inflation floating cashflow with notional N , defined on infla-  a i=s+1 tion index I (t), with initial observation date Ts (which may be in the past), e  6 X aTi next observation date I and final observation date Te T : + N ıi P (0;Ti ) (1 e )(R L(0;Ti 1;TDigitali )) a i=s+1 Â I (T ) Ã N e 1 I (T )  Matching inter-bucket correlations: three-factor Gaussian market s model. Theorem 1 shows that the regulatory add-on formula for anATM  a standard inflation compound cashflow with notional N , with initial swap can be recovered from a one-factor Hull-White model. To match the observation date Ts (which may be in the past), next observation date I and inter-bucket correlations of SA-CCR (see table D), we naturally move to a final observation Te 6 T : I (T ) three-factor Gaussian market model as follows: N e I (T )  we use maturity buckets defined by SA-CCR, M1 = [0; 1), M2 = [1; 5) s and M3 = [5; );  a standard interest rate compound cashflow, with start of compounding at 1 P  we define M (t) = 1 M , a map from positive numbers t to the Ts (which may be in the past), next reset date C , end of last compounding at i t Mi i Infopro 2 set of maturity buckets; Te and underlying index L(t; t + ı ) with tenor ; this cashflow is given by:  ZMk k = 1; 2; 3 we define three correlated Brownian motions t , , with the e 1 SA-CCR correlation structure, ie, dZM1 dZM2 = dZM2 dZM3 = 0:7 Y t t t t N (1 + ı L(Tk;Tk + ) M1 M3 and dZt dZt = 0:3. k=s We define the following three-factor Gaussian market model extension of Theorem 2 All elementary cashflows received (respectively, paid) at time T the Hull-White model, which allows us to recover the inter-bucket correla- should have an add-on contribution, using the supervisory formula: tions for zero-coupon bonds (with expiries T in a discrete set T1;:::;TN covering all business days up to horizon TN ), ie: AddOncashflow = ıSF(IR)d (IR)MF

 a (T t) M (T ) dP (t; T ) = rt P (t; T ) dt (1 e ) P (t; T ) dZ to the maturity bucket, corresponding to the payment date T . The following inputs
a


t should be used in the supervisory formula: where rt is the risk-free rate (defined as the continuously compounded rate of  a delta equal to 1 (respectively, 1); the shortestCopyright maturity zero-coupon bond). Now that we have a model identi-  a start date of 0 and an end date of T to compute the supervisory duration; fied from SA-CCR, we can move on to the second key element ofRSA-CCR: and cashflow decomposition for linear products.  an adjusted notional equal to the present value of the cashflow.

risk.net 3 Cutting edge: Regulation

E. Add-on components for the elementary cashflows listed in definition 1 Cashflow Hedging set Maturity bucket Effective notional Start End Delta Floating Rates, basis M (Tf ) (N + CF(T ))P (0;T ) 0 Tf 1 M (Tf + ) (N + CF(T ))P (0;T ) 0 Tf +  +1 Compound Rates, basis M (max(c ;Ts)) CF(T )P (0;T ) 0 max(c ;Ts) 1 M (Te) CF(T )P (0;T ) 0 Te +1 ı CMS Rates, basis M1 (N + CF(T ))P (0;T ) min(1;Ts) min(1;Te) 1 ı(Ts;Te) ı M2 (N + CF(T ))P (0;T ) min(max(1;Ts); 5) min(max(1;Te); 5) +1 ı(Ts;Te)) ı M3 (N + CF(T ))P (0;T ) max(5;Ts) max(5;Te) +1 ı(Ts;Te)) Inflation floating Inflation M (max(I ;Ts)) (N + CF(T ))P (0;T ) 0 M (max(I ;Ts)) 1 Te (N + CF(T ))P (0;T ) 0 Te +1 Inflation compound Inflation M (max(I ;Ts)) CF(T )P (0;T ) 0 M (max(I ;Ts)) 1 Services Te CF(T )P (0;T ) 0 Te +1 Hedging sets for rates cashflows are rates and basis per currency. Hedging sets for inflation cashflows are inflation per currency. Derivations of these values follow the floating example inour‘Example add-on derivation for a xibor coupon’ section. Details, including floating and CMS examples in full, are given in the accompanying technical appendix

