Basel Committee on Banking Supervision. Working Paper No 26. Foundations of the standardised approach for measuring counterparty credit risk exposures

Transcription

1 Basel Commttee on Bankng Supervson Workng Paper No 6 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures August 014 (rev. June 017)

2 The Workng Papers of the Basel Commttee on Bankng Supervson contan analyss carred out by experts of the Basel Commttee or ts workng groups. They may also reflect work carred out by one or more member nsttutons or by ts Secretarat. The subjects of the Workng Papers are of topcal nterest to supervsors and are techncal n character. The vews expressed n the Workng Papers are those of ther authors and do not represent the offcal vews of the Basel Commttee, ts member nsttutons or the BIS. Ths publcaton s avalable on the BIS webste ( Bank for Internatonal Settlements 017. All rghts reserved. Bref excerpts may be reproduced or translated provded the source s cted. ISSN

3 Contents Equaton Chapter 1 Secton 1Foundatons of the standardsed approach for measurng counterparty credt rsk exposures General structure of the SA-CCR Exposure at default Replacement cost Potental future exposure General framework for add-ons Assumptons and set-up Add-ons for unmargned nettng sets Add-ons for margned nettng sets Structure of add-on calculatons Trade-level add-ons Add-on aggregaton Add-on calculatons by asset class Interest rate Foregn exchange Credt Equtes Commodtes Multpler Calbraton Correlatons between IR maturty buckets Deltas for CDO References... 0 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

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5 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures Ths techncal paper explans modellng assumptons that were used n developng the standardsed approach for measurng counterparty credt rsk exposures (SA-CCR). The paper also clarfes certan aspects of the SA-CCR calbraton that are not dscussed n the fnal standard that was publshed n March 014 (revsed Aprl 014). 1 The language used to descrbe the SA-CCR n ths paper may dffer somewhat from the language used n the fnal standard. For example, the paper uses concepts that are not present n the fnal standard such as trade-level add-ons and sngle-factor subsets of hedgng sets. Furthermore, t does not use the concept of effectve notonal, whch s employed n the standard. The purpose of these adaptatons s to emphasse the common aggregaton framework that underpns the SA-CCR add-on formulas for dfferent asset classes. 1. General structure of the SA-CCR 1.1 Exposure at default The SA-CCR specfes exposure at default (EAD) measured at a nettng set level. The target EAD measure for a nettng set under the SA-CCR s the correspondng EAD measure under the Internal Model Method (IMM), gven by the product of multpler α (alpha) and Effectve Expected Postve Exposure (EEPE). Under the SA-CCR, the nettng-set-level EEPE s represented as the sum of two terms: the replacement cost (RC) and the potental future exposure (PFE). Thus, EAD usng SA-CCR s calculated va ( ) where the multpler alpha s set to the default IMM value, α = 1.4. EAD = α RC + PFE (1) Whle the Current Exposure Method (M) also represents exposure as the sum of the RC and the PFE terms, Equaton (1) dffers from EAD usng M n two mportant respects: The SA-CCR ncorporates the multpler alpha that (conceptually) converts EEPE nto a loan equvalent exposure (see ISDA-TBMA-LIBA (003); Canabarro, Pcoult and Wlde (003); and Wlde (005)). The M specfes RC and PFE only for the unmargned case, whle the SA-CCR ncludes formulatons of RC and PFE that dffer for margned and unmargned cases. The approach to developng the SA-CCR was to smply reflect the RC and PFE for partcular asset classes. RC represents a conservatve estmate of the amount the bank would lose f the counterparty were to default mmedately. It should be noted that margnng practces are becomng more complcated over tme and the approach to replacement cost reflects the dversty of margnng practces that are common n the market. The PFE component reflects ncreases n exposure that could occur over tme. PFE s related to volatlty that s observed n the asset class. 1 The fnal standard can be retreved from Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 1