Moreover, all non-fixed standard cashflows will have other add-on contri- We ignore convexity, so this setup includes timing mismatches such as a butions due to the volatility of their respective underlying indexes, calculated Libor rate paid in advance. using the supervisory formulas (with inputs as per table E) and allocated to the The forward value of the index is: appropriate maturity buckets and hedging sets (as stated in table E for received ı L(t; T ;T + ) = P (t; T )/P (t; T + ) 1 cashflows). f Mediaf f f f f CF(T ) is the projected cashflow, and M (t) the map that associates to t its P risk bucket and P (0;T ) the discount factor. Basis risk add-ons are allocated to We define the index-versus-discounting basis b via: the money market index  versus the discount hedging set. Pf (t; s) = Pb(t; s)P (t; s) Setting P (0;T ) = 1 would result in simplified add-ons in line with current regulatory stipulations Ignoring convexity, the coupon value is: In RSA-CCR add-on contributions for linear products are the aggregate V (t) = NP (t; T )(P (t; T )/P (t; T + ) 1) add-ons of the add-ons for each cashflow. This removes the SA-CCR issues f f f f mentioned above: lack of self-consistency, lack of appropriate risk sensitiv- DigitalApplying Itô’s lemma to V (t), we can see the instantaneous volatility of V (t) ity and dependence on economically equivalent confirmations. We formalise has three components, one from the floating rate index volatility, one from add-on calculations via cashflow decomposition in the definitions below. the cashflow’s present value and one from the basis risk volatility: Definition 2 A linear product has no optionality and is made of elemen- tary cashflows CF with payment dates T (i = 1; : : : ; n) and notional N . dVt = ( ) dt i i i


Definition 3 The total add-on of a portfolio of linear products is + N [ı L(t; Tf ;Tf + ) dP (t; T ) obtained from the aggregation of the individual add-ons of each cashflow that + P (t; T )Pb(t; Tf )/Pf (t; Tf + ) dP (t; Tf ) makes up the linear products. Add-on aggregation is performed according + P (t; T )Pf (t; Tf )/Pb(t; Tf + ) d(1/P (t; Tf + )) to SA-CCR. + P (t; T )P (t; Tf )/Pf (t; Tf + ) dPb(t; Tf ) Proposition 1 All portfolios of linear productsInfopro resulting in the same net + P (t; T )Pf (t; Tf )/P (t; Tf + ) d(1/Pb(t; Tf + ))] cashflows have the same aggregate add-ons under RSA-CCR. No dP dP terms are present above as these result in drift (dt) contribu- Proof By construction, the add-ons depend only on the net cashflows.   tions, not volatility (dW ) contributions.    Floating coupon: deterministic basis. Assuming here that the basis Propostion 1 is the property we are after: this ensures appropriate risk sen- curve is deterministic, we have for P (t; T ), P (t; Tf ) and P (t; Tf + ) the sitivity (no risk for no economic position, and vice versa) and no dependence following dynamics from the three-factor Gaussian market model: on economically equivalent confirmations.  M (T ) dP (t; T ) = ( ) dt (1 e a(T t))P (t; T ) dZ


a t Example add-on derivation for a xibor coupon  a(T t) M (Tf ) dP (t; Tf ) = ( ) dt (1 e )P (t; T ) dZt Here, we provide an example of how to derive the entries in table E for a float-


a  1 à  ing coupon on a deterministic index-versus-discounting basis. The stochastic a(Tf + t) d = ( ) dt + (1 e ) extension is availableCopyright online. P (t; Tf + )


a 1 M (T +) The floating rate cashflow is based on a money market index withfixing dZ f  P (t; T + ) t at Tf , tenor  (less than one year) and payment at T > , with notional N . f