6 1. Replacement cost For unmargned nettng sets, RC represents the loss that would occur f the counterparty defaulted mmedately. Suppose that a bank has a nettng set of trades wth a counterparty wth the current markto-market (MTM) value V. If the counterparty were to default mmedately, the loss for the bank would be equal to the greater of V and zero, so that RC s gven by max{ V ;0}. There may be cases when an unmargned nettng set s supported by collateral other than varaton margn. All such collateral s a form of ndependent collateral amount (ICA) that s posted at trade ncepton. Generally, a bank can both receve and post ICA. Receved ICA reduces the RC, as t can be used to offset losses n the event of a counterparty default. Posted ICA can be lost n the event of the counterparty default, unless t s segregated n a bankruptcy-remote account. Thus, the bank s net ICA poston, or NICA, for a nettng set s calculated va aggregatng all receved ICA wth postve sgn and all posted non-segregated ICA wth the negatve sgn. Snce all collateral n an unmargned nettng set has the form of ICA (e no varaton margn s exchanged), the entre collateral amount s gven by NICA. The RC s obtaned by subtractng the current cash-equvalent value of collateral avalable to the bank usng a one-year horzon,, from the nettng set MTM value: C (1 year) NoMargn RC = max{ V C (1 year);0} () For non-cash collateral, the cash-equvalent value C (1 year) s obtaned from the current MTM value of collateral C MTM va applcaton of an approprate supervsory harcut h (1 year) for a one-year tme horzon accordng to the general harcut formula: C CMTM [1 ht ( )] f CMTM > 0 () t = CMTM [1 + ht ( )] f CMTM < 0 The harcut for a gven tme horzon accounts for a possblty of an unfavourable change of collateral value over that tme horzon. For a margned nettng set, collateral generally conssts of two parts: ICA (also known as ntal margn) and varaton margn (VM) that s posted and returned dependng on the nettng set MTM (see, for example, Gregory (01, Chapter 5). The SA-CCR sets the RC for a margned nettng set equal to the maxmum of the current RC and a conservatve estmate of the future RC. The current RC s gven by RC = max{ V C (MPR);0} (4) Current Margn where C (MPR) s the cash-equvalent value of collateral obtaned from C MTM va applcaton of Equaton (3) wth tme horzon t set equal to the margn perod of rsk (MPR). 3 The future RC s nterpreted as the loss that could occur f the counterparty defaulted at an unknown tme pont wthn the one-year captal horzon. Because of the uncertanty of the default tme, the nettng set MTM value and, therefore, the true RC s not known. The SA-CCR conservatvely assumes that, at the tme of default, the nettng set MTM value s hgh enough to trgger a margn call. Ths would occur at the MTM level equal to the sum (3) See, for example, Pykhtn (011). 3 MPR represents the tme perod over whch exposure to counterparty may ncrease. For margned nettng sets, ths s the tme between the last margn call that the counterparty would respond to pror ts default and the closeout after the default. For unmargned trades, the tme perod s one year or the fnal maturty, consstent wth the one-year tme horzon generally used n the Basel accord. Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

7 of the threshold (TH) and the mnmum transfer amount (MTA). If there s ntal margn, the future RC s obtaned by subtractng NICA from the margn call trgger level, resultng n Future RCMargn = max{th + MTA NICA;0} (5) By takng the maxmum of Equatons (4) and (5) one obtans the RC for margned nettng sets: RC = max{ V C (MPR);TH + MTA NICA;0} (6) Margn Sometmes margn agreement thresholds are set at a very hgh level, whch would lead to unreasonably hgh values of the replacement cost and, therefore, the EEPE. Ths ssue s addressed n the SA-CCR by cappng the EEPE of a margned nettng set by the EEPE of an otherwse equvalent unmargned nettng set. 1.3 Potental future exposure The SA-CCR specfes the PFE as the product of the aggregate (e nettng-set-level) add-on and a multpler W () descrbed n Secton 5, dependent on the rato of the nettng set s current collateral-adjusted MTM value V C to the add-on tself: V C PFE = W AddOn aggregate AddOn aggregate The aggregate add-on s an estmate of the nettng set EEPE under the assumptons that no collateral s currently held or posted and that the current MTM values of all trades are zero. It s generally the case that PFE s hghest for at-the-money nettng sets. The multpler reduces the value of PFE when the collateral-adjusted MTM s negatve. (7). General framework for add-ons.1 Assumptons and set-up The nettng-set-level add-on represents a conservatve estmate of the EEPE of the nettng set under the followng assumptons: The current MTM value of each trade s zero (e trade s at-the-money, ATM). The bank nether holds nor has posted collateral for the nettng set. There are no cash flows wthn the captal horzon of one year. The evoluton process for MTM value of each trade follows arthmetc Brownan moton wth zero drft and fxed volatlty. These assumptons are needed to obtan the lnear dependence of the aggregate add-on on a sngle dynamc quantty the volatlty of the nettng set MTM value. The most mportant beneft of ths dependence s that one can aggregate add-ons from a trade level to a nettng-set level as f they were standard devatons. Furthermore, the SA-CCR calbraton can be reduced to makng assumptons on volatltes and correlatons of market rsk factors that drve trade MTM values. Usng these assumptons, the MTM value of trade at tme t can be represented as V() t = 1 { M t } σ tx (8) Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 3

8 where 1 {} s the ndcator functon of a Boolean varable (t takes a value of 1 f the argument s TRUE and value of 0 otherwse), M s the remanng maturty, X s a standard normal random varable. σ s the volatlty of the MTM value of trade and Suppose that we know the correlatons r j between random varables MTM value Vt () of a nettng set at tme t can be expressed va X and X j. Then, the V() t = σ () t ty (9) where Y s a standard normal random varable and σ () t s the annualsed normal volatlty of the nettng set at tme t gven by σ() t = 1{ } 1 r M t { M j t} j σσ j (10), j Note that, n spte of the assumpton of fxed annualsed volatlty for MTM value of each trade, the annualsed volatlty σ () t of the nettng set generally depends on tme t because trades that mature before tme t do not contrbute to σ () t. 1. Add-ons for unmargned nettng sets Expected exposure (EE) can be calculated for an unmargned nettng set: { } t = t ty = t t no-margn EE () E max σ(),0 ϕ(0) σ() where ϕ() s the standard normal probablty densty, so ϕ(0) = 1/ π. Ths value represents the average exposure of the nettng set at tme t. Cases where the nettng set value s negatve represent stuatons where the bank owes the counterparty and so the exposure s set to zero. Conceptually, the unmargned aggregate add-on should be defned as the EEPE calculated from the EE profle of Equaton (11). However, ths would requre calculatng σ () t at several tme ponts between zero and one year. To avod ths complexty, the consultatve paper BCBS (013) floored the remanng maturty of all trades by one year. 4 The maturty floor s equvalent to replacng σ () t wth σ (0) n Equaton (11) whenever t 1 year, whch allows one to calculate the EEPE from Equaton (11) n closed form: 1 year no-margn 1 no-margn AddOnaggregate = EE ( t) dt = ϕ(0) σ(0) 1 year 1 year 0 3 (11) (1) Note that Equaton (1) can be restated n terms of aggregaton of trade-level add-ons rather than aggregaton of trade-level volatltes: no-margn no-margn no-margn AddOnaggregate = rj AddOn AddOn j, j (13) where add-on of trade represents the EEPE of a nettng set consstng of only trade : 1 4 See Pykhtn (014) for analyss of the BCBS (013) proposals. 4 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