4 risk.net May 2019 Cutting edge: Regulation

F. Vanilla swaps with varying moneyness; percentage differences from the Gaussian H. Amortising swaps; percentage differences from the Gaussian market model market model shown in the last column shown in the last column Shifted Full Shifted HW1F LMM 3F approach HW1F LMM 3F RSA-CCR average average with RSA-CCR average average RSA-CCR with pay/ pay/ discount with pay/ pay/ with no market receive receive equal market receive receive Instrument SA-CCR discounting discount add-ons add-on Instrument SA-CCR to 1 discount add-ons add-on ATM swap 4% 3% 5% 0% 3,783,285 Amortising 2% 1% 5% 2% 5,996,321 ATM+100bp 1% 3% 5% 0% 3,984,416 10Y:5Y/100M; ATM+500bp 19% 1% 6% 0% 4,829,860 5Y/200M ATM 100bp 10% 3% 4% 0% 3,588,130 FRA replication 8% 1% 5% 2% 5,996,321 ATM500bp 37% 1% 6% 0% 2,875,001 5Y/200M and 4% 1% 5% 2% 5,996,321 forward start G. Vanilla swaps with different replications; percentage differences from HW1F 5Y5Y/100M shown in the last column 10Y/100M and 5% 1% Services5% 2% 5,996,321 Shifted 5Y/100M HW1F LMM 3F Amortising 100% 29% 17% 25% 352,986 RSA-CCR average average minus RSA-CCR with pay/ pay/ 10Y/150M with no market receive receive Instrument SA-CCR discounting discount add-ons add-on I. Zero-coupon swaps; percentage differences from the Gaussian market model ATM swap 4% 3% 5% 0% 3,783,285 FRA replication 9% 3% 5% 0% 3,783,285 shown in the last column ATM–FRAs inf% 0% 0% 0% 0 Full Shifted Split at 3Y 3% 3% 5% 0% 3,783,285 approach HW1F LMM 3F with RSA-CCR average average Mediadiscount with pay/ pay/ Hence: equal market receive receive Instrument SA-CCR to 1 discount add-ons add-on dVt = rt Vt dt Zero-coupon 2% 15% 3% 1% 3,999,402 ATM 10Y  a(T t) M (T ) (1 e )P (t; T )Nı L(t; T ;T + ) dZ a f f t Zero-coupon 9% 15% 3% 1% 4,341,650 ATM+100bp  a(T t) (1 e f ) 10Y a M (Tf ) Zero-coupon 27% 15% 2% 3% 5,368,783 P (t; T )N (1 + ı L(t; T ;T + )) dZ  f f t ATM+500bp  a(Tf + t) 10y + (1 e ) a DigitalZero-coupon 8% 15% 3% 1% 3,657,255 M (Tf +) ATM 100bp P (t; T )N (1 + ı L(t; Tf ;Tf + )) dZ (4)  t 10y From (4), we can see the floating rate cashflow add-ons bucket contributions Zero-coupon 72% 14% 3% 4% 2,290,554 ATM 500bp are as follows. 10y (1) Contribution from the present value of the cashflow:  maturity bucket, M (T ); completeness. Both RSA-CCR and SA-CCR can be considered approxima-  notional, NP (0;T )L(0;Tf ;Tf + );  duration, (1/a)(1 e aT ); tions to the simulation results. These examples demonstrate that RSA-CCR  delta, 1. has appropriate risk sensitivity compared with SA-CCR. In addition, from (2) Index volatility – fixing date contribution:Infopro the results of ambiguity resolution in RSA-CCR, for example, it is clear what  maturity bucket, M (Tf ); to do for zero-coupon swaps. Of course, by construction, RSA-CCR does not  effective notional, NP (0;T )(1 + ı L(0;Tf ;Tf + )); suffer from dependence on economically equivalent confirmations.  duration, (1/a)(1 e aTf );  Vanilla swaps with different moneyness (see table F). SA-CCR is insensi-  delta, 1. tive to moneyness, showing a 56% range of error with respect to the simula- (3) Index volatility – payment date contribution: tion, whereas the RSA-CCR error range is 2%.   maturity bucket, M (Tf + ); Vanilla swaps with different replications (see table G). Here, SA-CCR’s  effective notional, NP (0;T )(1 + L(0;Tf ;Tf + )); maximum error with respect to simulation is infinite as it gives non-zero risk  supervisory duration, (1/a)(1 e a(Tf +)); for a zero economic position. In contrast, RSA-CCR has a 5% range of error  delta, 1. and a maximum error of 5%. Further examples, including stochastic basis, can be found online.  Amortising/accreting vanilla swaps versus a combination of swaps (see table H). Here, the SA-CCR range of error is 98% (in one case, it gives a RSA-CCR performance versus SA-CCR zero add-on for non-zero risk). The range of error for RSA-CCR is 28% with Here, weCopyright compare RSA-CCR add-ons with SA-CCR add-ons as well as with no discounting, or 14% with market discounting versus the simulation. a simulation of the Gaussian market model (calibration details online). We  Zero-coupon swaps (see table I). The range of error for SA-CCR is 99%, also provide a reference comparison with a one-factor Hull-White model for compared with RSA-CCR at 3%.