9 AddOn = ϕ(0) σ 1 year (14) 3 no-margn However, applcaton of the one-year floor would result n unreasonably hgh trade-level addons for short-term trades. In partcular, short-term trades would have a capablty to offset long-term trades to a much greater extent that they should. 5 To prevent ths, the defnton of the aggregate add-on for unmargned nettng sets was kept n the form of Equaton (13), but the defnton of trade-level addon was changed to accommodate maturtes shorter than one year: where maturty factor = (15) 3 no-margn no-margn AddOn ϕ(0) σ 1 year MF no-margn MF s defned as no-margn mn{ M,1year} MF 1year = (16) scales down the volatlty of the trade MTM from one year to the trade remanng maturty trades wth M < 1year. M for the.3 Add-ons for margned nettng sets For margned nettng sets, the SA-CCR add-on s defned (n the sprt of the IMM Shortcut Method) as the expected ncrease of the nettng set MTM over the MPR. 6 Snce the portfolo s assumed to have the current MTM equal to zero, the add-on for a margned nettng set defned n ths manner reduces to the value of the EE of an otherwse equvalent unmargned nettng set at the tme pont equal to the MPR: margn no-margn AddOnaggregate EE (MPR) ϕ(0) σ(0) MPR = = (17) Smlarly, for a margned nettng set contanng only trade, the add-on s = (18) margn AddOn ϕ(0) σ MPR Thus, one can replace aggregaton of trade-level volatltes wth aggregaton of trade-level addons for both margned and unmargned nettng sets va Equaton (13) that we restate here for margned nettng sets: margn margn margn AddOnaggregate = rj AddOn AddOn j, j Fnally, trade-level margned add-ons n Equaton (18) can be restated n exactly the same form as trade-level non-margn add-ons n Equaton (15) wth dfferently defned maturty factors: = ϕ σ (19) 3 margn margn AddOn (0) 1 year MF 1 5 For example, to fully offset the rsk of a one-year FX forward, t would be suffcent to book a very short-term FX forward or an FX spot contract of the same notonal n the opposte drecton. 6 See BCBS (006) and BCBS (010). See also analyss n Gbson (005) where the IMM Shortcut Method was frst proposed. Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 5

10 margn 3 MPR MF 1year = (0) Thus, the entre dfference between margned and unmargned trade-level add-ons resdes n the maturty factors. Because of ths, there wll be no dstncton made between margned and unmargned nettng sets n the remander of ths paper when add-ons are dscussed. 3. Structure of add-on calculatons For the purpose of the add-on calculaton, each trade n the nettng set s assgned to at least one of fve asset classes: nterest rate (IR), foregn exchange (FX), credt, equtes and commodtes. The desgnaton should be made accordng to the nature of the prmary rsk factor (eg IR for most sngle currency IR swaps, FX for most cross-currency swaps, credt for most credt default swaps). For more complex trades, where t s dffcult to determne a sngle prmary rsk factor, bank supervsors may requre that trades be allocated to more than one asset class. Whle the SA-CCR add-on formulas are asset class-specfc, there are a number of common features for all asset classes. Most mportantly, nettng-set-level add-ons are calculated from trade-level add-ons va an aggregaton procedure based on the general prncples outlned n the prevous secton. 3.1 Trade-level add-ons The SA-CCR does not drectly operate wth trade volatltes. Instead, the drectonal add-on (trade) AddOn for each trade of asset class a s gven by the product of the followng four quanttes: drectonal delta ( a) ( ) δ, adjusted notonal d, supervsory factor SF a and maturty factor MF : (trade) ( a) ( a) AddOn δd SF MF = (1) where maturty factor MF s defned va Equaton (16) for unmargned trades and va Equaton (0) for margned trades. Equaton (1) s meant to be equvalent to Equatons (15) and (19) wth one dfference: add-ons n Equaton (1) can be negatve. Negatve add-ons are a means of accommodatng negatve correlatons r j n Equaton (13) wthout usng negatve correlatons explctly. Comparng Equaton (1) wth Equatons (15) and (19) reveals that the SA-CCR approxmates the trade-level volatlty va ( a) 3 SF ( a) σ = δ d () ϕ(0) where the frst factor (the rato) can be nterpreted as the standard devaton of the prmary rsk factor at the one-year horzon. The quanttes that appear n Equaton (1) are specfed n the fnal standard for the SA-CCR. They have the followng meanng: Drectonal delta δ : Delta serves two purposes: t specfes the drecton of the trade wth the respect to the prmary rsk factor (postve for long, and negatve for short) and serves as the scalng factor for trades that are non-lnear n the prmary rsk factor. Banks nternal deltas are not allowed; standardsed values are used nstead. Trades that are not optons or CDOs are assumed to be lnear n the underlyng rsk factor and have delta of unt magntude. For optons, banks should use the Black-Scholes formula for delta as provded n the fnal standard. For CDOs, 6 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

11 a standardsed formula provded n the standard text should be used wth nternal attachment and detachment ponts (ths formula s dscussed later n ths paper). Adjusted notonal d ( a) : Whle the defnton of adjusted notonal s asset class-specfc, one can generally state that adjusted notonal quantfes the sze of the trade. It s proportonal to ether a trade s notonal (as n the case of IR, FX and credt) or the current prce of the underlyng assets (as n the case of equty and commodty). For IR and credt dervatves, the adjusted notonal s also proportonal to the supervsory duraton. ( ) SF a : The supervsory factor s the supervsory value of EEPE of a nettng set consstng of a sngle lnear trade (e unt delta) of unt adjusted notonal belongng to the same subclass of asset class a as trade. Supervsory factors ncorporate the volatlty assumed by regulators for the prmary rsk factor. Whle the fnal standard specfes supervsory factors at specfc subclasses wthn each asset class, conceptually the SA-CCR structure s very flexble: the method can be easly made more or less granular wthout changng the structure of the approach. Supervsory factor 3. Add-on aggregaton Aggregaton of trade-level add-ons to the nettng set level s done by mposng a certan structure on the correlaton matrx r j n the aggregaton formula gven by Equaton (13). A key add-on aggregaton concept of the SA-CCR s the noton of a hedgng set. By defnton, a hedgng set s the largest collecton of trades of a gven asset class wthn a nettng set for whch nettng benefts are recognsed n the PFE add-on of the SA-CCR. No nettng s recognsed across hedgng sets, so a nettng-set-level add-on s calculated as a drect sum of the absolute values of hedgng-set-level add-ons: AddOn (no-margn) (HS) = (3) m AddOn m (HS) where AddOn m s the add-on for hedgng set m. Equaton (3) s equvalent to assumng perfect postve correlaton between hedgng set MTM values. Hedgng-set-level add-ons are obtaned va the followng two-step aggregaton process that recognses nettng wthn each hedgng set. All trades of a hedgng set are desgnated to sngle-factor subsets. It s assumed that all trades of a gven sngle-factor subset are drven by the same market factor, so full offset s allowed for such trades: AddOn (SFS) j = AddOn (4) SFS j (trade) (trade) where AddOn s the drectonal add-on (e add-ons for long and short trades have opposte sgns) for trade descrbed above and notaton SFS j should be nterpreted as all trades belongng to snglefactor subset j of a gven hedgng set. Sngle-factor subsets are further aggregated to a hedgng-set level under the assumpton that, wthn a hedgng set, rsk factors drvng sngle-factor subsets are mperfectly correlated: (HS) (SFS) (SFS) AddOn m = ρ jk AddOn j AddOn k (5) jk, HSm where notaton jk, HS m should be nterpreted as all pars of sngle-factor subsets belongng to the same hedgng set m of a gven asset class, and correlatons ρ jk are prescrbed by regulators. 1 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 7

12 The next secton descrbes asset-class-specfc aspects of the general framework outlned above. These aspects nclude the defnton of adjusted notonal along wth the specfcaton of hedgng sets, sngle-factor subsets and correlaton parameters. 4. Add-on calculatons by asset class 4.1 Interest rate Consder a fxed-for-floatng IR swap as the most common example of an IR dervatve. Suppose that swap payments start at tme S, end at tme E and refer to notonal N() t at tme t. In the contnuous lmt, swap MTM value can be wrtten as E swap [ ] max{ S, t} V ( t) = SR ( t) FR N ( τ)df( t, τ) dτ (6) where SR ( t ) s the swap rate at tme t, FR s the fxed rate and DF( t, τ ) s the dscount factor from tme τ to tme t. The SA-CCR assumes that all varablty of the swap MTM value results from changes of the swap rate, thus freezng the dscount factor n the ntegral. The IR supervsory factor s meant to capture the one-year volatlty of the swap rate, whle the adjusted notonal s meant to represent the value of the ntegral n Equaton (6) at tme t = 0. The SA-CCR approxmates the ntegral by settng the adjusted notonal equal to where the supervsory duraton gven by d = N SD (7) (IR ) N s the average of the swap notonal over the remanng lfe of the swap payments and SD s E exp( rs) exp( re) SD = exp( rt) dt = (8) r S wth the nterest rate set to r = Interest rate hedgng sets are specfed as all IR trades of a nettng set denomnated n the same currency. Three sngle-factor subsets of a hedgng set are defned va the range of the end date: (1) E 1 year ; () 1 year < E 5 years ; (3) E > 5 years. Thus, Equaton (4) specfes the add-ons for each of the three maturty buckets of a gven currency. Correlatons between the maturty buckets are set as follows: ρ1 = ρ3 = 70% and ρ 13 = 30% (calbraton of these correlatons s dscussed later n ths paper). Thus, n the case of IR, Equaton (5) becomes: (CCY) (MB) (MB) (MB) AddOn m = ( AddOn 1 ) + ( AddOn ) + ( AddOn3 ) (9) 1 (MB) (MB) (MB) (MB) (MB) (MB) +1.4 AddOn 1 AddOn AddOn AddOn AddOn1 AddOn 3 where the notaton CCY ndcates that the hedgng set s a currency and the notaton MB ndcates that the sngle-factor subsets are maturty buckets. 8 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