risk.net 5 Cutting edge: Regulation

 Forward starts (see table J). The range of error for SA-CCR is 68%, J. Forward starting swaps; percentage differences from the Gaussian market model compared with RSA-CCR at 1%. shown in the last column Shifted HW1F LMM 3F Conclusions RSA-CCR average average We recommend comparing SA-CCR with RSA-CCR in order to address RSA-CCR with pay/ pay/ observations on SA-CCR’s lack of self-consistency for linear trades, lack of with no market receive receive Instrument SA-CCR discounting discount add-ons add-on appropriate risk sensitivity (zero positions can have material add-ons; money- Fwd start ATM 2% 6% 3% 1% 3,668,070 ness is ignored), dependence on economically equivalent confirmations, and 1Y–10Y ambiguity of application for cases not explicitly described. This RSA-CCR Fwd start ATM 4% 6% 3% 1% 3,888,496 can guide banks in resolving ambiguities in their SA-CCR interpretation. 1Y–10Y +100bp To achieve consistent treatment of the foreign exchange risk factors within Fwd start ATM 22% 7% 3% 1% 4,772,075 the RSA-CCR cashflow decomposition approach, it is enough to consider 1Y–10Y only the currency pairs with respect to the domestic currency, ie, CCY/USD, +500bp Services as hedging sets. The add-on contribution to a forex asset class (hedging set Fwd start ATM 9% 6% 3% 1% 3,447,875 CCY/USD) for an elementary linear cashflow in the currency CCY (regard- 1Y–10Y 100bp less of the product type) is simply the present value of the cashflow con- Fwd start ATM 46% 6% 2% 2% 2,570,157 verted to the domestic currency, USD. Moreover, each cashflow has an inter- 1Y–10Y 500bp est rate add-on contribution as computed for each elementary cashflow by the RSA-CCR approach.  Mourad Berrahoui is the head of counterparty credit risk modelling at Lloyds expressed in this presentation are the personal views of the author and do not Banking Group in London. Othmane Islah is the head of quantitative research necessarily reflect the views or policies of current or previous employers. Not at Quantuply and a senior consultant for KVA/MVA at Lloyds Banking Group guaranteed fit for Mediaany purpose. Use at your own risk. in London. Chris Kenyon is the head of XVA quant modelling at MUFG Secu- Email: [email protected], rities EMEA plc, also in London. The authors would like to gratefully acknow- [email protected], ledge useful discussions with Lee McGinty and Crina Manolescu. The views [email protected].

REFERENCES

BCBS, 2013 BCBS, 2014b BCBS, 2018b Hull J and A White, 1990 Comments received on `The The standardised approach for Implementation of Basel standards Pricing interest-rate-derivative securities non-internal model method for measuring counterparty credit risk DigitalImplementation Report, November Review of Financial Studies 3(4), capitalising counterparty credit risk exposures 23, BIS, available at www.bis.org/ pages 573–592 exposures – consultative document' Standard, March 31, BIS, available bcbs/publ/d453.htm Comments, September 27, BIS, at www.bis.org/publ/bcbs279.htm OCC, 2018 available at www.bis.org/publ/ Berrahoui M, O Islah and Standardized approach for calculating bcbs254/comments.htm BCBS, 2018a C Kenyon, 2019 the exposure amount of derivative Frequently asked questions on the Technical appendix to `Making SA-CCR contracts BCBS, 2014a Basel III standardised approach for self-consistent and appropriately Federal Register 83(241), Monday, Foundations of the standardised measuring counterparty credit risk risk-sensitive by cashflow December 17, Proposed Rules, approach for measuring counterparty exposures decomposition in a 3-factor Gaussian Docket No. [R-1629 and RIN credit risk exposures FAQ, March 22, BIS, available at market model' 7100-AF22] Working Paper, August 28, BIS, www.bis.org/bcbs/publ/d438.htmInfopro Preprint, SSRN available at www.bis.org/publ/ bcbs_wp26.htm

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