13 4. Foregn exchange Consder a lnear FX trade such as an FX forward between a foregn currency and the domestc currency. Assumng that the forward maturty s greater than one year (smaller maturtes are accounted by maturty factors), volatlty of the forward s MTM value over the one-year horzon s ndependent of the forward s maturty and s gven by the product of the notonal of the foregn leg converted to the domestc currency usng the current FX spot rate and the one-year relatve volatlty of the FX spot rate. Ths example motvates specfyng the adjusted notonal accordng to 7 d (FX) = N (30) foregn foregn where N s the current value of the notonal of the foregn currency leg of trade measured n the domestc currency. If both legs of a trade are n dfferent foregn currences, a conservatve smplfcaton s appled: Equaton (30) should be calculated for both legs, and the maxmum should be chosen. An FX hedgng set s specfed as all trades of a nettng set referencng the same par of currences. It s assumed that all FX trades of the same hedgng set are drven by the same market factor, whch s the FX spot rate for the hedgng set s currency par. Thus, Equaton (4) aggregates all trades of a gven currency par, and Equaton (5) reduces to takng the absolute value of the sngle sngle-factor add-on: (CCY par) (SFS) AddOn m = AddOn 1. When delta for FX trades s calculated, the trade drecton wthn each currency par must be consstent (eg postve for all trades long GBP/EUR and negatve for all trades short GBP/EUR). 4.3 Credt MTM value of the most popular credt dervatve, sngle-name or ndex credt default swap (CDS), can be represented n the form smlar to Equaton (6): E CDS contr = max{ S, t} V ( t) CS ( t) CS N ( τ)df( t, τ)[1 EL ( t, τ)] dτ (31) contr where CS ( t ) s the credt spread at tme t, CS s the contractual credt spread and EL ( t, τ ) s the rsk-neutral expected loss due to default(s) between tme t and tme τ per unt of notonal. 8 The supervsory factors for credt dervatves account for the one-year volatlty of the credt spread. The adjusted notonal for credt dervatves s meant to represent the value of the ntegral n Equaton (31) at tme t = 0. It was decded to gnore the dfference between the ntegrals n Equatons (6) and (31) and set the adjusted notonal for credt dervatves equal to the one for IR dervatves gven by Equatons (7) and (8). All credt trades of a gven nettng set represent a sngle hedgng set. A sngle-factor subset s specfed as all credt trades referencng the same entty (whch can be ether a sngle name or an ndex). Thus, Equaton (4) aggregates all credt trades n a nettng set that reference the same entty. Aggregaton across enttes s done assumng that each entty s drven by a sngle systematc rsk factor and an dosyncratc rsk factor. The systematc factor loadng for entty j s quantfed va correlaton ρ between the full stochastc drver of credt spread of entty j and the systematc factor. Ths correlaton s set to 50% for enttes that are sngle names and to 80% for those that are ndces. The parwse correlaton ρρ and can have the followng values: between two dstnct enttes j and k s gven by the product j k j 7 One can show that ths specfcaton s approprate for another common example of FX lnear trades cross-currency swap. 8 In the case of sngle-name CDS, EL (, ) t τ s the rsk-neutral cumulatve probablty of default between t and τ. Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 9

14 5% between two sngle names; 40% between a sngle name and an ndex; 64% between two ndces. Applyng the sngle-factor assumpton to Equaton (5) results n the followng add-on aggregaton formula: AddOnCD = AddOn + 1 AddOn j j (Entty) (Entty) ρj j ( ρj ) ( j ) (3) The key features of credt dervatves that are captured are the maturty aspect and the bass rsk. The sngle-factor model reflects the fact that, although credt spreads n general move together, there can be consderable dosyncratc varaton n a sngle name that lmts the benefts of hedgng a long CDS wth one reference name wth a short CDS referencng a dfferent name. Ths strkes a balance between recognsng dversfcaton (whch s greatest when correlaton s zero) and hedgng of dssmlar names (whch s greatest when correlaton s one) Equtes For the smplest lnear equty dervatves, such as equty forwards, volatlty of the trade MTM value s equal to the product of the volatlty of the stock (ndex) prce and the number of unts of stock (ndex) referenced by the trade. The volatlty of the equty prce can be approxmated by the product of the current stock (ndex) prce and the relatve volatlty of the stock (ndex) prce. The supervsory factors for equty dervatves are meant to capture the relatve volatlty of the stock (ndex) prce, whle the adjusted notonal s set equal to the product to the current stock (ndex) prce and the number of unts referenced by the trade. Aggregaton of equty dervatves s done n exactly the same manner as aggregaton of credt dervatves. It s assumed that all trades n a nettng set referencng the same entty (sngle name or ndex) are drven by the same factor, so Equaton (4) s appled at an entty level. It s further assumed that stock prces and equty ndex values are drven by a sngle systematc factor wth the same correlaton values as for credt dervatves: 50% sngle names and to 80% for those that are ndces. Thus, Equaton (3) s used for aggregaton of equty dervatves across enttes. Ths desgn captures the tendency of equty markets to move together, but lmts the ablty of dfferent names to offset. Here agan, a balance was struck between recognsng dversfcaton benefts and hedgng benefts. 4.5 Commodtes For commodty dervatves, the arguments wth respect to the volatlty of the MTM value of lnear contracts are smlar to those used for equty dervatves above. The supervsory factors for commodty dervatves are meant to capture the relatve volatlty of the commodty prce, whle the adjusted notonal s defned as the product of the current unt prce (eg one barrel of ol) of the commodty referenced by the trade and the number of unts referenced by the trade. A commodty hedgng set s specfed as all trades of a nettng set referencng the same broad category of commodty: energy, metals, agrcultural, or other commodty. Sngle-factor subsets are specfed as a specfc commodty type: electrcty, ol, gas, nckel, corn. Aggregaton wthn commodty types s done va Equaton (4). Specfc commodty types are aggregated to a hedgng set level usng the same sngle-factor model that s used for credt and equty dervatves, but wth the correlaton parameter set to 40% for all commodty types. Ths results n the use of Equaton (3), but nterpretng j as a specfc commodty type wth ρ j set to 40% for all j. 10 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

15 5. Multpler Recall that a nettng-set-level add-on s an estmate of the nettng set PFE under the four assumptons stated n Secton.1 of ths paper: current MTM and collateral are equal to zero; there are no cash flows wthn the horzon; MTM follows zero-drft Brownan moton. Let us examne what would happen to the nettng set PFE when the frst two assumptons are relaxed, so that non-zero current MTM and collateral are allowed. Let us consder a margned nettng set. Generally, EE of a margned nettng set s charactersed by a full term structure calculated from 9 [ ] margn EE () t Emax{ Vt () Ct (),0} = (33) where Vt () s the MTM value of the nettng set at tme t and Ct () s the collateral avalable to the bank at tme t. The SA-CCR does not attempt to model collateral dynamcs through tme and smply takes a sngle pont t = MPR from the EE profle as the measure of exposure. Furthermore, the SA-CCR assumes that collateral s not exchanged durng the MPR, so only non-cash collateral can change value over the MPR due to volatlty of the collateral asset. The volatlty of non-cash collateral s not modelled ether: nstead, a harcut s appled to the current collateral value C to obtan a determnstc cash-equvalent C (MPR) value, as descrbed by Equaton (3). Thus, the SA-CCR target measure of exposure for margned nettng sets s MTM [ V C ] margn EE (MPR) E max{ (MPR) (MPR),0} = (34) Under the assumpton that a nettng set s MTM follows a drftless Brownan moton, the value of the MTM at the MPR can be descrbed va V(MPR) = V + σ (0) MPR Y (35) where σ (0) s the volatlty of the nettng set MTM value at tme 0 (e countng all trades n the nettng set) and Y s a standard normal random varable. 10 Substtutng Equaton (35) nto Equaton (34), calculatng the expectaton analytcally results n: [ V C ] V C (MPR) σ (0) MPR V C (MPR) σ (0) MPR margn EE (MPR) = (MPR) Φ + σ(0) MPR ϕ where Φ () s the standard normal cumulatve dstrbuton functon. The margned add-on n the form of Equaton (17) s obtaned from Equaton (36) by settng V C (MPR) = 0 : AddOn = ϕ(0) σ(0) MPR margn aggregate To obtan the PFE, one needs to subtract the current replacement cost from the rght-hand sde of Equaton (36). However, the SA-CCR does not gve any credt to PFE reducton when the replacement margn margn cost s postve (e PFE = AddOn aggregate whenever V C (MPR) 0 ). For the (36) V C (MPR) 0 < case, Equaton (36) produces the PFE because the current replacement cost s zero. 9 See, for example, Pykhtn (010). 10 See Secton. Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 11

16 To derve the SA-CCR multpler, we need to express the PFE n terms of add-on rather than volatlty. Ths s easly acheved by usng Equaton (17) to express σ (0) n terms of the margned add-on n Equaton (36): V C (MPR) = [ V C ] Φ ϕ margn aggregate margn AddOn aggregate V C(MPR) + ϕ ϕ(0) margn ϕ(0) AddOn aggregate margn PFE (MPR) (0) AddOn where Equaton (37) s vald only for the V C (MPR) < 0 case. (37) By defnton, the multpler s the rato of the PFE to the add-on. Combnng the postve and negatve collateralsed MTM cases and usng a shorthand notaton margn y = [ V C (MPR)] / AddOn, one obtans the model multpler: aggregate [ (0) y] ϕϕ Wmodel( y) = mn 1, yφ [ ϕ(0) y] + ϕ(0) Ths functon s shown by the dash-double-dotted curve n Fgure 1. The multpler n Equaton (38) s based on the assumpton that the nettng set future MTM value s normally dstrbuted. However, MTM values of real nettng sets are lkely to exhbt heaver tal behavour than the one of the normal dstrbuton. To account for ths possblty, a more conservatve multpler functon was chosen: ( ) mn 1, exp y Wexp y = where the factor of appears n the denomnator n order to match the slope of W ( y ) model at y = 0 functon s shown by the dash-dotted curve n Fgure 1. Whle the exponental multpler n Equaton (39) s sgnfcantly more conservatve than the model-based multpler n Equaton (38), concerns were rased that the multpler would stll approach zero wth nfnte overcollateralsaton. To address ths concern, a floor F was added to the exponental multpler n a manner that preserves the functon slope at the orgn: ( ) mn 1, (1 ) exp y WSA-CCR y = F + F (1 F) The sold curve n Fgure 1 shows ths functon wth F = 5% currently chosen n the SA-CCR. (38) (39). Ths (40) 1 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

17 Fgure 1: The dependence of multpler on ( V C ) / AddOn 1.0 Model Exponental Exponental wth 5% Floor 0.8 Multpler ( V C ) / AddOn To apply the normal approxmaton to unmargned nettng sets on a consstent bass, one has to calculate the EEPE as the tme average (between zero and one year) of the followng EE profle: V C() t V C() t EE() t = [ V C() t ] Φ + σ() t t ϕ σ() t t σ() t t Unfortunately, such averagng s possble n closed form only for the trval case V C () t = 0 that was used n Equaton (1). To crcumvent ths dffculty, the SA-CCR apples the same multpler functon n Equaton (40) to both margned and unmargned cases, wth functon argument y for nonmargned nettng sets defned n the same way as for margned nettng sets, but usng an unmargned no-margn aggregate add-on and one-year horzon for collateral harcut: y = [ V C (1year)] / AddOn aggregate The multpler formulaton recognses that the PFE porton of EEPE s smaller the further one moves away from an ATM nettng set. It s conservatve n ths regard n three ways. Frst, t ncorporates fat tals through the use of the exponental functon. Second, a floor s placed on the value of the multpler to ensure that some PFE s always recognsed. Thrd, the multpler only reduces PFE for negatve MTM values, but does not recognse possble reducton of PFE for postve MTM values. (41) 6. Calbraton Calbraton of supervsory factors and sngle-systematc-factor correlatons was done based on four calbraton exercses. Frst, the supervsory factors were calculated from asset class volatltes usng Equaton (). However, these ntal calbratons were based on volatlty estmates averaged from a wde varety of trades wthn an asset class. Then, these calbratons were compared to smulaton models of small portfolos of trades for each asset class. The smulaton models used were developed by supervsors Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 13

18 and banks own IMM models. Lastly, to ensure that the SA-CCR was applcable to large portfolos, a quanttatve mpact study was conducted where banks compared the results for ther own portfolos usng SA-CCR, IMM and M. The fnal calbraton consders the results of all of these exercses. The fnal parameter calbratons are provded n the fnal standards text. The rest of ths secton wll focus on two specfc aspects of the SA-CCR calbraton: correlatons between IR maturty buckets and deltas for CDO tranches. 6.1 Correlatons between IR maturty buckets Generally, the correlaton between MTM values of two IR trades referencng the same IR curve depends on four tme parameters: the start and end dates of ether trade as they are defned n Secton 4.1 of ths paper. However, ths four-dmensonal problem s not tractable wthn a smple non-model approach. To smplfy the problem, two tme dmensons were elmnated by assumng that the start date s equal to zero for all trades. 11 Now one can use hstorcal correlatons between swap rates of dfferent tenors as proxes for correlatons between MTM values of trades wth dfferent remanng maturtes. Hstorcal correlatons between weekly changes of swap rates of dfferent tenors (between one year and 30 years) for each of four major currences (USD, EUR, GBP and JPY) were calculated for the tme perod from January 1, 005 to December 31, 009. To fnd a parametrc functon of two tenors that adequately descrbes the hstorcal correlatons calculated for each currency, a two-step procedure was followed: Reducng the two tenor dmensons nto a sngle effectve dmenson: The data plotted as a functon of ths effectve dmenson should le on a smooth curve rather than be scattered. Data ponts n Fgure show the hstorcal correlatons as a functon of M M / mn{ M, M } for the four major currences, where k k M s the tenor of swap rate. One can see that the data ponts group along smooth lnes n all four plots, so ths combnaton of tenors s a good approxmaton for the effectve sngle dmenson. Fndng a smple parametrc fttng functon of the effectve dmenson: The sold curve n each of the panels of Fgure represents the functon gven by 1 ρ k = M Mk 1+ a mn{ M, Mk} wth the followng values of the parameters: ( a = 0.15 ; b = 0.5) for USD and ( a = 0.15 ; b = 0.7 ) for EUR, GBP and JPY. Snce values a = 0.15 and b = 0.7 used n Equaton (4) adequately descrbe three out of four major currences, t was decded to apply these values to all currences. One could apply Equaton (4) drectly to calculate correlatons between any par of IR trades of the same hedgng set (e the same currency). However, ths approach would nvolve a double summaton across the trades, whch could become problematc for large nettng sets. To overcome ths dffculty, t was decded to create maturty buckets and treat all trades wthn a gven maturty bucket as beng drven by a sngle factor. Under ths approach, the double summaton only apples to a small fxed number of buckets rather than to a potentally large number of ndvdual trades. The SA-CCR specfes three maturty buckets: (1) 0 1 year (mdpont s 0.5 year); () 1 5 years (mdpont s three years); (3) above fve years (mdpont s set to 18 years). Applyng Equaton (4) wth parameters a = 0.15 and b = 0.7 to the mdponts of these maturty buckets yelds the followng correlaton values: 68% between buckets 1 and b (4) 11 Snce real IR portfolos are domnated by ongong IR swaps, ths assumpton should not lead to large dstortons. 14 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

19 and and 3 and 8% between buckets 1 and 3. Roundng these numbers to 70% and 30%, respectvely, results n Equaton (9). Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 15

20 Fgure : Swap rate correlatons and the parametrc fttng functon for G4 currences 100% 90% USD USD swap rate correlatons 80% 70% 60% 50% 40% 30% 0% 10% 0% M M k / mn{ M, M k } 100% 90% EUR EUR swap rate correlatons 80% 70% 60% 50% 40% 30% 0% 10% 0% M M k / mn{ M, M k } 16 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

21 100% 90% GBP GBP swap rate correlatons 80% 70% 60% 50% 40% 30% 0% 10% 0% M M k / mn{ M, M k } 100% 90% JPY JPY swap rate correlatons 80% 70% 60% 50% 40% 30% 0% 10% 0% M M k / mn{ M, M k } Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 17

22 6. Deltas for CDO Let us consder a CDO structure of n tranches defned on a credt ndex. Tranche s specfed by the attachment pont P 1 and the detachment pont P (wth P 0 = 0 and P n = 1). The thckness of tranche δ P = P P (where = 1,..., n). A poston n all n tranches of a CDO structure s s gven by 1 economcally equvalent to a sngle poston n the underlyng ndex. The SA-CCR assumes that an ndex CDS and any CDO tranche referencng that ndex are drven by the same factor. Under ths assumpton, the add-on of a portfolo of long tranche postons (e bought protecton) s equal to the sum of the addons of ndvdual tranches, and the equvalence of the entre CDO captal structure to the underlyng ndex can be expressed va n δ( P 1, P) δp = 1 (43) = 1 where δ ( AD, ) s delta of a long poston n a tranche wth the attachment pont A and the detachment pont D wth respect to the credt spread of the underlyng ndex. In the lmt n, the tranches become nfntesmally thn, and the summaton n Equaton (43) transforms nto an ntegral: 1 g( P) dp = 1 (44) 0 Specfyng a sutable functon g() of a sngle argument that satsfes Equaton (44) would allow one to calculate delta of a tranche as the average of ths functon between the attachment pont A and the detachment pont D : D δ 1 ( A, D) = g( P) dp D A (45) Functon gp ( ) can be nterpreted as delta of an nfntesmally thn tranche (or, a tranchelet) wth the attachment pont P. Snce more senor tranches should have smaller add-ons than more junor tranches, functon g() should be monotoncally decreasng. It s extremely challengng to derve functon g() from the fundamentals. Instead, a benchmarkng exercse was used to calbrate ths functon for the SA-CCR. Banks partcpatng n the exercse were asked to calculate EEPE for all tranches referencng the CDX.NA.IG ndex. Then, the average across banks for each quantty was calculated (the hghest and the lowest reported values were excluded from the average) for both current and stressed states of the market. Ratos of the average tranche EEPE to the sum of average EEPE across all tranches are essentally IMM-mpled values of tranche delta; they were used for the calbraton. After tryng several smple parametrc functons, t was found that tranchelet delta gven by 1 A 1+ λ gp ( ) = (1 + λp) wth λ = 14 descrbes the IMM-mpled tranche deltas for CDX.NA.IG reasonably well. Fgure 3 shows ths functon along wth the value equal to one for the underlyng ndex. One can see from the plot that tranchelets wth the attachment pont below (above) the value of about 0.5% are treated more (less) (46) 1 The factor 1+ λ n the numerator s needed to satsfy Equaton (44). 18 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures

23 conservatvely than the ndex. Delta for any fnte tranche can be calculated as the average of the tranchelet delta functon between the attachment pont A and detachment pont D. For the functon gven by Equaton (46), ths averagng can be performed analytcally, resultng n 1+ λ δ ( AD, ) = (1 + λa) (1 + λd) (47) whch was chosen (wth λ = 14 ) as delta for CDO trades under the SA-CCR. Fgure 3: Delta assumed by the SA-CCR for nfntesmally thn tranches. Tranchelet Index g ( P ) % 0% 40% 60% 80% 100% P Foundatons of the standardsed approach for measurng counterparty credt rsk exposures 19

24 References Basel Commttee on Bankng Supervson (1988): Internatonal convergence of captal measurement and captal standards, July. (006): Internatonal convergence of captal measurement and captal standards: A revsed framework, June. (010): Basel III: A global regulatory framework for more reslent banks and bankng systems, December. (013): The non-nternal model method for captalsng counterparty credt rsk, consultatve document, June. Canabarro, E, E Pcoult and T Wlde (003): Analysng counterparty rsk, Rsk, September, pp 117. Gbson, M (005): Measurng counterparty credt exposure to a margned counterparty n M Pykhtn (ed), Counterparty credt rsk modellng, Rsk Books. Gregory J (01): Counterparty credt rsk and credt value adjustment, Wley. ISDA-TBMA-LIBA (003): Counterparty rsk treatment of OTC dervatves and securtes fnancng transactons, June, Pykhtn, M (010): Collateralsed credt exposure n E Canabarro (ed), Counterparty credt rsk, Rsk Books. (011): Counterparty rsk management and valuaton n T Belecky, D Brgo and F Patras (eds), Credt rsk fronters, Wley. (014): The non-nternal model method for counterparty credt rsk n E Canabarro and M Pykhtn (eds), Counterparty rsk management, Rsk Books. Wlde, T (005): Analytc methods for portfolo counterparty rsk n M Pykhtn (ed), Counterparty credt rsk modellng, Rsk Books. 0 Foundatons of the standardsed approach for measurng counterparty credt rsk exposures